Module

ResidueTheoremK_ratFunc_of_isAlgClosed.lean

Nyll.AlgebraicCurves.Differentials.Residues.ResidueTheoremK_ratFunc_of_isAlgClosed

594 declarations indexed from sourceView source ↗

Module documentation

Differentials, residues, and divisor correspondences

Canonical rational-function residues, partial-fraction generators, and coordinate independence establish the residue theorem in every characteristic.

Imports20

Nyll.AlgebraicCurves.Divisors.ClassGroup
Nyll.AlgebraicCurves.Places.Rational.Basic
Nyll.AlgebraicCurves.FunctionFields.CurveOver
Nyll.AlgebraicCurves.Differentials.Order
Nyll.AlgebraicCurves.Differentials.Uniformizer
Nyll.AlgebraicCurves.Cohomology.Adelic.Repartitions
Nyll.AlgebraicCurves.Cohomology.Adelic.Index
Nyll.AlgebraicCurves.Divisors.Canonical
Nyll.AlgebraicCurves.Differentials.Residues.Basic
Nyll.AlgebraicCurves.Divisors.Maps
Mathlib.RingTheory.AdicCompletion.Completeness
Mathlib.NumberTheory.RatFunc.Ostrowski
Mathlib.FieldTheory.RatFunc.AsPolynomial
Mathlib.Algebra.Polynomial.PartialFractions
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
Mathlib.RingTheory.Norm.Transitivity
Mathlib.Algebra.BigOperators.Field
Nyll.AlgebraicCurves.Places.OverDVR
Mathlib.Util.WithWeakNamespace

Imported by3

theorem

coeff_minpolyDiv_dim_sub_one

L359 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.FLT.EulerDualBasis.coeff_minpolyDiv_dim_sub_one

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.Module
open _root_.Algebra
namespace FLT
namespace EulerDualBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
  [FiniteDimensional K L] [Algebra.IsSeparable K L]
omit [FiniteDimensional K L] [Algebra.IsSeparable K L] in

Lean source · statement

theorem coeff_minpolyDiv_dim_sub_one (pb : PowerBasis K L) :
    (minpolyDiv K pb.gen).coeff (pb.dim - 1) = 1

theorem

traceDual_eq_one_div_of_val_eq

L366 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.FLT.EulerDualBasis.traceDual_eq_one_div_of_val_eq

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.Module
open _root_.Algebra
namespace FLT
namespace EulerDualBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
  [FiniteDimensional K L] [Algebra.IsSeparable K L]

Lean source · statement

theorem traceDual_eq_one_div_of_val_eq (pb : PowerBasis K L) {j : Fin pb.dim}
    (hj : (j : ℕ) = pb.dim - 1) :
    pb.basis.traceDual j = 1 / aeval pb.gen (derivative (minpoly K pb.gen))

theorem

trace_pow_div_aeval_derivative_minpoly_of_lt

L372 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.FLT.EulerDualBasis.trace_pow_div_aeval_derivative_minpoly_of_lt

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.Module
open _root_.Algebra
namespace FLT
namespace EulerDualBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
  [FiniteDimensional K L] [Algebra.IsSeparable K L]

Lean source · statement

theorem trace_pow_div_aeval_derivative_minpoly_of_lt (pb : PowerBasis K L) {k : ℕ}
    (hk : k < pb.dim - 1) :
    Algebra.trace K L
      (pb.gen ^ k / aeval pb.gen (derivative (minpoly K pb.gen))) = 0

theorem

trace_pow_div_aeval_derivative_minpoly_self

L385 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.FLT.EulerDualBasis.trace_pow_div_aeval_derivative_minpoly_self

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.Module
open _root_.Algebra
namespace FLT
namespace EulerDualBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
  [FiniteDimensional K L] [Algebra.IsSeparable K L]

Lean source · statement

theorem trace_pow_div_aeval_derivative_minpoly_self (pb : PowerBasis K L)
    (hd : 0 < pb.dim) :
    Algebra.trace K L
      (pb.gen ^ (pb.dim - 1) / aeval pb.gen (derivative (minpoly K pb.gen))) = 1

theorem

trace_root_pow_div_derivative_of_lt

L396 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.FLT.EulerDualBasis.trace_root_pow_div_derivative_of_lt

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.Module
open _root_.Algebra
namespace FLT
namespace EulerDualBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
  [FiniteDimensional K L] [Algebra.IsSeparable K L]
variable {F : Type*} [Field F] [CharZero F] {g : F[X]} [Fact (Irreducible g)]

Lean source · statement

theorem trace_root_pow_div_derivative_of_lt (hg : g.Monic) {k : ℕ}
    (hk : k < g.natDegree - 1) :
    Algebra.trace F (AdjoinRoot g)
      (AdjoinRoot.root g ^ k / aeval (AdjoinRoot.root g) (derivative g)) = 0

theorem

trace_root_pow_div_derivative_self

L409 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.FLT.EulerDualBasis.trace_root_pow_div_derivative_self

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.Module
open _root_.Algebra
namespace FLT
namespace EulerDualBasis
variable {K L : Type*} [Field K] [Field L] [Algebra K L]
  [FiniteDimensional K L] [Algebra.IsSeparable K L]
variable {F : Type*} [Field F] [CharZero F] {g : F[X]} [Fact (Irreducible g)]

Lean source · statement

theorem trace_root_pow_div_derivative_self (hg : g.Monic) (hd : 0 < g.natDegree) :
    Algebra.trace F (AdjoinRoot g)
      (AdjoinRoot.root g ^ (g.natDegree - 1) /
        aeval (AdjoinRoot.root g) (derivative g)) = 1

def

OrdDifferentialWellDefined

L471 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.OrdDifferentialWellDefined

Documentation

Differentials of two uniformizers differ by a coefficient of order zero.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}
variable (K F)

Lean source · statement

def OrdDifferentialWellDefined : Prop

theorem

ord_add_eq_min

L546 ↗

AlgebraicCurve.Place.ord_add_eq_min

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open scoped _root_.Polynomial.Bivariate
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_add_eq_min {f g : F} (hf : f ≠ 0) (hg : g ≠ 0)
  (h : v.ord f ≠ v.ord g) :
    v.ord (f + g) = min (v.ord f) (v.ord g)

abbrev

lg37_completion

L623 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37.lg37_completion

Documentation

The adic completion of the valuation subring at its maximal ideal.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace ModularCurve
namespace Lg37
open _root_.AlgebraicCurve
open ModularCurve.Ldgr36Si
open ModularCurve.Ldgr36Rc
open ModularCurve.Ldgr35Cs
section CompletionCarrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

abbrev lg37_completion (v : Place K F) : Type _

def

lg37_residueHat

L627 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37.lg37_residueHat

Documentation

The residue homomorphism from the completed valuation subring.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace ModularCurve
namespace Lg37
open _root_.AlgebraicCurve
open ModularCurve.Ldgr36Si
open ModularCurve.Ldgr36Rc
open ModularCurve.Ldgr35Cs
section CompletionCarrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

def lg37_residueHat (v : Place K F) :
    lg37_completion v →+* v.ResidueField

theorem

lg37_residueHat_algebraMap

L631 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37.lg37_residueHat_algebraMap

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace ModularCurve
namespace Lg37
open _root_.AlgebraicCurve
open ModularCurve.Ldgr36Si
open ModularCurve.Ldgr36Rc
open ModularCurve.Ldgr35Cs
section CompletionCarrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem lg37_residueHat_algebraMap (v : Place K F) (x : v.toValuationSubring) :
    lg37_residueHat v
        (algebraMap v.toValuationSubring (lg37_completion v) x)
      = IsLocalRing.residue v.toValuationSubring x

structure

Lg37CompletionSection

L637 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37.Lg37CompletionSection

Documentation

An algebra section of the residue map from the completed valuation subring.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace ModularCurve
namespace Lg37
open _root_.AlgebraicCurve
open ModularCurve.Ldgr36Si
open ModularCurve.Ldgr36Rc
open ModularCurve.Ldgr35Cs
section CompletionCarrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · full declaration

structure Lg37CompletionSection (v : Place K F) where
  /-- Lift residue field elements to the completion. -/
  lift : v.ResidueField →ₐ[K] lg37_completion v
  /-- The completed residue map is a left inverse of the lift. -/
  residueHat_lift : ∀ a, lg37_residueHat v (lift a) = a

theorem

ord_nonneg_of_mem

L700 ↗

AlgebraicCurve.Place.ord_nonneg_of_mem

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section SinglePlace
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_nonneg_of_mem {f : F} (hf :
  f ∈ v.toValuationSubring) : 0 ≤ v.ord f

theorem

mem_of_ord_nonneg

L717 ↗

AlgebraicCurve.Place.mem_of_ord_nonneg

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section SinglePlace
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_of_ord_nonneg {f : F} (hf : f ≠ 0) (h :
  0 ≤ v.ord f) :
    f ∈ v.toValuationSubring

theorem

mem_iff_ord_nonneg

L727 ↗

AlgebraicCurve.Place.mem_iff_ord_nonneg

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section SinglePlace
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_iff_ord_nonneg {f : F} (hf : f ≠ 0) :
    f ∈ v.toValuationSubring ↔ 0 ≤ v.ord f

theorem

exists_ord_pos

L732 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.exists_ord_pos

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section SinglePlace
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem exists_ord_pos : ∃ f : F, f ≠ 0 ∧ 0 < v.ord f

theorem

algebraMap_ne_zero

L746 ↗

AlgebraicCurve.Place.algebraMap_ne_zero

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']

Lean source · statement

theorem _root_.AlgebraicCurve.Place.algebraMap_ne_zero {f : F} (hf : f ≠ 0) :
  algebraMap F F' f ≠ 0

theorem

comap_algebraMap_ne_top

L753 ↗

AlgebraicCurve.Place.comap_algebraMap_ne_top

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']
open _root_.AlgebraicCurve.Place
variable (w : Place K F')

Lean source · statement

theorem _root_.AlgebraicCurve.Place.comap_algebraMap_ne_top [Algebra.IsIntegral F F'] :
    w.toValuationSubring.comap (algebraMap F F') ≠ ⊤

theorem

mem_comap_iff_ord_nonneg

L771 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.mem_comap_iff_ord_nonneg

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']
open _root_.AlgebraicCurve.Place
variable (w : Place K F')
open _root_.AlgebraicCurve.Place
variable {w} in

Lean source · statement

theorem mem_comap_iff_ord_nonneg {f : F} (hf : f ≠ 0) :
    f ∈ w.toValuationSubring.comap (algebraMap F F') ↔
      0 ≤ w.ord (algebraMap F F' f)

theorem

isUnit_mk_comap_iff

L778 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.isUnit_mk_comap_iff

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']
open _root_.AlgebraicCurve.Place
variable (w : Place K F')
open _root_.AlgebraicCurve.Place
variable {w} in
variable {w} in

Lean source · statement

theorem isUnit_mk_comap_iff {f : F} (hf : f ≠ 0)
    (hmem : f ∈ w.toValuationSubring.comap (algebraMap F F')) :
    IsUnit (⟨f, hmem⟩ : w.toValuationSubring.comap (algebraMap F F')) ↔
      w.ord (algebraMap F F' f) = 0

theorem

exists_ord_algebraMap_pos

L803 ↗

AlgebraicCurve.Place.exists_ord_algebraMap_pos

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']
open _root_.AlgebraicCurve.Place
variable (w : Place K F')
open _root_.AlgebraicCurve.Place
variable {w} in
variable {w} in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.exists_ord_algebraMap_pos [Algebra.IsIntegral F F'] :
    ∃ f : F, f ≠ 0 ∧ 0 < w.ord (algebraMap F F' f)

theorem

ramificationIndex_set_nonempty

L820 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.ramificationIndex_set_nonempty

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']
open _root_.AlgebraicCurve.Place
variable (w : Place K F')
open _root_.AlgebraicCurve.Place
variable {w} in
variable {w} in
open _root_.AlgebraicCurve.Place
variable [Algebra.IsIntegral F F']

Lean source · statement

theorem ramificationIndex_set_nonempty :
    {n : ℕ | 0 < n ∧ ∃ f : F, f ≠ 0 ∧ w.ord (algebraMap F F' f) = n}.Nonempty

theorem

isPrincipalIdealRing_comap

L825 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.isPrincipalIdealRing_comap

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Restrict
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F'] [Algebra F F']
open _root_.AlgebraicCurve.Place
variable (w : Place K F')
open _root_.AlgebraicCurve.Place
variable {w} in
variable {w} in
open _root_.AlgebraicCurve.Place
variable [Algebra.IsIntegral F F']

Lean source · statement

theorem isPrincipalIdealRing_comap :
    IsPrincipalIdealRing (w.toValuationSubring.comap (algebraMap F F'))

theorem

ord_add_eq_left

L907 ↗

AlgebraicCurve.Place.ord_add_eq_left

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve
section Ultrametric
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_add_eq_left {f g : F} (hf : f ≠ 0) (hg : g ≠ 0)
  (h : v.ord f < v.ord g) :
    v.ord (f + g) = v.ord f

theorem

ord_finitePlace_ne_zero_iff

L1010 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_finitePlace_ne_zero_iff

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem ord_finitePlace_ne_zero_iff {p : K[X]} (hp : Irreducible p) {q : K[X]} (hq : q ≠ 0) :
    (finitePlace K hp).ord (algebraMap K[X] (RatFunc K) q) ≠ 0 ↔ p ∣ q

instance

(anonymous instance)

L1019 ↗

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

instance : Nontrivial (MonoidWithZeroHom.valueGroup (MonoidWithZeroHom.ofClass
  (RatFunc.inftyValuation K)))

def

placeInfty

L1029 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.placeInfty

Documentation

The place at infinity of the rational function field.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · full declaration

def placeInfty : Place K (RatFunc K) where
  toValuationSubring := (RatFunc.inftyValuation K).valuationSubring
  algebraMap_mem' a := by
    rw [Valuation.mem_valuationSubring_iff]
    exact Valuation.IsTrivialOn.valuation_algebraMap_le_one (v := RatFunc.inftyValuation K) a
  ne_top' := by
    simp only [ne_eq, Valuation.valuationSubring_eq_top_iff, not_not]
    infer_instance
  isPrincipalIdealRing' :=
    (Valuation.valuationSubring_isDiscreteValuationRing
      (RatFunc.inftyValuation K)).toIsPrincipalIdealRing

theorem

placeInfty_toValuationSubring

L1041 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem placeInfty_toValuationSubring :
    (placeInfty K).toValuationSubring = (RatFunc.inftyValuation K).valuationSubring

theorem

inftyValuation_isEquiv_adicValuation

L1044 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.inftyValuation_isEquiv_adicValuation

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem inftyValuation_isEquiv_adicValuation :
    (RatFunc.inftyValuation K).IsEquiv (placeInfty K).adicValuation

theorem

placeInfty_ne_ofHeightOneSpectrum

L1048 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.placeInfty_ne_ofHeightOneSpectrum

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem placeInfty_ne_ofHeightOneSpectrum (w : HeightOneSpectrum K[X]) :
    placeInfty K ≠ Place.ofHeightOneSpectrum w

theorem

eq_ofHeightOneSpectrum_or_eq_placeInfty

L1061 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.eq_ofHeightOneSpectrum_or_eq_placeInfty

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}

Lean source · statement

theorem eq_ofHeightOneSpectrum_or_eq_placeInfty [DecidableEq (RatFunc K)]
    (v : Place K (RatFunc K)) :
    (∃ w : HeightOneSpectrum K[X], v = Place.ofHeightOneSpectrum w) ∨ v = placeInfty K

theorem

subsingleton_setOf_forall_ne_ofHeightOneSpectrum

L1071 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.subsingleton_setOf_forall_ne_ofHeightOneSpectrum

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}

Lean source · statement

theorem subsingleton_setOf_forall_ne_ofHeightOneSpectrum :
    {v : Place K (RatFunc K) |
      ∀ w : HeightOneSpectrum K[X], v ≠ Place.ofHeightOneSpectrum w}.Subsingleton

theorem

finite_setOf_valuation_ne_one

L1082 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.finite_setOf_valuation_ne_one

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}

Lean source · statement

theorem finite_setOf_valuation_ne_one {f : RatFunc K} (hf : f ≠ 0) :
    {w : HeightOneSpectrum K[X] | w.valuation (RatFunc K) f ≠ 1}.Finite

theorem

finite_setOf_ord_ne_zero

L1101 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.finite_setOf_ord_ne_zero

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}

Lean source · statement

theorem finite_setOf_ord_ne_zero {f : RatFunc K} (hf : f ≠ 0) :
    {v : Place K (RatFunc K) | v.ord f ≠ 0}.Finite

theorem

exists_sub_algebraMap_intDegree_neg

L1129 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.exists_sub_algebraMap_intDegree_neg

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section DegInfty
variable (K) [DecidableEq (RatFunc K)]

Lean source · statement

theorem exists_sub_algebraMap_intDegree_neg {x : RatFunc K}
    (hx : RatFunc.inftyValuation K x ≤ 1) :
    ∃ c : K, x - algebraMap K (RatFunc K) c = 0 ∨
      (x - algebraMap K (RatFunc K) c).intDegree < 0

theorem

deg_placeInfty

L1178 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.deg_placeInfty

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section DegInfty
variable (K) [DecidableEq (RatFunc K)]

Lean source · statement

theorem deg_placeInfty : (placeInfty K).deg = 1

theorem

exists_smul_eq_of_ne_zero

L1237 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_smul_eq_of_ne_zero

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
omit [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
  [Nontrivial Ω[F⁄K]] in

Lean source · statement

theorem exists_smul_eq_of_ne_zero (v : Place K F) {ω ω₀ : Ω[F⁄K]}
    (hω : ω ≠ 0) (hω₀ : ω₀ ≠ 0) : ∃ g : F, g ≠ 0 ∧ ω = g • ω₀

def

uniformizerSubring'

L1297 ↗

AlgebraicCurve.Place.uniformizerSubring'

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Transport
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

def _root_.AlgebraicCurve.Place.uniformizerSubring' : v.toValuationSubring

instance

instSMulCommClass_subring

L1301 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.instSMulCommClass_subring

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Transport
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

instance instSMulCommClass_subring : SMulCommClass K v.toValuationSubring F

theorem

kaehlerMap_subring_D_uniformizer

L1304 ↗

AlgebraicCurve.Place.kaehlerMap_subring_D_uniformizer

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Transport
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.kaehlerMap_subring_D_uniformizer :
    KaehlerDifferential.map K K v.toValuationSubring F
      (KaehlerDifferential.D K v.toValuationSubring v.uniformizerSubring') = v.dCoord

def

ValSubringKaehlerSpanTop

L1318 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringKaehlerSpanTop

Documentation

Each valuation subring has differentials spanned by its chosen uniformizer differential.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringKaehlerSpanTop : Prop

def

principalAdele

L1373 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.principalAdele

Documentation

The principal adele associated to a function.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
variable (K F) in

Lean source · full declaration

def principalAdele : F →ₗ[K] adeleSpace K F where
  toFun f := ⟨diagonalHom K F f, diagonal_mem_adeleSpace f⟩
  map_add' f g := Subtype.ext ((diagonalHom K F).map_add f g)
  map_smul' c f := Subtype.ext ((diagonalHom K F).map_smul c f)

def

P1PolynomialGenerators

L1389 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PolynomialGenerators

Documentation

The monomial generators of the polynomial part of a rational function.

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
variable (K F) in
section P1NamedRows
open scoped _root_.Polynomial
variable (K)
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]

Lean source · statement

def P1PolynomialGenerators : Set (RatFunc K)

def

P1PrincipalPartGenerators

L1392 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PrincipalPartGenerators

Documentation

The partial fraction atoms at finite places.

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
variable (K F) in
section P1NamedRows
open scoped _root_.Polynomial
variable (K)
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]

Lean source · statement

def P1PrincipalPartGenerators : Set (RatFunc K)

def

P1PartialFractionGenerators

L1397 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PartialFractionGenerators

Documentation

The polynomial and finite principal part generators of rational functions.

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
variable (K F) in
section P1NamedRows
open scoped _root_.Polynomial
variable (K)
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]

Lean source · statement

def P1PartialFractionGenerators : Set (RatFunc K)

theorem

gate_algebraMap_polynomial_mem_span_P1PolynomialGenerators

L1408 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.gate_algebraMap_polynomial_mem_span_P1PolynomialGenerators

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
variable (K F) in
section P1Gates
open scoped _root_.Polynomial

Lean source · statement

theorem gate_algebraMap_polynomial_mem_span_P1PolynomialGenerators (q : K[X]) :
    algebraMap K[X] (RatFunc K) q ∈ Submodule.span K (P1PolynomialGenerators K)

abbrev

mp72a102_t1_maximalIdealHat

L1459 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T1.mp72a102_t1_maximalIdealHat

Documentation

The ideal in the completion generated by the valuation subring maximal ideal.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T1
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr37Ch
open Mp72a101T2
open scoped _root_.Polynomial
attribute [local instance 2000] RatFunc.instAlgebraOfPolynomial
section Engine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

abbrev mp72a102_t1_maximalIdealHat (v : Place K F) :
    Ideal (lg37_completion v)

theorem

mp72a102_t1_isScalarTower

L1464 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T1.mp72a102_t1_isScalarTower

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T1
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr37Ch
open Mp72a101T2
open scoped _root_.Polynomial
attribute [local instance 2000] RatFunc.instAlgebraOfPolynomial
section Engine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t1_isScalarTower (v : Place K F) :
    IsScalarTower K v.toValuationSubring (lg37_completion v)

theorem

mp72a102_t1_isAdicComplete_maximalIdealHat

L1468 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T1.mp72a102_t1_isAdicComplete_maximalIdealHat

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T1
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr37Ch
open Mp72a101T2
open scoped _root_.Polynomial
attribute [local instance 2000] RatFunc.instAlgebraOfPolynomial
section Engine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t1_isAdicComplete_maximalIdealHat (v : Place K F) :
    IsAdicComplete (mp72a102_t1_maximalIdealHat v) (lg37_completion v)

theorem

mp72a102_t1_henselianRing_completion

L1476 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T1.mp72a102_t1_henselianRing_completion

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T1
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr37Ch
open Mp72a101T2
open scoped _root_.Polynomial
attribute [local instance 2000] RatFunc.instAlgebraOfPolynomial
section Engine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t1_henselianRing_completion (v : Place K F) :
    HenselianRing (lg37_completion v) (mp72a102_t1_maximalIdealHat v)

theorem

mp72a102_t1_maximalIdealHat_le_ker_residueHat

L1481 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T1.mp72a102_t1_maximalIdealHat_le_ker_residueHat

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T1
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr37Ch
open Mp72a101T2
open scoped _root_.Polynomial
attribute [local instance 2000] RatFunc.instAlgebraOfPolynomial
section Engine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t1_maximalIdealHat_le_ker_residueHat (v : Place K F) :
    mp72a102_t1_maximalIdealHat v ≤ RingHom.ker (lg37_residueHat v)

theorem

mp72a102_t1_exists_completion_root_of_residue_root

L1487 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T1.mp72a102_t1_exists_completion_root_of_residue_root

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T1
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr37Ch
open Mp72a101T2
open scoped _root_.Polynomial
attribute [local instance 2000] RatFunc.instAlgebraOfPolynomial
section Engine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t1_exists_completion_root_of_residue_root
    (v : Place K F) {p : K[X]} (hpmon : p.Monic) (hpsep : p.Separable)
    {rbar : v.ResidueField} (hrbar : Polynomial.aeval rbar p = 0) :
    ∃ α : lg37_completion v,
      Polynomial.aeval α p = 0 ∧ lg37_residueHat v α = rbar

theorem

mp72a102_t2_residueHat_algebraMap_base

L1595 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2.mp72a102_t2_residueHat_algebraMap_base

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T2
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr39Hf ModularCurve.Ldgr35Cs
open Mp72a101T2
open scoped _root_.Polynomial
section ResidueCalculus
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t2_residueHat_algebraMap_base (v : Place K F) (c : K) :
    lg37_residueHat v (algebraMap K (lg37_completion v) c)
      = algebraMap K v.ResidueField c

def

mp72a102_t2_residueHatAlgHom

L1605 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2.mp72a102_t2_residueHatAlgHom

Documentation

The completed residue homomorphism viewed as an algebra homomorphism.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace Mp72a102T2
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Ldgr39Hf ModularCurve.Ldgr35Cs
open Mp72a101T2
open scoped _root_.Polynomial
section ResidueCalculus
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · full declaration

def mp72a102_t2_residueHatAlgHom (v : Place K F) :
    lg37_completion v →ₐ[K] v.ResidueField where
  toRingHom := lg37_residueHat v
  commutes' := mp72a102_t2_residueHat_algebraMap_base v

instance

(anonymous instance)

L1665 ↗

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [Algebra.IsIntegral F F']
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (w : Place K F')

Lean source · full declaration

instance : IsLocalHom (restrictInclusion F w) where
  map_nonunit a ha := by
    have hord : w.ord (algebraMap F F' (a : F)) = 0 := by
      have h := w.ord_coe_unit ha.unit
      rwa [IsUnit.unit_spec, coe_restrictInclusion] at h
    have ha0 : (a : F) ≠ 0 := by
      rintro h0
      obtain ⟨b, hb⟩ := isUnit_iff_exists_inv.mp ha
      have hb' : algebraMap F F' (a : F) * (b : F') = 1 := congrArg Subtype.val hb
      rw [h0, _root_.map_zero, zero_mul] at hb'
      exact zero_ne_one hb'
    exact (isUnit_mk_comap_iff ha0 a.2).mpr hord

theorem

mk_mem_maximalIdeal_iff

L1716 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.mk_mem_maximalIdeal_iff

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)

Lean source · statement

theorem mk_mem_maximalIdeal_iff {f : F} (hf : f ∈ v.toValuationSubring) :
    (⟨f, hf⟩ : v.toValuationSubring) ∈ IsLocalRing.maximalIdeal v.toValuationSubring
      ↔ f = 0 ∨ 0 < v.ord f

def

uniformizerSubring

L1775 ↗

AlgebraicCurve.Place.uniformizerSubring

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)

Lean source · statement

def _root_.AlgebraicCurve.Place.uniformizerSubring : v.toValuationSubring

theorem

coe_uniformizerSubring

L1780 ↗

AlgebraicCurve.Place.coe_uniformizerSubring

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.coe_uniformizerSubring : (v.uniformizerSubring :
  F) = v.uniformizer

theorem

irreducible_uniformizerSubring

L1784 ↗

AlgebraicCurve.Place.irreducible_uniformizerSubring

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.irreducible_uniformizerSubring :
  Irreducible v.uniformizerSubring

theorem

uniformizerSubring_mem_maximalIdeal

L1789 ↗

AlgebraicCurve.Place.uniformizerSubring_mem_maximalIdeal

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.uniformizerSubring_mem_maximalIdeal :
    v.uniformizerSubring ∈ IsLocalRing.maximalIdeal v.toValuationSubring

theorem

uniformizer_mem

L1794 ↗

AlgebraicCurve.Place.uniformizer_mem

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.uniformizer_mem :
  v.uniformizer ∈ v.toValuationSubring

def

simplePoleSubmodule

L1799 ↗

AlgebraicCurve.Place.simplePoleSubmodule

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.simplePoleSubmodule : Submodule K F where
  carrier := {f | v.uniformizer * f ∈ v.toValuationSubring}
  add_mem' {f g} hf hg := by
    simpa only [Set.mem_ofPred_eq, mul_add] using add_mem hf hg
  zero_mem' := by simpa only [Set.mem_ofPred_eq, mul_zero] using zero_mem _
  smul_mem' c f hf := by
    change v.uniformizer * (c • f) ∈ v.toValuationSubring
    rw [Algebra.smul_def, mul_left_comm]
    exact mul_mem (v.algebraMap_mem' c) hf

theorem

mem_simplePoleSubmodule

L1811 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.mem_simplePoleSubmodule

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem mem_simplePoleSubmodule {f : F} :
    f ∈ v.simplePoleSubmodule ↔ v.uniformizer * f ∈ v.toValuationSubring

theorem

mem_simplePoleSubmodule_of_mem

L1814 ↗

AlgebraicCurve.Place.mem_simplePoleSubmodule_of_mem

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_simplePoleSubmodule_of_mem {f : F} (hf :
  f ∈ v.toValuationSubring) :
    f ∈ v.simplePoleSubmodule

def

simplePoleMulUniformizer

L1820 ↗

AlgebraicCurve.Place.simplePoleMulUniformizer

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.simplePoleMulUniformizer :
  v.simplePoleSubmodule →ₗ[K] v.toValuationSubring where
  toFun f := ⟨v.uniformizer * f.1, f.2⟩
  map_add' f g := Subtype.ext (mul_add _ _ _)
  map_smul' c f := Subtype.ext <| by
    change v.uniformizer * ((c • f : v.simplePoleSubmodule) : F)
        = ((c • ⟨v.uniformizer * f.1, f.2⟩ : v.toValuationSubring) : F)
    simp only [Algebra.smul_def, MulMemClass.coe_mul, v.coe_algebraMap, Submodule.coe_smul,
      mul_left_comm]

def

simplePoleResidueAux

L1831 ↗

AlgebraicCurve.Place.simplePoleResidueAux

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.simplePoleResidueAux :
  v.simplePoleSubmodule →ₗ[K] v.ResidueField

theorem

simplePoleResidueAux_apply

L1837 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.simplePoleResidueAux_apply

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem simplePoleResidueAux_apply (f : v.simplePoleSubmodule) :
    v.simplePoleResidueAux f = IsLocalRing.residue _ ⟨v.uniformizer * f.1, f.2⟩

theorem

simplePoleResidueAux_eq_zero_of_mem

L1840 ↗

AlgebraicCurve.Place.simplePoleResidueAux_eq_zero_of_mem

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.simplePoleResidueAux_eq_zero_of_mem {f : F} (hf :
  f ∈ v.toValuationSubring) :
    v.simplePoleResidueAux ⟨f, v.mem_simplePoleSubmodule_of_mem hf⟩ = 0

def

localResidueExtend

L1850 ↗

AlgebraicCurve.Place.localResidueExtend

Source context 33 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.localResidueExtend : F →ₗ[K] v.ResidueField

theorem

localResidueExtend_apply_of_mem

L1854 ↗

AlgebraicCurve.Place.localResidueExtend_apply_of_mem

Source context 34 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.localResidueExtend_apply_of_mem {f : F} (hf :
  f ∈ v.simplePoleSubmodule) :
    v.localResidueExtend f = v.simplePoleResidueAux ⟨f, hf⟩

def

localResidueDataOfExtend

L1861 ↗

AlgebraicCurve.Place.localResidueDataOfExtend

Source context 35 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.localResidueDataOfExtend : v.LocalResidueData where
  res := v.localResidueExtend
  res_of_mem f hf := by
    rw [v.localResidueExtend_apply_of_mem (v.mem_simplePoleSubmodule_of_mem hf),
      v.simplePoleResidueAux_eq_zero_of_mem hf]
  res_simplePole f hf := by
    rw [v.localResidueExtend_apply_of_mem hf, simplePoleResidueAux_apply]

instance

instHasLocalResidue

L1872 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.instHasLocalResidue

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · full declaration

instance instHasLocalResidue : HasLocalResidue K F where
  exists_data v := ⟨v.localResidueDataOfExtend⟩

theorem

gate_uniformizer_inv_mem_simplePoleSubmodule

L1878 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.gate_uniformizer_inv_mem_simplePoleSubmodule

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem gate_uniformizer_inv_mem_simplePoleSubmodule (v : Place K F) :
    v.uniformizer⁻¹ ∈ v.simplePoleSubmodule ∧ v.uniformizer⁻¹ ∉ v.toValuationSubring

theorem

ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one

L1925 ↗

ModularCurve.MilneAvAg9bRd14CubeRowCartierSliceStart.ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace ModularCurve
namespace MilneAvAg9bRd14CubeRowCartierSliceStart
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open scoped _root_.Polynomial
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd14CubeRowCartierSliceStart in
with_weak_namespace _root_.ModularCurve.MilneAvAg9bRd14CubeRowCartierSliceStart

Lean source · statement

theorem ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one
    {K F : Type*} [Field K] [Field F] [Algebra K F]
    (v : Place K F) (R : v.CanonicalLocalResidueDataK)
    {j : ℤ} (hj : j ≠ -1) :
    R.res (v.uniformizer ^ j) = 0

theorem

finiteResidue_of_deg_pos

L1972 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.finiteResidue_of_deg_pos

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section DegPos
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem finiteResidue_of_deg_pos (h : 0 < v.deg) : v.FiniteResidue

instance

instFiniteResidueOfHeightOneSpectrum

L1986 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instFiniteResidueOfHeightOneSpectrum

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

instance instFiniteResidueOfHeightOneSpectrum (w : HeightOneSpectrum K[X]) :
    (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).FiniteResidue

instance

instFiniteResiduePlaceInfty

L1993 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instFiniteResiduePlaceInfty

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

instance instFiniteResiduePlaceInfty [DecidableEq (RatFunc K)] :
    (placeInfty K).FiniteResidue

instance

instFiniteResidue

L1998 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instFiniteResidue

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K} in

Lean source · statement

instance instFiniteResidue (v : Place K (RatFunc K)) : v.FiniteResidue

theorem

algebraMap_polynomial_mem_of_ne_placeInfty'

L2056 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.algebraMap_polynomial_mem_of_ne_placeInfty'

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
section RatFuncIntegrality
open _root_.AlgebraicCurve.RationalFunctionField
variable (K)
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem algebraMap_polynomial_mem_of_ne_placeInfty'
    {v : Place K (RatFunc K)} (hv : v ≠ placeInfty K) (q : K[X]) :
    algebraMap K[X] (RatFunc K) q ∈ v.toValuationSubring

theorem

pow_X_mem_of_ne_placeInfty

L2063 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.pow_X_mem_of_ne_placeInfty

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
section RatFuncIntegrality
open _root_.AlgebraicCurve.RationalFunctionField
variable (K)
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem pow_X_mem_of_ne_placeInfty
    {v : Place K (RatFunc K)} (hv : v ≠ placeInfty K) (n : ℕ) :
    (RatFunc.X : RatFunc K) ^ n ∈ v.toValuationSubring

theorem

pow_X_mul_mem_of_ne_placeInfty

L2070 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.pow_X_mul_mem_of_ne_placeInfty

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
section RatFuncIntegrality
open _root_.AlgebraicCurve.RationalFunctionField
variable (K)
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem pow_X_mul_mem_of_ne_placeInfty
    {v : Place K (RatFunc K)} (hv : v ≠ placeInfty K) (n : ℕ)
    {g : RatFunc K} (hg : g ∈ v.toValuationSubring) :
    (RatFunc.X : RatFunc K) ^ n * g ∈ v.toValuationSubring

def

P1DifferentialCoeffRegularFinite

L2090 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1DifferentialCoeffRegularFinite

Documentation

The differential coefficient is regular at every finite place.

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
section P1PolynomialSubrows
open _root_.AlgebraicCurve.RationalFunctionField
variable (K)
variable [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]

Lean source · statement

def P1DifferentialCoeffRegularFinite {ω₀ : Ω[(RatFunc K)⁄K]} (_hω₀ : ω₀ ≠ 0) : Prop

abbrev

p1PrincipalPartAtom

L2110 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartAtom

Documentation

A rational function atom with a prescribed polynomial denominator power.

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
section P1PrincipalPartSubrows
open _root_.AlgebraicCurve.RationalFunctionField
variable (K)
variable [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]

Lean source · statement

abbrev p1PrincipalPartAtom (p c : K[X]) (m : ℕ) : RatFunc K

theorem

finitePlace_ne_placeInfty

L2118 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.finitePlace_ne_placeInfty

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable [HasCanonicalLocalResidueKStar K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasPrincipalDivisors K F]
section P1PrincipalPartSubrows
open _root_.AlgebraicCurve.RationalFunctionField
variable (K)
variable [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]
variable {K}
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F := RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
  [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem finitePlace_ne_placeInfty {p : K[X]} (hp : Irreducible p) :
    finitePlace K hp ≠ placeInfty K

theorem

ord_algebraMap_base

L2174 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.ord_algebraMap_base

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem ord_algebraMap_base (a : K) : v.ord (algebraMap K F a) = 0

theorem

exists_ord_finitePlace_eq_nsmul

L2193 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.exists_ord_finitePlace_eq_nsmul

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem exists_ord_finitePlace_eq_nsmul {p : K[X]} (hp : Irreducible p) {q : K[X]}
    (hq : q ≠ 0) :
    ∃ m : ℕ, (finitePlace K hp).ord (algebraMap K[X] (RatFunc K) q)
      = m * (finitePlace K hp).ord (algebraMap K[X] (RatFunc K) p)

theorem

ord_finitePlace_dvd

L2212 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_finitePlace_dvd

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem ord_finitePlace_dvd {p : K[X]} (hp : Irreducible p) {f : RatFunc K} (hf : f ≠ 0) :
    (finitePlace K hp).ord (algebraMap K[X] (RatFunc K) p) ∣ (finitePlace K hp).ord f

theorem

ord_finitePlace_self

L2226 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_finitePlace_self

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem ord_finitePlace_self {p : K[X]} (hp : Irreducible p) :
    (finitePlace K hp).ord (algebraMap K[X] (RatFunc K) p) = 1

theorem

ord_placeInfty_eq_zero_of_intDegree_eq_zero

L2244 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_eq_zero_of_intDegree_eq_zero

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_eq_zero_of_intDegree_eq_zero {f : RatFunc K} (hf : f ≠ 0)
    (h : f.intDegree = 0) : (placeInfty K).ord f = 0

theorem

ord_placeInfty_eq_intDegree_mul

L2251 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_eq_intDegree_mul

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_eq_intDegree_mul {f : RatFunc K} (hf : f ≠ 0) :
    (placeInfty K).ord f = f.intDegree * (placeInfty K).ord (RatFunc.X)

theorem

ord_placeInfty_X

L2288 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_X

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_X : (placeInfty K).ord (RatFunc.X : RatFunc K) = -1

theorem

ord_placeInfty_algebraMap'

L2307 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_algebraMap'

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
section PlaceInfty
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_algebraMap' {q : K[X]} (hq : q ≠ 0) :
    (placeInfty K).ord (algebraMap K[X] (RatFunc K) q) = -(q.natDegree : ℤ)

theorem

aeval_mem

L2394 ↗

AlgebraicCurve.Place.aeval_mem

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section SupportTransfer
variable {K F F' : Type*} [Field K] [Field F] [Field F'] [Algebra K F'] [Algebra F F']
  [FiniteDimensional F F']
variable (w : Place K F')
omit [FiniteDimensional F F'] in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.aeval_mem {Q : Polynomial F} {x : F'}
    (hcoeff : ∀ i, algebraMap F F' (Q.coeff i) ∈ w.toValuationSubring)
    (hx : x ∈ w.toValuationSubring) :
    Polynomial.aeval x Q ∈ w.toValuationSubring

theorem

exists_coeff_ord_ne_zero

L2402 ↗

AlgebraicCurve.Place.exists_coeff_ord_ne_zero

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section SupportTransfer
variable {K F F' : Type*} [Field K] [Field F] [Field F'] [Algebra K F'] [Algebra F F']
  [FiniteDimensional F F']
variable (w : Place K F')
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.exists_coeff_ord_ne_zero {f : F'} (hf : f ≠ 0)
  (hford : w.ord f ≠ 0) :
    ∃ i < (minpoly F f).natDegree, (minpoly F f).coeff i ≠ 0 ∧
      w.ord (algebraMap F F' ((minpoly F f).coeff i)) ≠ 0

def

chartHom

L2454 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.chartHom

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Chart
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Algebra R F] [IsFractionRing R F]
variable (w : Place K F)

Lean source · statement

def chartHom (hw : ∀ r : R, algebraMap R F r ∈ w.toValuationSubring) :
    R →+* w.toValuationSubring

theorem

inv_algebraMap_mem

L2459 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.inv_algebraMap_mem

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Chart
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Algebra R F] [IsFractionRing R F]
variable (w : Place K F)
omit [IsDedekindDomain R] [IsFractionRing R F] in

Lean source · statement

theorem inv_algebraMap_mem (hw : ∀ r : R, algebraMap R F r ∈ w.toValuationSubring)
    {s : R} (hs : IsUnit (chartHom w hw s)) :
    (algebraMap R F s)⁻¹ ∈ w.toValuationSubring

theorem

finite_setOf_ord_ne_zero_of_finiteDimensional

L2489 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.finite_setOf_ord_ne_zero_of_finiteDimensional

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Assembly
variable {K : Type*} [Field K] {F' : Type*} [Field F'] [Algebra K F']
  [Algebra (RatFunc K) F'] [IsScalarTower K (RatFunc K) F']
  [FiniteDimensional (RatFunc K) F'] [Algebra.IsSeparable (RatFunc K) F']

Lean source · statement

theorem finite_setOf_ord_ne_zero_of_finiteDimensional {f : F'} (hf : f ≠ 0) :
    {w : Place K F' | w.ord f ≠ 0}.Finite

def

uniformizerSubring''

L2536 ↗

AlgebraicCurve.Place.uniformizerSubring''

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

def _root_.AlgebraicCurve.Place.uniformizerSubring'' : v.toValuationSubring

def

polynomialAlgebra

L2540 ↗

AlgebraicCurve.Place.polynomialAlgebra

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.polynomialAlgebra :
  Algebra K[X] v.toValuationSubring

theorem

polynomialAlgebra_algebraMap_X

L2545 ↗

AlgebraicCurve.Place.polynomialAlgebra_algebraMap_X

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.polynomialAlgebra_algebraMap_X :
    let := v.polynomialAlgebra
    algebraMap K[X] v.toValuationSubring X = v.uniformizerSubring'' :=
  aeval_X _

theorem

polynomialIsScalarTower

L2551 ↗

AlgebraicCurve.Place.polynomialIsScalarTower

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.polynomialIsScalarTower :
    let := v.polynomialAlgebra
    IsScalarTower K K[X] v.toValuationSubring := by
  let := v.polynomialAlgebra
  refine IsScalarTower.of_algebraMap_eq fun a => ?_
  change _ = aeval v.uniformizerSubring'' (algebraMap K K[X] a)
  simp [Polynomial.algebraMap_eq, aeval_C]

theorem

range_mapBaseChange_le_span_D_uniformizer

L2560 ↗

AlgebraicCurve.Place.range_mapBaseChange_le_span_D_uniformizer

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.range_mapBaseChange_le_span_D_uniformizer :
    let := v.polynomialAlgebra
    let := v.polynomialIsScalarTower
    LinearMap.range (KaehlerDifferential.mapBaseChange K K[X] v.toValuationSubring)
      ≤ Submodule.span v.toValuationSubring
          {KaehlerDifferential.D K v.toValuationSubring v.uniformizerSubring''} := by
  let := v.polynomialAlgebra
  let := v.polynomialIsScalarTower
  change LinearMap.range (KaehlerDifferential.mapBaseChange K K[X] v.toValuationSubring)
    ≤ Submodule.span v.toValuationSubring
        {KaehlerDifferential.D K v.toValuationSubring v.uniformizerSubring''}
  rintro _ ⟨t, rfl⟩
  induction t with
  | add x y hx hy => rw [map_add]; exact add_mem hx hy
  | tmul b ω =>
    rw [KaehlerDifferential.mapBaseChange_tmul,
      show ω = (KaehlerDifferential.polynomialEquiv K ω) • (KaehlerDifferential.D K K[X] X) from
        ((KaehlerDifferential.polynomialEquiv K).symm_apply_apply ω).symm.trans
          (KaehlerDifferential.polynomialEquiv_symm K _),
      LinearMap.map_smul, KaehlerDifferential.map_D,
      show algebraMap K[X] v.toValuationSubring X = v.uniformizerSubring'' from aeval_X _,
      ← algebraMap_smul (A := v.toValuationSubring)]
    exact Submodule.smul_mem _ b
      (Submodule.smul_mem _ _ (Submodule.mem_span_singleton_self _))

theorem

range_mapBaseChange_eq_top_of_subsingleton

L2586 ↗

AlgebraicCurve.Place.range_mapBaseChange_eq_top_of_subsingleton

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.range_mapBaseChange_eq_top_of_subsingleton
    (hsub :
      let := v.polynomialAlgebra
      Subsingleton Ω[v.toValuationSubring⁄K[X]]) :
    let := v.polynomialAlgebra
    let := v.polynomialIsScalarTower
    LinearMap.range (KaehlerDifferential.mapBaseChange K K[X] v.toValuationSubring) = ⊤ := by
  let := v.polynomialAlgebra
  let := v.polynomialIsScalarTower
  change LinearMap.range (KaehlerDifferential.mapBaseChange K K[X] v.toValuationSubring) = ⊤
  rw [KaehlerDifferential.range_mapBaseChange]
  exact LinearMap.ker_eq_top.mpr (Subsingleton.elim _ _)

theorem

span_D_uniformizer_eq_top_of_subsingleton

L2600 ↗

AlgebraicCurve.Place.span_D_uniformizer_eq_top_of_subsingleton

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CotangentEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.span_D_uniformizer_eq_top_of_subsingleton
    (hsub :
      let := v.polynomialAlgebra
      Subsingleton Ω[v.toValuationSubring⁄K[X]]) :
    Submodule.span v.toValuationSubring
      {KaehlerDifferential.D K v.toValuationSubring v.uniformizerSubring''} = ⊤ := by
  refine top_le_iff.mp ?_
  rw [← v.range_mapBaseChange_eq_top_of_subsingleton hsub]
  exact v.range_mapBaseChange_le_span_D_uniformizer

def

ValSubringPolynomialFormallyUnramified

L2618 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringPolynomialFormallyUnramified

Documentation

Valuation subrings are formally unramified over their uniformizer polynomial algebras.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringPolynomialFormallyUnramified : Prop

theorem

valSubringKaehlerSpanTop_of_polynomialFormallyUnramified

L2625 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerSpanTop_of_polynomialFormallyUnramified

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerSpanTop_of_polynomialFormallyUnramified
    (hfu : ValSubringPolynomialFormallyUnramified K F) :
    ValSubringKaehlerSpanTop K F

theorem

exists_monic_irreducible_factorization

L2672 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_monic_irreducible_factorization

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.UniqueFactorizationMonoid
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K : Type*} [Field K]

Lean source · full declaration

theorem exists_monic_irreducible_factorization {d : K[X]} (hd : d.Monic) :
    ∃ (s : Finset K[X]) (n : K[X] → ℕ),
      (∀ p ∈ s, p.Monic) ∧ (∀ p ∈ s, Irreducible p) ∧
      (Set.Pairwise (s : Set K[X]) fun i j => IsCoprime i j) ∧
      d = ∏ p ∈ s, p ^ n p := by
  classical
  refine ⟨(normalizedFactors d).toFinset,
    fun p => (normalizedFactors d).count p, ?_, ?_, ?_, ?_⟩
  · intro p hp
    exact ((Polynomial.mem_normalizedFactors_iff hd.ne_zero).mp
      (Multiset.mem_toFinset.mp hp)).2.1
  · intro p hp
    exact irreducible_of_normalized_factor p (Multiset.mem_toFinset.mp hp)
  · intro p hp q hq hne
    have hp' := Multiset.mem_toFinset.mp hp
    have hq' := Multiset.mem_toFinset.mp hq
    rw [(irreducible_of_normalized_factor p hp').coprime_iff_not_dvd]
    exact fun hdvd => hne (normalizedFactors_eq_of_dvd d p hp' q hq' hdvd)
  · calc d = (normalizedFactors d).prod := by
            rw [prod_normalizedFactors_eq hd.ne_zero, hd.normalize_eq_self]
      _ = ∏ p ∈ (normalizedFactors d).toFinset, p ^ (normalizedFactors d).count p :=
            Finset.prod_multiset_count _

theorem

algebraMap_mem_span_P1PartialFractionGenerators

L2695 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.algebraMap_mem_span_P1PartialFractionGenerators

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.UniqueFactorizationMonoid
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K : Type*} [Field K]

Lean source · statement

theorem algebraMap_mem_span_P1PartialFractionGenerators (q : K[X]) :
    algebraMap K[X] (RatFunc K) q ∈ Submodule.span K (P1PartialFractionGenerators K)

theorem

div_pow_mem_span_P1PartialFractionGenerators

L2700 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.div_pow_mem_span_P1PartialFractionGenerators

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.UniqueFactorizationMonoid
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K : Type*} [Field K]

Lean source · statement

theorem div_pow_mem_span_P1PartialFractionGenerators
    {p c : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hdeg : c.degree < p.degree) {m : ℕ} (hm : 1 ≤ m) :
    algebraMap K[X] (RatFunc K) c / (algebraMap K[X] (RatFunc K) p) ^ m
      ∈ Submodule.span K (P1PartialFractionGenerators K)

theorem

ratFunc_mem_span_P1PartialFractionGenerators

L2707 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFunc_mem_span_P1PartialFractionGenerators

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.UniqueFactorizationMonoid
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K : Type*} [Field K]

Lean source · statement

theorem ratFunc_mem_span_P1PartialFractionGenerators (f : RatFunc K) :
    f ∈ Submodule.span K (P1PartialFractionGenerators K)

theorem

p1PartialFractionSpan_eq_top

L2735 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PartialFractionSpan_eq_top

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
open _root_.UniqueFactorizationMonoid
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
variable {K : Type*} [Field K]

Lean source · statement

theorem p1PartialFractionSpan_eq_top :
    Submodule.span K (P1PartialFractionGenerators K) = ⊤

def

residueFieldAdjoinRootEquiv

L2767 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Section.residueFieldAdjoinRootEquiv

Documentation

Present a residue field by adjoining its given primitive integral generator.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.IntermediateField
namespace ModularCurve
namespace KwNo6Section
section GenericIso
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

def residueFieldAdjoinRootEquiv {ᾱ : v.ResidueField} (hint : _root_.IsIntegral K ᾱ)
    (htop : K⟮ᾱ⟯ = ⊤) :
    AdjoinRoot (minpoly K ᾱ) ≃ₐ[K] v.ResidueField

theorem

residueFieldAdjoinRootEquiv_root

L2773 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Section.residueFieldAdjoinRootEquiv_root

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.IntermediateField
namespace ModularCurve
namespace KwNo6Section
section GenericIso
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem residueFieldAdjoinRootEquiv_root {ᾱ : v.ResidueField} (hint : _root_.IsIntegral K ᾱ)
    (htop : K⟮ᾱ⟯ = ⊤) :
    residueFieldAdjoinRootEquiv v hint htop (AdjoinRoot.root (minpoly K ᾱ)) = ᾱ

def

sectionOfPrimitiveRoot

L2787 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Section.sectionOfPrimitiveRoot

Documentation

Construct a section of the completed residue map from a lift of a primitive root.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.IntermediateField
namespace ModularCurve
namespace KwNo6Section
section GenericSection
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · full declaration

def sectionOfPrimitiveRoot {ᾱ : v.ResidueField} (hint : _root_.IsIntegral K ᾱ)
    (htop : K⟮ᾱ⟯ = ⊤) (αhat : lg37_completion v)
    (hroot : (aeval αhat) (minpoly K ᾱ) = 0)
    (hres : lg37_residueHat v αhat = ᾱ) :
    Lg37CompletionSection v where
  lift :=
    (AdjoinRoot.liftAlgHom (minpoly K ᾱ) (Algebra.ofId K (lg37_completion v)) αhat
      (by rw [aeval_def] at hroot; exact hroot)).comp
      (residueFieldAdjoinRootEquiv v hint htop).symm.toAlgHom
  residueHat_lift a := by
    have hres' : lg37_residueHat v αhat
        = residueFieldAdjoinRootEquiv v hint htop (AdjoinRoot.root (minpoly K ᾱ)) :=
      hres.trans (residueFieldAdjoinRootEquiv_root v hint htop).symm
    have hkey : (mp72a102_t2_residueHatAlgHom v).comp
          (AdjoinRoot.liftAlgHom (minpoly K ᾱ) (Algebra.ofId K (lg37_completion v)) αhat
            (by rw [aeval_def] at hroot; exact hroot))
        = (residueFieldAdjoinRootEquiv v hint htop).toAlgHom := by
      refine AdjoinRoot.algHom_ext ?_
      exact (congrArg (mp72a102_t2_residueHatAlgHom v)
        (AdjoinRoot.liftAlgHom_root (minpoly K ᾱ) (Algebra.ofId K (lg37_completion v)) αhat
          (by rw [aeval_def] at hroot; exact hroot))).trans hres'
    have h1 := DFunLike.congr_fun hkey ((residueFieldAdjoinRootEquiv v hint htop).symm a)
    rw [AlgHom.comp_apply] at h1
    have h2 : (residueFieldAdjoinRootEquiv v hint htop).toAlgHom
        ((residueFieldAdjoinRootEquiv v hint htop).symm a) = a :=
      (residueFieldAdjoinRootEquiv v hint htop).apply_symm_apply a
    exact h1.trans h2

def

poleSubmodule

L2850 ↗

AlgebraicCurve.Place.poleSubmodule

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)

Lean source · full declaration

def _root_.AlgebraicCurve.Place.poleSubmodule (n : ℕ) : Submodule K F where
  carrier := {f | v.uniformizer ^ n * f ∈ v.toValuationSubring}
  add_mem' {f g} hf hg := by
    simpa only [Set.mem_ofPred_eq, mul_add] using add_mem hf hg
  zero_mem' := by simpa only [Set.mem_ofPred_eq, mul_zero] using zero_mem _
  smul_mem' c f hf := by
    change v.uniformizer ^ n * (c • f) ∈ v.toValuationSubring
    rw [Algebra.smul_def, mul_left_comm]
    exact mul_mem (v.algebraMap_mem' c) hf

theorem

mem_poleSubmodule

L2862 ↗

AlgebraicCurve.Place.mem_poleSubmodule

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_poleSubmodule {n : ℕ} {f : F} :
    f ∈ v.poleSubmodule n ↔ v.uniformizer ^ n * f ∈ v.toValuationSubring

theorem

uniformizer_pow_ne_zero

L2866 ↗

AlgebraicCurve.Place.uniformizer_pow_ne_zero

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.uniformizer_pow_ne_zero (n : ℕ) :
  (v.uniformizer ^ n : F) ≠ 0

theorem

ord_uniformizer_pow

L2871 ↗

AlgebraicCurve.Place.ord_uniformizer_pow

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_uniformizer_pow (n : ℕ) : v.ord
  (v.uniformizer ^ n) = (n : ℤ)

theorem

coe_poleSubmodule_zero

L2877 ↗

AlgebraicCurve.Place.coe_poleSubmodule_zero

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.coe_poleSubmodule_zero :
    (v.poleSubmodule 0 : Set F) = (v.toValuationSubring : Set F)

theorem

poleSubmodule_one

L2883 ↗

AlgebraicCurve.Place.poleSubmodule_one

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.poleSubmodule_one : v.poleSubmodule 1 =
  v.simplePoleSubmodule

theorem

mem_poleSubmodule_iff_ord

L2888 ↗

AlgebraicCurve.Place.mem_poleSubmodule_iff_ord

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_poleSubmodule_iff_ord {n : ℕ} {f : F} (hf : f ≠ 0) :
    f ∈ v.poleSubmodule n ↔ -(n : ℤ) ≤ v.ord f

theorem

poleSubmodule_mono

L2895 ↗

AlgebraicCurve.Place.poleSubmodule_mono

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.poleSubmodule_mono : Monotone v.poleSubmodule

def

poleMulUniformizerPow

L2904 ↗

AlgebraicCurve.Place.poleMulUniformizerPow

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.poleMulUniformizerPow (n : ℕ) :
  v.poleSubmodule n →ₗ[K] v.toValuationSubring where
  toFun f := ⟨v.uniformizer ^ n * f.1, f.2⟩
  map_add' f g := Subtype.ext (mul_add _ _ _)
  map_smul' c f := Subtype.ext <| by
    change v.uniformizer ^ n * ((c • f : v.poleSubmodule n) : F)
        = ((c • ⟨v.uniformizer ^ n * f.1, f.2⟩ : v.toValuationSubring) : F)
    simp only [Algebra.smul_def, MulMemClass.coe_mul, v.coe_algebraMap, Submodule.coe_smul,
      mul_left_comm]

def

laurentTailCoeff

L2915 ↗

AlgebraicCurve.Place.laurentTailCoeff

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.laurentTailCoeff (n : ℕ) :
  v.poleSubmodule n →ₗ[K] v.ResidueField

theorem

laurentTailCoeff_apply

L2921 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.laurentTailCoeff_apply

Source context 33 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem laurentTailCoeff_apply {n : ℕ} (f : v.poleSubmodule n) :
    v.laurentTailCoeff n f = IsLocalRing.residue _ ⟨v.uniformizer ^ n * f.1, f.2⟩

theorem

laurentTail_remainder_mem_poleSubmodule

L2924 ↗

AlgebraicCurve.Place.laurentTail_remainder_mem_poleSubmodule

Source context 33 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.laurentTail_remainder_mem_poleSubmodule {n : ℕ} {f : F}
    (hf : f ∈ v.poleSubmodule (n + 1)) {c : v.toValuationSubring}
    (hc : IsLocalRing.residue _ c = v.laurentTailCoeff (n + 1) ⟨f, hf⟩) :
    f - (c : F) * (v.uniformizer ^ (n + 1))⁻¹ ∈ v.poleSubmodule n

theorem

localResidueData_res_eq_simplePoleResidueAux

L2950 ↗

AlgebraicCurve.Place.localResidueData_res_eq_simplePoleResidueAux

Source context 34 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.localResidueData_res_eq_simplePoleResidueAux (R :
  v.LocalResidueData) {f : F}
    (hf : f ∈ v.simplePoleSubmodule) : R.res f = v.simplePoleResidueAux ⟨f, hf⟩

theorem

gate_poleSubmodule_strictMono

L2958 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.gate_poleSubmodule_strictMono

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem gate_poleSubmodule_strictMono (v : Place K F) (n : ℕ) :
    (v.uniformizer ^ (n + 1))⁻¹ ∈ v.poleSubmodule (n + 1) ∧
      (v.uniformizer ^ (n + 1))⁻¹ ∉ v.poleSubmodule n

theorem

gate_localResidueData_uniformizer_inv

L2968 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.gate_localResidueData_uniformizer_inv

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem gate_localResidueData_uniformizer_inv (v : Place K F) (R : v.LocalResidueData) :
    R.res v.uniformizer⁻¹ = 1

theorem

mp72a102_t3_evalₐ_zero_depth

L3011 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_zero_depth

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section General
variable {R : Type*} [CommRing R] (I : Ideal R)

Lean source · statement

theorem mp72a102_t3_evalₐ_zero_depth (z : AdicCompletion I R) :
    AdicCompletion.evalₐ I 0 z = 0

theorem

mp72a102_t3_evalₐ_algebraMap

L3017 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_algebraMap

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section General
variable {R : Type*} [CommRing R] (I : Ideal R)

Lean source · statement

theorem mp72a102_t3_evalₐ_algebraMap (n : ℕ) (r : R) :
    AdicCompletion.evalₐ I n (algebraMap R (AdicCompletion I R) r)
      = Ideal.Quotient.mk (I ^ n) r

theorem

mp72a102_t3_evalₐ_factor

L3023 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_factor

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section General
variable {R : Type*} [CommRing R] (I : Ideal R)

Lean source · statement

theorem mp72a102_t3_evalₐ_factor {m n : ℕ} (hmn : m ≤ n) (z : AdicCompletion I R) :
    Ideal.Quotient.factor (Ideal.pow_le_pow_right hmn) (AdicCompletion.evalₐ I n z)
      = AdicCompletion.evalₐ I m z

theorem

mp72a102_t3_exists_rep_of_evalₐ_eq_zero

L3030 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_exists_rep_of_evalₐ_eq_zero

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section General
variable {R : Type*} [CommRing R] (I : Ideal R)

Lean source · statement

theorem mp72a102_t3_exists_rep_of_evalₐ_eq_zero {n : ℕ} {z : AdicCompletion I R}
    (h : AdicCompletion.evalₐ I n z = 0) :
    ∃ w ∈ I ^ n, AdicCompletion.evalₐ I (n + 1) z = Ideal.Quotient.mk (I ^ (n + 1)) w

theorem

mp72a102_t3_evalₐ_succ_mul_eq_zero

L3039 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_succ_mul_eq_zero

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section General
variable {R : Type*} [CommRing R] (I : Ideal R)

Lean source · statement

theorem mp72a102_t3_evalₐ_succ_mul_eq_zero {n : ℕ} {r : R} (hr : r ∈ I)
    {z : AdicCompletion I R} (h : AdicCompletion.evalₐ I n z = 0) :
    AdicCompletion.evalₐ I (n + 1) (algebraMap R (AdicCompletion I R) r * z) = 0

theorem

mp72a102_t3_evalₐ_one_eq_zero_of_evalOneₐ

L3047 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_one_eq_zero_of_evalOneₐ

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section General
variable {R : Type*} [CommRing R] (I : Ideal R)

Lean source · statement

theorem mp72a102_t3_evalₐ_one_eq_zero_of_evalOneₐ {z : AdicCompletion I R}
    (h : AdicCompletion.evalOneₐ I z = 0) : AdicCompletion.evalₐ I 1 z = 0

theorem

mp72a102_t3_evalDepth_add

L3061 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalDepth_add

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_evalDepth_add (v : Place K F) (n : ℕ)
    (x y : lg37_completion v) :
    AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n (x + y)
      = AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n x
        + AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n y

theorem

mp72a102_t3_evalDepth_mul

L3068 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalDepth_mul

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_evalDepth_mul (v : Place K F) (n : ℕ)
    (x y : lg37_completion v) :
    AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n (x * y)
      = AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n x
        * AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n y

theorem

mp72a102_t3_residueHat_congr_of_depth_one

L3075 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_residueHat_congr_of_depth_one

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_residueHat_congr_of_depth_one (v : Place K F)
    {x y : lg37_completion v}
    (h : AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) 1 x
        = AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) 1 y) :
    lg37_residueHat v x = lg37_residueHat v y

theorem

mp72a102_t3_evalDepth_one_eq_zero_of_residueHat

L3084 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalDepth_one_eq_zero_of_residueHat

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_evalDepth_one_eq_zero_of_residueHat (v : Place K F)
    {x : lg37_completion v} (hx : lg37_residueHat v x = 0) :
    AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) 1 x = 0

theorem

mp72a102_t3_eq_uniformizer_mul_of_mem_maximalIdeal

L3089 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_eq_uniformizer_mul_of_mem_maximalIdeal

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_eq_uniformizer_mul_of_mem_maximalIdeal (v : Place K F)
    {w : v.toValuationSubring} (hw : w ∈ maximalIdeal v.toValuationSubring) :
    ∃ u : v.toValuationSubring, w = v.uniformizerSubring * u

theorem

mp72a102_t3_exists_uniformizer_factor

L3095 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_exists_uniformizer_factor

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_exists_uniformizer_factor (v : Place K F)
    {x : lg37_completion v} (hx : lg37_residueHat v x = 0) (n : ℕ) :
    ∃ u : v.toValuationSubring,
      AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n
        (x - algebraMap v.toValuationSubring (lg37_completion v) v.uniformizerSubring
          * algebraMap v.toValuationSubring (lg37_completion v) u) = 0

theorem

mp72a102_t3_sigma_taylor_expansion

L3118 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_sigma_taylor_expansion

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace ModularCurve
namespace Mp72a102T3
open _root_.AlgebraicCurve
section Carrier
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a102_t3_sigma_taylor_expansion (v : Place K F)
    (S : Lg37CompletionSection v) (n : ℕ) (x : lg37_completion v) :
    ∃ a : ℕ → v.ResidueField,
      AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n
        (x - ∑ i ∈ Finset.range n, S.lift (a i)
          * algebraMap v.toValuationSubring (lg37_completion v)
              v.uniformizerSubring ^ i) = 0

theorem

p1PrincipalPartAtom_ne_zero

L3225 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartAtom_ne_zero

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section OrdIntegrand
variable [CharZero K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] [CharZero K]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem p1PrincipalPartAtom_ne_zero {p : K[X]} (hp : p ≠ 0) {c : K[X]} (hc : c ≠ 0)
    (m : ℕ) : p1PrincipalPartAtom K p c m ≠ 0

theorem

le_exp_neg_one_of_lt_one

L3305 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.le_exp_neg_one_of_lt_one

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve

Lean source · statement

theorem le_exp_neg_one_of_lt_one {x : ℤᵐ⁰} (hx : x < 1) : x ≤ exp (-1 : ℤ)

theorem

isUnit_algebraMap

L3319 ↗

AlgebraicCurve.Place.isUnit_algebraMap

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.isUnit_algebraMap {c : K} (hc : c ≠ 0) :
    IsUnit (algebraMap K v.toValuationSubring c)

theorem

adicValuation_algebraMap

L3324 ↗

AlgebraicCurve.Place.adicValuation_algebraMap

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.adicValuation_algebraMap {c : K} (hc : c ≠ 0) :
    v.adicValuation (algebraMap K F c) = 1

theorem

ord_algebraMap

L3330 ↗

AlgebraicCurve.Place.ord_algebraMap

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_algebraMap (c : K) : v.ord (algebraMap K F c) =
  0

theorem

ord_eq_neg_log_of_valuationSubring_eq

L3337 ↗

AlgebraicCurve.Place.ord_eq_neg_log_of_valuationSubring_eq

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_eq_neg_log_of_valuationSubring_eq (w :
  Valuation F ℤᵐ⁰)
    (hw : w.valuationSubring = v.toValuationSubring) {π : F} (hπ : w π = exp (-1 : ℤ))
    {f : F} (hf : f ≠ 0) : v.ord f = -log (w f)

theorem

ord_ofHeightOneSpectrum_eq_neg_log

L3375 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_ofHeightOneSpectrum_eq_neg_log

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem ord_ofHeightOneSpectrum_eq_neg_log (w : HeightOneSpectrum K[X]) {p : K[X]}
    (hp : p ≠ 0) (hw : w.asIdeal = Ideal.span {p}) {f : RatFunc K} (hf : f ≠ 0) :
    (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).ord f
      = -log (w.valuation (RatFunc K) f)

theorem

ord_ofHeightOneSpectrum_of_span

L3386 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_ofHeightOneSpectrum_of_span

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem ord_ofHeightOneSpectrum_of_span (w : HeightOneSpectrum K[X]) {p : K[X]}
    (hp : p ≠ 0) (hw : w.asIdeal = Ideal.span {p}) :
    (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).ord
      (algebraMap K[X] (RatFunc K) p) = 1

theorem

ord_ofHeightOneSpectrum_eq_zero_of_notMem

L3397 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_ofHeightOneSpectrum_eq_zero_of_notMem

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem ord_ofHeightOneSpectrum_eq_zero_of_notMem (w : HeightOneSpectrum K[X]) {q : K[X]}
    (hq : q ≠ 0) (hmem : q ∉ w.asIdeal) :
    (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).ord
      (algebraMap K[X] (RatFunc K) q) = 0

theorem

ord_placeInfty

L3409 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]
section PlaceInftyOrd
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty {f : RatFunc K} (hf : f ≠ 0) :
    (placeInfty K).ord f = -f.intDegree

theorem

ord_placeInfty_algebraMap

L3418 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_algebraMap

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]
section PlaceInftyOrd
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_algebraMap {q : K[X]} (hq : q ≠ 0) :
    (placeInfty K).ord (algebraMap K[X] (RatFunc K) q) = -(q.natDegree : ℤ)

theorem

single_add_single_apply_eq_ord

L3428 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.single_add_single_apply_eq_ord

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]
section IrreducibleDivisor
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem single_add_single_apply_eq_ord {p : K[X]} (hp : Irreducible p)
    (v : Place K (RatFunc K)) :
    (Finsupp.single (finitePlace K hp) (1 : ℤ)
        + Finsupp.single (placeInfty K) (-(p.natDegree : ℤ))) v
      = v.ord (algebraMap K[X] (RatFunc K) p)

theorem

degree_single_add_single

L3457 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.degree_single_add_single

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]
section IrreducibleDivisor
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem degree_single_add_single {p : K[X]} (hp : Irreducible p) :
    Divisor.degree (Finsupp.single (finitePlace K hp) (1 : ℤ)
        + Finsupp.single (placeInfty K) (-(p.natDegree : ℤ))) = 0

theorem

degree_eq_zero_of_forall_eq_ord_algebraMap

L3466 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.degree_eq_zero_of_forall_eq_ord_algebraMap

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem degree_eq_zero_of_forall_eq_ord_algebraMap (q : K[X]) :
    ∀ D : Divisor K (RatFunc K),
      (∀ v : Place K (RatFunc K), D v = v.ord (algebraMap K[X] (RatFunc K) q)) →
      Divisor.degree D = 0

def

principalDivisor

L3506 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.principalDivisor

Documentation

The principal divisor of a nonzero rational function.

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

def principalDivisor {f : RatFunc K} (hf : f ≠ 0) : Divisor K (RatFunc K)

theorem

principalDivisor_apply

L3510 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.principalDivisor_apply

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem principalDivisor_apply {f : RatFunc K} (hf : f ≠ 0) (v : Place K (RatFunc K)) :
    principalDivisor hf v = v.ord f

theorem

degree_eq_zero_of_forall_eq_ord

L3513 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.degree_eq_zero_of_forall_eq_ord

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem degree_eq_zero_of_forall_eq_ord {f : RatFunc K} (D : Divisor K (RatFunc K))
    (hD : ∀ v : Place K (RatFunc K), D v = v.ord f) : Divisor.degree D = 0

theorem

degree_principalDivisor

L3539 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.degree_principalDivisor

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

theorem degree_principalDivisor {f : RatFunc K} (hf : f ≠ 0) :
    Divisor.degree (principalDivisor hf) = 0

instance

instHasPrincipalDivisors

L3543 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instHasPrincipalDivisors

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · full declaration

instance instHasPrincipalDivisors : HasPrincipalDivisors K (RatFunc K) where
  exists_divisor _ hf := ⟨principalDivisor hf, fun _ => rfl, degree_principalDivisor hf⟩

theorem

D_ratFuncX_eq_neg_X_sq_smul_D_inv

L3576 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.D_ratFuncX_eq_neg_X_sq_smul_D_inv

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem D_ratFuncX_eq_neg_X_sq_smul_D_inv :
    KaehlerDifferential.D K (RatFunc K) RatFunc.X
      = (-(RatFunc.X : RatFunc K) ^ 2) •
          KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)⁻¹

theorem

ord_placeInfty_X_inv

L3583 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_X_inv

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_X_inv : (placeInfty K).ord (RatFunc.X : RatFunc K)⁻¹ = 1

theorem

ord_placeInfty_X_pow

L3586 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_X_pow

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_X_pow (n : ℕ) :
    (placeInfty K).ord ((RatFunc.X : RatFunc K) ^ n) = -(n : ℤ)

theorem

D_ratFuncX_inv_ne_zero

L3597 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.D_ratFuncX_inv_ne_zero

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section NonVanishing
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem D_ratFuncX_inv_ne_zero (hwd : OrdDifferentialWellDefined K (RatFunc K)) :
    KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)⁻¹ ≠ 0

theorem

differentialCoeff_placeInfty_D_X_eq

L3614 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_placeInfty_D_X_eq

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section DifferentialCoeff
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem differentialCoeff_placeInfty_D_X_eq :
    (placeInfty K).differentialCoeff (KaehlerDifferential.D K (RatFunc K) RatFunc.X)
      = (-(RatFunc.X : RatFunc K) ^ 2) *
          (placeInfty K).differentialCoeff
            (KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)⁻¹)

theorem

ord_differentialCoeff_placeInfty_D_X_inv_eq_zero

L3621 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_differentialCoeff_placeInfty_D_X_inv_eq_zero

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section DifferentialCoeff
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem ord_differentialCoeff_placeInfty_D_X_inv_eq_zero
    (hwd : OrdDifferentialWellDefined K (RatFunc K)) :
    (placeInfty K).ord
        ((placeInfty K).differentialCoeff
          (KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)⁻¹)) = 0

theorem

ordDifferential_placeInfty_D_ratFuncX

L3639 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferential_placeInfty_D_ratFuncX

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section OrdDifferential
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem ordDifferential_placeInfty_D_ratFuncX
    (hwd : OrdDifferentialWellDefined K (RatFunc K)) :
    (placeInfty K).ordDifferential
        (KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)) = -2

theorem

eq_ord_of_addHom_of_nonneg_iff

L3703 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.eq_ord_of_addHom_of_nonneg_iff

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section Uniqueness
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (w : Place K F)

Lean source · statement

theorem eq_ord_of_addHom_of_nonneg_iff (φ : F → ℤ)
    (hmul : ∀ x y, x ≠ 0 → y ≠ 0 → φ (x * y) = φ x + φ y)
    (hone : ∃ t, t ≠ 0 ∧ φ t = 1)
    (hiff : ∀ x, x ≠ 0 → (0 ≤ φ x ↔ x ∈ w.toValuationSubring))
    {x : F} (hx : x ≠ 0) : φ x = w.ord x

instance

(anonymous instance)

L3780 ↗

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section Setup
variable (v : Place K F)

Lean source · statement

instance (priority := 100) : IsScalarTower v.toValuationSubring F F'

theorem

neg_log_valuation_fiberCenter_eq_ord

L3795 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.neg_log_valuation_fiberCenter_eq_ord

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ValuationDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

theorem neg_log_valuation_fiberCenter_eq_ord (hw : w.restrict F = v) {x : F'} (hx : x ≠ 0) :
    -log ((fiberCenter F' v hw).valuation F' x) = w.ord x

theorem

le_ord_iff_mem_pow_fiberCenter

L3834 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.le_ord_iff_mem_pow_fiberCenter

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ValuationDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

theorem le_ord_iff_mem_pow_fiberCenter (hw : w.restrict F = v)
    {c : integralClosureAt F' v} (hc : c ≠ 0) (n : ℕ) :
    (n : ℤ) ≤ w.ord (algebraMap (integralClosureAt F' v) F' c) ↔
      c ∈ (fiberCenter F' v hw).asIdeal ^ n

theorem

ramificationIndex_eq_ramificationIdx_fiberCenter

L3857 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.ramificationIndex_eq_ramificationIdx_fiberCenter

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ValuationDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

theorem ramificationIndex_eq_ramificationIdx_fiberCenter (hw : w.restrict F = v) :
    w.ramificationIndex F =
      (IsLocalRing.maximalIdeal v.toValuationSubring).ramificationIdx'
        (fiberCenter F' v hw).asIdeal

def

toValuationSubringOfRestrictEq

L3890 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.toValuationSubringOfRestrictEq

Documentation

Map the integral closure into a valuation subring above the given place.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

def toValuationSubringOfRestrictEq (hw : w.restrict F = v) :
    integralClosureAt F' v →+* w.toValuationSubring

def

residueOfCenter

L3896 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.residueOfCenter

Documentation

The residue map on the integral closure at a place.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

def residueOfCenter (hw : w.restrict F = v) :
    integralClosureAt F' v →+* w.ResidueField

theorem

residueOfCenter_apply

L3901 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.residueOfCenter_apply

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}
omit [FiniteDimensional F F'] in

Lean source · statement

theorem residueOfCenter_apply (hw : w.restrict F = v) (c : integralClosureAt F' v) :
    residueOfCenter hw c = IsLocalRing.residue w.toValuationSubring
      (toValuationSubringOfRestrictEq hw c)

theorem

ker_residueOfCenter

L3905 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.ker_residueOfCenter

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

theorem ker_residueOfCenter (hw : w.restrict F = v) :
    RingHom.ker (residueOfCenter hw) = (fiberCenter F' v hw).asIdeal

theorem

surjective_residueOfCenter

L3911 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.surjective_residueOfCenter

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

theorem surjective_residueOfCenter (hw : w.restrict F = v) :
    Function.Surjective (residueOfCenter hw)

def

residueFieldEquivQuotientCenter

L3974 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.residueFieldEquivQuotientCenter

Documentation

Identify the quotient at the fibre center with the residue field above it.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

def residueFieldEquivQuotientCenter (hw : w.restrict F = v) :
    integralClosureAt F' v ⧸ (fiberCenter F' v hw).asIdeal ≃+* w.ResidueField

def

placeCongrEquiv

L3980 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.placeCongrEquiv

Documentation

Transport valuation subrings along equality of places.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · full declaration

def placeCongrEquiv {u u' : Place K F} (h : u = u') :
    u.toValuationSubring ≃+* u'.toValuationSubring where
  toFun x := ⟨x.1, h ▸ x.2⟩
  invFun x := ⟨x.1, h.symm ▸ x.2⟩
  left_inv _ := rfl
  right_inv _ := rfl
  map_mul' _ _ := rfl
  map_add' _ _ := rfl

def

restrictResidueFieldEquiv

L3990 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.restrictResidueFieldEquiv

Documentation

Identify residue fields using equality of a restricted place with the base place.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

def restrictResidueFieldEquiv (hw : w.restrict F = v) :
    (w.restrict F).ResidueField ≃+* IsLocalRing.ResidueField v.toValuationSubring

theorem

inertiaDeg_eq_inertiaDeg_fiberCenter

L3994 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.inertiaDeg_eq_inertiaDeg_fiberCenter

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']
attribute [local instance 0] valuationSubringAlgebra
section ResidueDictionary
variable {v : Place K F} {w : Place K F'}

Lean source · statement

theorem inertiaDeg_eq_inertiaDeg_fiberCenter (hw : w.restrict F = v) :
    w.inertiaDeg F =
      (fiberCenter F' v hw).asIdeal.inertiaDeg v.toValuationSubring

def

uniformizerSubring'''

L4068 ↗

AlgebraicCurve.Place.uniformizerSubring'''

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

def _root_.AlgebraicCurve.Place.uniformizerSubring''' :
  v.toValuationSubring

theorem

irreducible_uniformizerSubring'''

L4073 ↗

AlgebraicCurve.Place.irreducible_uniformizerSubring'''

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.irreducible_uniformizerSubring''' :
  Irreducible v.uniformizerSubring'''

theorem

maximalIdeal_eq_span_uniformizer

L4078 ↗

AlgebraicCurve.Place.maximalIdeal_eq_span_uniformizer

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.maximalIdeal_eq_span_uniformizer :
    IsLocalRing.maximalIdeal v.toValuationSubring = Ideal.span {v.uniformizerSubring'''}

theorem

D_polynomialAlgebra_uniformizer

L4083 ↗

AlgebraicCurve.Place.D_polynomialAlgebra_uniformizer

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.D_polynomialAlgebra_uniformizer :
    let := v.polynomialAlgebra
    KaehlerDifferential.D K[X] v.toValuationSubring v.uniformizerSubring''' = 0 := by
  let := v.polynomialAlgebra
  rw [show v.uniformizerSubring''' = algebraMap K[X] v.toValuationSubring X from
    (v.polynomialAlgebra_algebraMap_X).symm]
  exact (KaehlerDifferential.D K[X] v.toValuationSubring).map_algebraMap X

theorem

ker_algebraMap_residueField_eq_span

L4092 ↗

AlgebraicCurve.Place.ker_algebraMap_residueField_eq_span

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ker_algebraMap_residueField_eq_span :
    RingHom.ker (algebraMap v.toValuationSubring v.ResidueField)
      = Ideal.span {v.uniformizerSubring'''}

theorem

formallyUnramified_polynomial_residueField_of_isSeparable

L4099 ↗

AlgebraicCurve.Place.formallyUnramified_polynomial_residueField_of_isSeparable

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem
  _root_.AlgebraicCurve.Place.formallyUnramified_polynomial_residueField_of_isSeparable
    [Algebra.IsSeparable K (IsLocalRing.ResidueField v.toValuationSubring)] :
    let := v.polynomialAlgebra
    Algebra.FormallyUnramified K[X] (IsLocalRing.ResidueField v.toValuationSubring) := by
  let := v.polynomialAlgebra
  let := v.polynomialIsScalarTower
  have : Algebra.FormallyUnramified K (IsLocalRing.ResidueField v.toValuationSubring) :=
    Algebra.FormallyUnramified.of_isSeparable K _
  exact Algebra.FormallyUnramified.of_restrictScalars K K[X] _

theorem

subsingleton_residueKaehler_of_isSeparable

L4111 ↗

AlgebraicCurve.Place.subsingleton_residueKaehler_of_isSeparable

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.subsingleton_residueKaehler_of_isSeparable
    [Algebra.IsSeparable K (IsLocalRing.ResidueField v.toValuationSubring)] :
    let := v.polynomialAlgebra
    Subsingleton Ω[(IsLocalRing.ResidueField v.toValuationSubring)⁄K[X]] := by
  let := v.polynomialAlgebra
  have := v.formallyUnramified_polynomial_residueField_of_isSeparable
  exact Algebra.FormallyUnramified.subsingleton_kaehlerDifferential

theorem

subsingleton_residueTensor_polynomialKaehler_of_isSeparable

L4120 ↗

AlgebraicCurve.Place.subsingleton_residueTensor_polynomialKaehler_of_isSeparable

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem
  _root_.AlgebraicCurve.Place.subsingleton_residueTensor_polynomialKaehler_of_isSeparable
    [Algebra.IsSeparable K (IsLocalRing.ResidueField v.toValuationSubring)] :
    let := v.polynomialAlgebra
    Subsingleton (v.ResidueField ⊗[v.toValuationSubring] Ω[v.toValuationSubring⁄K[X]]) := by
  let := v.polynomialAlgebra
  let := v.polynomialIsScalarTower
  have := v.subsingleton_residueKaehler_of_isSeparable
  refine subsingleton_of_forall_eq 0 fun t => ?_
  obtain ⟨c, rfl⟩ :=
    ((KaehlerDifferential.exact_kerCotangentToTensor_mapBaseChange K[X]
      v.toValuationSubring v.ResidueField
      Ideal.Quotient.mk_surjective) t).mp (Subsingleton.elim _ _)
  obtain ⟨⟨x, hx⟩, rfl⟩ := Ideal.toCotangent_surjective _ c
  simp only [KaehlerDifferential.kerCotangentToTensor_toCotangent]
  rw [v.ker_algebraMap_residueField_eq_span] at hx
  induction hx using Submodule.span_induction with
  | mem y hy =>
    rcases hy with rfl
    rw [v.D_polynomialAlgebra_uniformizer, TensorProduct.tmul_zero]
  | zero => simp
  | add y z _ _ hy hz =>
    rw [map_add, TensorProduct.tmul_add, hy, hz, add_zero]
  | smul a y hy IH =>
    have hres : (algebraMap v.toValuationSubring v.ResidueField y) = 0 := by
      rw [IsLocalRing.ResidueField.algebraMap_eq, IsLocalRing.residue_eq_zero_iff,
        v.maximalIdeal_eq_span_uniformizer]
      exact hy
    rw [smul_eq_mul, Derivation.leibniz, TensorProduct.tmul_add,
      TensorProduct.tmul_smul, IH, smul_zero, zero_add,
      ← TensorProduct.smul_tmul, ← Algebra.algebraMap_eq_smul_one, hres,
      TensorProduct.zero_tmul]

theorem

finite_polynomialKaehler_of_finite_kaehler

L4154 ↗

AlgebraicCurve.Place.finite_polynomialKaehler_of_finite_kaehler

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.finite_polynomialKaehler_of_finite_kaehler
    [Module.Finite v.toValuationSubring Ω[v.toValuationSubring⁄K]] :
    let := v.polynomialAlgebra
    Module.Finite v.toValuationSubring Ω[v.toValuationSubring⁄K[X]] := by
  let := v.polynomialAlgebra
  let := v.polynomialIsScalarTower
  exact Module.Finite.of_surjective
    (KaehlerDifferential.map K K[X] v.toValuationSubring v.toValuationSubring)
    (KaehlerDifferential.map_surjective K K[X] v.toValuationSubring)

theorem

subsingleton_polynomialKaehler_of_isSeparable_of_finite

L4165 ↗

AlgebraicCurve.Place.subsingleton_polynomialKaehler_of_isSeparable_of_finite

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.subsingleton_polynomialKaehler_of_isSeparable_of_finite
    [Algebra.IsSeparable K (IsLocalRing.ResidueField v.toValuationSubring)]
    [Module.Finite v.toValuationSubring Ω[v.toValuationSubring⁄K]] :
    let := v.polynomialAlgebra
    Subsingleton Ω[v.toValuationSubring⁄K[X]] := by
  let := v.polynomialAlgebra
  have := v.finite_polynomialKaehler_of_finite_kaehler
  have := v.subsingleton_residueTensor_polynomialKaehler_of_isSeparable
  exact (IsLocalRing.subsingleton_tensorProduct (R := v.toValuationSubring)).mp ‹_›

theorem

isSeparable_residueField_of_charZero_of_finiteResidue

L4176 ↗

AlgebraicCurve.Place.isSeparable_residueField_of_charZero_of_finiteResidue

Source context 33 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.isSeparable_residueField_of_charZero_of_finiteResidue
    [CharZero K] [v.FiniteResidue] :
    Algebra.IsSeparable K (IsLocalRing.ResidueField v.toValuationSubring)

theorem

subsingleton_polynomialKaehler_of_charZero_of_finite

L4186 ↗

AlgebraicCurve.Place.subsingleton_polynomialKaehler_of_charZero_of_finite

Source context 34 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section ResidueEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.subsingleton_polynomialKaehler_of_charZero_of_finite
    [CharZero K] [v.FiniteResidue]
    [Module.Finite v.toValuationSubring Ω[v.toValuationSubring⁄K]] :
    let := v.polynomialAlgebra
    Subsingleton Ω[v.toValuationSubring⁄K[X]] := by
  have := v.isSeparable_residueField_of_charZero_of_finiteResidue
  exact v.subsingleton_polynomialKaehler_of_isSeparable_of_finite

def

ValSubringKaehlerFinite

L4202 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringKaehlerFinite

Documentation

The differentials of each valuation subring form a finite module.

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringKaehlerFinite : Prop

theorem

valSubringPolynomialFormallyUnramified_of_kaehlerFinite_of_charZero

L4207 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringPolynomialFormallyUnramified_of_kaehlerFinite_of_charZero

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringPolynomialFormallyUnramified_of_kaehlerFinite_of_charZero
    [CharZero K] [∀ v : Place K F, v.FiniteResidue]
    (hfin : ValSubringKaehlerFinite K F) :
    ValSubringPolynomialFormallyUnramified K F

theorem

valSubringKaehlerSpanTop_of_kaehlerFinite_of_charZero

L4215 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerSpanTop_of_kaehlerFinite_of_charZero

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
open scoped _root_.TensorProduct
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerSpanTop_of_kaehlerFinite_of_charZero
    [CharZero K] [∀ v : Place K F, v.FiniteResidue]
    (hfin : ValSubringKaehlerFinite K F) :
    ValSubringKaehlerSpanTop K F

theorem

exists_mem_poleSubmodule

L4271 ↗

AlgebraicCurve.Place.exists_mem_poleSubmodule

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.exists_mem_poleSubmodule (f : F) : ∃ n,
  f ∈ v.poleSubmodule n

theorem

uniformizer_pow_inv_mem_poleSubmodule_self

L4326 ↗

AlgebraicCurve.Place.uniformizer_pow_inv_mem_poleSubmodule_self

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.uniformizer_pow_inv_mem_poleSubmodule_self (m : ℕ) :
    (v.uniformizer ^ (m + 2))⁻¹ ∈ v.poleSubmodule (m + 2)

theorem

higherPoleMonomial_sum_mem_poleSubmodule

L4331 ↗

AlgebraicCurve.Place.higherPoleMonomial_sum_mem_poleSubmodule

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.higherPoleMonomial_sum_mem_poleSubmodule {s :
  Finset ℕ} {N : ℕ}
    (hs : ∀ m ∈ s, m < N) (c : ℕ → K) :
    ∑ m ∈ s, c m • (v.uniformizer ^ (m + 2))⁻¹ ∈ v.poleSubmodule (N + 1)

theorem

higherPoleMonomial_coeff_eq_zero_of_mem

L4341 ↗

AlgebraicCurve.Place.higherPoleMonomial_coeff_eq_zero_of_mem

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.higherPoleMonomial_coeff_eq_zero_of_mem (s :
  Finset ℕ) (c : ℕ → K)
    (hmem : ∑ m ∈ s, c m • (v.uniformizer ^ (m + 2))⁻¹ ∈ v.poleSubmodule 1) :
    ∀ m ∈ s, c m = 0

theorem

linearIndependent_higherPoleMonomial_mkQ

L4368 ↗

AlgebraicCurve.Place.linearIndependent_higherPoleMonomial_mkQ

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.linearIndependent_higherPoleMonomial_mkQ :
    LinearIndependent K
      (fun m : ℕ => (v.poleSubmodule 1).mkQ ((v.uniformizer ^ (m + 2))⁻¹)) := by
  rw [linearIndependent_iff']
  intro s c hsum m hm
  refine v.higherPoleMonomial_coeff_eq_zero_of_mem s c ?_ m hm
  have h := hsum
  simp only [← map_smul, ← map_sum] at h
  rwa [Submodule.mkQ_apply, Submodule.Quotient.mk_eq_zero] at h

def

higherPoleCorrectionAux

L4379 ↗

AlgebraicCurve.Place.higherPoleCorrectionAux

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.higherPoleCorrectionAux :
    Submodule.span K
        (Set.range fun m : ℕ => (v.poleSubmodule 1).mkQ ((v.uniformizer ^ (m + 2))⁻¹))
      →ₗ[K] v.ResidueField :=
  (Basis.span v.linearIndependent_higherPoleMonomial_mkQ).constr K
    (fun m => v.localResidueExtend ((v.uniformizer ^ (m + 2))⁻¹))

def

higherPoleCorrection

L4387 ↗

AlgebraicCurve.Place.higherPoleCorrection

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.higherPoleCorrection : F →ₗ[K] v.ResidueField

theorem

higherPoleCorrection_apply_of_mem

L4391 ↗

AlgebraicCurve.Place.higherPoleCorrection_apply_of_mem

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.higherPoleCorrection_apply_of_mem {f : F} (hf :
  f ∈ v.poleSubmodule 1) :
    v.higherPoleCorrection f = 0

theorem

higherPoleCorrection_uniformizer_pow_inv

L4399 ↗

AlgebraicCurve.Place.higherPoleCorrection_uniformizer_pow_inv

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.higherPoleCorrection_uniformizer_pow_inv (m : ℕ) :
    v.higherPoleCorrection ((v.uniformizer ^ (m + 2))⁻¹)
      = v.localResidueExtend ((v.uniformizer ^ (m + 2))⁻¹)

def

canonicalLocalResidueDataKOfExtend

L4415 ↗

AlgebraicCurve.Place.canonicalLocalResidueDataKOfExtend

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.canonicalLocalResidueDataKOfExtend :
  v.CanonicalLocalResidueDataK where
  res := v.localResidueExtend - v.higherPoleCorrection
  res_of_mem f hf := by
    have h1 : f ∈ v.poleSubmodule 1 :=
      v.poleSubmodule_one ▸ v.mem_simplePoleSubmodule_of_mem hf
    rw [LinearMap.sub_apply, v.higherPoleCorrection_apply_of_mem h1, sub_zero,
      v.localResidueExtend_apply_of_mem (v.mem_simplePoleSubmodule_of_mem hf),
      v.simplePoleResidueAux_eq_zero_of_mem hf]
  res_simplePole f hf := by
    have h1 : f ∈ v.poleSubmodule 1 := v.poleSubmodule_one ▸ hf
    rw [LinearMap.sub_apply, v.higherPoleCorrection_apply_of_mem h1, sub_zero,
      v.localResidueExtend_apply_of_mem hf, simplePoleResidueAux_apply]
  res_higherPoleMonomial n hn := by
    obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hn
    rw [show 1 + m + 1 = m + 2 from by omega, LinearMap.sub_apply,
      v.higherPoleCorrection_uniformizer_pow_inv m, sub_self]

theorem

res_algebraMap_mul_uniformizer_pow_inv

L4492 ↗

AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_algebraMap_mul_uniformizer_pow_inv

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
namespace CanonicalLocalResidueDataK
open scoped _root_.AlgebraicCurve.Place.CanonicalLocalResidueDataK in

Lean source · statement

theorem
  _root_.AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_algebraMap_mul_uniformizer_pow_inv
    (R : v.CanonicalLocalResidueDataK) {n : ℕ} (hn : 1 ≤ n) (c : K) :
    R.res ((algebraMap K F c) * (v.uniformizer ^ (n + 1))⁻¹) = 0

theorem

gate_canonicalLocalResidueDataK_uniformizer_inv

L4504 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.gate_canonicalLocalResidueDataK_uniformizer_inv

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem gate_canonicalLocalResidueDataK_uniformizer_inv (v : Place K F)
    (R : v.CanonicalLocalResidueDataK) : R.res v.uniformizer⁻¹ = 1

theorem

ag9b15u_eq_zero_or_one_le_ord_of_residue_eq_zero

L4545 ↗

ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed.ag9b15u_eq_zero_or_one_le_ord_of_residue_eq_zero

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace ModularCurve
namespace MilneAvAg9bRd15UnitNormalFormLaurentSeed
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open scoped _root_.Polynomial
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed in
with_weak_namespace _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed

Lean source · statement

theorem ag9b15u_eq_zero_or_one_le_ord_of_residue_eq_zero
    {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
    {g : F} (hg : g ∈ v.toValuationSubring)
    (h0 : IsLocalRing.residue _ (⟨g, hg⟩ : v.toValuationSubring) = 0) :
    g = 0 ∨ 1 ≤ v.ord g

theorem

ag9b15u_exists_unit_normal_form_of_surj

L4571 ↗

ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed.ag9b15u_exists_unit_normal_form_of_surj

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace ModularCurve
namespace MilneAvAg9bRd15UnitNormalFormLaurentSeed
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open scoped _root_.Polynomial
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
open _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
  (ag9b15u_eq_zero_or_one_le_ord_of_residue_eq_zero)
in
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed in
with_weak_namespace _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed

Lean source · statement

theorem ag9b15u_exists_unit_normal_form_of_surj
    {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    {w : F} (hw0 : w ≠ 0) (hw : v.ord w = 0) :
    ∃ (c : K) (s : F), c ≠ 0 ∧ (s = 0 ∨ 1 ≤ v.ord s) ∧
      w = algebraMap K F c * (1 + s)

theorem

ag9b15u_exists_K_truncation_of_mem_poleSubmodule

L4622 ↗

ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed.ag9b15u_exists_K_truncation_of_mem_poleSubmodule

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace ModularCurve
namespace MilneAvAg9bRd15UnitNormalFormLaurentSeed
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open scoped _root_.Polynomial
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed in
with_weak_namespace _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed

Lean source · statement

theorem ag9b15u_exists_K_truncation_of_mem_poleSubmodule
    {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (n : ℕ) {f : F} (hf : f ∈ v.poleSubmodule n) :
    ∃ c : ℕ → K,
      f - ∑ j ∈ Finset.range n, algebraMap K F (c j) * (v.uniformizer ^ (j + 1))⁻¹
        ∈ v.toValuationSubring

theorem

ag9b15u_differentialCoeff_D_unit_mul_uniformizer

L4665 ↗

ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed.ag9b15u_differentialCoeff_D_unit_mul_uniformizer

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace ModularCurve
namespace MilneAvAg9bRd15UnitNormalFormLaurentSeed
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open scoped _root_.Polynomial
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed in
with_weak_namespace _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed

Lean source · statement

theorem ag9b15u_differentialCoeff_D_unit_mul_uniformizer
    {K F : Type*} [Field K] [Field F] [Algebra K F]
    (v : Place K F) [v.DCoordGenerates] [Nontrivial Ω[F⁄K]] (w : F) :
    v.differentialCoeff (KaehlerDifferential.D K F (w * v.uniformizer))
      = w + v.uniformizer * v.differentialCoeff (KaehlerDifferential.D K F w)

def

aCoeff

L4710 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff

Documentation

A Taylor coefficient in the completed valuation subring for the chosen residue section.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

def aCoeff (v : Place K F) (S : Lg37CompletionSection v)
    (n : ℕ) (x : lg37_completion v) : v.ResidueField

theorem

mp72a103_t2_evalDepth_of_uniformizer_mul

L4743 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a103T2.mp72a103_t2_evalDepth_of_uniformizer_mul

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace Mp72a103T2
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Mp72a102T3
open Mp72a102T2
section Calculus
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a103_t2_evalDepth_of_uniformizer_mul (v : Place K F) {n : ℕ}
    {z : lg37_completion v}
    (h : AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) (n + 1)
        (algebraMap v.toValuationSubring (lg37_completion v)
            v.uniformizerSubring * z) = 0) :
    AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n z = 0

theorem

mp72a103_t2_lift_eq_zero_of_depth_one

L4768 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a103T2.mp72a103_t2_lift_eq_zero_of_depth_one

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace Mp72a103T2
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Mp72a102T3
open Mp72a102T2
section Calculus
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a103_t2_lift_eq_zero_of_depth_one (v : Place K F)
    (S : Lg37CompletionSection v) {c : v.ResidueField}
    (h : AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) 1 (S.lift c) = 0) :
    c = 0

theorem

mp72a103_t2_taylor_coeff_eq_zero_of_depth

L4776 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a103T2.mp72a103_t2_taylor_coeff_eq_zero_of_depth

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.IsLocalRing
open _root_.Polynomial
open scoped _root_.Polynomial
namespace Mp72a103T2
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open ModularCurve.Lg37 ModularCurve.Mp72a102T3
open Mp72a102T2
section Calculus
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem mp72a103_t2_taylor_coeff_eq_zero_of_depth (v : Place K F)
    (S : Lg37CompletionSection v) :
    ∀ (n : ℕ) (a : ℕ → v.ResidueField),
      AdicCompletion.evalₐ (maximalIdeal v.toValuationSubring) n
        (∑ i ∈ Finset.range n, S.lift (a i)
          * algebraMap v.toValuationSubring (lg37_completion v)
              v.uniformizerSubring ^ i) = 0 →
      ∀ i < n, a i = 0

abbrev

dX

L4887 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.dX

Documentation

The differential of the rational function field coordinate.

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

abbrev dX : Ω[(RatFunc K)⁄K]

theorem

aeval_ratFuncX_eq_algebraMap

L4889 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.aeval_ratFuncX_eq_algebraMap

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem aeval_ratFuncX_eq_algebraMap (q : K[X]) :
    aeval (RatFunc.X : RatFunc K) q = algebraMap K[X] (RatFunc K) q

theorem

D_algebraMap_polynomial

L4896 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.D_algebraMap_polynomial

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem D_algebraMap_polynomial (q : K[X]) :
    KaehlerDifferential.D K (RatFunc K) (algebraMap K[X] (RatFunc K) q)
      = algebraMap K[X] (RatFunc K) q.derivative • dX K

theorem

denom_sq_smul_D_eq

L4902 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.denom_sq_smul_D_eq

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem denom_sq_smul_D_eq (f : RatFunc K) :
    (algebraMap K[X] (RatFunc K) f.denom) ^ 2 • KaehlerDifferential.D K (RatFunc K) f
      = algebraMap K[X] (RatFunc K)
          (f.num.derivative * f.denom - f.num * f.denom.derivative) • dX K

theorem

span_dX_eq_top

L4920 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.span_dX_eq_top

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem span_dX_eq_top :
    Submodule.span (RatFunc K) {dX K} = ⊤

theorem

dX_ne_zero

L4934 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.dX_ne_zero

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem dX_ne_zero [Nontrivial Ω[(RatFunc K)⁄K]] : dX K ≠ 0

theorem

not_dvd_derivative_of_sq_not_dvd

L4942 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.not_dvd_derivative_of_sq_not_dvd

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem not_dvd_derivative_of_sq_not_dvd {p : K[X]} (hp : Irreducible p) (hsep : p.Separable)
    {q : K[X]} (hpq : p ∣ q) (hpq2 : ¬ p ^ 2 ∣ q) : ¬ p ∣ q.derivative

theorem

not_dvd_derivative_of_ord_eq_one

L4957 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.not_dvd_derivative_of_ord_eq_one

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}

Lean source · statement

theorem not_dvd_derivative_of_ord_eq_one {w : HeightOneSpectrum K[X]} {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) (hsep : p.Separable)
    {q : K[X]} (hq : q ≠ 0)
    (hord : (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).ord
      (algebraMap K[X] (RatFunc K) q) = 1) :
    ¬ p ∣ q.derivative

theorem

uniformizer_ne_zero'

L4998 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.uniformizer_ne_zero'

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section NumDenom
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem uniformizer_ne_zero' : (v).uniformizer ≠ 0

theorem

ord_algebraMap_denom_uniformizer_eq_zero

L5001 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_algebraMap_denom_uniformizer_eq_zero

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section NumDenom
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ord_algebraMap_denom_uniformizer_eq_zero {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) :
    (v).ord (algebraMap K[X] (RatFunc K) (v).uniformizer.denom) = 0

theorem

ord_algebraMap_num_uniformizer_eq_one

L5030 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_algebraMap_num_uniformizer_eq_one

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section NumDenom
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ord_algebraMap_num_uniformizer_eq_one {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) :
    (v).ord (algebraMap K[X] (RatFunc K) (v).uniformizer.num) = 1

theorem

not_dvd_num'denom_sub_numdenom'

L5044 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.not_dvd_num'denom_sub_numdenom'

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section NumDenom
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem not_dvd_num'denom_sub_numdenom' {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) (hsep : p.Separable) :
    ¬ p ∣ ((v).uniformizer.num.derivative * (v).uniformizer.denom
            - (v).uniformizer.num * (v).uniformizer.denom.derivative)

theorem

ord_differentialCoeff_dX_ofHeightOneSpectrum

L5081 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_differentialCoeff_dX_ofHeightOneSpectrum

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section PerPlace
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ord_differentialCoeff_dX_ofHeightOneSpectrum {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) (hsep : p.Separable) :
    (v).ord ((v).differentialCoeff (dX K)) = 0

theorem

differentialCoeff_dX_mem_ofHeightOneSpectrum

L5116 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_dX_mem_ofHeightOneSpectrum

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section PerPlace
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem differentialCoeff_dX_mem_ofHeightOneSpectrum {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) (hsep : p.Separable) :
    (v).differentialCoeff (dX K) ∈ (v).toValuationSubring

theorem

p1DifferentialCoeffRegularFinite_dX

L5135 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1DifferentialCoeffRegularFinite_dX

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]
variable {K}
section Discharge
variable (K)
variable [CharZero K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F := RatFunc K)]
  [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem p1DifferentialCoeffRegularFinite_dX :
    P1DifferentialCoeffRegularFinite K (dX_ne_zero K)

theorem

ord_ofHeightOneSpectrum_irreducible_eq_zero_of_ne

L5181 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_ofHeightOneSpectrum_irreducible_eq_zero_of_ne

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ord_ofHeightOneSpectrum_irreducible_eq_zero_of_ne
    (w : HeightOneSpectrum K[X]) {p : K[X]} (hp : Irreducible p)
    (hne : w ≠ heightOneSpectrumOfIrreducible K hp) :
    (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).ord
      (algebraMap K[X] (RatFunc K) p) = 0

theorem

ord_algebraMap_irreducible_eq_zero_of_ne

L5194 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_algebraMap_irreducible_eq_zero_of_ne

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)

Lean source · statement

theorem ord_algebraMap_irreducible_eq_zero_of_ne
    {p : K[X]} (hp : Irreducible p) {v : Place K (RatFunc K)}
    (hvp : v ≠ finitePlace K hp) (hvinf : v ≠ placeInfty K) :
    v.ord (algebraMap K[X] (RatFunc K) p) = 0

theorem

inv_algebraMap_pow_mem_of_ne_finitePlace

L5203 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.inv_algebraMap_pow_mem_of_ne_finitePlace

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)

Lean source · statement

theorem inv_algebraMap_pow_mem_of_ne_finitePlace
    {p : K[X]} (hp : Irreducible p) {v : Place K (RatFunc K)}
    (hvp : v ≠ finitePlace K hp) (hvinf : v ≠ placeInfty K) (m : ℕ) :
    ((algebraMap K[X] (RatFunc K) p) ^ m)⁻¹ ∈ v.toValuationSubring

theorem

p1PrincipalPartAtom_mem_of_ne_finitePlace

L5213 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartAtom_mem_of_ne_finitePlace

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)

Lean source · statement

theorem p1PrincipalPartAtom_mem_of_ne_finitePlace
    {p : K[X]} (hp : Irreducible p) (c : K[X]) (m : ℕ) {v : Place K (RatFunc K)}
    (hvp : v ≠ finitePlace K hp) (hvinf : v ≠ placeInfty K) :
    p1PrincipalPartAtom K p c m ∈ v.toValuationSubring

theorem

ord_placeInfty_p1PrincipalPartAtom

L5231 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_p1PrincipalPartAtom

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)

Lean source · statement

theorem ord_placeInfty_p1PrincipalPartAtom
    {p : K[X]} (hp : p ≠ 0) {c : K[X]} (hc : c ≠ 0) (m : ℕ) :
    (placeInfty K).ord (p1PrincipalPartAtom K p c m)
      = (m : ℤ) * p.natDegree - c.natDegree

theorem

one_le_ord_placeInfty_p1PrincipalPartAtom

L5244 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.one_le_ord_placeInfty_p1PrincipalPartAtom

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)

Lean source · statement

theorem one_le_ord_placeInfty_p1PrincipalPartAtom
    {p : K[X]} (hp : Irreducible p) {c : K[X]} (hc : c ≠ 0) {m : ℕ}
    (hdeg : c.degree < p.degree) (hm : 1 ≤ m) :
    1 ≤ (placeInfty K).ord (p1PrincipalPartAtom K p c m)

def

KaehlerRankOne

L5301 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.KaehlerRankOne

Documentation

The function field differential module is free of rank one.

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def KaehlerRankOne : Prop

theorem

of_finiteResidue_of_kaehlerRankOne

L5310 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.IsCurveOver.of_finiteResidue_of_kaehlerRankOne

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}
open scoped _root_.AlgebraicCurve.IsCurveOver in

Lean source · full declaration

theorem IsCurveOver.of_finiteResidue_of_kaehlerRankOne
    [HasPrincipalDivisors K F] [∀ v : Place K F, v.FiniteResidue]
    (hΩ : KaehlerRankOne K F) : IsCurveOver K F where
  finiteResidue v := Place.FiniteResidue.finite (v := v)
  kaehler_free_rank_one := hΩ

theorem

isCurveOver_of_kaehlerRankOne

L5323 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.isCurveOver_of_kaehlerRankOne

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K]

Lean source · statement

theorem isCurveOver_of_kaehlerRankOne (hΩ : KaehlerRankOne K (RatFunc K)) :
    IsCurveOver K (RatFunc K)

def

RamificationInertiaIdentity

L5358 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RamificationInertiaIdentity

Documentation

The degree sum over places above a base place equals the field extension degree contribution.

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Identity
variable (K F F' : Type*) [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [Algebra.IsIntegral F F']

Lean source · statement

def RamificationInertiaIdentity : Prop

theorem

degree_eq_sum_support

L5377 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Divisor.degree_eq_sum_support

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Divisor
open _root_.AlgebraicCurve.Divisor
open _root_.AlgebraicCurve.Divisor
section Pullback
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [Algebra.IsIntegral F F']

Lean source · statement

theorem degree_eq_sum_support (D : Divisor K F) :
    degree D = ∑ v ∈ D.support, D v * (v.deg : ℤ)

theorem

degree_eq_finrank_mul_of_forall_eq_ord_algebraMap

L5382 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Divisor.degree_eq_finrank_mul_of_forall_eq_ord_algebraMap

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Divisor
open _root_.AlgebraicCurve.Divisor
open _root_.AlgebraicCurve.Divisor
section Pullback
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [Algebra.IsIntegral F F']

Lean source · statement

theorem degree_eq_finrank_mul_of_forall_eq_ord_algebraMap
    (H : RamificationInertiaIdentity K F F') {g : F}
    {D' : Divisor K F'} (hD' : ∀ w : Place K F', D' w = w.ord (algebraMap F F' g))
    {D : Divisor K F} (hD : ∀ v : Place K F, D v = v.ord g) :
    degree D' = (Module.finrank F F' : ℤ) * degree D

theorem

ord_prod

L5429 ↗

AlgebraicCurve.Place.ord_prod

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Divisor
open _root_.AlgebraicCurve.Divisor
open _root_.AlgebraicCurve.Divisor
section Galois
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_prod {ι : Type*} (v : Place K F') (s : Finset ι)
    (g : ι → F') (hg : ∀ i ∈ s, g i ≠ 0) :
    v.ord (∏ i ∈ s, g i) = ∑ i ∈ s, v.ord (g i)

theorem

sum_smul_apply_eq_ord_prod

L5443 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Divisor.sum_smul_apply_eq_ord_prod

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Divisor
open _root_.AlgebraicCurve.Divisor
open _root_.AlgebraicCurve.Divisor
section Galois
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
variable [FiniteDimensional F F']

Lean source · statement

theorem sum_smul_apply_eq_ord_prod {f : F'} (hf : f ≠ 0)
    {D : Divisor K F'} (hD : ∀ w : Place K F', D w = w.ord f) (w : Place K F') :
    (∑ σ : F' ≃ₐ[F] F', (AlgEquiv.restrictScalars K σ) • D) w
      = w.ord (∏ σ : F' ≃ₐ[F] F', σ f)

theorem

degree_eq_zero_of_isGalois

L5456 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Divisor.degree_eq_zero_of_isGalois

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Divisor
open _root_.AlgebraicCurve.Divisor
open _root_.AlgebraicCurve.Divisor
section Galois
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
variable [FiniteDimensional F F']

Lean source · statement

theorem degree_eq_zero_of_isGalois [IsGalois F F'] [HasPrincipalDivisors K F]
    (H : RamificationInertiaIdentity K F F')
    {f : F'} {D : Divisor K F'} (hD : ∀ w : Place K F', D w = w.ord f) :
    degree D = 0

def

principalDivisorOf

L5507 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.principalDivisorOf

Documentation

The principal divisor of a nonzero function in the extension field.

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] {F' : Type*} [Field F'] [Algebra K F']
  [Algebra (RatFunc K) F'] [IsScalarTower K (RatFunc K) F']
  [FiniteDimensional (RatFunc K) F'] [Algebra.IsSeparable (RatFunc K) F']

Lean source · statement

def principalDivisorOf {f : F'} (hf : f ≠ 0) : Divisor K F'

theorem

degree_eq_zero_of_forall_eq_ord_of_isGalois

L5511 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.degree_eq_zero_of_forall_eq_ord_of_isGalois

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] {F' : Type*} [Field F'] [Algebra K F']
  [Algebra (RatFunc K) F'] [IsScalarTower K (RatFunc K) F']
  [FiniteDimensional (RatFunc K) F'] [Algebra.IsSeparable (RatFunc K) F']

Lean source · statement

theorem degree_eq_zero_of_forall_eq_ord_of_isGalois [IsGalois (RatFunc K) F']
    (H : RamificationInertiaIdentity K (RatFunc K) F')
    {f : F'} {D : Divisor K F'} (hD : ∀ w : Place K F', D w = w.ord f) :
    Divisor.degree D = 0

theorem

hasPrincipalDivisors_of_isGalois

L5517 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.hasPrincipalDivisors_of_isGalois

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] {F' : Type*} [Field F'] [Algebra K F']
  [Algebra (RatFunc K) F'] [IsScalarTower K (RatFunc K) F']
  [FiniteDimensional (RatFunc K) F'] [Algebra.IsSeparable (RatFunc K) F']

Lean source · full declaration

theorem hasPrincipalDivisors_of_isGalois [IsGalois (RatFunc K) F']
    (_H : RamificationInertiaIdentity K (RatFunc K) F') :
    HasPrincipalDivisors K F' where
  exists_divisor _ hf :=
    ⟨principalDivisorOf hf, fun _ => rfl,
      degree_eq_zero_of_forall_eq_ord_of_isGalois _H fun _ => rfl⟩

theorem

surjective_algebraMap_residueField_placeInfty

L5566 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.surjective_algebraMap_residueField_placeInfty

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section SurjectivePlaceInfty
variable (K) [DecidableEq (RatFunc K)]

Lean source · statement

theorem surjective_algebraMap_residueField_placeInfty :
    Function.Surjective (algebraMap K (placeInfty K).ResidueField)

def

CanonicalLocalResidueKDifferentialCoordIndep

L5582 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.CanonicalLocalResidueKDifferentialCoordIndep

Documentation

Higher pole monomials have zero residue in any uniformizer differential coordinate.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section NamedRow
variable (K F)
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]

Lean source · statement

def CanonicalLocalResidueKDifferentialCoordIndep : Prop

theorem

ratFuncX_inv_pow_inv

L5602 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncX_inv_pow_inv

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section XInvCoordinate
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem ratFuncX_inv_pow_inv (m : ℕ) :
    (((RatFunc.X : RatFunc K)⁻¹) ^ m)⁻¹ = (RatFunc.X : RatFunc K) ^ m

theorem

X_pow_mul_differentialCoeff_D_X_eq_neg

L5606 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.X_pow_mul_differentialCoeff_D_X_eq_neg

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section XInvCoordinate
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem X_pow_mul_differentialCoeff_D_X_eq_neg (n : ℕ) :
    (RatFunc.X : RatFunc K) ^ n
        * (placeInfty K).differentialCoeff (KaehlerDifferential.D K (RatFunc K) RatFunc.X)
      = -((placeInfty K).differentialCoeff
            (KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)⁻¹)
          * (((RatFunc.X : RatFunc K)⁻¹) ^ (n + 2))⁻¹)

theorem

canonicalLocalResidueDataK_res_X_pow_mul_D_X_of_coordIndep

L5622 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_res_X_pow_mul_D_X_of_coordIndep

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section Bridge
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem canonicalLocalResidueDataK_res_X_pow_mul_D_X_of_coordIndep
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (R : (placeInfty K).CanonicalLocalResidueDataK) (n : ℕ) :
    R.res ((RatFunc.X : RatFunc K) ^ n
        * (placeInfty K).differentialCoeff
            (KaehlerDifferential.D K (RatFunc K) RatFunc.X)) = 0

theorem

canonicalLocalResidueDataK_kaehlerResidueTerm_X_pow_of_coordIndep

L5632 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_kaehlerResidueTerm_X_pow_of_coordIndep

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section Bridge
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem canonicalLocalResidueDataK_kaehlerResidueTerm_X_pow_of_coordIndep
    (_hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (R : (placeInfty K).CanonicalLocalResidueDataK) (n : ℕ) :
    Algebra.trace K (placeInfty K).ResidueField
        (R.res (diagonalHom K (RatFunc K) ((RatFunc.X : RatFunc K) ^ n) (placeInfty K)
          * (placeInfty K).differentialCoeff
              (KaehlerDifferential.D K (RatFunc K) RatFunc.X))) = 0

theorem

finite_kaehler_of_essFiniteType

L5690 ↗

AlgebraicCurve.Place.finite_kaehler_of_essFiniteType

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section EftEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.finite_kaehler_of_essFiniteType
  [Algebra.EssFiniteType K v.toValuationSubring] :
    Module.Finite v.toValuationSubring Ω[v.toValuationSubring⁄K]

def

ValSubringEssFiniteType

L5703 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringEssFiniteType

Documentation

Every valuation subring is essentially of finite type over the base field.

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringEssFiniteType : Prop

theorem

valSubringKaehlerFinite_of_essFiniteType

L5708 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerFinite_of_essFiniteType

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerFinite_of_essFiniteType
    (heft : ValSubringEssFiniteType K F) :
    ValSubringKaehlerFinite K F

theorem

aCoeff_zero

L5755 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_zero

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_zero (v : Place K F) (S : Lg37CompletionSection v)
    (x : lg37_completion v) :
    aCoeff v S 0 x = lg37_residueHat v x

theorem

aCoeff_add

L5767 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_add

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_add (v : Place K F) (S : Lg37CompletionSection v) (m : ℕ)
    (x y : lg37_completion v) :
    aCoeff v S m (x + y) = aCoeff v S m x + aCoeff v S m y

theorem

aCoeff_smul

L5817 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_smul

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_smul (v : Place K F) (S : Lg37CompletionSection v) (m : ℕ) (c : K)
    (x : lg37_completion v) :
    aCoeff v S m (algebraMap K (lg37_completion v) c * x)
      = algebraMap K v.ResidueField c * aCoeff v S m x

theorem

aCoeff_shift

L5876 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_shift

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_shift (v : Place K F) (S : Lg37CompletionSection v) (n : ℕ)
    (x : lg37_completion v) :
    aCoeff v S (n + 1)
        (algebraMap v.toValuationSubring (lg37_completion v) v.uniformizerSubring * x)
      = aCoeff v S n x

theorem

aCoeff_shift_pow

L5943 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_shift_pow

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_shift_pow (v : Place K F) (S : Lg37CompletionSection v) (n k : ℕ)
    (x : lg37_completion v) :
    aCoeff v S (n + k)
        ((algebraMap v.toValuationSubring (lg37_completion v) v.uniformizerSubring) ^ k
          * x)
      = aCoeff v S n x

theorem

aCoeff_one_eq_zero

L5960 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_one_eq_zero

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_one_eq_zero (v : Place K F) (S : Lg37CompletionSection v)
    {n : ℕ} (hn : 1 ≤ n) : aCoeff v S n 1 = 0

theorem

clearPow_mem

L6002 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.clearPow_mem

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem clearPow_mem (v : Place K F) (f : F) {N : ℕ} (hN : (-v.ord f).toNat ≤ N) :
    v.uniformizer ^ (N + 1) * f ∈ v.toValuationSubring

def

clearedHat

L6013 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.clearedHat

Documentation

Embed a function in the completion after clearing its pole with a uniformizer power.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

def clearedHat (v : Place K F) (f : F) {N : ℕ} (hN : (-v.ord f).toNat ≤ N) :
    lg37_completion v

def

resStar

L6019 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar

Documentation

The residue coefficient extracted from the pole-cleared completed function.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

def resStar (v : Place K F) (S : Lg37CompletionSection v) (f : F) :
    v.ResidueField

theorem

aCoeff_clearedHat_of_le

L6023 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_clearedHat_of_le

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem aCoeff_clearedHat_of_le (v : Place K F) (S : Lg37CompletionSection v) (f : F)
    {N : ℕ} (hN : (-v.ord f).toNat ≤ N) :
    aCoeff v S N (clearedHat v f hN) = resStar v S f

theorem

resStar_add

L6038 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_add

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem resStar_add (v : Place K F) (S : Lg37CompletionSection v) (f g : F) :
    resStar v S (f + g) = resStar v S f + resStar v S g

theorem

resStar_smul

L6054 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_smul

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem resStar_smul (v : Place K F) (S : Lg37CompletionSection v) (c : K) (f : F) :
    resStar v S (c • f) = c • resStar v S f

def

resStarₗ

L6070 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStarₗ

Documentation

The residue coefficient map as a linear map over the base field.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · full declaration

def resStarₗ (v : Place K F) (S : Lg37CompletionSection v) :
    F →ₗ[K] v.ResidueField where
  toFun := resStar v S
  map_add' := resStar_add v S
  map_smul' c f := by rw [RingHom.id_apply]; exact resStar_smul v S c f

theorem

resStar_simplePole

L6076 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_simplePole

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem resStar_simplePole (v : Place K F) (S : Lg37CompletionSection v) (f : F)
    (hf : v.uniformizer * f ∈ v.toValuationSubring) :
    resStar v S f = IsLocalRing.residue _ ⟨v.uniformizer * f, hf⟩

theorem

resStar_of_mem

L6096 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_of_mem

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem resStar_of_mem (v : Place K F) (S : Lg37CompletionSection v) {f : F}
    (hf : f ∈ v.toValuationSubring) : resStar v S f = 0

theorem

resStar_higherPoleMonomial

L6104 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_higherPoleMonomial

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem resStar_higherPoleMonomial (v : Place K F) (S : Lg37CompletionSection v)
    {n : ℕ} (hn : 1 ≤ n) : resStar v S (v.uniformizer ^ (n + 1))⁻¹ = 0

def

canonicalLocalResidueDataKStar

L6126 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.KwNo6Pin.canonicalLocalResidueDataKStar

Documentation

Canonical local residue data constructed from the completed residue section.

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.Mp72a102T3 Mp72a103T2 Mp72a102T1
open ModularCurve.KwNo6Section
namespace ModularCurve
namespace KwNo6Pin
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · full declaration

def canonicalLocalResidueDataKStar (v : Place K F)
    (S : Lg37CompletionSection v) : v.CanonicalLocalResidueDataK where
  res := resStarₗ v S
  res_of_mem _f hf := resStar_of_mem v S hf
  res_simplePole f hf := resStar_simplePole v S f hf
  res_higherPoleMonomial _n hn := resStar_higherPoleMonomial v S hn

theorem

natDegree_numDenomWronskian_lt

L6171 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.natDegree_numDenomWronskian_lt

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section WronskianBound
variable {K : Type*} [Field K]

Lean source · statement

theorem natDegree_numDenomWronskian_lt {n d : K[X]} (hn : n ≠ 0) (hd : d ≠ 0)
    (hW : n.derivative * d - n * d.derivative ≠ 0) :
    (n.derivative * d - n * d.derivative).natDegree < n.natDegree + d.natDegree

def

ratFuncDXCoeff

L6193 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff

Documentation

The rational function derivative given by the quotient rule.

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeff
variable (K : Type*) [Field K]

Lean source · statement

def ratFuncDXCoeff (f : RatFunc K) : RatFunc K

theorem

ratFuncDXCoeff_def

L6197 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_def

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeff
variable (K : Type*) [Field K]

Lean source · statement

theorem ratFuncDXCoeff_def (f : RatFunc K) :
    ratFuncDXCoeff K f
      = algebraMap K[X] (RatFunc K) (f.num.derivative * f.denom - f.num * f.denom.derivative)
        / (algebraMap K[X] (RatFunc K) f.denom) ^ 2

theorem

D_eq_ratFuncDXCoeff_smul_dX

L6202 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.D_eq_ratFuncDXCoeff_smul_dX

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeff
variable (K : Type*) [Field K]

Lean source · statement

theorem D_eq_ratFuncDXCoeff_smul_dX (f : RatFunc K) :
    KaehlerDifferential.D K (RatFunc K) f = ratFuncDXCoeff K f • dX K

theorem

wronskian_ne_zero_of_ratFuncDXCoeff_ne_zero

L6209 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.wronskian_ne_zero_of_ratFuncDXCoeff_ne_zero

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeff
variable (K : Type*) [Field K]

Lean source · statement

theorem wronskian_ne_zero_of_ratFuncDXCoeff_ne_zero {f : RatFunc K}
    (h : ratFuncDXCoeff K f ≠ 0) :
    f.num.derivative * f.denom - f.num * f.denom.derivative ≠ 0

theorem

ord_algebraMap_denom_eq_zero_of_ord_eq_one

L6224 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_algebraMap_denom_eq_zero_of_ord_eq_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section FinitePlaces
variable {K : Type*} [Field K]
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ord_algebraMap_denom_eq_zero_of_ord_eq_one {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) {f : RatFunc K}
    (hf : f ≠ 0) (hord : (v).ord f = 1) :
    (v).ord (algebraMap K[X] (RatFunc K) f.denom) = 0

theorem

ord_algebraMap_num_eq_one_of_ord_eq_one

L6252 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_algebraMap_num_eq_one_of_ord_eq_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section FinitePlaces
variable {K : Type*} [Field K]
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ord_algebraMap_num_eq_one_of_ord_eq_one {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) {f : RatFunc K}
    (hf : f ≠ 0) (hord : (v).ord f = 1) :
    (v).ord (algebraMap K[X] (RatFunc K) f.num) = 1

theorem

not_dvd_wronskian_of_ord_eq_one

L6265 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.not_dvd_wronskian_of_ord_eq_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section FinitePlaces
variable {K : Type*} [Field K]
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem not_dvd_wronskian_of_ord_eq_one {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) (hsep : p.Separable)
    {f : RatFunc K} (hf : f ≠ 0) (hord : (v).ord f = 1) :
    ¬ p ∣ (f.num.derivative * f.denom - f.num * f.denom.derivative)

theorem

ratFuncDXCoeff_ne_zero_and_ord_eq_zero_of_ord_eq_one

L6292 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_ne_zero_and_ord_eq_zero_of_ord_eq_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section FinitePlaces
variable {K : Type*} [Field K]
variable {w : HeightOneSpectrum K[X]}
local notation "v" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ratFuncDXCoeff_ne_zero_and_ord_eq_zero_of_ord_eq_one {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) (hsep : p.Separable)
    {f : RatFunc K} (hf : f ≠ 0) (hord : (v).ord f = 1) :
    ratFuncDXCoeff K f ≠ 0 ∧ (v).ord (ratFuncDXCoeff K f) = 0

theorem

ord_placeInfty_ratFuncDXCoeff_ge

L6324 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_ratFuncDXCoeff_ge

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftySide
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_placeInfty_ratFuncDXCoeff_ge {g : RatFunc K} (hg : g ≠ 0)
    (hW : g.num.derivative * g.denom - g.num * g.denom.derivative ≠ 0) :
    (placeInfty K).ord g + 1 ≤ (placeInfty K).ord (ratFuncDXCoeff K g)

theorem

exists_dXCoeff_ord_ge_two_of_ord_placeInfty_eq_zero

L6350 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_dXCoeff_ord_ge_two_of_ord_placeInfty_eq_zero

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftySide
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem exists_dXCoeff_ord_ge_two_of_ord_placeInfty_eq_zero {g : RatFunc K}
    (hg : g ≠ 0) (hord : (placeInfty K).ord g = 0) :
    ∃ e : RatFunc K, (e = 0 ∨ 2 ≤ (placeInfty K).ord e) ∧
      KaehlerDifferential.D K (RatFunc K) g = e • dX K

theorem

exists_unit_dXCoeff_of_ord_placeInfty_eq_neg_one

L6380 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_unit_dXCoeff_of_ord_placeInfty_eq_neg_one

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftySide
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem exists_unit_dXCoeff_of_ord_placeInfty_eq_neg_one {h : RatFunc K}
    (hh : h ≠ 0) (hord : (placeInfty K).ord h = -1) :
    ∃ b : RatFunc K, b ≠ 0 ∧ (placeInfty K).ord b = 0 ∧
      KaehlerDifferential.D K (RatFunc K) h = b • dX K

theorem

exists_dXCoeff_ord_two_of_ord_placeInfty_eq_one

L6423 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_dXCoeff_ord_two_of_ord_placeInfty_eq_one

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftySide
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem exists_dXCoeff_ord_two_of_ord_placeInfty_eq_one {f : RatFunc K}
    (hf : f ≠ 0) (hord : (placeInfty K).ord f = 1) :
    ∃ e : RatFunc K, e ≠ 0 ∧ (placeInfty K).ord e = 2 ∧
      KaehlerDifferential.D K (RatFunc K) f = e • dX K

theorem

exists_ord_zero_smul_of_smul_dX_eq

L6445 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_ord_zero_smul_of_smul_dX_eq

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Glue
variable {K : Type*} [Field K]

Lean source · statement

theorem exists_ord_zero_smul_of_smul_dX_eq {v : Place K (RatFunc K)} {π π' c c' : RatFunc K}
    (hc : c ≠ 0) (hc' : c' ≠ 0) (hord : v.ord c' = v.ord c)
    (hD : KaehlerDifferential.D K (RatFunc K) π = c • dX K)
    (hD' : KaehlerDifferential.D K (RatFunc K) π' = c' • dX K) :
    ∃ u : RatFunc K, v.ord u = 0 ∧
      KaehlerDifferential.D K (RatFunc K) π' = u • KaehlerDifferential.D K (RatFunc K) π

theorem

ordDifferentialWellDefined_ratFunc

L6458 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferentialWellDefined_ratFunc

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField

Lean source · statement

theorem ordDifferentialWellDefined_ratFunc (K : Type*) [Field K] [CharZero K] :
    OrdDifferentialWellDefined K (RatFunc K)

theorem

two_le_ord_placeInfty_p1PrincipalPartAtom

L6530 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.two_le_ord_placeInfty_p1PrincipalPartAtom

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)

Lean source · statement

theorem two_le_ord_placeInfty_p1PrincipalPartAtom
    {p : K[X]} (hp : Irreducible p) {c : K[X]} (hc : c ≠ 0) {m : ℕ}
    (hdeg : c.degree < p.degree) (hm : 2 ≤ m) :
    2 ≤ (placeInfty K).ord (p1PrincipalPartAtom K p c m)

theorem

not_dvd_of_degree_lt

L6551 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.not_dvd_of_degree_lt

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem not_dvd_of_degree_lt {p c : K[X]} (hc : c ≠ 0)
    (hdeg : c.degree < p.degree) : ¬ p ∣ c

theorem

ord_finitePlace_of_degree_lt

L6556 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_finitePlace_of_degree_lt

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ord_finitePlace_of_degree_lt {p : K[X]} (hp : Irreducible p) {c : K[X]} (hc : c ≠ 0)
    (hdeg : c.degree < p.degree) :
    (finitePlace K hp).ord (algebraMap K[X] (RatFunc K) c) = 0

theorem

ord_finitePlace_p1PrincipalPartAtom

L6563 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_finitePlace_p1PrincipalPartAtom

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ord_finitePlace_p1PrincipalPartAtom {p : K[X]} (hp : Irreducible p)
    {c : K[X]} (hc : c ≠ 0) (hdeg : c.degree < p.degree) (m : ℕ) :
    (finitePlace K hp).ord (p1PrincipalPartAtom K p c m) = -(m : ℤ)

def

P1DifferentialCoeffUnitFinite

L6614 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1DifferentialCoeffUnitFinite

Documentation

The differential coefficient has order zero at every finite place.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)
section UnitFinite
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]

Lean source · statement

def P1DifferentialCoeffUnitFinite {ω₀ : Ω[(RatFunc K)⁄K]} (_hω₀ : ω₀ ≠ 0) : Prop

theorem

p1DifferentialCoeffRegularFinite_of_unitFinite

L6621 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1DifferentialCoeffRegularFinite_of_unitFinite

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] [DecidableEq (RatFunc K)]
variable (K)
section UnitFinite
variable [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F :=
  RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
variable [HasPrincipalDivisors K (RatFunc K)]
variable {K}
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [HasCanonicalDivisor (K := K) (F := RatFunc K)]
  [Nontrivial Ω[(RatFunc K)⁄K]] [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem p1DifferentialCoeffRegularFinite_of_unitFinite
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0)
    (hunit : P1DifferentialCoeffUnitFinite K hω₀) :
    P1DifferentialCoeffRegularFinite K hω₀

def

kaehlerPolynomialBasis

L6660 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerPolynomialBasis

Documentation

The basis of polynomial differentials induced by the coordinate differential.

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IntermediateField
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.TensorProduct
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K : Type*) [Field K]

Lean source · statement

def kaehlerPolynomialBasis : Basis Unit K[X] Ω[K[X]⁄K]

instance

instFormallyEtalePolynomialRatFunc

L6663 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.instFormallyEtalePolynomialRatFunc

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IntermediateField
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.TensorProduct
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K : Type*) [Field K]

Lean source · statement

instance instFormallyEtalePolynomialRatFunc : Algebra.FormallyEtale K[X] (RatFunc K)

def

kaehlerRatFuncBasis

L6667 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerRatFuncBasis

Documentation

The basis of rational function differentials obtained by localizing polynomial differentials.

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IntermediateField
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.TensorProduct
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K : Type*) [Field K]

Lean source · statement

def kaehlerRatFuncBasis : Basis Unit (RatFunc K) Ω[(RatFunc K)⁄K]

theorem

kaehlerRankOne_ratFunc

L6671 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerRankOne_ratFunc

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IntermediateField
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.TensorProduct
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K : Type*) [Field K]

Lean source · statement

theorem kaehlerRankOne_ratFunc : KaehlerRankOne K (RatFunc K)

instance

instIsCurveOverRatFunc

L6675 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.instIsCurveOverRatFunc

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IntermediateField
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.TensorProduct
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K : Type*) [Field K]

Lean source · statement

instance instIsCurveOverRatFunc : IsCurveOver K (RatFunc K)

theorem

sum_ramificationIndex_mul_inertiaDeg_of_forall_mem_iff

L6726 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg_of_forall_mem_iff

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
@[expose] public section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
attribute [local instance 0] valuationSubringAlgebra
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']

Lean source · statement

theorem sum_ramificationIndex_mul_inertiaDeg_of_forall_mem_iff
    (v : Place K F) (s : Finset (Place K F'))
    (hs : ∀ w : Place K F', w ∈ s ↔ w.restrict F = v) :
    ∑ w ∈ s, (w.ramificationIndex F : ℤ) * (w.inertiaDeg F : ℤ)
      = (Module.finrank F F' : ℤ)

theorem

sum_ramificationIndex_mul_deg_of_forall_mem_iff

L6761 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.sum_ramificationIndex_mul_deg_of_forall_mem_iff

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
@[expose] public section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
attribute [local instance 0] valuationSubringAlgebra
variable {K F F' : Type*} [Field K] [Field F] [Field F']
  [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
  [FiniteDimensional F F'] [Algebra.IsSeparable F F']

Lean source · statement

theorem sum_ramificationIndex_mul_deg_of_forall_mem_iff
    (v : Place K F) (s : Finset (Place K F'))
    (hs : ∀ w : Place K F', w ∈ s ↔ w.restrict F = v) :
    ∑ w ∈ s, (w.ramificationIndex F : ℤ) * (w.deg : ℤ)
      = (Module.finrank F F' : ℤ) * (v.deg : ℤ)

theorem

ramificationInertiaIdentity_of_finiteDimensional

L6782 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ramificationInertiaIdentity_of_finiteDimensional

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
@[expose] public section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve

Lean source · statement

theorem ramificationInertiaIdentity_of_finiteDimensional
    (K F F' : Type*) [Field K] [Field F] [Field F']
    [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
    [FiniteDimensional F F'] [Algebra.IsSeparable F F'] :
    RamificationInertiaIdentity K F F'

instance

instHasPrincipalDivisorsOfIsGalois

L6798 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instHasPrincipalDivisorsOfIsGalois

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
@[expose] public section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K] {F' : Type*} [Field F'] [Algebra K F']
  [Algebra (RatFunc K) F'] [IsScalarTower K (RatFunc K) F']
  [FiniteDimensional (RatFunc K) F'] [Algebra.IsSeparable (RatFunc K) F']

Lean source · statement

instance (priority := 50) instHasPrincipalDivisorsOfIsGalois [IsGalois (RatFunc K) F'] :
    HasPrincipalDivisors K F'

theorem

derivative_monic_natDegree_one_eq_one

L6854 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.derivative_monic_natDegree_one_eq_one

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section PCoordinateIdentity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem derivative_monic_natDegree_one_eq_one {p : K[X]} (hpmon : p.Monic)
    (hdeg : p.natDegree = 1) : derivative p = 1

theorem

D_algebraMap_polynomial_degOne

L6861 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.D_algebraMap_polynomial_degOne

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section PCoordinateIdentity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem D_algebraMap_polynomial_degOne {p : K[X]} (hpmon : p.Monic) (hdeg : p.natDegree = 1) :
    KaehlerDifferential.D K (RatFunc K) (algebraMap K[X] (RatFunc K) p) = dX K

theorem

eq_C_coeff_zero_of_degree_lt_degOne

L6869 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.eq_C_coeff_zero_of_degree_lt_degOne

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section PCoordinateIdentity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem eq_C_coeff_zero_of_degree_lt_degOne {p c : K[X]} (hpirr : Irreducible p)
    (hdeg : p.natDegree = 1) (hcdeg : c.degree < p.degree) : c = C (c.coeff 0)

theorem

p1PrincipalPartAtom_mul_differentialCoeff_dX_degOne

L6878 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartAtom_mul_differentialCoeff_dX_degOne

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section PCoordinateIdentity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem p1PrincipalPartAtom_mul_differentialCoeff_dX_degOne
    {p c : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p) (hdeg : p.natDegree = 1)
    (hcdeg : c.degree < p.degree) (m : ℕ) :
    p1PrincipalPartAtom K p c m * (finitePlace K hpirr).differentialCoeff (dX K)
      = c.coeff 0 • ((finitePlace K hpirr).differentialCoeff
            (KaehlerDifferential.D K (RatFunc K) (algebraMap K[X] (RatFunc K) p))
          * ((algebraMap K[X] (RatFunc K) p) ^ m)⁻¹)

def

P1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg

L6900 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg

Documentation

Higher pole atoms at finite places of degree at least two have zero canonical residue.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section NamedHigherDeg
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

def P1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg : Prop

theorem

p1FinitePlaceCanonicalResidueAtom_of_coordIndep_degOne

L6914 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceCanonicalResidueAtom_of_coordIndep_degOne

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section DegOneDischarge
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem p1FinitePlaceCanonicalResidueAtom_of_coordIndep_degOne
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    {p c : K[X]} {m : ℕ} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hcdeg : c.degree < p.degree) (hm : 2 ≤ m) (hdeg : p.natDegree = 1)
    (R : (finitePlace K hpirr).CanonicalLocalResidueDataK) :
    R.res (p1PrincipalPartAtom K p c m * (finitePlace K hpirr).differentialCoeff (dX K)) = 0

theorem

p1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg_of_algClosed

L6945 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg_of_algClosed

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
section AlgClosed
variable [IsAlgClosed K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg_of_algClosed :
    P1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg K

theorem

ofPrime_congr

L6983 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ValuationSubring.ofPrime_congr

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace ValuationSubring
open _root_.ValuationSubring
open _root_.ValuationSubring
variable {F : Type*} [Field F]

Lean source · statement

theorem ofPrime_congr {R : ValuationSubring F} {P Q : Ideal R}
    [P.IsPrime] [Q.IsPrime] (h : P = Q) : R.ofPrime P = R.ofPrime Q

theorem

eq_top_of_idealOfLE_eq_bot

L6987 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ValuationSubring.eq_top_of_idealOfLE_eq_bot

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace ValuationSubring
open _root_.ValuationSubring
open _root_.ValuationSubring
variable {F : Type*} [Field F]

Lean source · statement

theorem eq_top_of_idealOfLE_eq_bot {R S : ValuationSubring F} (h : R ≤ S)
    (hbot : idealOfLE R S h = ⊥) : S = ⊤

theorem

idealOfLE_ne_bot_of_ne_top

L6992 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ValuationSubring.idealOfLE_ne_bot_of_ne_top

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace ValuationSubring
open _root_.ValuationSubring
open _root_.ValuationSubring
variable {F : Type*} [Field F]

Lean source · statement

theorem idealOfLE_ne_bot_of_ne_top {R S : ValuationSubring F} (h : R ≤ S)
    (hS : S ≠ ⊤) : idealOfLE R S h ≠ ⊥

theorem

eq_of_isDiscreteValuationRing_of_le

L6996 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ValuationSubring.eq_of_isDiscreteValuationRing_of_le

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace ValuationSubring
open _root_.ValuationSubring
open _root_.ValuationSubring
variable {F : Type*} [Field F]

Lean source · statement

theorem eq_of_isDiscreteValuationRing_of_le {R S : ValuationSubring F}
    [IsDiscreteValuationRing R] (h : R ≤ S) (hS : S ≠ ⊤) : R = S

def

modelInclusion

L7021 ↗

AlgebraicCurve.Place.modelInclusion

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))

Lean source · statement

def _root_.AlgebraicCurve.Place.modelInclusion :
  A →+* v.toValuationSubring

theorem

coe_modelInclusion

L7026 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.coe_modelInclusion

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem coe_modelInclusion (a : A) :
    ((v.modelInclusion A hA a : v.toValuationSubring) : F) = (a : F)

def

modelAlgebra

L7029 ↗

AlgebraicCurve.Place.modelAlgebra

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.modelAlgebra :
  Algebra A v.toValuationSubring

def

centerIdeal

L7034 ↗

AlgebraicCurve.Place.centerIdeal

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

def _root_.AlgebraicCurve.Place.centerIdeal : Ideal A

instance

centerIdeal_isPrime

L7039 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.centerIdeal_isPrime

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

instance centerIdeal_isPrime : (v.centerIdeal A hA).IsPrime

theorem

isUnit_modelInclusion_of_not_mem_centerIdeal

L7043 ↗

AlgebraicCurve.Place.isUnit_modelInclusion_of_not_mem_centerIdeal

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.isUnit_modelInclusion_of_not_mem_centerIdeal {s : A}
    (hs : s ∉ v.centerIdeal A hA) :
    IsUnit (v.modelInclusion A hA s)

theorem

isUnit_modelInclusion_of_mem_primeCompl

L7052 ↗

AlgebraicCurve.Place.isUnit_modelInclusion_of_mem_primeCompl

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.isUnit_modelInclusion_of_mem_primeCompl (s :
  (v.centerIdeal A hA).primeCompl) :
    IsUnit (v.modelInclusion A hA (s : A))

theorem

modelAlgebra_isScalarTower

L7058 ↗

AlgebraicCurve.Place.modelAlgebra_isScalarTower

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.modelAlgebra_isScalarTower :
    let := v.modelAlgebra A hA
    IsScalarTower K A v.toValuationSubring := by
  let := v.modelAlgebra A hA
  refine IsScalarTower.of_algebraMap_eq fun c => ?_
  apply Subtype.ext
  simp only [coe_modelInclusion, RingHom.algebraMap_toAlgebra]
  rfl

theorem

essFiniteType_of_finiteType_isLocalization_centerIdeal

L7068 ↗

AlgebraicCurve.Place.essFiniteType_of_finiteType_isLocalization_centerIdeal

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterIdeal
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
variable (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.essFiniteType_of_finiteType_isLocalization_centerIdeal
    [Algebra.FiniteType K A]
    (hloc : let := v.modelAlgebra A hA
            IsLocalization.AtPrime v.toValuationSubring (v.centerIdeal A hA)) :
    Algebra.EssFiniteType K v.toValuationSubring := by
  let := v.modelAlgebra A hA
  have := v.modelAlgebra_isScalarTower A hA
  have : Algebra.EssFiniteType A v.toValuationSubring :=
    Algebra.EssFiniteType.of_isLocalization _ (v.centerIdeal A hA).primeCompl
  exact Algebra.EssFiniteType.comp K A v.toValuationSubring

def

ValSubringDedekindModel

L7087 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringDedekindModel

Documentation

Each valuation subring is a localization of a finite type model at its center.

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringDedekindModel : Prop

theorem

valSubringEssFiniteType_of_dedekindModel

L7096 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringEssFiniteType_of_dedekindModel

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringEssFiniteType_of_dedekindModel
    (hmodel : ValSubringDedekindModel K F) :
    ValSubringEssFiniteType K F

theorem

valSubringKaehlerFinite_of_dedekindModel

L7102 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerFinite_of_dedekindModel

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.CongruenceSubgroup
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Polynomial
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerFinite_of_dedekindModel
    (hmodel : ValSubringDedekindModel K F) :
    ValSubringKaehlerFinite K F

theorem

completionSection_nonempty_generic

L7150 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.completionSection_nonempty_generic

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.KwNo6Section ModularCurve.KwNo6Pin
open Mp72a102T1
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem completionSection_nonempty_generic (v : Place K F)
    [FiniteDimensional K v.ResidueField] [Algebra.IsSeparable K v.ResidueField] :
    Nonempty (Lg37CompletionSection v)

instance

instHasCanonicalLocalResidueKStar

L7163 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.instHasCanonicalLocalResidueKStar

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.Polynomial
open _root_.IsLocalRing
open _root_.AlgebraicCurve
open ModularCurve.Lg37 ModularCurve.KwNo6Section ModularCurve.KwNo6Pin
open Mp72a102T1
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable (K F) in

Lean source · full declaration

instance instHasCanonicalLocalResidueKStar [IsCurveOver K F] [PerfectField K] :
    HasCanonicalLocalResidueKStar K F where
  dataKStar v :=
    have : FiniteDimensional K v.ResidueField := Place.FiniteResidue.finite
    have : Algebra.IsSeparable K v.ResidueField :=
      Algebra.IsAlgebraic.isSeparable_of_perfectField
    canonicalLocalResidueDataKStar v
      (Classical.choice (completionSection_nonempty_generic v))

theorem

ordDifferentialWellDefined_ratFunc_of_perfectField

L7199 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferentialWellDefined_ratFunc_of_perfectField

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField

Lean source · statement

theorem ordDifferentialWellDefined_ratFunc_of_perfectField (K : Type*) [Field K]
    [PerfectField K] :
    OrdDifferentialWellDefined K (RatFunc K)

theorem

p1DifferentialCoeffUnitFinite_dX

L7290 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1DifferentialCoeffUnitFinite_dX

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section UnitFinite
variable [CharZero K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1DifferentialCoeffUnitFinite_dX :
    P1DifferentialCoeffUnitFinite K (dX_ne_zero K)

theorem

ord_finitePlace_mOneAtom_mul_differentialCoeff

L7364 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_finitePlace_mOneAtom_mul_differentialCoeff

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section MOneSimplePoleBounds
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

theorem ord_finitePlace_mOneAtom_mul_differentialCoeff
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0)
    (hunit : ∀ v : Place K (RatFunc K), v ≠ placeInfty K → v.ord (v.differentialCoeff ω₀) = 0)
    {p : K[X]} (hp : Irreducible p) {c : K[X]} (hc : c ≠ 0)
    (hdeg : c.degree < p.degree) :
    (finitePlace K hp).ord
        (p1PrincipalPartAtom K p c 1 * (finitePlace K hp).differentialCoeff ω₀) = -1

theorem

ord_placeInfty_mOneAtom_mul_differentialCoeff_ge_neg_one

L7381 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_mOneAtom_mul_differentialCoeff_ge_neg_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section MOneSimplePoleBounds
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

theorem ord_placeInfty_mOneAtom_mul_differentialCoeff_ge_neg_one
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0)
    (hInfty : (placeInfty K).ordDifferential ω₀ = -2)
    {p : K[X]} (hp : Irreducible p) {c : K[X]} (hc : c ≠ 0)
    (hdeg : c.degree < p.degree) :
    -1 ≤ (placeInfty K).ord
        (p1PrincipalPartAtom K p c 1 * (placeInfty K).differentialCoeff ω₀)

theorem

p1MOneAtom_mul_differentialCoeff_mem_simplePole_finitePlace

L7399 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1MOneAtom_mul_differentialCoeff_mem_simplePole_finitePlace

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section MOneSimplePoleBounds
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

theorem p1MOneAtom_mul_differentialCoeff_mem_simplePole_finitePlace
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0)
    (hunit : ∀ v : Place K (RatFunc K), v ≠ placeInfty K → v.ord (v.differentialCoeff ω₀) = 0)
    {p : K[X]} (hp : Irreducible p) (c : K[X]) (hdeg : c.degree < p.degree) :
    p1PrincipalPartAtom K p c 1 * (finitePlace K hp).differentialCoeff ω₀
      ∈ (finitePlace K hp).simplePoleSubmodule

theorem

p1MOneAtom_mul_differentialCoeff_mem_simplePole_placeInfty

L7424 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1MOneAtom_mul_differentialCoeff_mem_simplePole_placeInfty

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section MOneSimplePoleBounds
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

theorem p1MOneAtom_mul_differentialCoeff_mem_simplePole_placeInfty
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0)
    (hInfty : (placeInfty K).ordDifferential ω₀ = -2)
    {p : K[X]} (hp : Irreducible p) (c : K[X]) (hdeg : c.degree < p.degree) :
    p1PrincipalPartAtom K p c 1 * (placeInfty K).differentialCoeff ω₀
      ∈ (placeInfty K).simplePoleSubmodule

def

P1PrincipalPartMOneSimplePoleCancel

L7459 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PrincipalPartMOneSimplePoleCancel

Documentation

The traced finite and infinite residues of a simple principal part atom cancel.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section MOneSimplePoleRow
variable (K)
variable [DecidableEq (RatFunc K)]
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

def P1PrincipalPartMOneSimplePoleCancel {ω₀ : Ω[(RatFunc K)⁄K]} (_hω₀ : ω₀ ≠ 0) : Prop

instance

instHasPrincipalDivisorsRatFuncSelf

L7538 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.instHasPrincipalDivisorsRatFuncSelf

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.Polynomial
open _root_.IntermediateField
open _root_.AlgebraicCurve
namespace ModularCurve
section Corollaries
variable (K : Type*) [Field K]

Lean source · statement

instance instHasPrincipalDivisorsRatFuncSelf : HasPrincipalDivisors K (RatFunc K)

def

P1FinitePlaceCanonicalResidueAtomMGeTwoTrace

L7577 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1FinitePlaceCanonicalResidueAtomMGeTwoTrace

Documentation

The traced residue of each higher pole atom at a finite place vanishes.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section TraceRow
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

def P1FinitePlaceCanonicalResidueAtomMGeTwoTrace : Prop

def

P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg

L7585 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg

Documentation

The traced residue of higher pole atoms at finite places of degree at least two vanishes.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section TraceRow
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

def P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg : Prop

theorem

p1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg_of_higherDeg

L7601 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg_of_higherDeg

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section Monotonicity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem p1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg_of_higherDeg
    (h : P1FinitePlaceCanonicalResidueAtomMGeTwoHigherDeg K) :
    P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K

theorem

p1FinitePlaceCanonicalResidueAtomTrace_of_coordIndep_degOne

L7616 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceCanonicalResidueAtomTrace_of_coordIndep_degOne

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section DegOneDischarge
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem p1FinitePlaceCanonicalResidueAtomTrace_of_coordIndep_degOne
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    {p c : K[X]} {m : ℕ} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hcdeg : c.degree < p.degree) (hm : 2 ≤ m) (hdeg : p.natDegree = 1)
    (R : (finitePlace K hpirr).CanonicalLocalResidueDataK) :
    Algebra.trace K (finitePlace K hpirr).ResidueField
        (R.res (p1PrincipalPartAtom K p c m
          * (finitePlace K hpirr).differentialCoeff (dX K))) = 0

theorem

p1FinitePlaceCanonicalResidueAtomMGeTwoTrace_of_coordIndep_higherDeg

L7629 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceCanonicalResidueAtomMGeTwoTrace_of_coordIndep_higherDeg

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section DegOneDischarge
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in

Lean source · statement

theorem p1FinitePlaceCanonicalResidueAtomMGeTwoTrace_of_coordIndep_higherDeg
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K) :
    P1FinitePlaceCanonicalResidueAtomMGeTwoTrace K

theorem

p1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg_of_algClosed

L7666 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg_of_algClosed

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section AlgClosed
variable [IsAlgClosed K] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [DecidableEq (RatFunc K)]
  [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]] in
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg_of_algClosed :
    P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K

theorem

inv_mem_of_not_mem_centerIdeal

L7703 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.inv_mem_of_not_mem_centerIdeal

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterNonzero
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))

Lean source · statement

theorem inv_mem_of_not_mem_centerIdeal {a : A} (ha : a ∉ v.centerIdeal A hA) :
    ((a : F))⁻¹ ∈ v.toValuationSubring

theorem

div_mem_of_not_mem_centerIdeal

L7717 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.div_mem_of_not_mem_centerIdeal

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterNonzero
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))

Lean source · statement

theorem div_mem_of_not_mem_centerIdeal (r : A) {a : A}
    (ha : a ∉ v.centerIdeal A hA) :
    (r : F) / (a : F) ∈ v.toValuationSubring

theorem

centerIdeal_ne_bot_of_isFractionRing

L7724 ↗

AlgebraicCurve.Place.centerIdeal_ne_bot_of_isFractionRing

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section CenterNonzero
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))

Lean source · statement

theorem _root_.AlgebraicCurve.Place.centerIdeal_ne_bot_of_isFractionRing
  [IsFractionRing A F] :
    v.centerIdeal A hA ≠ ⊥

def

centerLocalizationSubalgebra

L7748 ↗

AlgebraicCurve.Place.centerLocalizationSubalgebra

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))

Lean source · statement

def _root_.AlgebraicCurve.Place.centerLocalizationSubalgebra
  [IsFractionRing A F] :
    Subalgebra A F

instance

centerLocalizationSubalgebra_isLocalization

L7755 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.centerLocalizationSubalgebra_isLocalization

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place

Lean source · statement

instance centerLocalizationSubalgebra_isLocalization [IsFractionRing A F] :
    IsLocalization.AtPrime (v.centerLocalizationSubalgebra A hA) (v.centerIdeal A hA)

instance

centerLocalizationSubalgebra_isDomain

L7759 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.centerLocalizationSubalgebra_isDomain

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place

Lean source · statement

instance centerLocalizationSubalgebra_isDomain [IsFractionRing A F] :
    IsDomain (v.centerLocalizationSubalgebra A hA)

theorem

centerLocalizationSubalgebra_isDiscreteValuationRing

L7763 ↗

AlgebraicCurve.Place.centerLocalizationSubalgebra_isDiscreteValuationRing

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.centerLocalizationSubalgebra_isDiscreteValuationRing
    [IsDedekindDomain A] [IsFractionRing A F] :
    IsDiscreteValuationRing (v.centerLocalizationSubalgebra A hA)

instance

centerLocalizationSubalgebra_isFractionRing

L7770 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.centerLocalizationSubalgebra_isFractionRing

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

instance centerLocalizationSubalgebra_isFractionRing [IsFractionRing A F] :
    IsFractionRing (v.centerLocalizationSubalgebra A hA) F

theorem

centerLocalizationSubalgebra_subset

L7774 ↗

AlgebraicCurve.Place.centerLocalizationSubalgebra_subset

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.centerLocalizationSubalgebra_subset
  [IsFractionRing A F] :
    ((v.centerLocalizationSubalgebra A hA : Subalgebra A F) : Set F)
      ⊆ (v.toValuationSubring : Set F)

def

centerLocalizationValuationSubring

L7784 ↗

AlgebraicCurve.Place.centerLocalizationValuationSubring

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

def _root_.AlgebraicCurve.Place.centerLocalizationValuationSubring
    [IsDedekindDomain A] [IsFractionRing A F] : ValuationSubring F where
  toSubring := (v.centerLocalizationSubalgebra A hA).toSubring
  mem_or_inv_mem' x := by
    have := v.centerLocalizationSubalgebra_isDiscreteValuationRing A hA
    rcases ValuationRing.isInteger_or_isInteger (v.centerLocalizationSubalgebra A hA) x with
      ⟨a, ha⟩ | ⟨a, ha⟩
    · exact Or.inl (ha ▸ a.2)
    · exact Or.inr (ha ▸ a.2)

theorem

coe_centerLocalizationValuationSubring

L7795 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.coe_centerLocalizationValuationSubring

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem coe_centerLocalizationValuationSubring [IsDedekindDomain A] [IsFractionRing A F] :
    ((v.centerLocalizationValuationSubring A hA : ValuationSubring F) : Set F)
      = ((v.centerLocalizationSubalgebra A hA : Subalgebra A F) : Set F)

instance

centerLocalizationValuationSubring_isDiscreteValuationRing

L7799 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.centerLocalizationValuationSubring_isDiscreteValuationRing

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

instance centerLocalizationValuationSubring_isDiscreteValuationRing
    [IsDedekindDomain A] [IsFractionRing A F] :
    IsDiscreteValuationRing (v.centerLocalizationValuationSubring A hA)

theorem

centerLocalizationValuationSubring_le

L7804 ↗

AlgebraicCurve.Place.centerLocalizationValuationSubring_le

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.centerLocalizationValuationSubring_le
  [IsDedekindDomain A] [IsFractionRing A F] :
    v.centerLocalizationValuationSubring A hA ≤ v.toValuationSubring

theorem

centerLocalizationValuationSubring_eq

L7810 ↗

AlgebraicCurve.Place.centerLocalizationValuationSubring_eq

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.centerLocalizationValuationSubring_eq
  [IsDedekindDomain A] [IsFractionRing A F] :
    v.centerLocalizationValuationSubring A hA = v.toValuationSubring

theorem

coe_centerLocalizationSubalgebra_eq

L7817 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.coe_centerLocalizationSubalgebra_eq

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem coe_centerLocalizationSubalgebra_eq [IsDedekindDomain A] [IsFractionRing A F] :
    ((v.centerLocalizationSubalgebra A hA : Subalgebra A F) : Set F)
      = (v.toValuationSubring : Set F)

theorem

mem_centerLocalizationSubalgebra_of_mem

L7823 ↗

AlgebraicCurve.Place.mem_centerLocalizationSubalgebra_of_mem

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_centerLocalizationSubalgebra_of_mem
  [IsDedekindDomain A] [IsFractionRing A F]
    (x : F) (hx : x ∈ v.toValuationSubring) :
    x ∈ (v.centerLocalizationSubalgebra A hA : Subalgebra A F)

theorem

isLocalization_centerIdeal_of_isDedekindDomain

L7830 ↗

AlgebraicCurve.Place.isLocalization_centerIdeal_of_isDedekindDomain

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section LocalizationSubalgebra
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
  (A : Subalgebra K F) (hA : (A : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.isLocalization_centerIdeal_of_isDedekindDomain
    [IsDedekindDomain A] [IsFractionRing A F] :
    let := v.modelAlgebra A hA
    IsLocalization.AtPrime v.toValuationSubring (v.centerIdeal A hA) := by
  let := v.modelAlgebra A hA
  refine (isLocalization_iff _ _).mpr ⟨?_, ?_, ?_⟩
  · exact v.isUnit_modelInclusion_of_mem_primeCompl A hA
  · intro x
    have hxP : (x : F) ∈ (v.centerLocalizationSubalgebra A hA : Subalgebra A F) :=
      v.mem_centerLocalizationSubalgebra_of_mem A hA (x : F) x.2
    obtain ⟨a, s, hs, hxeq⟩ := hxP
    refine ⟨⟨a, ⟨s, hs⟩⟩, Subtype.ext ?_⟩
    change (x : F) * ((s : A) : F) = ((a : A) : F)
    have halg_s : algebraMap A F s = ((s : A) : F) := rfl
    have halg_a : algebraMap A F a = ((a : A) : F) := rfl
    have hs0 : algebraMap A F s ≠ 0 := by
      intro hF
      have hsne : s ≠ 0 := fun h => (h ▸ hs) (Ideal.zero_mem _)
      exact hsne (Subtype.ext (p := (· ∈ A)) (halg_s ▸ hF))
    rw [← halg_s, ← halg_a, hxeq, mul_assoc, inv_mul_cancel₀ hs0, mul_one]
  · intro a₁ a₂ h
    have hF : (a₁ : F) = (a₂ : F) :=
      congrArg (Subtype.val : v.toValuationSubring → F) h
    exact ⟨1, by rw [show a₁ = a₂ from Subtype.ext hF]⟩

def

ValSubringDedekindFractionModel

L7863 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringDedekindFractionModel

Documentation

Each valuation subring contains a finite type Dedekind model of the function field.

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringDedekindFractionModel : Prop

theorem

valSubringDedekindModel_of_dedekindFractionModel

L7870 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringDedekindModel_of_dedekindFractionModel

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringDedekindModel_of_dedekindFractionModel
    (hfrac : ValSubringDedekindFractionModel K F) :
    ValSubringDedekindModel K F

theorem

valSubringKaehlerFinite_of_dedekindFractionModel

L7876 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerFinite_of_dedekindFractionModel

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerFinite_of_dedekindFractionModel
    (hfrac : ValSubringDedekindFractionModel K F) :
    ValSubringKaehlerFinite K F

theorem

aeval_root_eq_sum_range

L7935 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.aeval_root_eq_sum_range

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EulerGeneral
variable [CharZero K] {p : K[X]} [Fact (Irreducible p)]
omit [DecidableEq (RatFunc K)] [CharZero K] in

Lean source · statement

theorem aeval_root_eq_sum_range {c : K[X]} {d : ℕ} (hd : c.natDegree < d) :
    (aeval (AdjoinRoot.root p) c : AdjoinRoot p)
      = ∑ k ∈ Finset.range d, c.coeff k • AdjoinRoot.root p ^ k

theorem

trace_adjoinRoot_mk_div_mk_derivative_of_degree_lt

L7942 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_adjoinRoot_mk_div_mk_derivative_of_degree_lt

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EulerGeneral
variable [CharZero K] {p : K[X]} [Fact (Irreducible p)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem trace_adjoinRoot_mk_div_mk_derivative_of_degree_lt (hpmon : p.Monic)
    {c : K[X]} (hdeg : c.degree < p.degree) :
    Algebra.trace K (AdjoinRoot p)
        (AdjoinRoot.mk p c / AdjoinRoot.mk p (derivative p))
      = c.coeff (p.natDegree - 1)

def

finitePlaceResidueFieldAlgEquivAdjoinRoot

L7977 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.finitePlaceResidueFieldAlgEquivAdjoinRoot

Documentation

Present the residue field of a finite place by adjoining a root of its polynomial.

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section AlgEquiv
variable {p : K[X]} (hpirr : Irreducible p)

Lean source · statement

def finitePlaceResidueFieldAlgEquivAdjoinRoot :
    AdjoinRoot p ≃ₐ[K] (finitePlace K hpirr).ResidueField

theorem

finitePlaceResidueFieldAlgEquivAdjoinRoot_mk

L7982 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.finitePlaceResidueFieldAlgEquivAdjoinRoot_mk

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section AlgEquiv
variable {p : K[X]} (hpirr : Irreducible p)
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem finitePlaceResidueFieldAlgEquivAdjoinRoot_mk (q : K[X]) :
    finitePlaceResidueFieldAlgEquivAdjoinRoot K hpirr (AdjoinRoot.mk p q)
      = IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) q,
          algebraMap_mem_ofHeightOneSpectrum K _ q⟩

theorem

trace_finitePlace_residueField_eq_trace_adjoinRoot

L7995 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_finitePlace_residueField_eq_trace_adjoinRoot

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section TraceTransport
variable {p : K[X]} (hpirr : Irreducible p)
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem trace_finitePlace_residueField_eq_trace_adjoinRoot
    (y : (finitePlace K hpirr).ResidueField) :
    Algebra.trace K (finitePlace K hpirr).ResidueField y
      = Algebra.trace K (AdjoinRoot p)
          ((finitePlaceResidueFieldAlgEquivAdjoinRoot K hpirr).symm y)

def

P1FinitePlaceSimplePoleResidueAdjoinRootValue

L8012 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1FinitePlaceSimplePoleResidueAdjoinRootValue

Documentation

The residue of a simple pole atom equals its numerator divided by the derivative in the root field.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section BridgeRows
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

def P1FinitePlaceSimplePoleResidueAdjoinRootValue : Prop

def

P1PlaceInftySimplePoleResidueEulerValue

L8022 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PlaceInftySimplePoleResidueEulerValue

Documentation

The traced residue at infinity equals minus the relevant numerator coefficient.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section BridgeRows
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]

Lean source · statement

def P1PlaceInftySimplePoleResidueEulerValue : Prop

theorem

trace_finitePlace_simplePoleResidue_of_adjoinRootValue

L8041 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_finitePlace_simplePoleResidue_of_adjoinRootValue

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section FinitePlaceTrace
variable [CharZero K]
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates]
omit [DecidableEq (RatFunc K)] [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)]
    [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] in

Lean source · statement

theorem trace_finitePlace_simplePoleResidue_of_adjoinRootValue
    (hbridge : P1FinitePlaceSimplePoleResidueAdjoinRootValue K)
    {p c : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p) (hdeg : c.degree < p.degree)
    (hfmem : p1PrincipalPartAtom K p c 1 * (finitePlace K hpirr).differentialCoeff (dX K)
        ∈ (finitePlace K hpirr).simplePoleSubmodule) :
    Algebra.trace K (finitePlace K hpirr).ResidueField
        ((finitePlace K hpirr).simplePoleResidueAux ⟨_, hfmem⟩)
      = c.coeff (p.natDegree - 1)

theorem

p1PrincipalPartMOneSimplePoleCancel_of_eulerBridge

L8066 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartMOneSimplePoleCancel_of_eulerBridge

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section HigherDegReduction
variable [CharZero K]
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)]
    [∀ (v : Place K (RatFunc K)), v.DCoordGenerates]
    [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem p1PrincipalPartMOneSimplePoleCancel_of_eulerBridge
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0) (heq : ω₀ = dX K)
    (hfin : P1FinitePlaceSimplePoleResidueAdjoinRootValue K)
    (hinf : P1PlaceInftySimplePoleResidueEulerValue K) :
    P1PrincipalPartMOneSimplePoleCancel K hω₀

theorem

isIntegrallyClosed_toValuationSubring

L8114 ↗

AlgebraicCurve.Place.isIntegrallyClosed_toValuationSubring

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section IntegrallyClosed
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.isIntegrallyClosed_toValuationSubring :
    IsIntegrallyClosed v.toValuationSubring

theorem

mem_valuationSubring_of_isIntegral

L8118 ↗

AlgebraicCurve.Place.mem_valuationSubring_of_isIntegral

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section IntegrallyClosed
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_valuationSubring_of_isIntegral {x : F}
    (hx : _root_.IsIntegral v.toValuationSubring x) :
    x ∈ v.toValuationSubring

theorem

modelAlgebra_isScalarTower_top

L8128 ↗

AlgebraicCurve.Place.modelAlgebra_isScalarTower_top

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section IntegrallyClosed
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (B : Subalgebra K F) (hB : (B : Set F) ⊆ (v.toValuationSubring : Set F))
include hB in

Lean source · full declaration

theorem _root_.AlgebraicCurve.Place.modelAlgebra_isScalarTower_top :
    let := v.modelAlgebra B hB
    IsScalarTower B v.toValuationSubring F := by
  let := v.modelAlgebra B hB
  refine IsScalarTower.of_algebraMap_eq fun b => ?_
  rfl

theorem

mem_valuationSubring_of_isIntegral_subalgebra

L8137 ↗

AlgebraicCurve.Place.mem_valuationSubring_of_isIntegral_subalgebra

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section IntegrallyClosed
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (B : Subalgebra K F) (hB : (B : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
include hB in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_valuationSubring_of_isIntegral_subalgebra {x : F}
    (hx : _root_.IsIntegral B x) :
    x ∈ v.toValuationSubring

theorem

integralClosure_subset_valuationSubring

L8146 ↗

AlgebraicCurve.Place.integralClosure_subset_valuationSubring

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
section IntegrallyClosed
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (B : Subalgebra K F) (hB : (B : Set F) ⊆ (v.toValuationSubring : Set F))
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
include hB in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.integralClosure_subset_valuationSubring :
    ((integralClosure B F : Subalgebra B F) : Set F)
      ⊆ (v.toValuationSubring : Set F)

def

IsAffineChart

L8161 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.IsAffineChart

Documentation

A finite type Dedekind subalgebra whose fraction field is the function field.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TwoCharts
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def IsAffineChart (A : Subalgebra K F) : Prop

def

ValSubringTwoAffineCharts

L8165 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ValSubringTwoAffineCharts

Documentation

Two affine charts together cover all valuation subrings of the function field.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TwoCharts
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def ValSubringTwoAffineCharts : Prop

theorem

valSubringDedekindFractionModel_of_twoAffineCharts

L8172 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringDedekindFractionModel_of_twoAffineCharts

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TwoCharts
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringDedekindFractionModel_of_twoAffineCharts
    (h : ValSubringTwoAffineCharts K F) :
    ValSubringDedekindFractionModel K F

theorem

valSubringKaehlerFinite_of_twoAffineCharts

L8180 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerFinite_of_twoAffineCharts

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TwoCharts
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerFinite_of_twoAffineCharts
    (h : ValSubringTwoAffineCharts K F) :
    ValSubringKaehlerFinite K F

def

toKSubalgebra

L8211 ↗

AlgebraicCurve.Place.toKSubalgebra

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Gates
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
namespace Place
open scoped _root_.AlgebraicCurve.Place in

Lean source · full declaration

def _root_.AlgebraicCurve.Place.toKSubalgebra (v :
  Place K F) : Subalgebra K F where
  __ := v.toValuationSubring.toSubring
  algebraMap_mem' c := v.algebraMap_mem' c

theorem

coe_toKSubalgebra

L8220 ↗

AlgebraicCurve.Place.coe_toKSubalgebra

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Gates
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
open _root_.AlgebraicCurve
namespace Place
open scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.coe_toKSubalgebra (v : Place K F) :
    (v.toKSubalgebra : Set F) = (v.toValuationSubring : Set F)

theorem

gate_adjoin_subset_valuationSubring_of_mem

L8225 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.gate_adjoin_subset_valuationSubring_of_mem

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Gates
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve

Lean source · statement

theorem gate_adjoin_subset_valuationSubring_of_mem (v : Place K F) {t : F}
    (ht : t ∈ v.toValuationSubring) :
    ((Algebra.adjoin K {t} : Subalgebra K F) : Set F)
      ⊆ (v.toValuationSubring : Set F)

theorem

simplePoleResidueAux_mul_of_mem

L8274 ↗

AlgebraicCurve.Place.simplePoleResidueAux_mul_of_mem

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.simplePoleResidueAux_mul_of_mem {a : F} (ha :
  a ∈ v.toValuationSubring)
    {f : F} (hf : f ∈ v.simplePoleSubmodule)
    (haf : a * f ∈ v.simplePoleSubmodule) :
    v.simplePoleResidueAux ⟨a * f, haf⟩
      = IsLocalRing.residue _ ⟨a, ha⟩ * v.simplePoleResidueAux ⟨f, hf⟩

theorem

mul_mem_simplePoleSubmodule_of_mem

L8285 ↗

AlgebraicCurve.Place.mul_mem_simplePoleSubmodule_of_mem

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mul_mem_simplePoleSubmodule_of_mem {a : F} (ha :
  a ∈ v.toValuationSubring)
    {f : F} (hf : f ∈ v.simplePoleSubmodule) :
    a * f ∈ v.simplePoleSubmodule

theorem

denom_notMem_of_mem_ofHeightOneSpectrum

L8301 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.denom_notMem_of_mem_ofHeightOneSpectrum

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section DerivativeRegular
variable (K)
variable [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem denom_notMem_of_mem_ofHeightOneSpectrum (w : HeightOneSpectrum K[X]) {f : RatFunc K}
    (hf : f ∈ (Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w).toValuationSubring) :
    f.denom ∉ w.asIdeal

theorem

differentialCoeff_add'

L8331 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_add'

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section DerivativeRegular
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem differentialCoeff_add' (v : Place K (RatFunc K)) (ω₁ ω₂ : Ω[(RatFunc K)⁄K]) :
    v.differentialCoeff (ω₁ + ω₂) = v.differentialCoeff ω₁ + v.differentialCoeff ω₂

theorem

differentialCoeff_D_algebraMap_polynomial

L8337 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_D_algebraMap_polynomial

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section DerivativeRegular
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem differentialCoeff_D_algebraMap_polynomial (v : Place K (RatFunc K)) (q : K[X]) :
    v.differentialCoeff (D K (RatFunc K) (algebraMap K[X] (RatFunc K) q))
      = algebraMap K[X] (RatFunc K) q.derivative * v.differentialCoeff (dX K)

theorem

differentialCoeff_D_mem_finitePlace

L8343 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_D_mem_finitePlace

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section DerivativeRegular
variable (K)
variable [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem differentialCoeff_D_mem_finitePlace [CharZero K] {p : K[X]} (hpirr : Irreducible p)
    {f : RatFunc K} (hf : f ∈ (finitePlace K hpirr).toValuationSubring) :
    (finitePlace K hpirr).differentialCoeff (D K (RatFunc K) f)
      ∈ (finitePlace K hpirr).toValuationSubring

theorem

uniformizer_div_mem_finitePlace

L8383 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.uniformizer_div_mem_finitePlace

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section LeibnizCore
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] [CharZero K] [∀ (v : Place K (RatFunc K)), v.DCoordGenerates]
  [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem uniformizer_div_mem_finitePlace {p : K[X]} (hpirr : Irreducible p) :
    (finitePlace K hpirr).uniformizer / algebraMap K[X] (RatFunc K) p
      ∈ (finitePlace K hpirr).toValuationSubring

theorem

one_sub_uniformizer_div_mul_differentialCoeff_D

L8395 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.one_sub_uniformizer_div_mul_differentialCoeff_D

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section LeibnizCore
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem one_sub_uniformizer_div_mul_differentialCoeff_D
    {p : K[X]} (hpirr : Irreducible p) :
    (1 : RatFunc K) - (finitePlace K hpirr).uniformizer / algebraMap K[X] (RatFunc K) p
        * (finitePlace K hpirr).differentialCoeff
            (D K (RatFunc K) (algebraMap K[X] (RatFunc K) p))
      = algebraMap K[X] (RatFunc K) p
          * (finitePlace K hpirr).differentialCoeff
              (D K (RatFunc K) ((finitePlace K hpirr).uniformizer
                / algebraMap K[X] (RatFunc K) p))

theorem

uniformizer_div_mul_differentialCoeff_D_mem_finitePlace

L8428 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.uniformizer_div_mul_differentialCoeff_D_mem_finitePlace

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section LeibnizCore
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem uniformizer_div_mul_differentialCoeff_D_mem_finitePlace
    {p : K[X]} (hpirr : Irreducible p) :
    (finitePlace K hpirr).uniformizer / algebraMap K[X] (RatFunc K) p
        * (finitePlace K hpirr).differentialCoeff
            (D K (RatFunc K) (algebraMap K[X] (RatFunc K) p))
      ∈ (finitePlace K hpirr).toValuationSubring

theorem

residue_uniformizer_div_mul_differentialCoeff_D_eq_one

L8445 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.residue_uniformizer_div_mul_differentialCoeff_D_eq_one

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section LeibnizCore
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem residue_uniformizer_div_mul_differentialCoeff_D_eq_one
    {p : K[X]} (hpirr : Irreducible p) :
    IsLocalRing.residue _ ⟨_, uniformizer_div_mul_differentialCoeff_D_mem_finitePlace K hpirr⟩
      = (1 : (finitePlace K hpirr).ResidueField)

theorem

residue_algebraMap_derivative_ne_zero

L8495 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.residue_algebraMap_derivative_ne_zero

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section BridgeDischarge
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K
  (RatFunc K)] [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem residue_algebraMap_derivative_ne_zero {p : K[X]} (hpirr : Irreducible p) :
    IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) p.derivative,
        algebraMap_mem_ofHeightOneSpectrum K _ p.derivative⟩
      ≠ (0 : (finitePlace K hpirr).ResidueField)

theorem

simplePoleResidueAux_finitePlace_p1PrincipalPartAtom_mOne

L8510 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.simplePoleResidueAux_finitePlace_p1PrincipalPartAtom_mOne

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section BridgeDischarge
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem simplePoleResidueAux_finitePlace_p1PrincipalPartAtom_mOne
    {p c : K[X]} (hpirr : Irreducible p)
    (hfmem : p1PrincipalPartAtom K p c 1 * (finitePlace K hpirr).differentialCoeff (dX K)
        ∈ (finitePlace K hpirr).simplePoleSubmodule) :
    (finitePlace K hpirr).simplePoleResidueAux ⟨_, hfmem⟩
      = IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) c,
            algebraMap_mem_ofHeightOneSpectrum K _ c⟩
        / IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) p.derivative,
            algebraMap_mem_ofHeightOneSpectrum K _ p.derivative⟩

theorem

p1FinitePlaceSimplePoleResidueAdjoinRootValue_dX

L8567 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1FinitePlaceSimplePoleResidueAdjoinRootValue_dX

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section BridgeDischarge
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [HasLocalResidue K (RatFunc K)] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1FinitePlaceSimplePoleResidueAdjoinRootValue_dX :
    P1FinitePlaceSimplePoleResidueAdjoinRootValue K

theorem

p1PrincipalPartMOneSimplePoleCancel_of_inftyBridge

L8596 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartMOneSimplePoleCancel_of_inftyBridge

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section ComposedEngine
variable (K)
variable [DecidableEq (RatFunc K)] [CharZero K]
variable [HasCanonicalDivisor (K := K) (F := RatFunc K)] [HasLocalResidue K (RatFunc K)]
  [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalDivisor (K := K) (F := RatFunc K)]
    [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1PrincipalPartMOneSimplePoleCancel_of_inftyBridge
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0) (heq : ω₀ = dX K)
    (hinf : P1PlaceInftySimplePoleResidueEulerValue K) :
    P1PrincipalPartMOneSimplePoleCancel K hω₀

theorem

isPrincipalIdealRing_adjoin

L8631 ↗

Transcendental.isPrincipalIdealRing_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AdjoinRingProperties
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem _root_.Transcendental.isPrincipalIdealRing_adjoin {t : F} (ht :
  Transcendental K t) :
    IsPrincipalIdealRing ↥(Algebra.adjoin K ({t} : Set F))

theorem

isDedekindDomain_adjoin

L8639 ↗

Transcendental.isDedekindDomain_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AdjoinRingProperties
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem _root_.Transcendental.isDedekindDomain_adjoin {t : F} (ht : Transcendental K t) :
    IsDedekindDomain ↥(Algebra.adjoin K ({t} : Set F))

theorem

isIntegrallyClosed_adjoin

L8644 ↗

Transcendental.isIntegrallyClosed_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AdjoinRingProperties
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem _root_.Transcendental.isIntegrallyClosed_adjoin {t : F} (ht : Transcendental K t) :
    IsIntegrallyClosed ↥(Algebra.adjoin K ({t} : Set F))

theorem

isNoetherianRing_adjoin

L8649 ↗

Transcendental.isNoetherianRing_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AdjoinRingProperties
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem _root_.Transcendental.isNoetherianRing_adjoin {t : F} (ht : Transcendental K t) :
    IsNoetherianRing ↥(Algebra.adjoin K ({t} : Set F))

theorem

adjoin_singleton

L8654 ↗

Algebra.FiniteType.adjoin_singleton

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AdjoinRingProperties
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem _root_.Algebra.FiniteType.adjoin_singleton (t : F) :
    Algebra.FiniteType K ↥(Algebra.adjoin K ({t} : Set F))

theorem

inv

L8658 ↗

Transcendental.inv

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AdjoinRingProperties
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem _root_.Transcendental.inv {t : F} (ht : Transcendental K t) :
    Transcendental K t⁻¹

def

transcendentalChart

L8669 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.transcendentalChart

Documentation

The integral closure in F of the K-subalgebra generated by t, regarded as a K-subalgebra.

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TranscendentalChart
variable (K : Type*) {F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

def transcendentalChart (t : F) : Subalgebra K F

theorem

coe_transcendentalChart

L8672 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.coe_transcendentalChart

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TranscendentalChart
variable (K : Type*) {F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem coe_transcendentalChart (t : F) :
    (transcendentalChart K t : Set F)
      = ((integralClosure (Algebra.adjoin K ({t} : Set F)) F : Subalgebra _ F) : Set F)

theorem

transcendentalChart_subset_valuationSubring_of_mem

L8679 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.transcendentalChart_subset_valuationSubring_of_mem

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TranscendentalChart
variable (K : Type*) {F : Type*} [Field K] [Field F] [Algebra K F]
variable {K}

Lean source · statement

theorem transcendentalChart_subset_valuationSubring_of_mem (v : Place K F)
    {t : F} (ht : t ∈ v.toValuationSubring) :
    (transcendentalChart K t : Set F) ⊆ (v.toValuationSubring : Set F)

theorem

transcendentalChart_subset_or_inv_subset

L8686 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.transcendentalChart_subset_or_inv_subset

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section TranscendentalChart
variable (K : Type*) {F : Type*} [Field K] [Field F] [Algebra K F]
variable {K}

Lean source · statement

theorem transcendentalChart_subset_or_inv_subset (v : Place K F) (t : F) :
    (transcendentalChart K t : Set F) ⊆ (v.toValuationSubring : Set F)
      ∨ (transcendentalChart K t⁻¹ : Set F) ⊆ (v.toValuationSubring : Set F)

theorem

isScalarTower_integralClosure_subalgebra

L8703 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isScalarTower_integralClosure_subalgebra

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section
variable (B : Subalgebra K F)

Lean source · statement

theorem isScalarTower_integralClosure_subalgebra :
    IsScalarTower K B ↥(integralClosure B F)

theorem

finiteType_integralClosure_of_moduleFinite

L8708 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.finiteType_integralClosure_of_moduleFinite

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
section
variable (B : Subalgebra K F)

Lean source · statement

theorem finiteType_integralClosure_of_moduleFinite (hft : Algebra.FiniteType K B)
    (hfin : Module.Finite B ↥(integralClosure B F)) :
    Algebra.FiniteType K ↥(integralClosure B F)

theorem

isDedekindDomain_integralClosure_adjoin

L8717 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isDedekindDomain_integralClosure_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isDedekindDomain_integralClosure_adjoin {t : F} (ht : Transcendental K t)
    [FiniteDimensional K⟮t⟯ F] [Algebra.IsSeparable K⟮t⟯ F] :
    IsDedekindDomain ↥(integralClosure (Algebra.adjoin K ({t} : Set F)) F)

theorem

isFractionRing_integralClosure_adjoin

L8723 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isFractionRing_integralClosure_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isFractionRing_integralClosure_adjoin (t : F) [FiniteDimensional K⟮t⟯ F] :
    IsFractionRing ↥(integralClosure (Algebra.adjoin K ({t} : Set F)) F) F

theorem

moduleFinite_integralClosure_adjoin

L8729 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.moduleFinite_integralClosure_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem moduleFinite_integralClosure_adjoin {t : F} (ht : Transcendental K t)
    [FiniteDimensional K⟮t⟯ F] [Algebra.IsSeparable K⟮t⟯ F] :
    Module.Finite ↥(Algebra.adjoin K ({t} : Set F))
      ↥(integralClosure (Algebra.adjoin K ({t} : Set F)) F)

theorem

finiteType_integralClosure_adjoin

L8738 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.finiteType_integralClosure_adjoin

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem finiteType_integralClosure_adjoin {t : F} (ht : Transcendental K t)
    [FiniteDimensional K⟮t⟯ F] [Algebra.IsSeparable K⟮t⟯ F] :
    Algebra.FiniteType K ↥(integralClosure (Algebra.adjoin K ({t} : Set F)) F)

theorem

isAffineChart_restrictScalars_integralClosure

L8744 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isAffineChart_restrictScalars_integralClosure

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isAffineChart_restrictScalars_integralClosure (B : Subalgebra K F)
    (hft : Algebra.FiniteType K ↥(integralClosure B F))
    (hded : IsDedekindDomain ↥(integralClosure B F))
    (hfrac : IsFractionRing ↥(integralClosure B F) F) :
    IsAffineChart K F ((integralClosure B F).restrictScalars K)

theorem

isAffineChart_transcendentalChart

L8751 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isAffineChart_transcendentalChart

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section IsAffineChartEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isAffineChart_transcendentalChart {t : F} (ht : Transcendental K t)
    [FiniteDimensional K⟮t⟯ F] [Algebra.IsSeparable K⟮t⟯ F] :
    IsAffineChart K F (transcendentalChart K t)

def

HasSeparatingTranscendental

L8767 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.HasSeparatingTranscendental

Documentation

A transcendental element and its inverse give finite separable presentations of the function field.

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SeparatingTranscendental
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def HasSeparatingTranscendental : Prop

theorem

valSubringTwoAffineCharts_of_hasSeparatingTranscendental

L8774 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringTwoAffineCharts_of_hasSeparatingTranscendental

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SeparatingTranscendental
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringTwoAffineCharts_of_hasSeparatingTranscendental
    (h : HasSeparatingTranscendental K F) :
    ValSubringTwoAffineCharts K F

theorem

valSubringKaehlerFinite_of_hasSeparatingTranscendental

L8783 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerFinite_of_hasSeparatingTranscendental

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IsLocalRing
open _root_.IntermediateField
open _root_.IntermediateField.algebraAdjoinAdjoin
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SeparatingTranscendental
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerFinite_of_hasSeparatingTranscendental
    (h : HasSeparatingTranscendental K F) :
    ValSubringKaehlerFinite K F

theorem

ord_placeInfty_mOneAtom_mul_differentialCoeff_dX

L8848 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_mOneAtom_mul_differentialCoeff_dX

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section LowDegRegular
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem ord_placeInfty_mOneAtom_mul_differentialCoeff_dX
    (hInfty : (placeInfty K).ord ((placeInfty K).differentialCoeff (dX K)) = -2)
    {p c : K[X]} (hp : p ≠ 0) (hc : c ≠ 0) :
    (placeInfty K).ord
        (p1PrincipalPartAtom K p c 1 * (placeInfty K).differentialCoeff (dX K))
      = (p.natDegree : ℤ) - c.natDegree - 2

theorem

p1MOneAtom_mul_differentialCoeff_dX_mem_placeInfty

L8863 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1MOneAtom_mul_differentialCoeff_dX_mem_placeInfty

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section LowDegRegular
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1MOneAtom_mul_differentialCoeff_dX_mem_placeInfty
    (hInfty : (placeInfty K).ord ((placeInfty K).differentialCoeff (dX K)) = -2)
    {p c : K[X]} (hp : p ≠ 0) (hcdeg : c = 0 ∨ c.natDegree + 2 ≤ p.natDegree) :
    p1PrincipalPartAtom K p c 1 * (placeInfty K).differentialCoeff (dX K)
      ∈ (placeInfty K).toValuationSubring

theorem

trace_placeInfty_simplePoleResidue_mOne_of_lowDeg

L8892 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_placeInfty_simplePoleResidue_mOne_of_lowDeg

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section LowDegValue
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem trace_placeInfty_simplePoleResidue_mOne_of_lowDeg
    (hInfty : (placeInfty K).ord ((placeInfty K).differentialCoeff (dX K)) = -2)
    {p c : K[X]} (hpirr : Irreducible p)
    (hcdeg : c = 0 ∨ c.natDegree + 2 ≤ p.natDegree)
    (himem : p1PrincipalPartAtom K p c 1 * (placeInfty K).differentialCoeff (dX K)
        ∈ (placeInfty K).simplePoleSubmodule) :
    Algebra.trace K (placeInfty K).ResidueField
        ((placeInfty K).simplePoleResidueAux ⟨_, himem⟩)
      = -(c.coeff (p.natDegree - 1))

def

P1PlaceInftySimplePoleResidueEulerValueTopDeg

L8916 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PlaceInftySimplePoleResidueEulerValueTopDeg

Documentation

The residue formula at infinity for a numerator of degree one below its denominator.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section NamedCarriers
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

def P1PlaceInftySimplePoleResidueEulerValueTopDeg : Prop

def

P1PlaceInftySimplePoleResidueEulerValueMonomial

L8925 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PlaceInftySimplePoleResidueEulerValueMonomial

Documentation

The residue at infinity of the highest principal part monomial is minus one.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section NamedCarriers
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

def P1PlaceInftySimplePoleResidueEulerValueMonomial : Prop

theorem

p1PrincipalPartAtom_add_mul_differentialCoeff

L8942 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartAtom_add_mul_differentialCoeff

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in
omit [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem p1PrincipalPartAtom_add_mul_differentialCoeff (p c₁ c₂ : K[X]) (m : ℕ)
    (v : Place K (RatFunc K)) (ω : Ω[(RatFunc K)⁄K]) :
    p1PrincipalPartAtom K p (c₁ + c₂) m * v.differentialCoeff ω
      = p1PrincipalPartAtom K p c₁ m * v.differentialCoeff ω
        + p1PrincipalPartAtom K p c₂ m * v.differentialCoeff ω

theorem

p1PrincipalPartAtom_C_mul_mul_differentialCoeff

L8956 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartAtom_C_mul_mul_differentialCoeff

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [DecidableEq (RatFunc K)] in
omit [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem p1PrincipalPartAtom_C_mul_mul_differentialCoeff (p c : K[X]) (a : K) (m : ℕ)
    (v : Place K (RatFunc K)) (ω : Ω[(RatFunc K)⁄K]) :
    p1PrincipalPartAtom K p (C a * c) m * v.differentialCoeff ω
      = a • (p1PrincipalPartAtom K p c m * v.differentialCoeff ω)

theorem

trace_placeInfty_simplePoleResidueAux_mOne_add

L8969 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_placeInfty_simplePoleResidueAux_mOne_add

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem trace_placeInfty_simplePoleResidueAux_mOne_add
    {p c₁ c₂ : K[X]} {ω : Ω[(RatFunc K)⁄K]}
    (h1 : p1PrincipalPartAtom K p c₁ 1 * (placeInfty K).differentialCoeff ω
        ∈ (placeInfty K).simplePoleSubmodule)
    (h2 : p1PrincipalPartAtom K p c₂ 1 * (placeInfty K).differentialCoeff ω
        ∈ (placeInfty K).simplePoleSubmodule)
    (h12 : p1PrincipalPartAtom K p (c₁ + c₂) 1 * (placeInfty K).differentialCoeff ω
        ∈ (placeInfty K).simplePoleSubmodule) :
    Algebra.trace K (placeInfty K).ResidueField
        ((placeInfty K).simplePoleResidueAux ⟨_, h12⟩)
      = Algebra.trace K (placeInfty K).ResidueField
          ((placeInfty K).simplePoleResidueAux ⟨_, h1⟩)
        + Algebra.trace K (placeInfty K).ResidueField
            ((placeInfty K).simplePoleResidueAux ⟨_, h2⟩)

theorem

trace_placeInfty_simplePoleResidueAux_mOne_C_mul

L8989 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_placeInfty_simplePoleResidueAux_mOne_C_mul

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem trace_placeInfty_simplePoleResidueAux_mOne_C_mul
    {p c : K[X]} {a : K} {ω : Ω[(RatFunc K)⁄K]}
    (hc : p1PrincipalPartAtom K p c 1 * (placeInfty K).differentialCoeff ω
        ∈ (placeInfty K).simplePoleSubmodule)
    (hac : p1PrincipalPartAtom K p (C a * c) 1 * (placeInfty K).differentialCoeff ω
        ∈ (placeInfty K).simplePoleSubmodule) :
    Algebra.trace K (placeInfty K).ResidueField
        ((placeInfty K).simplePoleResidueAux ⟨_, hac⟩)
      = a * Algebra.trace K (placeInfty K).ResidueField
          ((placeInfty K).simplePoleResidueAux ⟨_, hc⟩)

theorem

p1PlaceInftySimplePoleResidueEulerValue_of_topDeg

L9004 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PlaceInftySimplePoleResidueEulerValue_of_topDeg

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1PlaceInftySimplePoleResidueEulerValue_of_topDeg
    (hInfty : (placeInfty K).ord ((placeInfty K).differentialCoeff (dX K)) = -2)
    (htop : P1PlaceInftySimplePoleResidueEulerValueTopDeg K) :
    P1PlaceInftySimplePoleResidueEulerValue K

theorem

p1PlaceInftySimplePoleResidueEulerValueTopDeg_of_monomial

L9016 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PlaceInftySimplePoleResidueEulerValueTopDeg_of_monomial

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1PlaceInftySimplePoleResidueEulerValueTopDeg_of_monomial
    (hInfty : (placeInfty K).ord ((placeInfty K).differentialCoeff (dX K)) = -2)
    (hmono : P1PlaceInftySimplePoleResidueEulerValueMonomial K) :
    P1PlaceInftySimplePoleResidueEulerValueTopDeg K

theorem

p1PlaceInftySimplePoleResidueEulerValue_of_monomial

L9068 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PlaceInftySimplePoleResidueEulerValue_of_monomial

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section EraseLeadLinearity
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1PlaceInftySimplePoleResidueEulerValue_of_monomial
    (hInfty : (placeInfty K).ord ((placeInfty K).differentialCoeff (dX K)) = -2)
    (hmono : P1PlaceInftySimplePoleResidueEulerValueMonomial K) :
    P1PlaceInftySimplePoleResidueEulerValue K

theorem

p1PrincipalPartMOneSimplePoleCancel_of_inftyMonomial

L9087 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartMOneSimplePoleCancel_of_inftyMonomial

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section ComposedEngine
variable [CharZero K]
variable [HasCanonicalDivisor (K := K) (F := RatFunc K)] [HasLocalResidue K (RatFunc K)]
  [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalDivisor (K := K) (F := RatFunc K)]
    [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1PrincipalPartMOneSimplePoleCancel_of_inftyMonomial
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0) (heq : ω₀ = dX K)
    (hmono : P1PlaceInftySimplePoleResidueEulerValueMonomial K) :
    P1PrincipalPartMOneSimplePoleCancel K hω₀

theorem

adjoin_simple_inv

L9120 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.adjoin_simple_inv

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem adjoin_simple_inv (t : F) : K⟮t⁻¹⟯ = K⟮t⟯

theorem

algebraMap_comp_equivOfEq

L9127 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.algebraMap_comp_equivOfEq

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem algebraMap_comp_equivOfEq {S T : IntermediateField K F} (h : S = T) :
    (algebraMap T F).comp ((IntermediateField.equivOfEq h).toRingEquiv : S →+* T)
      = ((RingEquiv.refl F : F ≃+* F) : F →+* F).comp (algebraMap S F)

theorem

finiteDimensional_of_eq

L9132 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.finiteDimensional_of_eq

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem finiteDimensional_of_eq {S T : IntermediateField K F} (h : S = T)
    [FiniteDimensional ↥S F] : FiniteDimensional ↥T F

theorem

isSeparable_of_eq

L9137 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.isSeparable_of_eq

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isSeparable_of_eq {S T : IntermediateField K F} (h : S = T)
    [Algebra.IsSeparable ↥S F] : Algebra.IsSeparable ↥T F

theorem

finiteDimensional_adjoin_inv

L9142 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.finiteDimensional_adjoin_inv

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem finiteDimensional_adjoin_inv (t : F) [FiniteDimensional K⟮t⟯ F] :
    FiniteDimensional K⟮t⁻¹⟯ F

theorem

isSeparable_adjoin_inv

L9146 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.isSeparable_adjoin_inv

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isSeparable_adjoin_inv (t : F) [Algebra.IsSeparable K⟮t⟯ F] :
    Algebra.IsSeparable K⟮t⁻¹⟯ F

theorem

isSeparable_of_charZero_of_finiteDimensional

L9160 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isSeparable_of_charZero_of_finiteDimensional

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section CharZeroSeparable
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isSeparable_of_charZero_of_finiteDimensional [CharZero K]
    (S : IntermediateField K F) [FiniteDimensional ↥S F] :
    Algebra.IsSeparable ↥S F

theorem

isSeparable_adjoin_of_charZero_of_finiteDimensional

L9166 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isSeparable_adjoin_of_charZero_of_finiteDimensional

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section CharZeroSeparable
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isSeparable_adjoin_of_charZero_of_finiteDimensional [CharZero K] (t : F)
    [FiniteDimensional K⟮t⟯ F] : Algebra.IsSeparable K⟮t⟯ F

def

HasSeparatingTranscendentalCore

L9177 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.HasSeparatingTranscendentalCore

Documentation

A transcendental element gives a finite extension presentation of the function field.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SeparatingTranscendentalCore
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def HasSeparatingTranscendentalCore : Prop

theorem

hasSeparatingTranscendental_of_core

L9182 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.hasSeparatingTranscendental_of_core

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SeparatingTranscendentalCore
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem hasSeparatingTranscendental_of_core [CharZero K]
    (h : HasSeparatingTranscendentalCore K F) :
    HasSeparatingTranscendental K F

theorem

valSubringKaehlerFinite_of_core

L9192 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.valSubringKaehlerFinite_of_core

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SeparatingTranscendentalCore
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem valSubringKaehlerFinite_of_core [CharZero K]
    (h : HasSeparatingTranscendentalCore K F) :
    ValSubringKaehlerFinite K F

theorem

ord_placeInfty_X_pow_natDegree_div

L9255 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_placeInfty_X_pow_natDegree_div

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioResidue

Lean source · statement

theorem ord_placeInfty_X_pow_natDegree_div {p : K[X]} (hp : p ≠ 0) :
    (placeInfty K).ord
        (algebraMap K[X] (RatFunc K) (X ^ p.natDegree) / algebraMap K[X] (RatFunc K) p) = 0

theorem

X_pow_natDegree_div_mem_placeInfty

L9268 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.X_pow_natDegree_div_mem_placeInfty

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioResidue

Lean source · statement

theorem X_pow_natDegree_div_mem_placeInfty {p : K[X]} (hp : p ≠ 0) :
    algebraMap K[X] (RatFunc K) (X ^ p.natDegree) / algebraMap K[X] (RatFunc K) p
      ∈ (placeInfty K).toValuationSubring

theorem

degree_X_pow_natDegree_sub_lt_of_monic

L9280 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.degree_X_pow_natDegree_sub_lt_of_monic

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioResidue
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem degree_X_pow_natDegree_sub_lt_of_monic {p : K[X]} (hpmon : p.Monic) :
    ((X : K[X]) ^ p.natDegree - p).degree < p.degree

theorem

residue_placeInfty_X_pow_natDegree_div_monic_eq_one

L9287 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.residue_placeInfty_X_pow_natDegree_div_monic_eq_one

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioResidue

Lean source · statement

theorem residue_placeInfty_X_pow_natDegree_div_monic_eq_one {p : K[X]} (hpmon : p.Monic) :
    IsLocalRing.residue _ ⟨_, X_pow_natDegree_div_mem_placeInfty K hpmon.ne_zero⟩
      = (1 : (placeInfty K).ResidueField)

def

P1PlaceInftySimplePoleResidueEulerValueX

L9335 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.P1PlaceInftySimplePoleResidueEulerValueX

Documentation

The residue at infinity of the reciprocal coordinate differential is minus one.

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section NamedCarrier
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

def P1PlaceInftySimplePoleResidueEulerValueX : Prop

theorem

p1MonomialAtom_eq_monicRatio_mul

L9351 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1MonomialAtom_eq_monicRatio_mul

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioReduction
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] [∀ (v : Place K (RatFunc K)),
  v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem p1MonomialAtom_eq_monicRatio_mul {p : K[X]} (hpdeg : 1 ≤ p.natDegree) :
    p1PrincipalPartAtom K p (X ^ (p.natDegree - 1)) 1 * (placeInfty K).differentialCoeff (dX K)
      = (algebraMap K[X] (RatFunc K) (X ^ p.natDegree) / algebraMap K[X] (RatFunc K) p)
        * (p1PrincipalPartAtom K (X : K[X]) 1 1
            * (placeInfty K).differentialCoeff (dX K))

theorem

p1XInvAtom_mul_differentialCoeff_dX_mem_simplePole_placeInfty

L9370 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1XInvAtom_mul_differentialCoeff_dX_mem_simplePole_placeInfty

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioReduction
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1XInvAtom_mul_differentialCoeff_dX_mem_simplePole_placeInfty
    (hInfty : (placeInfty K).ordDifferential (dX K) = -2) :
    p1PrincipalPartAtom K (X : K[X]) 1 1 * (placeInfty K).differentialCoeff (dX K)
      ∈ (placeInfty K).simplePoleSubmodule

theorem

simplePoleResidueAux_placeInfty_monomial_eq_X

L9378 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.simplePoleResidueAux_placeInfty_monomial_eq_X

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioReduction
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem simplePoleResidueAux_placeInfty_monomial_eq_X
    (hInfty : (placeInfty K).ordDifferential (dX K) = -2)
    {p : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p)
    (himem : p1PrincipalPartAtom K p (X ^ (p.natDegree - 1)) 1
          * (placeInfty K).differentialCoeff (dX K)
        ∈ (placeInfty K).simplePoleSubmodule) :
    (placeInfty K).simplePoleResidueAux ⟨_, himem⟩
      = (placeInfty K).simplePoleResidueAux
          ⟨_, p1XInvAtom_mul_differentialCoeff_dX_mem_simplePole_placeInfty K hInfty⟩

theorem

p1PlaceInftySimplePoleResidueEulerValueMonomial_of_X

L9401 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PlaceInftySimplePoleResidueEulerValueMonomial_of_X

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section MonicRatioReduction
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1PlaceInftySimplePoleResidueEulerValueMonomial_of_X
    (hInfty : (placeInfty K).ordDifferential (dX K) = -2)
    (hX : P1PlaceInftySimplePoleResidueEulerValueX K) :
    P1PlaceInftySimplePoleResidueEulerValueMonomial K

theorem

p1PrincipalPartMOneSimplePoleCancel_of_inftyX

L9421 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartMOneSimplePoleCancel_of_inftyX

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
section ComposedEngine
variable [CharZero K]
variable [HasCanonicalDivisor (K := K) (F := RatFunc K)] [HasLocalResidue K (RatFunc K)]
  [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalDivisor (K := K) (F := RatFunc K)]
    [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1PrincipalPartMOneSimplePoleCancel_of_inftyX
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0) (heq : ω₀ = dX K)
    (hX : P1PlaceInftySimplePoleResidueEulerValueX K) :
    P1PrincipalPartMOneSimplePoleCancel K hω₀

theorem

finiteDimensional_top

L9461 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.finiteDimensional_top

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K E : Type*} [Field K] [Field E] [Algebra K E]

Lean source · statement

theorem finiteDimensional_top : FiniteDimensional ↥(⊤ : IntermediateField K E) E

theorem

adjoin_simple_gen_eq_top

L9465 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.adjoin_simple_gen_eq_top

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K E : Type*} [Field K] [Field E] [Algebra K E]

Lean source · statement

theorem adjoin_simple_gen_eq_top (α : E) :
    K⟮AdjoinSimple.gen K α⟯ = (⊤ : IntermediateField K ↥K⟮α⟯)

theorem

transcendental_gen

L9470 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.transcendental_gen

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K E : Type*} [Field K] [Field E] [Algebra K E]

Lean source · statement

theorem transcendental_gen {α : E} (hα : Transcendental K α) :
    Transcendental K (AdjoinSimple.gen K α)

def

IsPurelyTranscendentalSimple

L9487 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.IsPurelyTranscendentalSimple

Documentation

The function field is generated by a single transcendental element.

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section PurelyTranscendentalSimple
variable (K F : Type*) [Field K] [Field F] [Algebra K F]

Lean source · statement

def IsPurelyTranscendentalSimple : Prop

theorem

hasSeparatingTranscendentalCore_of_isPurelyTranscendentalSimple

L9492 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.hasSeparatingTranscendentalCore_of_isPurelyTranscendentalSimple

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section PurelyTranscendentalSimple
variable (K F : Type*) [Field K] [Field F] [Algebra K F]
variable {K F}

Lean source · statement

theorem hasSeparatingTranscendentalCore_of_isPurelyTranscendentalSimple
    (h : IsPurelyTranscendentalSimple K F) :
    HasSeparatingTranscendentalCore K F

theorem

isPurelyTranscendentalSimple_adjoin_simple

L9506 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isPurelyTranscendentalSimple_adjoin_simple

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section SelfAdjoinInstance
variable {K E : Type*} [Field K] [Field E] [Algebra K E]

Lean source · statement

theorem isPurelyTranscendentalSimple_adjoin_simple {α : E} (hα : Transcendental K α) :
    IsPurelyTranscendentalSimple K ↥K⟮α⟯

def

CanonicalLocalResidueKSimplePoleCoordIndep

L9586 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.CanonicalLocalResidueKSimplePoleCoordIndep

Documentation

The reciprocal uniformizer differential has residue one in every uniformizer coordinate.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section NamedRow
variable (K F)
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]

Lean source · statement

def CanonicalLocalResidueKSimplePoleCoordIndep : Prop

theorem

p1XAtom_mul_differentialCoeff_dX_eq_neg

L9604 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1XAtom_mul_differentialCoeff_dX_eq_neg

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section XInvCoordinate
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1XAtom_mul_differentialCoeff_dX_eq_neg :
    p1PrincipalPartAtom K (X : K[X]) 1 1 * (placeInfty K).differentialCoeff (dX K)
      = -((placeInfty K).differentialCoeff
            (KaehlerDifferential.D K (RatFunc K) (RatFunc.X : RatFunc K)⁻¹)
          * ((RatFunc.X : RatFunc K)⁻¹)⁻¹)

theorem

trace_placeInfty_residueField_one

L9627 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.trace_placeInfty_residueField_one

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section InftyXDischarge
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [∀ (v : Place K (RatFunc K)), v.DCoordGenerates] [Nontrivial Ω[RatFunc K⁄K]] in

Lean source · statement

theorem trace_placeInfty_residueField_one :
    Algebra.trace K (placeInfty K).ResidueField 1 = 1

theorem

simplePoleResidueAux_placeInfty_p1XAtom_eq_neg_one

L9633 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.simplePoleResidueAux_placeInfty_p1XAtom_eq_neg_one

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section InftyXDischarge
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem simplePoleResidueAux_placeInfty_p1XAtom_eq_neg_one
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K))
    (himem : p1PrincipalPartAtom K (X : K[X]) 1 1 * (placeInfty K).differentialCoeff (dX K)
        ∈ (placeInfty K).simplePoleSubmodule) :
    (placeInfty K).simplePoleResidueAux ⟨_, himem⟩ = -1

theorem

p1PlaceInftySimplePoleResidueEulerValueX_of_simplePoleCoordIndep

L9644 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PlaceInftySimplePoleResidueEulerValueX_of_simplePoleCoordIndep

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section InftyXDischarge
variable (K) [DecidableEq (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]

Lean source · statement

theorem p1PlaceInftySimplePoleResidueEulerValueX_of_simplePoleCoordIndep
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K)) :
    P1PlaceInftySimplePoleResidueEulerValueX K

theorem

p1PrincipalPartMOneSimplePoleCancel_of_simplePoleCoordIndep

L9663 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1PrincipalPartMOneSimplePoleCancel_of_simplePoleCoordIndep

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section ComposedEngine
variable (K) [DecidableEq (RatFunc K)] [CharZero K]
variable [HasCanonicalDivisor (K := K) (F := RatFunc K)] [HasLocalResidue K (RatFunc K)]
  [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [∀ v : Place K (RatFunc K), v.DCoordGenerates] [Nontrivial Ω[(RatFunc K)⁄K]]
omit [HasCanonicalDivisor (K := K) (F := RatFunc K)]
    [HasLocalResidue K (RatFunc K)]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem p1PrincipalPartMOneSimplePoleCancel_of_simplePoleCoordIndep
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    {ω₀ : Ω[(RatFunc K)⁄K]} (hω₀ : ω₀ ≠ 0) (heq : ω₀ = dX K)
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K)) :
    P1PrincipalPartMOneSimplePoleCancel K hω₀

theorem

adjoin_simple_map_algHom'

L9702 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.adjoin_simple_map_algHom'

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F F' : Type*} [Field K] [Field F] [Field F'] [Algebra K F] [Algebra K F']

Lean source · statement

theorem adjoin_simple_map_algHom' (e : F →ₐ[K] F') (t : F) :
    (K⟮t⟯).map e = K⟮e t⟯

theorem

adjoin_simple_eq_top_of_algEquiv

L9706 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.IntermediateField.adjoin_simple_eq_top_of_algEquiv

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace IntermediateField
open _root_.IntermediateField
open _root_.IntermediateField
variable {K F F' : Type*} [Field K] [Field F] [Field F'] [Algebra K F] [Algebra K F']

Lean source · statement

theorem adjoin_simple_eq_top_of_algEquiv (e : F ≃ₐ[K] F') {t : F}
    (htop : K⟮t⟯ = (⊤ : IntermediateField K F)) :
    K⟮e t⟯ = (⊤ : IntermediateField K F')

theorem

congr

L9724 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.IsPurelyTranscendentalSimple.congr

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Congr
variable {K F F' : Type*} [Field K] [Field F] [Field F'] [Algebra K F] [Algebra K F']

Lean source · statement

theorem IsPurelyTranscendentalSimple.congr (e : F ≃ₐ[K] F')
    (h : IsPurelyTranscendentalSimple K F) : IsPurelyTranscendentalSimple K F'

theorem

isPurelyTranscendentalSimple_ratFunc

L9739 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isPurelyTranscendentalSimple_ratFunc

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section RatFuncInstance
variable (K : Type*) [Field K]

Lean source · statement

theorem isPurelyTranscendentalSimple_ratFunc :
    IsPurelyTranscendentalSimple K (RatFunc K)

theorem

isPurelyTranscendentalSimple_of_ratFuncAlgEquiv

L9751 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.isPurelyTranscendentalSimple_of_ratFuncAlgEquiv

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Iff
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem isPurelyTranscendentalSimple_of_ratFuncAlgEquiv
    (h : Nonempty (RatFunc K ≃ₐ[K] F)) : IsPurelyTranscendentalSimple K F

theorem

dCoordGenerates_of_valSubringKaehlerSpanTop

L9814 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.dCoordGenerates_of_valSubringKaehlerSpanTop

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AbstractEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
open scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem Place.dCoordGenerates_of_valSubringKaehlerSpanTop
    (hspan : ValSubringKaehlerSpanTop K F) (v : Place K F) :
    v.DCoordGenerates

theorem

dCoordGenerates_of_valSubringKaehlerFinite_of_charZero

L9836 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.dCoordGenerates_of_valSubringKaehlerFinite_of_charZero

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section AbstractEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
open scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem Place.dCoordGenerates_of_valSubringKaehlerFinite_of_charZero
    [CharZero K] [∀ v : Place K F, v.FiniteResidue]
    (hfin : ValSubringKaehlerFinite K F) (v : Place K F) :
    v.DCoordGenerates

theorem

dCoordGenerates_of_isPurelyTranscendentalSimple

L9851 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.dCoordGenerates_of_isPurelyTranscendentalSimple

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section ChainComposites
variable {K F F' : Type*} [Field K] [Field F] [Field F']
variable [Algebra K F] [Algebra K F'] [Algebra F F'] [IsScalarTower K F F']
open scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem Place.dCoordGenerates_of_isPurelyTranscendentalSimple
    [CharZero K] [∀ v : Place K F, v.FiniteResidue]
    (h : IsPurelyTranscendentalSimple K F) (v : Place K F) :
    v.DCoordGenerates

theorem

gate_dCoordGenerates_of_ratFunc_sat

L9877 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.gate_dCoordGenerates_of_ratFunc_sat

Source context 14 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
open _root_.IntermediateField
namespace ModularCurve
open _root_.AlgebraicCurve
variable (N : ℕ) [NeZero N]

Lean source · statement

theorem gate_dCoordGenerates_of_ratFunc_sat
    {K F : Type*} [Field K] [Field F] [Algebra K F]
    [CharZero K] [∀ v : Place K F, v.FiniteResidue]
    (hRat : Nonempty (RatFunc K ≃ₐ[K] F)) (v : Place K F) :
    v.DCoordGenerates

instance

instDCoordGenerates

L9914 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instDCoordGenerates

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open scoped _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable {K : Type*} [Field K]

Lean source · statement

instance (priority := low) instDCoordGenerates [CharZero K] (v : Place K (RatFunc K)) :
    v.DCoordGenerates

theorem

exists_smul_dX_eq

L9983 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_smul_dX_eq

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]

Lean source · statement

theorem exists_smul_dX_eq (ω : Ω[(RatFunc K)⁄K]) : ∃ c : RatFunc K, c • dX K = ω

theorem

ordDifferential_dX_of_ne_placeInfty

L9992 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferential_dX_of_ne_placeInfty

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section ProfileValues
variable [CharZero K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ordDifferential_dX_of_ne_placeInfty {v : Place K (RatFunc K)}
    (hv : v ≠ placeInfty K) : v.ordDifferential (dX K) = 0

theorem

ordDifferential_dX_placeInfty

L9997 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferential_dX_placeInfty

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section ProfileValues
variable [CharZero K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ordDifferential_dX_placeInfty :
    (placeInfty K).ordDifferential (dX K) = -2

theorem

exists_divisor_smul_dX

L10002 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_divisor_smul_dX

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section ProfileValues
variable [CharZero K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem exists_divisor_smul_dX {c : RatFunc K} (hc : c ≠ 0) :
    ∃ D : Divisor K (RatFunc K),
      (∀ v : Place K (RatFunc K), D v = v.ordDifferential (c • dX K)) ∧
        Divisor.degree D = -2

instance

instHasCanonicalDivisorRatFunc

L10024 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.instHasCanonicalDivisorRatFunc

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section Instance
variable [CharZero K]

Lean source · full declaration

instance instHasCanonicalDivisorRatFunc :
    HasCanonicalDivisor (K := K) (F := RatFunc K) where
  exists_divisor ω hω := by
    classical
    obtain ⟨c, hc⟩ := exists_smul_dX_eq K ω
    have hc0 : c ≠ 0 := by
      rintro rfl
      rw [zero_smul] at hc
      exact hω hc.symm
    obtain ⟨D, hD, -⟩ := exists_divisor_smul_dX K hc0
    refine ⟨D, fun v => ?_⟩
    rw [← hc]
    exact hD v

theorem

D_pow_succ_inv

L10097 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.D_pow_succ_inv

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KaehlerToolbox
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem D_pow_succ_inv (f : F) (m : ℕ) :
    KaehlerDifferential.D K F ((f ^ (m + 1))⁻¹)
      = (-((m + 1 : ℕ) : F) * (f ^ (m + 2))⁻¹) • KaehlerDifferential.D K F f

theorem

differentialCoeff_add''

L10113 ↗

AlgebraicCurve.Place.differentialCoeff_add''

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KaehlerToolbox
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.differentialCoeff_add'' [v.DCoordGenerates]
  [Nontrivial Ω[F⁄K]] (ω₁ ω₂ : Ω[F⁄K]) :
    v.differentialCoeff (ω₁ + ω₂) = v.differentialCoeff ω₁ + v.differentialCoeff ω₂

theorem

differentialCoeff_D_uniformizer_pow_inv

L10121 ↗

AlgebraicCurve.Place.differentialCoeff_D_uniformizer_pow_inv

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KaehlerToolbox
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F)
open _root_.AlgebraicCurve.Place
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.differentialCoeff_D_uniformizer_pow_inv
  [v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
    (m : ℕ) :
    v.differentialCoeff (KaehlerDifferential.D K F ((v.uniformizer ^ (m + 1))⁻¹))
      = -((m + 1 : ℕ) : F) * (v.uniformizer ^ (m + 2))⁻¹

theorem

ratFuncDXCoeff_eq_of_D_eq_smul_dX

L10138 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_eq_of_D_eq_smul_dX

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeffIntegrality
variable {K : Type*} [Field K]

Lean source · statement

theorem ratFuncDXCoeff_eq_of_D_eq_smul_dX {f e : RatFunc K}
    (h : KaehlerDifferential.D K (RatFunc K) f = e • dX K) : ratFuncDXCoeff K f = e

theorem

ratFuncDXCoeff_zero

L10146 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_zero

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeffIntegrality
variable {K : Type*} [Field K]

Lean source · statement

theorem ratFuncDXCoeff_zero : ratFuncDXCoeff K (0 : RatFunc K) = 0

theorem

ord_algebraMap_denom_eq_zero_of_ord_nonneg

L10155 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ord_algebraMap_denom_eq_zero_of_ord_nonneg

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeffIntegrality
variable {K : Type*} [Field K]
section FinitePlaces
variable {w : HeightOneSpectrum K[X]}
local notation "vw" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ord_algebraMap_denom_eq_zero_of_ord_nonneg {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) {f : RatFunc K}
    (hf : f ≠ 0) (hord : 0 ≤ (vw).ord f) :
    (vw).ord (algebraMap K[X] (RatFunc K) f.denom) = 0

theorem

ratFuncDXCoeff_eq_zero_or_ord_nonneg_of_ord_nonneg

L10183 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_eq_zero_or_ord_nonneg_of_ord_nonneg

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeffIntegrality
variable {K : Type*} [Field K]
section FinitePlaces
variable {w : HeightOneSpectrum K[X]}
local notation "vw" => Place.ofHeightOneSpectrum (K := K) (F := RatFunc K) w

Lean source · statement

theorem ratFuncDXCoeff_eq_zero_or_ord_nonneg_of_ord_nonneg {p : K[X]}
    (hp : Irreducible p) (hwp : w.asIdeal = Ideal.span {p}) {g : RatFunc K}
    (hg : g ≠ 0) (hord : 0 ≤ (vw).ord g) :
    ratFuncDXCoeff K g = 0 ∨ 0 ≤ (vw).ord (ratFuncDXCoeff K g)

theorem

ratFuncDXCoeff_ne_zero_and_ord_placeInfty_eq_two_of_ord_eq_one

L10220 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_ne_zero_and_ord_placeInfty_eq_two_of_ord_eq_one

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeffIntegrality
variable {K : Type*} [Field K]
section PlaceInftySide
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ratFuncDXCoeff_ne_zero_and_ord_placeInfty_eq_two_of_ord_eq_one
    {f : RatFunc K} (hf : f ≠ 0) (hord : (placeInfty K).ord f = 1) :
    ratFuncDXCoeff K f ≠ 0 ∧ (placeInfty K).ord (ratFuncDXCoeff K f) = 2

theorem

ratFuncDXCoeff_eq_zero_or_two_le_ord_placeInfty_of_ord_nonneg

L10227 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_eq_zero_or_two_le_ord_placeInfty_of_ord_nonneg

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section DXCoeffIntegrality
variable {K : Type*} [Field K]
section PlaceInftySide
variable [DecidableEq (RatFunc K)]

Lean source · statement

theorem ratFuncDXCoeff_eq_zero_or_two_le_ord_placeInfty_of_ord_nonneg
    {g : RatFunc K} (hg : g ≠ 0) (hord : 0 ≤ (placeInfty K).ord g) :
    ratFuncDXCoeff K g = 0 ∨ 2 ≤ (placeInfty K).ord (ratFuncDXCoeff K g)

theorem

ratFuncDXCoeff_uniformizer_ne_zero_and_ord_le

L10252 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ratFuncDXCoeff_uniformizer_ne_zero_and_ord_le

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section LemmaA
variable {K : Type*} [Field K] [CharZero K]

Lean source · statement

theorem ratFuncDXCoeff_uniformizer_ne_zero_and_ord_le (v : Place K (RatFunc K))
    {g : RatFunc K} (hg : g ∈ v.toValuationSubring) :
    ratFuncDXCoeff K v.uniformizer ≠ 0 ∧
      (ratFuncDXCoeff K g = 0 ∨
        v.ord (ratFuncDXCoeff K v.uniformizer) ≤ v.ord (ratFuncDXCoeff K g))

theorem

differentialCoeff_D_eq_ratFuncDXCoeff_div

L10285 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_D_eq_ratFuncDXCoeff_div

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section LemmaA
variable {K : Type*} [Field K] [CharZero K]

Lean source · statement

theorem differentialCoeff_D_eq_ratFuncDXCoeff_div (v : Place K (RatFunc K)) (f : RatFunc K)
    (hπ : ratFuncDXCoeff K v.uniformizer ≠ 0) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) f)
      = ratFuncDXCoeff K f / ratFuncDXCoeff K v.uniformizer

theorem

differentialCoeff_D_mem_of_mem

L10294 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.differentialCoeff_D_mem_of_mem

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section LemmaA
variable {K : Type*} [Field K] [CharZero K]

Lean source · statement

theorem differentialCoeff_D_mem_of_mem (v : Place K (RatFunc K))
    {g : RatFunc K} (hg : g ∈ v.toValuationSubring) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) g) ∈ v.toValuationSubring

theorem

res_differentialCoeff_D_of_mem_poleSubmodule

L10321 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_differentialCoeff_D_of_mem_poleSubmodule

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ExactDifferentialEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F) [v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
omit [HasCanonicalLocalResidueKStar K F] in
open scoped _root_.AlgebraicCurve.Place.CanonicalLocalResidueDataK in

Lean source · statement

theorem CanonicalLocalResidueDataK.res_differentialCoeff_D_of_mem_poleSubmodule
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (hint : ∀ h : F, h ∈ v.toValuationSubring →
      v.differentialCoeff (KaehlerDifferential.D K F h) ∈ v.toValuationSubring)
    (R : v.CanonicalLocalResidueDataK) (N : ℕ) {g : F} (hg : g ∈ v.poleSubmodule N) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K F g)) = 0

theorem

res_differentialCoeff_D_of_surj

L10362 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_differentialCoeff_D_of_surj

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ExactDifferentialEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F) [v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
omit [HasCanonicalLocalResidueKStar K F] in
open scoped _root_.AlgebraicCurve.Place.CanonicalLocalResidueDataK in

Lean source · statement

theorem CanonicalLocalResidueDataK.res_differentialCoeff_D_of_surj
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (hint : ∀ h : F, h ∈ v.toValuationSubring →
      v.differentialCoeff (KaehlerDifferential.D K F h) ∈ v.toValuationSubring)
    (R : v.CanonicalLocalResidueDataK) (g : F) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K F g)) = 0

theorem

res_differentialCoeff_D_mul_pow_inv_of_surj

L10374 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_differentialCoeff_D_mul_pow_inv_of_surj

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ExactDifferentialEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F) [v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
omit [HasCanonicalLocalResidueKStar K F] in
open scoped _root_.AlgebraicCurve.Place.CanonicalLocalResidueDataK in

Lean source · statement

theorem CanonicalLocalResidueDataK.res_differentialCoeff_D_mul_pow_inv_of_surj [CharZero K]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (hint : ∀ h : F, h ∈ v.toValuationSubring →
      v.differentialCoeff (KaehlerDifferential.D K F h) ∈ v.toValuationSubring)
    (R : v.CanonicalLocalResidueDataK) (π' : F) {n : ℕ} (hn : 1 ≤ n) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K F π') * ((π') ^ (n + 1))⁻¹) = 0

theorem

res_differentialCoeff_D_mul_inv_of_integral

L10402 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.Place.CanonicalLocalResidueDataK.res_differentialCoeff_D_mul_inv_of_integral

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ExactDifferentialEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable (v : Place K F) [v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
omit [HasCanonicalLocalResidueKStar K F] in
open scoped _root_.AlgebraicCurve.Place.CanonicalLocalResidueDataK in

Lean source · statement

theorem CanonicalLocalResidueDataK.res_differentialCoeff_D_mul_inv_of_integral
    (hint : ∀ h : F, h ∈ v.toValuationSubring →
      v.differentialCoeff (KaehlerDifferential.D K F h) ∈ v.toValuationSubring)
    (R : v.CanonicalLocalResidueDataK) {π' : F} (hπ' : v.ord π' = 1) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K F π') * (π')⁻¹) = 1

theorem

canonicalLocalResidueDataK_res_differentialCoeff_D_mul_pow_inv_ratFunc

L10450 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_res_differentialCoeff_D_mul_pow_inv_ratFunc

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section RatFuncClauses
variable {K : Type*} [Field K] [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem canonicalLocalResidueDataK_res_differentialCoeff_D_mul_pow_inv_ratFunc
    (v : Place K (RatFunc K)) (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (π' : RatFunc K) (R : v.CanonicalLocalResidueDataK) {n : ℕ} (hn : 1 ≤ n) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) π')
      * ((π') ^ (n + 1))⁻¹) = 0

theorem

canonicalLocalResidueDataK_res_differentialCoeff_D_mul_inv_ratFunc

L10459 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_res_differentialCoeff_D_mul_inv_ratFunc

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section RatFuncClauses
variable {K : Type*} [Field K] [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem canonicalLocalResidueDataK_res_differentialCoeff_D_mul_inv_ratFunc
    (v : Place K (RatFunc K)) {π' : RatFunc K} (hπ' : v.ord π' = 1)
    (R : v.CanonicalLocalResidueDataK) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) π') * (π')⁻¹) = 1

theorem

canonicalLocalResidueKSimplePoleCoordIndep_ratFunc

L10468 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueKSimplePoleCoordIndep_ratFunc

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField

Lean source · statement

theorem canonicalLocalResidueKSimplePoleCoordIndep_ratFunc (K : Type*) [Field K] [CharZero K]
    [HasCanonicalLocalResidueKStar K (RatFunc K)] :
    CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K)

theorem

canonicalLocalResidueDataK_res_differentialCoeff_D_mul_pow_inv_placeInfty

L10479 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_res_differentialCoeff_D_mul_pow_inv_placeInfty

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftyConsumers
variable (K : Type*) [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem canonicalLocalResidueDataK_res_differentialCoeff_D_mul_pow_inv_placeInfty
    (π' : RatFunc K) (R : (placeInfty K).CanonicalLocalResidueDataK) {n : ℕ} (hn : 1 ≤ n) :
    R.res ((placeInfty K).differentialCoeff (KaehlerDifferential.D K (RatFunc K) π')
      * ((π') ^ (n + 1))⁻¹) = 0

theorem

canonicalLocalResidueDataK_res_X_pow_mul_D_X

L10487 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_res_X_pow_mul_D_X

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftyConsumers
variable (K : Type*) [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem canonicalLocalResidueDataK_res_X_pow_mul_D_X
    (R : (placeInfty K).CanonicalLocalResidueDataK) (n : ℕ) :
    R.res ((RatFunc.X : RatFunc K) ^ n
        * (placeInfty K).differentialCoeff
            (KaehlerDifferential.D K (RatFunc K) RatFunc.X)) = 0

theorem

canonicalLocalResidueDataK_kaehlerResidueTerm_X_pow

L10497 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueDataK_kaehlerResidueTerm_X_pow

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PlaceInftyConsumers
variable (K : Type*) [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem canonicalLocalResidueDataK_kaehlerResidueTerm_X_pow
    (R : (placeInfty K).CanonicalLocalResidueDataK) (n : ℕ) :
    Algebra.trace K (placeInfty K).ResidueField
        (R.res (diagonalHom K (RatFunc K) ((RatFunc.X : RatFunc K) ^ n) (placeInfty K)
          * (placeInfty K).differentialCoeff
              (KaehlerDifferential.D K (RatFunc K) RatFunc.X))) = 0

theorem

surjective_algebraMap_residueField_of_deg_eq_one

L10512 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.surjective_algebraMap_residueField_of_deg_eq_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section AlgClosedDischarge
variable (K : Type*) [Field K] [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem surjective_algebraMap_residueField_of_deg_eq_one {F : Type*} [Field F] [Algebra K F]
    (v : Place K F) (hdeg : v.deg = 1) :
    Function.Surjective (algebraMap K v.ResidueField)

theorem

surjective_algebraMap_residueField_ratFunc_of_isAlgClosed

L10521 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.surjective_algebraMap_residueField_ratFunc_of_isAlgClosed

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section AlgClosedDischarge
variable (K : Type*) [Field K] [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)]
omit [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem surjective_algebraMap_residueField_ratFunc_of_isAlgClosed [IsAlgClosed K]
    (v : Place K (RatFunc K)) :
    Function.Surjective (algebraMap K v.ResidueField)

theorem

canonicalLocalResidueKDifferentialCoordIndep_ratFunc_of_isAlgClosed

L10537 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.canonicalLocalResidueKDifferentialCoordIndep_ratFunc_of_isAlgClosed

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section AlgClosedDischarge
variable (K : Type*) [Field K] [CharZero K] [HasCanonicalLocalResidueKStar K (RatFunc K)]
variable [IsAlgClosed K]
omit [HasCanonicalLocalResidueKStar K (RatFunc K)] in

Lean source · statement

theorem canonicalLocalResidueKDifferentialCoordIndep_ratFunc_of_isAlgClosed :
    CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K)

theorem

exists_ne_zero_smul_dX_of_uniformizer

L10575 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_ne_zero_smul_dX_of_uniformizer

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerfectDCoord
variable (K : Type*) [Field K] [PerfectField K]

Lean source · statement

theorem exists_ne_zero_smul_dX_of_uniformizer (v : Place K (RatFunc K)) :
    ∃ c : RatFunc K, c ≠ 0 ∧
      KaehlerDifferential.D K (RatFunc K) v.uniformizer = c • dX K

instance

instDCoordGeneratesPerfectField

L10594 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.instDCoordGeneratesPerfectField

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerfectDCoord
variable (K : Type*) [Field K] [PerfectField K]
namespace RationalFunctionField
open _root_.AlgebraicCurve.RationalFunctionField
open _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField

Lean source · statement

instance (priority := low) instDCoordGeneratesPerfectField (v : Place K (RatFunc K)) :
    v.DCoordGenerates

theorem

p1DifferentialCoeffUnitFinite_dX_of_perfectField

L10615 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1DifferentialCoeffUnitFinite_dX_of_perfectField

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section PerfectProfileValues
variable [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem p1DifferentialCoeffUnitFinite_dX_of_perfectField :
    P1DifferentialCoeffUnitFinite K (dX_ne_zero K)

theorem

p1DifferentialCoeffRegularFinite_dX_of_perfectField

L10624 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p1DifferentialCoeffRegularFinite_dX_of_perfectField

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section PerfectProfileValues
variable [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem p1DifferentialCoeffRegularFinite_dX_of_perfectField :
    P1DifferentialCoeffRegularFinite K (dX_ne_zero K)

theorem

ordDifferential_dX_of_ne_placeInfty_of_perfectField

L10629 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferential_dX_of_ne_placeInfty_of_perfectField

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section PerfectProfileValues
variable [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ordDifferential_dX_of_ne_placeInfty_of_perfectField {v : Place K (RatFunc K)}
    (hv : v ≠ placeInfty K) : v.ordDifferential (dX K) = 0

theorem

ordDifferential_dX_placeInfty_of_perfectField

L10634 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ordDifferential_dX_placeInfty_of_perfectField

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section PerfectProfileValues
variable [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ordDifferential_dX_placeInfty_of_perfectField :
    (placeInfty K).ordDifferential (dX K) = -2

theorem

exists_divisor_smul_dX_of_perfectField

L10640 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.exists_divisor_smul_dX_of_perfectField

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section PerfectProfileValues
variable [PerfectField K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem exists_divisor_smul_dX_of_perfectField {c : RatFunc K} (hc : c ≠ 0) :
    ∃ D : Divisor K (RatFunc K),
      (∀ v : Place K (RatFunc K), D v = v.ordDifferential (c • dX K)) ∧
        Divisor.degree D = -2

instance

instHasCanonicalDivisorRatFuncPerfectField

L10663 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.instHasCanonicalDivisorRatFuncPerfectField

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.KaehlerDifferential
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Profile
variable (K : Type*) [Field K]
section PerfectInstance
variable [PerfectField K]

Lean source · full declaration

instance instHasCanonicalDivisorRatFuncPerfectField :
    HasCanonicalDivisor (K := K) (F := RatFunc K) where
  exists_divisor ω hω := by
    classical
    obtain ⟨c, hc⟩ := exists_smul_dX_eq K ω
    have hc0 : c ≠ 0 := by
      rintro rfl
      rw [zero_smul] at hc
      exact hω hc.symm
    obtain ⟨D, hD, -⟩ := exists_divisor_smul_dX_of_perfectField K hc0
    refine ⟨D, fun v => ?_⟩
    rw [← hc]
    exact hD v

theorem

ag9b13t_res_differentialCoeff_D_mul_pow_inv_of_surj_of_natCast_ne_zero

L10733 ↗

ModularCurve.MilneAvAg9bRd13T2CoordIndepChar3.ag9b13t_res_differentialCoeff_D_mul_pow_inv_of_surj_of_natCast_ne_zero

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace ModularCurve
namespace MilneAvAg9bRd13T2CoordIndepChar3
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
open scoped _root_.Polynomial
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.MilneAvAg9bRd13T2CoordIndepChar3 in
with_weak_namespace _root_.ModularCurve.MilneAvAg9bRd13T2CoordIndepChar3

Lean source · statement

theorem ag9b13t_res_differentialCoeff_D_mul_pow_inv_of_surj_of_natCast_ne_zero
    {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
    (v : Place K F) [v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (hint : ∀ h : F, h ∈ v.toValuationSubring →
      v.differentialCoeff (KaehlerDifferential.D K F h) ∈ v.toValuationSubring)
    (R : v.CanonicalLocalResidueDataK) (π' : F) {n : ℕ} (hn : 1 ≤ n)
    (hnK : (n : K) ≠ 0) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K F π') * ((π') ^ (n + 1))⁻¹) = 0

theorem

residueTheoremK_placeInfty_clause_X_pow

L10837 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.residueTheoremK_placeInfty_clause_X_pow

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve.RationalFunctionField
section RatFuncPlaceInftyClause
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem residueTheoremK_placeInfty_clause_X_pow
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK) (n : ℕ) :
    kaehlerResidueTermKFam Rfam (KaehlerDifferential.D K (RatFunc K) RatFunc.X)
      (diagonalHom K (RatFunc K) ((RatFunc.X : RatFunc K) ^ n)) (placeInfty K) = 0

theorem

ag9b12c_trace_root_pow_div_derivative_of_lt_of_perfectField

L10898 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_trace_root_pow_div_derivative_of_lt_of_perfectField

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section EulerPerfectField
variable {p : K[X]} [Fact (Irreducible p)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_trace_root_pow_div_derivative_of_lt_of_perfectField (hpmon : p.Monic)
    {k : ℕ} (hk : k < p.natDegree - 1) :
    Algebra.trace K (AdjoinRoot p)
      (AdjoinRoot.root p ^ k / aeval (AdjoinRoot.root p) (derivative p)) = 0

theorem

ag9b12c_trace_root_pow_div_derivative_self_of_perfectField

L10912 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_trace_root_pow_div_derivative_self_of_perfectField

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section EulerPerfectField
variable {p : K[X]} [Fact (Irreducible p)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_trace_root_pow_div_derivative_self_of_perfectField (hpmon : p.Monic)
    (hd : 0 < p.natDegree) :
    Algebra.trace K (AdjoinRoot p)
      (AdjoinRoot.root p ^ (p.natDegree - 1) /
        aeval (AdjoinRoot.root p) (derivative p)) = 1

theorem

ag9b12c_aeval_root_eq_sum_range

L10927 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_aeval_root_eq_sum_range

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section EulerPerfectField
variable {p : K[X]} [Fact (Irreducible p)]
omit [PerfectField K] [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_aeval_root_eq_sum_range {c : K[X]} {d : ℕ} (hd : c.natDegree < d) :
    (aeval (AdjoinRoot.root p) c : AdjoinRoot p)
      = ∑ k ∈ Finset.range d, c.coeff k • AdjoinRoot.root p ^ k

theorem

ag9b12c_trace_adjoinRoot_mk_div_mk_derivative_of_perfectField

L10934 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_trace_adjoinRoot_mk_div_mk_derivative_of_perfectField

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section EulerPerfectField
variable {p : K[X]} [Fact (Irreducible p)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_trace_adjoinRoot_mk_div_mk_derivative_of_perfectField (hpmon : p.Monic)
    {c : K[X]} (hdeg : c.degree < p.degree) :
    Algebra.trace K (AdjoinRoot p)
        (AdjoinRoot.mk p c / AdjoinRoot.mk p (derivative p))
      = c.coeff (p.natDegree - 1)

theorem

ag9b12c_differentialCoeff_add

L10967 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_differentialCoeff_add

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section DerivativeRegular
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_differentialCoeff_add (v : Place K (RatFunc K))
    (ω₁ ω₂ : Ω[(RatFunc K)⁄K]) :
    v.differentialCoeff (ω₁ + ω₂) = v.differentialCoeff ω₁ + v.differentialCoeff ω₂

theorem

ag9b12c_differentialCoeff_D_mem_finitePlace

L10974 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_differentialCoeff_D_mem_finitePlace

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section DerivativeRegular
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_differentialCoeff_D_mem_finitePlace {p : K[X]} (hpirr : Irreducible p)
    {f : RatFunc K} (hf : f ∈ (finitePlace K hpirr).toValuationSubring) :
    (finitePlace K hpirr).differentialCoeff (D K (RatFunc K) f)
      ∈ (finitePlace K hpirr).toValuationSubring

theorem

ag9b12c_uniformizer_div_mem_finitePlace

L11010 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_uniformizer_div_mem_finitePlace

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section LeibnizCore
omit [PerfectField K] [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_uniformizer_div_mem_finitePlace {p : K[X]} (hpirr : Irreducible p) :
    (finitePlace K hpirr).uniformizer / algebraMap K[X] (RatFunc K) p
      ∈ (finitePlace K hpirr).toValuationSubring

theorem

ag9b12c_one_sub_uniformizer_div_mul_differentialCoeff_D

L11022 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_one_sub_uniformizer_div_mul_differentialCoeff_D

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section LeibnizCore
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_one_sub_uniformizer_div_mul_differentialCoeff_D
    {p : K[X]} (hpirr : Irreducible p) :
    (1 : RatFunc K) - (finitePlace K hpirr).uniformizer / algebraMap K[X] (RatFunc K) p
        * (finitePlace K hpirr).differentialCoeff
            (D K (RatFunc K) (algebraMap K[X] (RatFunc K) p))
      = algebraMap K[X] (RatFunc K) p
          * (finitePlace K hpirr).differentialCoeff
              (D K (RatFunc K) ((finitePlace K hpirr).uniformizer
                / algebraMap K[X] (RatFunc K) p))

theorem

ag9b12c_uniformizer_div_mul_differentialCoeff_D_mem_finitePlace

L11055 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_uniformizer_div_mul_differentialCoeff_D_mem_finitePlace

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section LeibnizCore
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_uniformizer_div_mul_differentialCoeff_D_mem_finitePlace
    {p : K[X]} (hpirr : Irreducible p) :
    (finitePlace K hpirr).uniformizer / algebraMap K[X] (RatFunc K) p
        * (finitePlace K hpirr).differentialCoeff
            (D K (RatFunc K) (algebraMap K[X] (RatFunc K) p))
      ∈ (finitePlace K hpirr).toValuationSubring

theorem

ag9b12c_residue_uniformizer_div_mul_differentialCoeff_D_eq_one

L11071 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_residue_uniformizer_div_mul_differentialCoeff_D_eq_one

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section LeibnizCore
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_residue_uniformizer_div_mul_differentialCoeff_D_eq_one
    {p : K[X]} (hpirr : Irreducible p) :
    IsLocalRing.residue _
        ⟨_, ag9b12c_uniformizer_div_mul_differentialCoeff_D_mem_finitePlace K hpirr⟩
      = (1 : (finitePlace K hpirr).ResidueField)

theorem

ag9b12c_residue_algebraMap_derivative_ne_zero

L11118 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_residue_algebraMap_derivative_ne_zero

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section BridgeDischarge
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_residue_algebraMap_derivative_ne_zero {p : K[X]} (hpirr : Irreducible p) :
    IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) p.derivative,
        algebraMap_mem_ofHeightOneSpectrum K _ p.derivative⟩
      ≠ (0 : (finitePlace K hpirr).ResidueField)

theorem

ag9b12c_simplePoleResidueAux_finitePlace_p1PrincipalPartAtom_mOne

L11133 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_simplePoleResidueAux_finitePlace_p1PrincipalPartAtom_mOne

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section BridgeDischarge
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_simplePoleResidueAux_finitePlace_p1PrincipalPartAtom_mOne
    {p c : K[X]} (hpirr : Irreducible p)
    (hfmem : p1PrincipalPartAtom K p c 1 * (finitePlace K hpirr).differentialCoeff (dX K)
        ∈ (finitePlace K hpirr).simplePoleSubmodule) :
    (finitePlace K hpirr).simplePoleResidueAux ⟨_, hfmem⟩
      = IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) c,
            algebraMap_mem_ofHeightOneSpectrum K _ c⟩
        / IsLocalRing.residue _ ⟨algebraMap K[X] (RatFunc K) p.derivative,
            algebraMap_mem_ofHeightOneSpectrum K _ p.derivative⟩

theorem

ag9b12c_p1FinitePlaceSimplePoleResidueAdjoinRootValue_of_perfectField

L11188 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_p1FinitePlaceSimplePoleResidueAdjoinRootValue_of_perfectField

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section BridgeDischarge
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_p1FinitePlaceSimplePoleResidueAdjoinRootValue_of_perfectField :
    P1FinitePlaceSimplePoleResidueAdjoinRootValue K

theorem

ag9b12c_trace_finitePlace_simplePoleResidue_mOne_of_perfectField

L11209 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_trace_finitePlace_simplePoleResidue_mOne_of_perfectField

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section TraceValue
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem ag9b12c_trace_finitePlace_simplePoleResidue_mOne_of_perfectField
    {p c : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p) (hdeg : c.degree < p.degree)
    (hfmem : p1PrincipalPartAtom K p c 1 * (finitePlace K hpirr).differentialCoeff (dX K)
        ∈ (finitePlace K hpirr).simplePoleSubmodule) :
    Algebra.trace K (finitePlace K hpirr).ResidueField
        ((finitePlace K hpirr).simplePoleResidueAux ⟨_, hfmem⟩)
      = c.coeff (p.natDegree - 1)

theorem

ag9b12c_p1MOneSimplePoleCancel_dX_of_inftyEulerValue_of_perfectField

L11222 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b12c_p1MOneSimplePoleCancel_dX_of_inftyEulerValue_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
section TraceValue

Lean source · statement

theorem ag9b12c_p1MOneSimplePoleCancel_dX_of_inftyEulerValue_of_perfectField
    (hinf : P1PlaceInftySimplePoleResidueEulerValue K) :
    P1PrincipalPartMOneSimplePoleCancel K (dX_ne_zero K)

theorem

ag9b13e_ratFuncDXCoeff_uniformizer_ne_zero_and_ord_le_of_perfectField

L11265 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_ratFuncDXCoeff_uniformizer_ne_zero_and_ord_le_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section LemmaAPerfect
variable {K : Type*} [Field K] [PerfectField K]

Lean source · statement

theorem ag9b13e_ratFuncDXCoeff_uniformizer_ne_zero_and_ord_le_of_perfectField
    (v : Place K (RatFunc K)) {g : RatFunc K} (hg : g ∈ v.toValuationSubring) :
    ratFuncDXCoeff K v.uniformizer ≠ 0 ∧
      (ratFuncDXCoeff K g = 0 ∨
        v.ord (ratFuncDXCoeff K v.uniformizer) ≤ v.ord (ratFuncDXCoeff K g))

theorem

ag9b13e_differentialCoeff_D_eq_ratFuncDXCoeff_div_of_perfectField

L11300 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_differentialCoeff_D_eq_ratFuncDXCoeff_div_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section LemmaAPerfect
variable {K : Type*} [Field K] [PerfectField K]

Lean source · statement

theorem ag9b13e_differentialCoeff_D_eq_ratFuncDXCoeff_div_of_perfectField
    (v : Place K (RatFunc K)) (f : RatFunc K)
    (hπ : ratFuncDXCoeff K v.uniformizer ≠ 0) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) f)
      = ratFuncDXCoeff K f / ratFuncDXCoeff K v.uniformizer

theorem

ag9b13e_differentialCoeff_D_mem_of_mem_of_perfectField

L11310 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_differentialCoeff_D_mem_of_mem_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section LemmaAPerfect
variable {K : Type*} [Field K] [PerfectField K]

Lean source · statement

theorem ag9b13e_differentialCoeff_D_mem_of_mem_of_perfectField (v : Place K (RatFunc K))
    {g : RatFunc K} (hg : g ∈ v.toValuationSubring) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) g) ∈ v.toValuationSubring

theorem

ag9b13e_canonicalLocalResidueKSimplePoleCoordIndep_ratFunc_of_perfectField

L11330 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_canonicalLocalResidueKSimplePoleCoordIndep_ratFunc_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section SimplePoleRowPerfect
variable (K : Type*) [Field K] [PerfectField K]

Lean source · statement

theorem ag9b13e_canonicalLocalResidueKSimplePoleCoordIndep_ratFunc_of_perfectField :
    CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K)

theorem

ag9b13e_p1PlaceInftySimplePoleResidueEulerValueX_of_perfectField

L11342 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_p1PlaceInftySimplePoleResidueEulerValueX_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ChainPerfect
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ag9b13e_p1PlaceInftySimplePoleResidueEulerValueX_of_perfectField :
    P1PlaceInftySimplePoleResidueEulerValueX K

theorem

ag9b13e_p1PlaceInftySimplePoleResidueEulerValueMonomial_of_perfectField

L11347 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_p1PlaceInftySimplePoleResidueEulerValueMonomial_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ChainPerfect
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ag9b13e_p1PlaceInftySimplePoleResidueEulerValueMonomial_of_perfectField :
    P1PlaceInftySimplePoleResidueEulerValueMonomial K

theorem

ag9b13e_p1PlaceInftySimplePoleResidueEulerValue_of_perfectField

L11353 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.ag9b13e_p1PlaceInftySimplePoleResidueEulerValue_of_perfectField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ChainPerfect
variable (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ag9b13e_p1PlaceInftySimplePoleResidueEulerValue_of_perfectField :
    P1PlaceInftySimplePoleResidueEulerValue K

def

kaehlerResidueFunctionalK

L11411 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK

Documentation

The global residue functional obtained by restricting the differential Weil map to principal adeles.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KernelEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
variable [HasPrincipalDivisors K F]

Lean source · statement

def kaehlerResidueFunctionalK (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) : F →ₗ[K] K

theorem

kaehlerResidueFunctionalK_eq_finsum

L11416 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_eq_finsum

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KernelEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
variable [HasPrincipalDivisors K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem kaehlerResidueFunctionalK_eq_finsum
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK) {ω : Ω[F⁄K]} (hω : ω ≠ 0) (f : F) :
    kaehlerResidueFunctionalK Rfam hω f
      = ∑ᶠ v, kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v

theorem

kaehlerResidueFunctionalK_eq_zero_of_term_zero_compl

L11422 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_eq_zero_of_term_zero_compl

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KernelEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
variable [HasPrincipalDivisors K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem kaehlerResidueFunctionalK_eq_zero_of_term_zero_compl
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK) {ω : Ω[F⁄K]} (hω : ω ≠ 0) {f : F}
    {T : Finset (Place K F)}
    (hcompl : ∀ v ∉ T, kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v = 0)
    (hsum : ∑ v ∈ T, kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v = 0) :
    kaehlerResidueFunctionalK Rfam hω f = 0

theorem

kaehlerResidueFunctionalK_eq_zero_of_singleton

L11434 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_eq_zero_of_singleton

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KernelEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
variable [HasPrincipalDivisors K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem kaehlerResidueFunctionalK_eq_zero_of_singleton
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK) {ω : Ω[F⁄K]} (hω : ω ≠ 0) {f : F}
    {v₀ : Place K F}
    (hcompl : ∀ v, v ≠ v₀ → kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v = 0)
    (hv₀ : kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v₀ = 0) :
    kaehlerResidueFunctionalK Rfam hω f = 0

theorem

kaehlerResidueFunctionalK_eq_zero_of_pair

L11447 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_eq_zero_of_pair

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section KernelEngine
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
variable [HasPrincipalDivisors K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem kaehlerResidueFunctionalK_eq_zero_of_pair
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK) {ω : Ω[F⁄K]} (hω : ω ≠ 0) {f : F}
    {v₀ v₁ : Place K F} (hne : v₀ ≠ v₁)
    (hcompl : ∀ v, v ≠ v₀ → v ≠ v₁ → kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v = 0)
    (hsum : kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v₀
      + kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v₁ = 0) :
    kaehlerResidueFunctionalK Rfam hω f = 0

theorem

kaehlerResidueTermKFam_eq_of_mem_simplePoleSubmodule

L11469 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueTermKFam_eq_of_mem_simplePoleSubmodule

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section SimplePoleSeam
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem kaehlerResidueTermKFam_eq_of_mem_simplePoleSubmodule
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} {f : F} {v : Place K F}
    (hmem : f * v.differentialCoeff ω ∈ v.simplePoleSubmodule) :
    kaehlerResidueTermKFam Rfam ω (diagonalHom K F f) v
      = Algebra.trace K v.ResidueField
          (v.simplePoleResidueAux ⟨f * v.differentialCoeff ω, hmem⟩)

theorem

kaehlerResidueTermKFam_smul_diagonal

L11488 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueTermKFam_smul_diagonal

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section SmulReduction
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem kaehlerResidueTermKFam_smul_diagonal
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    (g : F) (ω : Ω[F⁄K]) (f : F) (v : Place K F) :
    kaehlerResidueTermKFam Rfam (g • ω) (diagonalHom K F f) v
      = kaehlerResidueTermKFam Rfam ω (diagonalHom K F (g * f)) v

theorem

weilOfKaehlerK_smul_diagonal

L11500 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.weilOfKaehlerK_smul_diagonal

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section SmulReduction
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [∀ v : Place K F, v.DCoordGenerates] [Nontrivial Ω[F⁄K]]
variable [HasCanonicalDivisor (K := K) (F := F)]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem weilOfKaehlerK_smul_diagonal [HasPrincipalDivisors K F]
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK) {g : F} {ω : Ω[F⁄K]}
    (hω : ω ≠ 0) (hgω : g • ω ≠ 0) (f : F) :
    weilOfKaehlerK Rfam hgω ⟨diagonalHom K F f, diagonal_mem_adeleSpace f⟩
      = weilOfKaehlerK Rfam hω
          ⟨diagonalHom K F (g * f), diagonal_mem_adeleSpace (g * f)⟩

theorem

kaehlerResidueFunctionalK_dX_X_pow

L11517 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_dX_X_pow

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueFunctionalK_dX_X_pow
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK) (n : ℕ) :
    kaehlerResidueFunctionalK Rfam (dX_ne_zero K) ((RatFunc.X : RatFunc K) ^ n) = 0

theorem

kaehlerResidueTermKFam_atom_eq_zero_of_ne_of_ne

L11531 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueTermKFam_atom_eq_zero_of_ne_of_ne

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueTermKFam_atom_eq_zero_of_ne_of_ne
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    {p : K[X]} (hpirr : Irreducible p) (c : K[X]) (m : ℕ) {v : Place K (RatFunc K)}
    (hvp : v ≠ finitePlace K hpirr) (hvinf : v ≠ placeInfty K) :
    kaehlerResidueTermKFam Rfam (dX K)
      (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c m)) v = 0

theorem

kaehlerResidueTermKFam_atom_placeInfty_eq_zero_of_two_le

L11544 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueTermKFam_atom_placeInfty_eq_zero_of_two_le

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueTermKFam_atom_placeInfty_eq_zero_of_two_le
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    {p : K[X]} (hpirr : Irreducible p) (c : K[X]) {m : ℕ}
    (hdeg : c.degree < p.degree) (hm : 2 ≤ m) :
    kaehlerResidueTermKFam Rfam (dX K)
      (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c m)) (placeInfty K) = 0

theorem

kaehlerResidueTermKFam_atom_finitePlace_eq_zero_of_two_le

L11571 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueTermKFam_atom_finitePlace_eq_zero_of_two_le

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [CharZero K] [DecidableEq (RatFunc K)] [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueTermKFam_atom_finitePlace_eq_zero_of_two_le
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K)
    {p c : K[X]} {m : ℕ} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hdeg : c.degree < p.degree) (hm : 2 ≤ m) :
    kaehlerResidueTermKFam Rfam (dX K)
      (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c m)) (finitePlace K hpirr) = 0

theorem

kaehlerResidueTermKFam_atom_mOne_two_place_sum

L11584 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueTermKFam_atom_mOne_two_place_sum

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueTermKFam_atom_mOne_two_place_sum
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K))
    {p c : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p) (hdeg : c.degree < p.degree) :
    kaehlerResidueTermKFam Rfam (dX K)
        (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c 1)) (finitePlace K hpirr)
      + kaehlerResidueTermKFam Rfam (dX K)
          (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c 1)) (placeInfty K) = 0

theorem

kaehlerResidueFunctionalK_dX_atom

L11603 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_dX_atom

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueFunctionalK_dX_atom
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K)
    {p c : K[X]} {m : ℕ} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hdeg : c.degree < p.degree) (hm : 1 ≤ m) :
    kaehlerResidueFunctionalK Rfam (dX_ne_zero K) (p1PrincipalPartAtom K p c m) = 0

theorem

kaehlerResidueFunctionalK_dX_eq_zero

L11625 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.kaehlerResidueFunctionalK_dX_eq_zero

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem kaehlerResidueFunctionalK_dX_eq_zero
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K)
    (h : RatFunc K) :
    kaehlerResidueFunctionalK Rfam (dX_ne_zero K) h = 0

theorem

residueTheoremK_ratFunc_of_subrows

L11657 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.residueTheoremK_ratFunc_of_subrows

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Engine
variable {K : Type*} [Field K] [CharZero K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem residueTheoremK_ratFunc_of_subrows
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hsp : CanonicalLocalResidueKSimplePoleCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K) :
    ResidueTheoremK K (RatFunc K)

theorem

residueTheoremK_ratFunc_of_isAlgClosed_main

L11680 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.residueTheoremK_ratFunc_of_isAlgClosed_main

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section AlgClosedDischarge
variable (K : Type*) [Field K] [CharZero K] [IsAlgClosed K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem residueTheoremK_ratFunc_of_isAlgClosed_main : ResidueTheoremK K (RatFunc K)

theorem

p0n21_rtk_p1MOneSimplePoleCancel_dX_of_perfectField

L11732 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_p1MOneSimplePoleCancel_dX_of_perfectField

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section MOnePerfect

Lean source · statement

theorem p0n21_rtk_p1MOneSimplePoleCancel_dX_of_perfectField
    (K : Type*) [Field K] [PerfectField K] [DecidableEq (RatFunc K)] :
    P1PrincipalPartMOneSimplePoleCancel K (dX_ne_zero K)

theorem

p0n21_rtk_kaehlerResidueFunctionalK_dX_X_pow

L11746 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueFunctionalK_dX_X_pow

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueFunctionalK_dX_X_pow
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K)) (n : ℕ) :
    kaehlerResidueFunctionalK Rfam (dX_ne_zero K) ((RatFunc.X : RatFunc K) ^ n) = 0

theorem

p0n21_rtk_kaehlerResidueTermKFam_atom_eq_zero_of_ne_of_ne

L11765 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueTermKFam_atom_eq_zero_of_ne_of_ne

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueTermKFam_atom_eq_zero_of_ne_of_ne
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    {p : K[X]} (hpirr : Irreducible p) (c : K[X]) (m : ℕ) {v : Place K (RatFunc K)}
    (hvp : v ≠ finitePlace K hpirr) (hvinf : v ≠ placeInfty K) :
    kaehlerResidueTermKFam Rfam (dX K)
      (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c m)) v = 0

theorem

p0n21_rtk_kaehlerResidueTermKFam_atom_placeInfty_eq_zero_of_two_le

L11778 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueTermKFam_atom_placeInfty_eq_zero_of_two_le

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueTermKFam_atom_placeInfty_eq_zero_of_two_le
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    {p : K[X]} (hpirr : Irreducible p) (c : K[X]) {m : ℕ}
    (hdeg : c.degree < p.degree) (hm : 2 ≤ m) :
    kaehlerResidueTermKFam Rfam (dX K)
      (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c m)) (placeInfty K) = 0

theorem

p0n21_rtk_kaehlerResidueTermKFam_atom_finitePlace_eq_zero_of_two_le

L11805 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueTermKFam_atom_finitePlace_eq_zero_of_two_le

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [PerfectField K] [DecidableEq (RatFunc K)] [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueTermKFam_atom_finitePlace_eq_zero_of_two_le
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K)
    {p c : K[X]} {m : ℕ} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hdeg : c.degree < p.degree) (hm : 2 ≤ m) :
    kaehlerResidueTermKFam Rfam (dX K)
      (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c m)) (finitePlace K hpirr) = 0

theorem

p0n21_rtk_kaehlerResidueTermKFam_atom_mOne_two_place_sum

L11818 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueTermKFam_atom_mOne_two_place_sum

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [HasPrincipalDivisors K (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueTermKFam_atom_mOne_two_place_sum
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    {p c : K[X]} (hpmon : p.Monic) (hpirr : Irreducible p) (hdeg : c.degree < p.degree) :
    kaehlerResidueTermKFam Rfam (dX K)
        (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c 1)) (finitePlace K hpirr)
      + kaehlerResidueTermKFam Rfam (dX K)
          (diagonalHom K (RatFunc K) (p1PrincipalPartAtom K p c 1)) (placeInfty K) = 0

theorem

p0n21_rtk_kaehlerResidueFunctionalK_dX_atom

L11835 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueFunctionalK_dX_atom

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueFunctionalK_dX_atom
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K)
    {p c : K[X]} {m : ℕ} (hpmon : p.Monic) (hpirr : Irreducible p)
    (hdeg : c.degree < p.degree) (hm : 1 ≤ m) :
    kaehlerResidueFunctionalK Rfam (dX_ne_zero K) (p1PrincipalPartAtom K p c m) = 0

theorem

p0n21_rtk_kaehlerResidueFunctionalK_dX_eq_zero

L11857 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_kaehlerResidueFunctionalK_dX_eq_zero

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section PerGenerator
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
variable [HasPrincipalDivisors K (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_kaehlerResidueFunctionalK_dX_eq_zero
    (Rfam : ∀ v : Place K (RatFunc K), v.CanonicalLocalResidueDataK)
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K)
    (h : RatFunc K) :
    kaehlerResidueFunctionalK Rfam (dX_ne_zero K) h = 0

theorem

p0n21_rtk_residueTheoremK_ratFunc_of_subrows_charFree

L11889 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_residueTheoremK_ratFunc_of_subrows_charFree

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section EngineCharFree
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_residueTheoremK_ratFunc_of_subrows_charFree
    (hwd : OrdDifferentialWellDefined K (RatFunc K))
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K) :
    ResidueTheoremK K (RatFunc K)

theorem

p0n21_rtk_residueTheoremK_ratFunc_of_coordIndep_higherDeg_of_perfectField

L11905 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_residueTheoremK_ratFunc_of_coordIndep_higherDeg_of_perfectField

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section EngineCharFree
variable {K : Type*} [Field K] [PerfectField K] [DecidableEq (RatFunc K)]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem p0n21_rtk_residueTheoremK_ratFunc_of_coordIndep_higherDeg_of_perfectField
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K))
    (hHigh5 : P1FinitePlaceCanonicalResidueAtomMGeTwoTraceHigherDeg K) :
    ResidueTheoremK K (RatFunc K)

theorem

p0n21_rtk_residueTheoremK_ratFunc_of_isAlgClosed_of_coordIndep

L11917 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_residueTheoremK_ratFunc_of_isAlgClosed_of_coordIndep

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section AlgClosedAnyChar

Lean source · statement

theorem p0n21_rtk_residueTheoremK_ratFunc_of_isAlgClosed_of_coordIndep
    (K : Type*) [Field K] [IsAlgClosed K]
    (hcoord : CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K)) :
    ResidueTheoremK K (RatFunc K)

theorem

p0n21_rtk_surjective_algebraMap_residueField_of_deg_eq_one

L11935 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_surjective_algebraMap_residueField_of_deg_eq_one

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ResidualLocation

Lean source · statement

theorem p0n21_rtk_surjective_algebraMap_residueField_of_deg_eq_one
    {K F : Type*} [Field K] [Field F] [Algebra K F]
    (v : Place K F) (hdeg : v.deg = 1) :
    Function.Surjective (algebraMap K v.ResidueField)

theorem

p0n21_rtk_surjective_algebraMap_residueField_ratFunc_of_isAlgClosed

L11944 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n21_rtk_surjective_algebraMap_residueField_ratFunc_of_isAlgClosed

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section ResidualLocation

Lean source · statement

theorem p0n21_rtk_surjective_algebraMap_residueField_ratFunc_of_isAlgClosed
    (K : Type*) [Field K] [IsAlgClosed K] (v : Place K (RatFunc K)) :
    Function.Surjective (algebraMap K v.ResidueField)

theorem

p0n22_cpf_uniformizer_pow_mul_mem_cases

L11994 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_uniformizer_pow_mul_mem_cases

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_uniformizer_pow_mul_mem_cases
    (v : Place K (RatFunc K))
    {σ : RatFunc K} (hσ : σ ∈ v.toValuationSubring) {k : ℕ} (hk : 1 ≤ k) :
    v.uniformizer ^ k * σ = 0 ∨ 1 ≤ v.ord (v.uniformizer ^ k * σ)

theorem

p0n22_cpf_one_add_ne_zero

L12006 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_one_add_ne_zero

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_one_add_ne_zero
    (v : Place K (RatFunc K))
    {g : RatFunc K} (hg : g = 0 ∨ 1 ≤ v.ord g) :
    (1 : RatFunc K) + g ≠ 0

theorem

p0n22_cpf_ord_one_add_eq_zero

L12023 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_ord_one_add_eq_zero

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_ord_one_add_eq_zero
    (v : Place K (RatFunc K))
    {g : RatFunc K} (hg : g = 0 ∨ 1 ≤ v.ord g) :
    v.ord ((1 : RatFunc K) + g) = 0

theorem

p0n22_cpf_one_add_inv_mem

L12039 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_one_add_inv_mem

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_one_add_inv_mem
    (v : Place K (RatFunc K))
    {g : RatFunc K} (hg : g = 0 ∨ 1 ≤ v.ord g) :
    ((1 : RatFunc K) + g)⁻¹ ∈ v.toValuationSubring

theorem

p0n22_cpf_logDeriv_mul

L12047 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_logDeriv_mul

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_logDeriv_mul
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    {w₁ w₂ : RatFunc K} (h1 : w₁ ≠ 0) (h2 : w₂ ≠ 0) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) (w₁ * w₂))
        * (w₁ * w₂)⁻¹
      = v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w₁) * w₁⁻¹
        + v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w₂)
            * w₂⁻¹

theorem

p0n22_cpf_logDeriv_mem

L12066 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_logDeriv_mem

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]

Lean source · statement

theorem p0n22_cpf_logDeriv_mem
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    {w : RatFunc K} (hw0 : w ≠ 0) (hword : v.ord w = 0) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w) * w⁻¹
      ∈ v.toValuationSubring

theorem

p0n22_cpf_differentialCoeff_D_one_add_algebraMap_mul_pow

L12079 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_differentialCoeff_D_one_add_algebraMap_mul_pow

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_differentialCoeff_D_one_add_algebraMap_mul_pow
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]] (a : K) (k : ℕ) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
        ((1 : RatFunc K)
          + algebraMap K (RatFunc K) a * v.uniformizer ^ k))
      = algebraMap K (RatFunc K) a
        * (((k : ℕ) : RatFunc K) * v.uniformizer ^ (k - 1))

theorem

p0n22_cpf_differentialCoeff_D_one_add_pow_mul

L12095 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_differentialCoeff_D_one_add_pow_mul

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_differentialCoeff_D_one_add_pow_mul
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]] (σ : RatFunc K) (k : ℕ) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
        ((1 : RatFunc K) + v.uniformizer ^ k * σ))
      = v.uniformizer ^ k
          * v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) σ)
        + σ * (((k : ℕ) : RatFunc K) * v.uniformizer ^ (k - 1))

theorem

p0n22_cpf_exists_residue_lift_decomp

L12118 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_exists_residue_lift_decomp

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in
open _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
  (ag9b15u_eq_zero_or_one_le_ord_of_residue_eq_zero)
in

Lean source · statement

theorem p0n22_cpf_exists_residue_lift_decomp
    (v : Place K (RatFunc K))
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    {r : RatFunc K} (hr : r ∈ v.toValuationSubring) :
    ∃ (d : K) (r' : RatFunc K), r' ∈ v.toValuationSubring ∧
      r = algebraMap K (RatFunc K) d + v.uniformizer * r'

theorem

p0n22_cpf_exists_elementary_peel

L12156 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_exists_elementary_peel

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_exists_elementary_peel
    (v : Place K (RatFunc K))
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    {k : ℕ} (hk : 1 ≤ k) {σ : RatFunc K} (hσ : σ ∈ v.toValuationSubring) :
    ∃ (a : K) (τ : RatFunc K), τ ∈ v.toValuationSubring ∧
      (1 : RatFunc K) + v.uniformizer ^ k * σ
        = ((1 : RatFunc K)
              + algebraMap K (RatFunc K) a * v.uniformizer ^ k)
          * ((1 : RatFunc K) + v.uniformizer ^ (k + 1) * τ)

theorem

p0n22_cpf_exists_laurent_expansion_one

L12192 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_exists_laurent_expansion_one

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in
open _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
  (ag9b15u_exists_K_truncation_of_mem_poleSubmodule)
in

Lean source · statement

theorem p0n22_cpf_exists_laurent_expansion_one
    (v : Place K (RatFunc K))
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (f : RatFunc K) :
    ∃ (s : Finset ℤ) (c : ℤ → K) (r : RatFunc K),
      r ∈ v.toValuationSubring ∧
      f = (∑ j ∈ s, c j • v.uniformizer ^ j) + v.uniformizer * r

theorem

p0n22_cpf_res_geom_core

L12231 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_geom_core

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in
open _root_.ModularCurve.MilneAvAg9bRd14CubeRowCartierSliceStart
  (ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one)
in

Lean source · full declaration

theorem p0n22_cpf_res_geom_core
    (v : Place K (RatFunc K)) (R : v.CanonicalLocalResidueDataK)
    (a : K) {k q : ℕ} (hk : 1 ≤ k) (hq : 1 ≤ q) :
    R.res (algebraMap K (RatFunc K) a * v.uniformizer ^ (k - 1)
        * ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
        * (v.uniformizer ^ q)⁻¹)
      = if k ∣ q
        then algebraMap K v.ResidueField (a * (-a) ^ (q / k - 1))
        else 0 := by
  have hπ0 : (v.uniformizer : RatFunc K) ≠ 0 := v.uniformizer_ne_zero
  have hgcase : algebraMap K (RatFunc K) a * v.uniformizer ^ k = 0
      ∨ 1 ≤ v.ord (algebraMap K (RatFunc K) a * v.uniformizer ^ k) := by
    rw [mul_comm]
    exact p0n22_cpf_uniformizer_pow_mul_mem_cases v (v.algebraMap_mem' a) hk
  have hu0 : (1 : RatFunc K)
      + algebraMap K (RatFunc K) a * v.uniformizer ^ k ≠ 0 :=
    p0n22_cpf_one_add_ne_zero v hgcase
  have huinv : ((1 : RatFunc K)
      + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
      ∈ v.toValuationSubring :=
    p0n22_cpf_one_add_inv_mem v hgcase
  have hgeom : ((1 : RatFunc K)
      + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
      = (∑ t ∈ Finset.range q,
          (-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ t)
        + (-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ q
          * ((1 : RatFunc K)
              + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹ := by
    refine (eq_inv_of_mul_eq_one_left ?_).symm
    rw [add_mul, mul_assoc, inv_mul_cancel₀ hu0, mul_one]
    have hgs := geom_sum_mul
      (-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) q
    linear_combination -hgs
  rw [hgeom, mul_add, add_mul, map_add]
  have htail : R.res ((algebraMap K (RatFunc K) a * v.uniformizer ^ (k - 1)
      * ((-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ q
          * ((1 : RatFunc K)
              + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹))
      * (v.uniformizer ^ q)⁻¹) = 0 := by
    apply R.res_of_mem
    have hqle : q ≤ k * q := Nat.le_of_dvd (Nat.mul_pos hk hq) (dvd_mul_left q k)
    have hpowsplit : (v.uniformizer : RatFunc K) ^ (k * q)
        = v.uniformizer ^ (k * q - q) * v.uniformizer ^ q := by
      rw [← pow_add, Nat.sub_add_cancel hqle]
    have hxq : (-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ q
        = algebraMap K (RatFunc K) ((-a) ^ q)
          * (v.uniformizer ^ (k * q - q) * v.uniformizer ^ q) := by
      rw [map_pow, _root_.map_neg,
        show -(algebraMap K (RatFunc K) a * v.uniformizer ^ k)
            = -algebraMap K (RatFunc K) a * v.uniformizer ^ k from
          (neg_mul _ _).symm,
        mul_pow, ← pow_mul, hpowsplit]
    have hcc : (v.uniformizer : RatFunc K) ^ q
        * ((v.uniformizer : RatFunc K) ^ q)⁻¹ = 1 :=
      mul_inv_cancel₀ (pow_ne_zero _ hπ0)
    have heq : (algebraMap K (RatFunc K) a * v.uniformizer ^ (k - 1)
        * ((-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ q
            * ((1 : RatFunc K)
                + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹))
        * (v.uniformizer ^ q)⁻¹
        = algebraMap K (RatFunc K) a * v.uniformizer ^ (k - 1)
          * (algebraMap K (RatFunc K) ((-a) ^ q)
              * v.uniformizer ^ (k * q - q)
              * ((1 : RatFunc K)
                  + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹) := by
      rw [hxq]
      linear_combination (algebraMap K (RatFunc K) a
        * v.uniformizer ^ (k - 1)
        * (algebraMap K (RatFunc K) ((-a) ^ q)
            * v.uniformizer ^ (k * q - q)
            * ((1 : RatFunc K)
                + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹)) * hcc
    rw [heq]
    exact mul_mem (mul_mem (v.algebraMap_mem' a) (pow_mem v.uniformizer_mem _))
      (mul_mem (mul_mem (v.algebraMap_mem' _) (pow_mem v.uniformizer_mem _)) huinv)
  have hterm : (algebraMap K (RatFunc K) a * v.uniformizer ^ (k - 1)
      * (∑ t ∈ Finset.range q,
          (-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ t))
      * (v.uniformizer ^ q)⁻¹
      = ∑ t ∈ Finset.range q, (a * (-a) ^ t)
          • v.uniformizer ^ ((k : ℤ) * ((t : ℤ) + 1) - 1 - (q : ℤ)) := by
    rw [Finset.mul_sum, Finset.sum_mul]
    refine Finset.sum_congr rfl fun t _ => ?_
    have hxt : (-(algebraMap K (RatFunc K) a * v.uniformizer ^ k)) ^ t
        = algebraMap K (RatFunc K) ((-a) ^ t)
          * v.uniformizer ^ (k * t) := by
      rw [map_pow, _root_.map_neg,
        show -(algebraMap K (RatFunc K) a * v.uniformizer ^ k)
            = -algebraMap K (RatFunc K) a * v.uniformizer ^ k from
          (neg_mul _ _).symm,
        mul_pow, ← pow_mul]
    rw [hxt, Algebra.smul_def, map_mul, map_pow, _root_.map_neg,
      show (k : ℤ) * ((t : ℤ) + 1) - 1 - (q : ℤ)
          = ((k - 1 : ℕ) : ℤ) + ((k * t : ℕ) : ℤ) + -((q : ℕ) : ℤ) from by
        push_cast [Nat.cast_sub hk]
        ring,
      zpow_add₀ hπ0, zpow_add₀ hπ0, zpow_natCast, zpow_natCast, _root_.zpow_neg, zpow_natCast]
    ring
  rw [hterm, map_sum]
  simp only [map_smul]
  by_cases hdvd : k ∣ q
  · rw [ite_eq_left hdvd, htail, add_zero]
    obtain ⟨e, rfl⟩ := hdvd
    rcases Nat.eq_zero_or_pos e with rfl | he1
    · simp at hq
    · rw [Nat.mul_div_cancel_left e hk]
      have he_lt : e - 1 < k * e := by
        have h1 : e ≤ k * e := Nat.le_of_dvd (Nat.mul_pos hk he1) (dvd_mul_left e k)
        omega
      have hother : ∀ t ∈ Finset.range (k * e), t ≠ e - 1 →
          (a * (-a) ^ t) • R.res ((v.uniformizer : RatFunc K)
            ^ ((k : ℤ) * ((t : ℤ) + 1) - 1 - ((k * e : ℕ) : ℤ))) = 0 := by
        intro t _ htne
        have hne : (k : ℤ) * ((t : ℤ) + 1) - 1 - ((k * e : ℕ) : ℤ) ≠ -1 := by
          intro hcontra
          have h2 : (k : ℤ) * ((t : ℤ) + 1) = (k : ℤ) * (e : ℤ) := by
            push_cast at hcontra
            linarith
          have hk0 : (k : ℤ) ≠ 0 := by omega
          have h3 : (t : ℤ) + 1 = (e : ℤ) := mul_left_cancel₀ hk0 h2
          omega
        rw [ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one
          v R hne, smul_zero]
      rw [Finset.sum_eq_single_of_mem (e - 1) (Finset.mem_range.mpr he_lt) hother,
        show (k : ℤ) * (((e - 1 : ℕ) : ℤ) + 1) - 1 - ((k * e : ℕ) : ℤ) = -1 from by
          push_cast [Nat.cast_sub he1]
          ring,
        _root_.zpow_neg, zpow_one, gate_canonicalLocalResidueDataK_uniformizer_inv v R,
        ← Algebra.algebraMap_eq_smul_one]
  · rw [ite_eq_right hdvd, htail, add_zero]
    refine Finset.sum_eq_zero fun t _ => ?_
    have hne : (k : ℤ) * ((t : ℤ) + 1) - 1 - (q : ℤ) ≠ -1 := by
      intro hcontra
      have h2 : (k : ℤ) * ((t : ℤ) + 1) = (q : ℤ) := by linarith
      have h3 : k * (t + 1) = q := by exact_mod_cast h2
      exact hdvd ⟨t + 1, h3.symm⟩
    rw [ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one
      v R hne, smul_zero]

theorem

p0n22_cpf_res_logDeriv_elementary

L12372 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_logDeriv_elementary

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in

Lean source · full declaration

theorem p0n22_cpf_res_logDeriv_elementary
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (R : v.CanonicalLocalResidueDataK) (a : K) {k q : ℕ} (hk : 1 ≤ k) (hq : 1 ≤ q) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
          ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k))
        * ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
        * (v.uniformizer ^ q)⁻¹)
      = ((k : ℕ) : K) • (if k ∣ q
          then algebraMap K v.ResidueField (a * (-a) ^ (q / k - 1))
          else 0) := by
  rw [p0n22_cpf_differentialCoeff_D_one_add_algebraMap_mul_pow v a k]
  have hsplit : algebraMap K (RatFunc K) a
        * (((k : ℕ) : RatFunc K) * v.uniformizer ^ (k - 1))
        * ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
        * (v.uniformizer ^ q)⁻¹
      = ((k : ℕ) : K) • (algebraMap K (RatFunc K) a
          * v.uniformizer ^ (k - 1)
          * ((1 : RatFunc K)
              + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
          * (v.uniformizer ^ q)⁻¹) := by
    rw [Algebra.smul_def, map_natCast]
    ring
  rw [hsplit, map_smul, p0n22_cpf_res_geom_core v R a hk hq]

theorem

p0n22_cpf_res_logDeriv_highLevel_vanish

L12400 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_logDeriv_highLevel_vanish

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]

Lean source · statement

theorem p0n22_cpf_res_logDeriv_highLevel_vanish
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (R : v.CanonicalLocalResidueDataK)
    {σ : RatFunc K} (hσ : σ ∈ v.toValuationSubring)
    {k q : ℕ} (hbig : q + 1 ≤ k) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
          ((1 : RatFunc K) + v.uniformizer ^ k * σ))
        * ((1 : RatFunc K) + v.uniformizer ^ k * σ)⁻¹
        * (v.uniformizer ^ q)⁻¹) = 0

theorem

p0n22_cpf_row_integrand_eq_pow_mul_dlog

L12448 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_row_integrand_eq_pow_mul_dlog

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharFreeToolkit
variable {K : Type*} [Field K] [PerfectField K]
omit [PerfectField K] in
open _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
  (ag9b15u_differentialCoeff_D_unit_mul_uniformizer)
in

Lean source · statement

theorem p0n22_cpf_row_integrand_eq_pow_mul_dlog
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    {w : RatFunc K} (hw0 : w ≠ 0) (k : ℕ) :
    v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
        (w * v.uniformizer))
        * ((w * v.uniformizer) ^ (k + 1))⁻¹
      = (w ^ k * v.uniformizer ^ k)⁻¹
        * ((v.uniformizer)⁻¹
            + v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w)
                * w⁻¹)

theorem

p0n22_cpf_res_logDeriv_elementary_cartier

L12477 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_logDeriv_elementary_cartier

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharPCartier
variable {p : ℕ} [hp : Fact p.Prime]
variable {K : Type*} [Field K] [PerfectField K] [hKp : CharP K p]
omit [PerfectField K] in

Lean source · statement

theorem p0n22_cpf_res_logDeriv_elementary_cartier
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (R : v.CanonicalLocalResidueDataK) (a : K) {k n : ℕ} (hk : 1 ≤ k) (hn : 1 ≤ n) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
          ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k))
        * ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
        * (v.uniformizer ^ (p * n))⁻¹)
      = (R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
          ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k))
        * ((1 : RatFunc K)
            + algebraMap K (RatFunc K) a * v.uniformizer ^ k)⁻¹
        * (v.uniformizer ^ n)⁻¹)) ^ p

theorem

p0n22_cpf_res_logDeriv_principalUnit_cartier_aux

L12529 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_logDeriv_principalUnit_cartier_aux

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharPCartier
variable {p : ℕ} [hp : Fact p.Prime]
variable {K : Type*} [Field K] [PerfectField K] [hKp : CharP K p]

Lean source · statement

theorem p0n22_cpf_res_logDeriv_principalUnit_cartier_aux
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (R : v.CanonicalLocalResidueDataK) {n : ℕ} (hn : 1 ≤ n) :
    ∀ d k : ℕ, 1 ≤ k → p * n + 1 ≤ k + d →
      ∀ σ : RatFunc K, σ ∈ v.toValuationSubring →
        R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
              ((1 : RatFunc K) + v.uniformizer ^ k * σ))
            * ((1 : RatFunc K) + v.uniformizer ^ k * σ)⁻¹
            * (v.uniformizer ^ (p * n))⁻¹)
          = (R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
              ((1 : RatFunc K) + v.uniformizer ^ k * σ))
            * ((1 : RatFunc K) + v.uniformizer ^ k * σ)⁻¹
            * (v.uniformizer ^ n)⁻¹)) ^ p

theorem

p0n22_cpf_res_zpow_mul_logDeriv_cartier

L12575 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_zpow_mul_logDeriv_cartier

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharPCartier
variable {p : ℕ} [hp : Fact p.Prime]
variable {K : Type*} [Field K] [PerfectField K] [hKp : CharP K p]
open _root_.ModularCurve.MilneAvAg9bRd15UnitNormalFormLaurentSeed
  (ag9b15u_exists_unit_normal_form_of_surj)
in

Lean source · statement

theorem p0n22_cpf_res_zpow_mul_logDeriv_cartier
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (R : v.CanonicalLocalResidueDataK)
    {w : RatFunc K} (hw0 : w ≠ 0) (hword : v.ord w = 0) (j : ℤ) :
    R.res (v.uniformizer ^ ((p : ℤ) * j)
        * (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w)
            * w⁻¹))
      = (R.res (v.uniformizer ^ j
        * (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w)
            * w⁻¹))) ^ p

theorem

p0n22_cpf_res_pow_mul_dlog_unit

L12640 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_pow_mul_dlog_unit

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharPCartier
variable {p : ℕ} [hp : Fact p.Prime]
variable {K : Type*} [Field K] [PerfectField K] [hKp : CharP K p]
open _root_.ModularCurve.MilneAvAg9bRd14CubeRowCartierSliceStart
  (ag9b14c_res_uniformizer_zpow_eq_zero_of_ne_neg_one)
in

Lean source · statement

theorem p0n22_cpf_res_pow_mul_dlog_unit
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (R : v.CanonicalLocalResidueDataK)
    {w : RatFunc K} (hw0 : w ≠ 0) (hword : v.ord w = 0) (f : RatFunc K) :
    R.res (f ^ p * ((v.uniformizer)⁻¹
        + v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w)
            * w⁻¹))
      = (R.res (f * ((v.uniformizer)⁻¹
        + v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) w)
            * w⁻¹))) ^ p

theorem

p0n22_cpf_res_row_at_unit_mul_uniformizer

L12751 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_row_at_unit_mul_uniformizer

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharPCartier
variable {p : ℕ} [hp : Fact p.Prime]
variable {K : Type*} [Field K] [PerfectField K] [hKp : CharP K p]
include hp hKp in
open _root_.ModularCurve.MilneAvAg9bRd13T2CoordIndepChar3
  (ag9b13t_res_differentialCoeff_D_mul_pow_inv_of_surj_of_natCast_ne_zero)
in

Lean source · statement

theorem p0n22_cpf_res_row_at_unit_mul_uniformizer
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (R : v.CanonicalLocalResidueDataK)
    {w : RatFunc K} (hw0 : w ≠ 0) (hword : v.ord w = 0) :
    ∀ n : ℕ, 1 ≤ n →
      R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K)
          (w * v.uniformizer))
        * ((w * v.uniformizer) ^ (n + 1))⁻¹) = 0

theorem

p0n22_cpf_res_differentialCoeff_D_mul_pow_inv_of_surj

L12797 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_res_differentialCoeff_D_mul_pow_inv_of_surj

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section CharPCartier
variable {p : ℕ} [hp : Fact p.Prime]
variable {K : Type*} [Field K] [PerfectField K] [hKp : CharP K p]
include hp hKp in

Lean source · statement

theorem p0n22_cpf_res_differentialCoeff_D_mul_pow_inv_of_surj
    (v : Place K (RatFunc K)) [v.DCoordGenerates]
    [Nontrivial Ω[(RatFunc K)⁄K]]
    (hsurj : Function.Surjective (algebraMap K v.ResidueField))
    (π' : RatFunc K) (hπ' : v.ord π' = 1)
    (R : v.CanonicalLocalResidueDataK) {n : ℕ} (hn : 1 ≤ n) :
    R.res (v.differentialCoeff (KaehlerDifferential.D K (RatFunc K) π')
      * ((π') ^ (n + 1))⁻¹) = 0

theorem

p0n22_cpf_canonicalLocalResidueKDifferentialCoordIndep_ratFunc_of_isAlgClosed

L12825 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_canonicalLocalResidueKDifferentialCoordIndep_ratFunc_of_isAlgClosed

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Headlines

Lean source · statement

theorem p0n22_cpf_canonicalLocalResidueKDifferentialCoordIndep_ratFunc_of_isAlgClosed
    (K : Type*) [Field K] [IsAlgClosed K] (p : ℕ) [Fact p.Prime] [CharP K p] :
    CanonicalLocalResidueKDifferentialCoordIndep K (RatFunc K)

theorem

p0n22_cpf_residueTheoremK_ratFunc_of_isAlgClosed_of_charP

L12833 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_residueTheoremK_ratFunc_of_isAlgClosed_of_charP

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Headlines

Lean source · statement

theorem p0n22_cpf_residueTheoremK_ratFunc_of_isAlgClosed_of_charP
    (K : Type*) [Field K] [IsAlgClosed K] (p : ℕ) [Fact p.Prime] [CharP K p]
    :
    ResidueTheoremK K (RatFunc K)

theorem

p0n22_cpf_residueTheoremK_ratFunc_of_isAlgClosed_main

L12841 ↗

AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.p0n22_cpf_residueTheoremK_ratFunc_of_isAlgClosed_main

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve
open ResidueTheoremK_ratFunc_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Lg37
open ResidueTheoremK_ratFunc_of_isAlgClosed.ModularCurve.Mp72a102T3
open AlgebraicCurves.Differentials.Residues.Proofs.ResidueTheoremK_ratFunc_of_isAlgClosed.Mp72a102T2
section
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.KaehlerDifferential
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
open scoped _root_.Polynomial
open _root_.AlgebraicCurve.RationalFunctionField
section Headlines

Lean source · statement

theorem p0n22_cpf_residueTheoremK_ratFunc_of_isAlgClosed_main
    (K : Type*) [Field K] [IsAlgClosed K] :
    ResidueTheoremK K (RatFunc K)

theorem