Module

FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.lean

Nyll.AlgebraicCurves.Cohomology.RiemannRoch.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed

281 declarations indexed from sourceView source ↗

Module documentation

Function-field Riemann-Roch from the residue theorem over algebraically closed constants. Transcendence towers, pole corrections and residue pairings supply the genus and duality arguments.

Imports21

Nyll.AlgebraicCurves.Divisors.ClassGroup
Nyll.AlgebraicCurves.Divisors.Maps
Nyll.AlgebraicCurves.Places.OverDVR
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
Mathlib.FieldTheory.RatFunc.AsPolynomial
Mathlib.FieldTheory.RatFunc.Degree
Mathlib.FieldTheory.RatFunc.Valuation
Mathlib.NumberTheory.RatFunc.Ostrowski
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
Mathlib.RingTheory.AdicCompletion.Algebra
Mathlib.RingTheory.AdicCompletion.Completeness
Nyll.AlgebraicCurves.Cohomology.RiemannRoch
Mathlib.Algebra.Polynomial.Basis
Mathlib.Util.WithWeakNamespace

Imported by2

abbrev

lg37_completion

L320 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37.lg37_completion

Documentation

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

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L342 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37.lg37_residueHat

Documentation

The residue homomorphism from the completed valuation subring.

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L364 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37.lg37_residueHat_algebraMap

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L388 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37.Lg37CompletionSection

Documentation

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

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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
  /-- The algebra lift from the residue field to the completed valuation subring. -/
  lift : v.ResidueField →ₐ[K] lg37_completion v
  /-- The residue map composed with the algebra lift is the identity. -/
  residueHat_lift : ∀ a, lg37_residueHat v (lift a) = a

theorem

ord_nonneg_of_mem

L422 ↗

AlgebraicCurve.Place.ord_nonneg_of_mem

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L439 ↗

AlgebraicCurve.Place.mem_of_ord_nonneg

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L449 ↗

AlgebraicCurve.Place.mem_iff_ord_nonneg

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L454 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.exists_ord_pos

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L468 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.algebraMap_ne_zero

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 algebraMap_ne_zero {f : F} (hf : f ≠ 0) : algebraMap F F' f ≠ 0

theorem

comap_algebraMap_ne_top

L473 ↗

AlgebraicCurve.Place.comap_algebraMap_ne_top

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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']
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

L491 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_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.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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']
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

L498 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_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.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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']
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

L523 ↗

AlgebraicCurve.Place.exists_ord_algebraMap_pos

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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']
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

L540 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_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.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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']
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

L545 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.isPrincipalIdealRing_comap

Source context 28 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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']
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

degree_eq_sum_support

L619 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Divisor.degree_eq_sum_support

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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)

instance

(anonymous instance)

L672 ↗

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L682 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.placeInfty

Documentation

The place at infinity of the rational function field.

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L694 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L697 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.inftyValuation_isEquiv_adicValuation

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

eq_ofHeightOneSpectrum_or_eq_placeInfty

L705 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.eq_ofHeightOneSpectrum_or_eq_placeInfty

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

exists_sub_algebraMap_intDegree_neg

L725 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.exists_sub_algebraMap_intDegree_neg

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L774 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.deg_placeInfty

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

def

uniformizerSubring

L835 ↗

AlgebraicCurve.Place.uniformizerSubring

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L840 ↗

AlgebraicCurve.Place.coe_uniformizerSubring

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L844 ↗

AlgebraicCurve.Place.irreducible_uniformizerSubring

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L849 ↗

AlgebraicCurve.Place.uniformizerSubring_mem_maximalIdeal

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L854 ↗

AlgebraicCurve.Place.uniformizer_mem

Source context 28 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L859 ↗

AlgebraicCurve.Place.simplePoleSubmodule

Source context 29 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L871 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.mem_simplePoleSubmodule

Source context 30 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L874 ↗

AlgebraicCurve.Place.mem_simplePoleSubmodule_of_mem

Source context 30 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L880 ↗

AlgebraicCurve.Place.simplePoleMulUniformizer

Source context 31 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L891 ↗

AlgebraicCurve.Place.simplePoleResidueAux

Source context 32 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L897 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.simplePoleResidueAux_apply

Source context 33 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L900 ↗

AlgebraicCurve.Place.simplePoleResidueAux_eq_zero_of_mem

Source context 33 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L910 ↗

AlgebraicCurve.Place.localResidueExtend

Source context 34 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L914 ↗

AlgebraicCurve.Place.localResidueExtend_apply_of_mem

Source context 35 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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⟩

abbrev

mp72a102_t1_maximalIdealHat

L964 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T1.mp72a102_t1_maximalIdealHat

Documentation

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

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L969 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T1.mp72a102_t1_isScalarTower

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L973 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T1.mp72a102_t1_isAdicComplete_maximalIdealHat

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L981 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T1.mp72a102_t1_henselianRing_completion

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L986 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T1.mp72a102_t1_maximalIdealHat_le_ker_residueHat

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L992 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T1.mp72a102_t1_exists_completion_root_of_residue_root

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1092 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2.mp72a102_t2_residueHat_algebraMap_base

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1121 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2.mp72a102_t2_residueHatAlgHom

Documentation

The completed residue homomorphism viewed as an algebra homomorphism.

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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)

L1188 ↗

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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, map_zero, zero_mul] at hb'
      exact zero_ne_one hb'
    exact (isUnit_mk_comap_iff ha0 a.2).mpr hord

theorem

isUnit_mk_of_ord_eq_zero

L1227 ↗

AlgebraicCurve.Place.isUnit_mk_of_ord_eq_zero

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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.isUnit_mk_of_ord_eq_zero {f : F} (hf : f ≠ 0) (h :
    v.ord f = 0) :
    IsUnit (⟨f, v.mem_of_ord_nonneg hf h.ge⟩ : v.toValuationSubring)

theorem

residue_ne_zero_of_ord_eq_zero

L1239 ↗

AlgebraicCurve.Place.residue_ne_zero_of_ord_eq_zero

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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.residue_ne_zero_of_ord_eq_zero {f : F} (hf : f ≠ 0) (h
    : v.ord f = 0) :
    residue v.toValuationSubring (⟨f, v.mem_of_ord_nonneg hf h.ge⟩ : v.toValuationSubring)
      ≠ 0

def

IsRational

L1246 ↗

AlgebraicCurve.Place.IsRational

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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

def _root_.AlgebraicCurve.Place.IsRational : Prop

def

residueInv

L1250 ↗

AlgebraicCurve.Place.residueInv

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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

def _root_.AlgebraicCurve.Place.residueInv : v.ResidueField → K

theorem

algebraMap_residueInv

L1254 ↗

AlgebraicCurve.Place.algebraMap_residueInv

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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.algebraMap_residueInv (hv : v.IsRational) (x :
    v.ResidueField) :
    algebraMap K v.ResidueField (v.residueInv x) = x

def

evalAt

L1262 ↗

AlgebraicCurve.Place.evalAt

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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_.AlgebraicCurves.Cohomology.RiemannRoch.Proofs in
open _root_.Classical FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Classical in

Lean source · statement

def _root_.AlgebraicCurve.Place.evalAt (f : F) : K

theorem

evalAt_of_mem

L1268 ↗

AlgebraicCurve.Place.evalAt_of_mem

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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.evalAt_of_mem {f : F} (hf : f ∈ v.toValuationSubring) :
    v.evalAt f = v.residueInv (residue v.toValuationSubring ⟨f, hf⟩)

theorem

algebraMap_evalAt

L1273 ↗

AlgebraicCurve.Place.algebraMap_evalAt

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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.algebraMap_evalAt (hv : v.IsRational) {f : F} (hf : f
    ∈ v.toValuationSubring) :
    algebraMap K v.ResidueField (v.evalAt f)
      = residue v.toValuationSubring (⟨f, hf⟩ : v.toValuationSubring)

theorem

evalAt_ne_zero

L1281 ↗

AlgebraicCurve.Place.evalAt_ne_zero

Source context 28 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.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

theorem _root_.AlgebraicCurve.Place.evalAt_ne_zero (hv : v.IsRational) {f : F} (hf : f ≠
    0) (h : v.ord f = 0) :
    v.evalAt f ≠ 0

theorem

degree_nonneg_of_nonneg

L1325 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Divisor.degree_nonneg_of_nonneg

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.CongruenceSubgroup
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {D E : Divisor K F}
open scoped _root_.AlgebraicCurve.Divisor in

Lean source · statement

theorem Divisor.degree_nonneg_of_nonneg [IsCurveOver K F]
    {D : Divisor K F} (hD : ∀ v, 0 ≤ D v) : 0 ≤ Divisor.degree D

theorem

lSpace_eq_bot_of_degree_neg

L1342 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.lSpace_eq_bot_of_degree_neg

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.CongruenceSubgroup
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {D E : Divisor K F}

Lean source · statement

theorem lSpace_eq_bot_of_degree_neg [IsCurveOver K F]
    (hD : Divisor.degree D < 0) : LSpace D = ⊥

theorem

ell_eq_zero_of_degree_neg

L1370 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.ell_eq_zero_of_degree_neg

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.CongruenceSubgroup
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {D E : Divisor K F}

Lean source · statement

theorem ell_eq_zero_of_degree_neg [IsCurveOver K F] (hD : Divisor.degree D < 0) :
    ell D = 0

theorem

ord_placeInfty_eq_zero_of_intDegree_eq_zero

L1398 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_eq_zero_of_intDegree_eq_zero

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1425 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_eq_intDegree_mul

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1482 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_placeInfty_X

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

def

chartHom

L1525 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.chartHom

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1547 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.inv_algebraMap_mem

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

def

poleSubmodule

L1591 ↗

AlgebraicCurve.Place.poleSubmodule

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1603 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.mem_poleSubmodule

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 mem_poleSubmodule {n : ℕ} {f : F} :
    f ∈ v.poleSubmodule n ↔ v.uniformizer ^ n * f ∈ v.toValuationSubring

theorem

uniformizer_pow_ne_zero

L1606 ↗

AlgebraicCurve.Place.uniformizer_pow_ne_zero

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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.uniformizer_pow_ne_zero (n : ℕ) : (v.uniformizer ^ n :
    F) ≠ 0

theorem

ord_uniformizer_pow

L1611 ↗

AlgebraicCurve.Place.ord_uniformizer_pow

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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.ord_uniformizer_pow (n : ℕ) : v.ord (v.uniformizer ^
    n) = (n : ℤ)

theorem

poleSubmodule_one

L1618 ↗

AlgebraicCurve.Place.poleSubmodule_one

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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.poleSubmodule_one : v.poleSubmodule 1 =
    v.simplePoleSubmodule

theorem

mem_poleSubmodule_iff_ord

L1623 ↗

AlgebraicCurve.Place.mem_poleSubmodule_iff_ord

Source context 28 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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.mem_poleSubmodule_iff_ord {n : ℕ} {f : F} (hf : f ≠ 0) :
    f ∈ v.poleSubmodule n ↔ -(n : ℤ) ≤ v.ord f

theorem

poleSubmodule_mono

L1630 ↗

AlgebraicCurve.Place.poleSubmodule_mono

Source context 29 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_mono : Monotone v.poleSubmodule

theorem

gate_poleSubmodule_strictMono

L1641 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.gate_poleSubmodule_strictMono

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

mp72a102_t3_evalₐ_zero_depth

L1683 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_zero_depth

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1689 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_algebraMap

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1695 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_factor

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1702 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_exists_rep_of_evalₐ_eq_zero

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1711 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_succ_mul_eq_zero

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1719 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalₐ_one_eq_zero_of_evalOneₐ

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1733 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalDepth_add

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1740 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalDepth_mul

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1747 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_residueHat_congr_of_depth_one

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1756 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_evalDepth_one_eq_zero_of_residueHat

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1761 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_eq_uniformizer_mul_of_mem_maximalIdeal

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1767 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_exists_uniformizer_factor

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L1790 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3.mp72a102_t3_sigma_taylor_expansion

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

def

residueFieldAdjoinRootEquiv

L1877 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Section.residueFieldAdjoinRootEquiv

Documentation

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

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 : IsIntegral K ᾱ)
    (htop : K⟮ᾱ⟯ = ⊤) :
    AdjoinRoot (minpoly K ᾱ) ≃ₐ[K] v.ResidueField

theorem

residueFieldAdjoinRootEquiv_root

L1893 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Section.residueFieldAdjoinRootEquiv_root

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.IntermediateField
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 : IsIntegral K ᾱ)
    (htop : K⟮ᾱ⟯ = ⊤) :
    residueFieldAdjoinRootEquiv v hint htop (AdjoinRoot.root (minpoly K ᾱ)) = ᾱ

def

sectionOfPrimitiveRoot

L1911 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Section.sectionOfPrimitiveRoot

Documentation

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

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.IntermediateField
open _root_.AlgebraicCurve
open _root_.Polynomial
open _root_.IntermediateField
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 : 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

theorem

ord_sub_evalAt_pos

L1957 ↗

AlgebraicCurve.Place.ord_sub_evalAt_pos

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
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 _root_.AlgebraicCurve.Place.ord_sub_evalAt_pos (hrat : v.IsRational) {f : F} (hf :
    f ∈ v.toValuationSubring)
    (hne : f - algebraMap K F (v.evalAt f) ≠ 0) :
    0 < v.ord (f - algebraMap K F (v.evalAt f))

theorem

isRational_of_deg_eq_one

L2006 ↗

AlgebraicCurve.Place.isRational_of_deg_eq_one

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Polynomial
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.isRational_of_deg_eq_one (h : v.deg = 1) :
    v.IsRational

theorem

mk_mem_maximalIdeal_iff

L2041 ↗

AlgebraicCurve.Place.mk_mem_maximalIdeal_iff

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 _root_.AlgebraicCurve.Place.mk_mem_maximalIdeal_iff {f : F} (hf : f ∈
    v.toValuationSubring) :
    (⟨f, hf⟩ : v.toValuationSubring) ∈ IsLocalRing.maximalIdeal v.toValuationSubring
      ↔ f = 0 ∨ 0 < v.ord f

theorem

one_le_deg

L2083 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.one_le_deg

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem one_le_deg [IsCurveOver K F] (P : Place K F) : 1 ≤ P.deg

def

lSpaceShiftEquiv

L2101 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.lSpaceShiftEquiv

Documentation

Multiply by the inverse function to shift the divisor defining a Riemann-Roch space.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · full declaration

def lSpaceShiftEquiv (D : Divisor K F) {g : F} (hg : g ≠ 0)
    {Dg : Divisor K F} (hDg : ∀ v, Dg v = v.ord g) :
    LSpace D ≃ₗ[K] LSpace (D + Dg) where
  toFun h := ⟨(h : F) * g⁻¹, by
    rw [mem_lSpace_iff_ord]
    rcases eq_or_ne (h : F) 0 with hh0 | hh0
    · left; rw [hh0, zero_mul]
    · right; intro v
      rw [Finsupp.add_apply, hDg, v.ord_mul hh0 (inv_ne_zero hg), v.ord_inv]
      have := (mem_lSpace_iff_ord.mp h.2).resolve_left hh0 v
      linarith⟩
  invFun k := ⟨(k : F) * g, by
    rw [mem_lSpace_iff_ord]
    rcases eq_or_ne (k : F) 0 with hk0 | hk0
    · left; rw [hk0, zero_mul]
    · right; intro v
      have := (mem_lSpace_iff_ord.mp k.2).resolve_left hk0 v
      rw [Finsupp.add_apply, hDg] at this
      rw [v.ord_mul hk0 hg]
      linarith⟩
  map_add' h₁ h₂ := by apply Subtype.ext; push_cast; ring
  map_smul' c h := by
    apply Subtype.ext
    simp only [RingHom.id_apply, SetLike.val_smul, Algebra.smul_def]
    ring
  left_inv h := by
    apply Subtype.ext
    change (h : F) * g⁻¹ * g = (h : F)
    rw [mul_assoc, inv_mul_cancel₀ hg, mul_one]
  right_inv k := by
    apply Subtype.ext
    change (k : F) * g * g⁻¹ = (k : F)
    rw [mul_assoc, mul_inv_cancel₀ hg, mul_one]

theorem

eq_ord_of_addHom_of_nonneg_iff

L2188 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.eq_ord_of_addHom_of_nonneg_iff

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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)

L2265 ↗

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2280 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.neg_log_valuation_fiberCenter_eq_ord

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2319 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.le_ord_iff_mem_pow_fiberCenter

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2342 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.ramificationIndex_eq_ramificationIdx_fiberCenter

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2375 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.toValuationSubringOfRestrictEq

Documentation

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

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2381 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.residueOfCenter

Documentation

The residue map on the integral closure at a place.

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2386 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.residueOfCenter_apply

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2390 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.ker_residueOfCenter

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2396 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.surjective_residueOfCenter

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2459 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.residueFieldEquivQuotientCenter

Documentation

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

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2465 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.placeCongrEquiv

Documentation

Transport valuation subrings along equality of places.

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2475 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.restrictResidueFieldEquiv

Documentation

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

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2479 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.inertiaDeg_eq_inertiaDeg_fiberCenter

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

theorem

sum_ramificationIndex_mul_inertiaDeg

L2520 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.sum_ramificationIndex_mul_inertiaDeg

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 Assembly
variable (v : Place K F)

Lean source · statement

theorem sum_ramificationIndex_mul_inertiaDeg [HasPrincipalDivisors K F'] :
    ∑ w ∈ v.fiber F', (w.ramificationIndex F : ℤ) * (w.inertiaDeg F : ℤ)
      = (Module.finrank F F' : ℤ)

instance

instSumRamificationInertia

L2557 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.instSumRamificationInertia

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.IsLocalRing
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve

Lean source · statement

instance instSumRamificationInertia {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'] [HasPrincipalDivisors K F'] :
    SumRamificationInertia K F F'

def

aCoeff

L2582 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff

Documentation

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

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2606 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a103T2.mp72a103_t2_evalDepth_of_uniformizer_mul

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2647 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a103T2.mp72a103_t2_lift_eq_zero_of_depth_one

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L2671 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a103T2.mp72a103_t2_taylor_coeff_eq_zero_of_depth

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

theorem

eq_algebraMap_of_forall_ord_nonneg

L2748 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.Place.eq_algebraMap_of_forall_ord_nonneg

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
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 [HasPrincipalDivisors K F]

Lean source · statement

theorem eq_algebraMap_of_forall_ord_nonneg (v₀ : Place K F) (hrat : v₀.IsRational)
    (hdeg : v₀.deg ≠ 0) {g : F} (hg : g ≠ 0) (hord : ∀ v : Place K F, 0 ≤ v.ord g) :
    ∃ c : K, g = algebraMap K F c

theorem

constantsAreBase_of_exists_isRational

L2781 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.constantsAreBase_of_exists_isRational

Source context 15 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem constantsAreBase_of_exists_isRational [HasPrincipalDivisors K F]
    (v₀ : Place K F) (hrat : v₀.IsRational) (hdeg : v₀.deg ≠ 0) :
    ConstantsAreBase K F

theorem

constantsAreBase_of_deg_eq_one

L2801 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.constantsAreBase_of_deg_eq_one

Source context 15 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]

Lean source · statement

theorem constantsAreBase_of_deg_eq_one [HasPrincipalDivisors K F]
    (v₀ : Place K F) (hdeg : v₀.deg = 1) :
    ConstantsAreBase K F

def

nestedComapMapMkQEquiv

L2831 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Submodule.nestedComapMapMkQEquiv

Documentation

Identify the nested quotient image with the corresponding quotient of the middle submodule.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.IsDedekindDomain
open _root_.WithZero
namespace Submodule
open _root_.Submodule
open _root_.Submodule
variable {K X : Type*} [Field K] [AddCommGroup X] [Module K X]

Lean source · statement

def nestedComapMapMkQEquiv {A B C : Submodule K X} (_hAB : A ≤ B) (hBC : B ≤ C) :
    ((B.comap C.subtype).map (A.comap C.subtype).mkQ : Submodule K _)
      ≃ₗ[K] B ⧸ A.comap B.subtype

theorem

finrank_quotient_chain

L2859 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Submodule.finrank_quotient_chain

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsLocalRing
open _root_.Module.IsLocalRing
open _root_.IsDedekindDomain
open _root_.WithZero
namespace Submodule
open _root_.Submodule
open _root_.Submodule
variable {K X : Type*} [Field K] [AddCommGroup X] [Module K X]

Lean source · statement

theorem finrank_quotient_chain {A B C : Submodule K X} (hAB : A ≤ B) (hBC : B ≤ C)
    [hCB : Module.Finite K (C ⧸ B.comap C.subtype)]
    [hBA : Module.Finite K (B ⧸ A.comap B.subtype)] :
    Module.Finite K (C ⧸ A.comap C.subtype) ∧
      finrank K (C ⧸ A.comap C.subtype)
        = finrank K (C ⧸ B.comap C.subtype) + finrank K (B ⧸ A.comap B.subtype)

def

adeleBddQuotSingleEquivResidueField

L2887 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleBddQuotSingleEquivResidueField

Documentation

Identify the one-place bounded adele quotient with the residue field at that place.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

def adeleBddQuotSingleEquivResidueField (D : Divisor K F) (P : Place K F) :
    (adeleBdd D ⧸ (adeleBdd (D - Finsupp.single P 1)).comap (adeleBdd D).subtype)
      ≃ₗ[K] P.ResidueField

theorem

finrank_adeleBdd_quotient_single

L2998 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.finrank_adeleBdd_quotient_single

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem finrank_adeleBdd_quotient_single [IsCurveOver K F] (D : Divisor K F) (P : Place K F) :
    Module.Finite K
        (adeleBdd D ⧸ (adeleBdd (D - Finsupp.single P 1)).comap (adeleBdd D).subtype) ∧
      finrank K
          (adeleBdd D ⧸ (adeleBdd (D - Finsupp.single P 1)).comap (adeleBdd D).subtype)
        = P.deg

theorem

finrank_adeleBdd_quotient

L3008 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.finrank_adeleBdd_quotient

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem finrank_adeleBdd_quotient [IsCurveOver K F] {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    Module.Finite K (adeleBdd D₂ ⧸ (adeleBdd D₁).comap (adeleBdd D₂).subtype) ∧
      (finrank K (adeleBdd D₂ ⧸ (adeleBdd D₁).comap (adeleBdd D₂).subtype) : ℤ)
        = Divisor.degree D₂ - Divisor.degree D₁

theorem

module_finite_adeleBdd_quotient

L3069 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.module_finite_adeleBdd_quotient

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem module_finite_adeleBdd_quotient [IsCurveOver K F] {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    Module.Finite K (adeleBdd D₂ ⧸ (adeleBdd D₁).comap (adeleBdd D₂).subtype)

theorem

finrank_adeleBdd_quotient_eq

L3073 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.finrank_adeleBdd_quotient_eq

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem finrank_adeleBdd_quotient_eq [IsCurveOver K F] {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    (finrank K (adeleBdd D₂ ⧸ (adeleBdd D₁).comap (adeleBdd D₂).subtype) : ℤ)
      = Divisor.degree D₂ - Divisor.degree D₁

def

lSpaceQuotientToAdeleBddQuotient

L3082 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.lSpaceQuotientToAdeleBddQuotient

Documentation

The map induced by the diagonal function embedding between the Riemann-Roch and bounded adele quotients.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

def lSpaceQuotientToAdeleBddQuotient {D₁ D₂ : Divisor K F} (_hD : D₁ ≤ D₂) :
    (LSpace D₂ ⧸ (LSpace D₁).comap (LSpace D₂).subtype)
      →ₗ[K] (adeleBdd D₂ ⧸ (adeleBdd D₁).comap (adeleBdd D₂).subtype)

theorem

lSpaceQuotientToAdeleBddQuotient_injective

L3093 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.lSpaceQuotientToAdeleBddQuotient_injective

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem lSpaceQuotientToAdeleBddQuotient_injective {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    Function.Injective (lSpaceQuotientToAdeleBddQuotient hD)

theorem

ell_sub_ell_le_degree_sub_degree

L3106 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.ell_sub_ell_le_degree_sub_degree

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem ell_sub_ell_le_degree_sub_degree [IsCurveOver K F] {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂)
    [FiniteDimensional K (LSpace D₂)] :
    (ell D₂ : ℤ) - (ell D₁ : ℤ) ≤ Divisor.degree D₂ - Divisor.degree D₁

theorem

adeleBdd_inf_sup_globalSub

L3125 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleBdd_inf_sup_globalSub

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]

Lean source · statement

theorem adeleBdd_inf_sup_globalSub {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    adeleBdd D₂ ⊓ (adeleBdd D₁ ⊔ globalSub K F)
      = adeleBdd D₁ ⊔ adeleBdd D₂ ⊓ globalSub K F

theorem

weilDualityAdelic_of_residueRows

L3150 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilDualityAdelic_of_residueRows

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.IsDedekindDomain
open _root_.WithZero
open _root_.Module
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 weilDualityAdelic_of_residueRows (hΩℓ : WeilOmegaEllAgrees K F) :
    WeilDualityAdelic K F

theorem

comap_subtype_sup_of_le_of_le

L3177 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Submodule.comap_subtype_sup_of_le_of_le

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace Submodule
open _root_.Submodule
open _root_.Submodule
variable {K X : Type*} [Field K] [AddCommGroup X] [Module K X]

Lean source · statement

theorem comap_subtype_sup_of_le_of_le {S T C : Submodule K X} (hS : S ≤ C) (hT : T ≤ C) :
    S.comap C.subtype ⊔ T.comap C.subtype = (S ⊔ T).comap C.subtype

theorem

globalSub_le_adeleSpace

L3192 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.globalSub_le_adeleSpace

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem globalSub_le_adeleSpace [HasPrincipalDivisors K F] :
    globalSub K F ≤ adeleSpace K F

theorem

adeleBdd_sup_globalSub_le_adeleSpace

L3197 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleBdd_sup_globalSub_le_adeleSpace

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem adeleBdd_sup_globalSub_le_adeleSpace [HasPrincipalDivisors K F] (D : Divisor K F) :
    adeleBdd D ⊔ globalSub K F ≤ adeleSpace K F

theorem

adeleBddPrincipal_eq_comap

L3201 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleBddPrincipal_eq_comap

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem adeleBddPrincipal_eq_comap [HasPrincipalDivisors K F] (D : Divisor K F) :
    adeleBddPrincipal K F D = (adeleBdd D ⊔ globalSub K F).comap (adeleSpace K F).subtype

theorem

finiteDimensional_lSpace

L3210 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.finiteDimensional_lSpace

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section FiniteLSpace
variable [IsCurveOver K F] [Nonempty (Place K F)]
omit [Nonempty (Place K F)] in

Lean source · statement

theorem finiteDimensional_lSpace [hL0 : FiniteDimensional K (LSpace (0 : Divisor K F))]
    (D : Divisor K F) : FiniteDimensional K (LSpace D)

theorem

range_lSpaceQuotientToAdeleBddQuotient

L3230 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.range_lSpaceQuotientToAdeleBddQuotient

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem range_lSpaceQuotientToAdeleBddQuotient {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    LinearMap.range (lSpaceQuotientToAdeleBddQuotient hD)
      = ((adeleBdd D₁ ⊔ globalSub K F).comap (adeleBdd D₂).subtype).map
          ((adeleBdd D₁).comap (adeleBdd D₂).subtype).mkQ

theorem

finrank_adeleBddSup_quotient

L3271 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.finrank_adeleBddSup_quotient

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem finrank_adeleBddSup_quotient [IsCurveOver K F] [Nonempty (Place K F)]
    {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) [hL : FiniteDimensional K (LSpace D₂)] :
    Module.Finite K (↥(adeleBdd D₂ ⊔ globalSub K F)
        ⧸ (adeleBdd D₁ ⊔ globalSub K F).comap (adeleBdd D₂ ⊔ globalSub K F).subtype) ∧
      (finrank K (↥(adeleBdd D₂ ⊔ globalSub K F)
          ⧸ (adeleBdd D₁ ⊔ globalSub K F).comap (adeleBdd D₂ ⊔ globalSub K F).subtype) : ℤ)
        = (Divisor.degree D₂ - ell D₂) - (Divisor.degree D₁ - ell D₁)

theorem

adeleBddSup_eq_of_degree_sub_ell_eq

L3342 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleBddSup_eq_of_degree_sub_ell_eq

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section Index
variable [IsCurveOver K F] [Nonempty (Place K F)]

Lean source · statement

theorem adeleBddSup_eq_of_degree_sub_ell_eq {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂)
    [FiniteDimensional K (LSpace D₂)]
    (heq : Divisor.degree D₂ - ell D₂ = Divisor.degree D₁ - ell D₁) :
    adeleBdd D₂ ⊔ globalSub K F = adeleBdd D₁ ⊔ globalSub K F

theorem

eq_of_ge

L3370 ↗

AlgebraicCurve.RiemannGenusReachedAt.eq_of_ge

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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 RiemannGenusReachedAt
open scoped _root_.AlgebraicCurve.RiemannGenusReachedAt in

Lean source · statement

theorem _root_.AlgebraicCurve.RiemannGenusReachedAt.eq_of_ge [IsCurveOver K F] [Nonempty
    (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    {D : Divisor K F} (hD : D₀ ≤ D) :
    Divisor.degree D - ell D = γ - 1

theorem

adeleSpace_eq_of_genusReached

L3385 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleSpace_eq_of_genusReached

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
open _root_.AlgebraicCurve

Lean source · statement

theorem adeleSpace_eq_of_genusReached [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀) :
    adeleSpace K F = adeleBdd D₀ ⊔ globalSub K F

theorem

indexOfSpecialty_eq_of_genusReached

L3402 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.indexOfSpecialty_eq_of_genusReached

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
open _root_.AlgebraicCurve

Lean source · statement

theorem indexOfSpecialty_eq_of_genusReached [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀) (D : Divisor K F) :
    Module.Finite K (adeleSpace K F ⧸ adeleBddPrincipal K F D) ∧
      (indexOfSpecialty D : ℤ) = (ell D : ℤ) - (Divisor.degree D + 1 - γ)

theorem

riemannIndexFormula_of_genusReached

L3449 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.riemannIndexFormula_of_genusReached

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
open _root_.AlgebraicCurve

Lean source · statement

theorem riemannIndexFormula_of_genusReached
    (hg : ∀ [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)],
      RiemannGenusReached K F (genus K F)) :
    RiemannIndexFormula K F

theorem

gate_riemannInequality_of_genusReached

L3458 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.gate_riemannInequality_of_genusReached

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
open _root_.AlgebraicCurve

Lean source · statement

theorem gate_riemannInequality_of_genusReached [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀) (D : Divisor K F) :
    Divisor.degree D + 1 - γ ≤ (ell D : ℤ)

theorem

adeleSpaceMul_add

L3489 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleSpaceMul_add

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem adeleSpaceMul_add [HasPrincipalDivisors K F] (f g : F) :
    adeleSpaceMul K F (f + g) = adeleSpaceMul K F f + adeleSpaceMul K F g

theorem

adeleSpaceMul_zero

L3495 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.adeleSpaceMul_zero

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem adeleSpaceMul_zero [HasPrincipalDivisors K F] :
    adeleSpaceMul K F (0 : F) = 0

theorem

weilSmul_add_left

L3501 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilSmul_add_left

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem weilSmul_add_left [HasPrincipalDivisors K F] (f g : F)
    (φ : Module.Dual K (adeleSpace K F)) :
    weilSmul K F (f + g) φ = weilSmul K F f φ + weilSmul K F g φ

theorem

weilSmul_zero_left

L3507 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilSmul_zero_left

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem weilSmul_zero_left [HasPrincipalDivisors K F]
    (φ : Module.Dual K (adeleSpace K F)) :
    weilSmul K F (0 : F) φ = 0

theorem

weilSmul_neg_left

L3513 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilSmul_neg_left

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem weilSmul_neg_left [HasPrincipalDivisors K F] (f : F)
    (φ : Module.Dual K (adeleSpace K F)) :
    weilSmul K F (-f) φ = -weilSmul K F f φ

theorem

weilSmul_sub_left

L3520 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilSmul_sub_left

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem weilSmul_sub_left [HasPrincipalDivisors K F] (f g : F)
    (φ : Module.Dual K (adeleSpace K F)) :
    weilSmul K F (f - g) φ = weilSmul K F f φ - weilSmul K F g φ

theorem

weilSmul_eq_zero_iff_left

L3525 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilSmul_eq_zero_iff_left

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem weilSmul_eq_zero_iff_left [HasPrincipalDivisors K F]
    {f : F} {φ : Module.Dual K (adeleSpace K F)} (hφ : φ ≠ 0) :
    weilSmul K F f φ = 0 ↔ f = 0

theorem

weilSmul_left_injective

L3532 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilSmul_left_injective

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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 · full declaration

theorem weilSmul_left_injective [HasPrincipalDivisors K F]
    {φ : Module.Dual K (adeleSpace K F)} (hφ : φ ≠ 0) :
    Function.Injective (fun f : F => weilSmul K F f φ) := by
  intro f g hfg
  have hsub : weilSmul K F (f - g) φ = 0 := by
    rw [weilSmul_sub_left, sub_eq_zero]; exact hfg
  exact sub_eq_zero.mp ((weilSmul_eq_zero_iff_left hφ).mp hsub)

def

doubleResiduePairing

L3542 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.doubleResiduePairing

Documentation

The sum of the two residue pairings on the product of the corresponding Riemann-Roch spaces.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

def doubleResiduePairing [HasPrincipalDivisors K F] (W D : Divisor K F)
    {φ μ : Module.Dual K (adeleSpace K F)}
    (hφ : φ ∈ omegaSpace W) (hμ : μ ∈ omegaSpace W) :
    LSpace (W - D) × LSpace (W - D) →ₗ[K] omegaSpace D

theorem

doubleResiduePairing_apply_coe

L3548 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.doubleResiduePairing_apply_coe

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem doubleResiduePairing_apply_coe [HasPrincipalDivisors K F] (W D : Divisor K F)
    {φ μ : Module.Dual K (adeleSpace K F)}
    (hφ : φ ∈ omegaSpace W) (hμ : μ ∈ omegaSpace W) (g h : LSpace (W - D)) :
    (doubleResiduePairing K F W D hφ hμ (g, h) : Module.Dual K (adeleSpace K F))
      = weilSmul K F (g : F) φ + weilSmul K F (h : F) μ

theorem

doubleResiduePairing_injective

L3556 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.doubleResiduePairing_injective

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem doubleResiduePairing_injective [HasPrincipalDivisors K F] (W D : Divisor K F)
    {φ μ : Module.Dual K (adeleSpace K F)}
    (hφ : φ ∈ omegaSpace W) (hμ : μ ∈ omegaSpace W)
    (hφ0 : φ ≠ 0) (hindep : ∀ f : F, μ ≠ weilSmul K F f φ) :
    Function.Injective (doubleResiduePairing K F W D hφ hμ)

theorem

two_mul_ell_le_indexOfSpecialty

L3583 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.two_mul_ell_le_indexOfSpecialty

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem two_mul_ell_le_indexOfSpecialty [HasPrincipalDivisors K F] (W D : Divisor K F)
    {φ μ : Module.Dual K (adeleSpace K F)}
    (hφ : φ ∈ omegaSpace W) (hμ : μ ∈ omegaSpace W)
    (hφ0 : φ ≠ 0) (hindep : ∀ f : F, μ ≠ weilSmul K F f φ)
    [Module.Finite K (omegaSpace (K := K) (F := F) D)]
    [Module.Finite K (LSpace (K := K) (F := F) (W - D))] :
    2 * ell (W - D) ≤ indexOfSpecialty D

theorem

degree_add_one_sub_genus_le_ell_of_riemannIndexFormula

L3596 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.degree_add_one_sub_genus_le_ell_of_riemannIndexFormula

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem degree_add_one_sub_genus_le_ell_of_riemannIndexFormula
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
    (hRI : RiemannIndexFormula K F) (D : Divisor K F) :
    Divisor.degree D + 1 - (genus K F : ℤ) ≤ (ell D : ℤ)

theorem

omegaSpace_finite_of_riemannIndexFormula

L3604 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.omegaSpace_finite_of_riemannIndexFormula

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem omegaSpace_finite_of_riemannIndexFormula
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
    (hRI : RiemannIndexFormula K F) {D : Divisor K F}
    (hpos : 0 < (ell D : ℤ) - Divisor.degree D - 1 + (genus K F : ℤ)) :
    Module.Finite K (omegaSpace (K := K) (F := F) D)

theorem

lSpace_finite_of_riemannIndexFormula

L3618 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.lSpace_finite_of_riemannIndexFormula

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem lSpace_finite_of_riemannIndexFormula
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
    (hRI : RiemannIndexFormula K F) {D : Divisor K F}
    (hpos : 0 < Divisor.degree D + 1 - (genus K F : ℤ)) :
    Module.Finite K (LSpace (K := K) (F := F) D)

theorem

exists_weilSmul_eq_of_riemannIndexFormula

L3627 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.exists_weilSmul_eq_of_riemannIndexFormula

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem exists_weilSmul_eq_of_riemannIndexFormula
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)] [Nonempty (Place K F)]
    (hRI : RiemannIndexFormula K F)
    {φ μ : Module.Dual K (adeleSpace K F)} {W : Divisor K F}
    (hφ : φ ∈ omegaSpace W) (hμ : μ ∈ omegaSpace W) (hφ0 : φ ≠ 0) :
    ∃ f : F, μ = weilSmul K F f φ

theorem

weilDifferentialRankOne_of_riemannIndexFormula

L3671 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilDifferentialRankOne_of_riemannIndexFormula

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem weilDifferentialRankOne_of_riemannIndexFormula
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)] [Nonempty (Place K F)]
    (hRI : RiemannIndexFormula K F) :
    WeilDifferentialRankOne K F

def

ResiduePairingSurjective

L3719 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.ResiduePairingSurjective

Documentation

Surjectivity of the residue pairing for a nonzero differential functional with maximal divisor.

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
  [Nontrivial Ω[F⁄K]]
variable (K F)
variable {K F}
variable (K F)
variable {K F}
variable (K F)

Lean source · statement

def ResiduePairingSurjective : Prop

theorem

omegaSpace_finite_of_genusReached

L3751 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.omegaSpace_finite_of_genusReached

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section Finiteness
variable [IsCurveOver K F] [Nonempty (Place K F)]
  [FiniteDimensional K (LSpace (0 : Divisor K F))]

Lean source · statement

theorem omegaSpace_finite_of_genusReached
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀) (D : Divisor K F) :
    Module.Finite K (omegaSpace (K := K) (F := F) D)

theorem

exists_weilSmul_eq_of_genusReached

L3765 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.exists_weilSmul_eq_of_genusReached

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section ParametricRankOne
variable [IsCurveOver K F] [Nonempty (Place K F)]
  [FiniteDimensional K (LSpace (0 : Divisor K F))]

Lean source · statement

theorem exists_weilSmul_eq_of_genusReached
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    {φ μ : Module.Dual K (adeleSpace K F)} {W : Divisor K F}
    (hφ : φ ∈ omegaSpace W) (hμ : μ ∈ omegaSpace W) (hφ0 : φ ≠ 0) :
    ∃ f : F, μ = weilSmul K F f φ

theorem

weilDifferentialRankOne_of_genusReached

L3801 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilDifferentialRankOne_of_genusReached

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section ParametricRankOne
variable [IsCurveOver K F] [Nonempty (Place K F)]
  [FiniteDimensional K (LSpace (0 : Divisor K F))]

Lean source · statement

theorem weilDifferentialRankOne_of_genusReached
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀) :
    WeilDifferentialRankOne K F

theorem

residuePairing_surjective_of_rankOne_max

L3814 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.residuePairing_surjective_of_rankOne_max

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem residuePairing_surjective_of_rankOne_max [HasPrincipalDivisors K F]
    (hRankOne : WeilDifferentialRankOne K F)
    {W : Divisor K F} {φ : Module.Dual K (adeleSpace K F)}
    (hφ : φ ∈ omegaSpace W) (hφ0 : φ ≠ 0)
    (hWmax : ∀ E : Divisor K F, φ ∈ omegaSpace E → E ≤ W) (D : Divisor K F) :
    Function.Surjective (residuePairing K F W D hφ)

theorem

stichtenothGenus_nonneg

L3853 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.stichtenothGenus_nonneg

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section Identification
variable [IsCurveOver K F] [Nonempty (Place K F)]
  [FiniteDimensional K (LSpace (0 : Divisor K F))]
  [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
  [HasCanonicalLocalResidueKStar K F]
omit [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
  [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem stichtenothGenus_nonneg
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    (hC : ConstantsAreBase K F) : 0 ≤ γ

theorem

genus_eq_degree_div

L3863 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.genus_eq_degree_div

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
section Identification
variable [IsCurveOver K F] [Nonempty (Place K F)]
  [FiniteDimensional K (LSpace (0 : Divisor K F))]
  [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
  [HasCanonicalLocalResidueKStar K F]
omit [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
  [∀ v : Place K F, v.DCoordGenerates] [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem genus_eq_degree_div :
    ∃ (ω₀ : Ω[F⁄K]) (hω₀ : ω₀ ≠ 0),
      genus K F = (Divisor.degree (canonicalDivisorOf hω₀) + 2).toNat / 2

theorem

uniformizer_pow_inv_mem_poleSubmodule_self

L3903 ↗

AlgebraicCurve.Place.uniformizer_pow_inv_mem_poleSubmodule_self

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3908 ↗

AlgebraicCurve.Place.higherPoleMonomial_sum_mem_poleSubmodule

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3918 ↗

AlgebraicCurve.Place.higherPoleMonomial_coeff_eq_zero_of_mem

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3945 ↗

AlgebraicCurve.Place.linearIndependent_higherPoleMonomial_mkQ

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3956 ↗

AlgebraicCurve.Place.higherPoleCorrectionAux

Source context 28 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3964 ↗

AlgebraicCurve.Place.higherPoleCorrection

Source context 29 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3968 ↗

AlgebraicCurve.Place.higherPoleCorrection_apply_of_mem

Source context 30 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3976 ↗

AlgebraicCurve.Place.higherPoleCorrection_uniformizer_pow_inv

Source context 31 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L3992 ↗

AlgebraicCurve.Place.canonicalLocalResidueDataKOfExtend

Source context 32 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

aCoeff_zero

L4048 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_zero

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4060 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_add

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4110 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_smul

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4169 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_shift

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4236 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_shift_pow

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4253 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_one_eq_zero

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4295 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.clearPow_mem

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4306 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.clearedHat

Documentation

Embed a function in the completion after clearing its pole with a uniformizer power.

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4312 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar

Documentation

The residue coefficient extracted from the pole-cleared completed function.

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4316 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.aCoeff_clearedHat_of_le

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4331 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_add

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4347 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_smul

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4363 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStarₗ

Documentation

The residue coefficient map as a linear map over the base field.

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4369 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_simplePole

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4389 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_of_mem

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4397 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.resStar_higherPoleMonomial

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

L4419 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.KwNo6Pin.canonicalLocalResidueDataKStar

Documentation

Canonical local residue data constructed from the completed residue section.

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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

mem_lSpace_of_weilSmul_mem_omegaSpace

L4450 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.mem_lSpace_of_weilSmul_mem_omegaSpace

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem mem_lSpace_of_weilSmul_mem_omegaSpace [HasPrincipalDivisors K F]
    {W D : Divisor K F} {φ : Module.Dual K (adeleSpace K F)}
    (hWmax : ∀ E : Divisor K F, φ ∈ omegaSpace E → E ≤ W)
    {f : F} (hfφ : weilSmul K F f φ ∈ omegaSpace D) :
    f ∈ LSpace (W - D)

theorem

residuePairing_surjective

L4484 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.residuePairing_surjective

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem residuePairing_surjective [HasPrincipalDivisors K F]
    (hRk1 : WeilDifferentialRankOne K F) (W D : Divisor K F)
    {φ : Module.Dual K (adeleSpace K F)} (hφ : φ ∈ omegaSpace W) (hφ0 : φ ≠ 0)
    (hWmax : ∀ E : Divisor K F, φ ∈ omegaSpace E → E ≤ W) :
    Function.Surjective (residuePairing K F W D hφ)

theorem

residuePairingSurjective_of_weilDifferentialRankOne

L4508 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.residuePairingSurjective_of_weilDifferentialRankOne

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem residuePairingSurjective_of_weilDifferentialRankOne [HasPrincipalDivisors K F]
    (hRk1 : WeilDifferentialRankOne K F) : ResiduePairingSurjective K F

theorem

residuePairingSurjective_of_riemannIndexFormula

L4524 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.residuePairingSurjective_of_riemannIndexFormula

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem residuePairingSurjective_of_riemannIndexFormula
    [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)] [Nonempty (Place K F)]
    (hRI : RiemannIndexFormula K F) : ResiduePairingSurjective K F

theorem

weilOfKaehlerK_mem_omegaSpace_of_residueTheoremK

L4557 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.weilOfKaehlerK_mem_omegaSpace_of_residueTheoremK

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 _root_.AlgebraicCurve.RationalFunctionField
section CanonicalKResidueTerm
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
section SupportMachinery
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
variable (K F)
variable {K F}
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem weilOfKaehlerK_mem_omegaSpace_of_residueTheoremK [HasPrincipalDivisors K F]
    (hRT : ResidueTheoremK K F) (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    weilOfKaehlerK Rfam hω ∈ omegaSpace (canonicalDivisorOf hω)

def

simplePoleProbe

L4623 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.simplePoleProbe

Documentation

A single-place adele probing the residue coefficient of the specified differential.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]
section SimplePoleProbe
variable [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
  [Nontrivial Ω[F⁄K]]

Lean source · statement

def simplePoleProbe {ω : Ω[F⁄K]} (_hω : ω ≠ 0) (v : Place K F)
    (c : v.toValuationSubring) : adeleSpace K F

theorem

simplePoleProbe_mem_adeleBdd

L4630 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.simplePoleProbe_mem_adeleBdd

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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]
section SimplePoleProbe
variable [HasCanonicalLocalResidueKStar K F]
variable [HasCanonicalDivisor (K := K) (F := F)] [∀ v : Place K F, v.DCoordGenerates]
  [Nontrivial Ω[F⁄K]]
open _root_.AlgebraicCurves.Cohomology.RiemannRoch.Proofs in
open scoped _root_.Classical FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Classical in
omit [HasCanonicalLocalResidueKStar K F] [Nontrivial Ω[F⁄K]] in

Lean source · statement

theorem simplePoleProbe_mem_adeleBdd {ω : Ω[F⁄K]} (hω : ω ≠ 0) (v : Place K F)
    (c : v.toValuationSubring) {D : Divisor K F} (hDv : canonicalDivisorOf hω v < D v) :
    (simplePoleProbe hω v c : Place K F → F) ∈ adeleBdd D

theorem

exists_trace_residue_ne_zero

L4658 ↗

AlgebraicCurve.Place.exists_trace_residue_ne_zero

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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 scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.exists_trace_residue_ne_zero [HasSeparableResidue K F]
    (v : Place K F) :
    ∃ c : v.toValuationSubring,
      Algebra.trace K v.ResidueField (IsLocalRing.residue _ c) ≠ 0

theorem

exists_riemannGenusReachedAt_of_bounded

L4714 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.exists_riemannGenusReachedAt_of_bounded

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem exists_riemannGenusReachedAt_of_bounded
    [IsCurveOver K F] [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (hbdd : RiemannGenusBounded K F) :
    ∃ (γ : ℤ) (D₀ : Divisor K F), RiemannGenusReachedAt γ D₀

theorem

stichtenothGenusExists_of_bounded

L4742 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.stichtenothGenusExists_of_bounded

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem stichtenothGenusExists_of_bounded
    [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (hbdd : RiemannGenusBounded K F) :
    StichtenothGenusExists K F

theorem

completionSection_nonempty_generic

L4767 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.completionSection_nonempty_generic

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
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)

theorem

mul_mem_lSpace_add

L4799 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.mul_mem_lSpace_add

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem mul_mem_lSpace_add {D E : Divisor K F} {f g : F}
    (hf : f ∈ LSpace D) (hg : g ∈ LSpace E) :
    f * g ∈ LSpace (D + E)

theorem

pow_mem_lSpace_nsmul

L4812 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.pow_mem_lSpace_nsmul

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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

theorem pow_mem_lSpace_nsmul {B : Divisor K F} {f : F} (hf : f ∈ LSpace B) (j : ℕ) :
    f ^ j ∈ LSpace (j • B)

structure

PoleDivisorPackage

L4821 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage

Documentation

A function with a controlled effective pole divisor and a family independent over its powers.

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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 · full declaration

structure PoleDivisorPackage (K F : Type*) [Field K] [Field F] [Algebra K F] where
  /-- The function whose powers generate the prescribed independent family. -/
  x : F
  /-- The divisor bounding the poles of the chosen function. -/
  B : Divisor K F
  /-- The pole bound divisor is effective. -/
  hB_eff : 0 ≤ B
  /-- The function belongs to the Riemann-Roch space of its pole bound. -/
  hx_mem : x ∈ LSpace B
  /-- The degree of the pole bound divisor. -/
  n : ℕ
  /-- The degree of the pole bound divisor is positive. -/
  hn_pos : 0 < n
  /-- The degree of the pole bound divisor equals the chosen natural number. -/
  degB_eq : Divisor.degree B = n
  /-- The common multiple of the pole divisor bounding the auxiliary family. -/
  c : ℕ
  /-- The auxiliary family of functions. -/
  u : Fin n → F
  /-- Each auxiliary function lies in the common bounded Riemann-Roch space. -/
  hu_mem : ∀ i, u i ∈ LSpace (c • B)
  /-- The products of coordinate powers with the auxiliary family are linearly independent. -/
  lin_indep : LinearIndependent K (fun p : ℕ × Fin n => x ^ p.1 * u p.2)

def

HasPoleDivisorPackage

L4847 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.HasPoleDivisorPackage

Documentation

Existence of a function and independent family with the specified pole divisor bounds.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

def HasPoleDivisorPackage : Prop

theorem

pow_mul_u_mem_lSpace

L4853 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.pow_mul_u_mem_lSpace

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in
namespace PoleDivisorPackage
variable (P : PoleDivisorPackage K F)

Lean source · statement

theorem pow_mul_u_mem_lSpace (j : ℕ) (i : Fin P.n) :
    P.x ^ j * P.u i ∈ LSpace ((j + P.c) • P.B)

theorem

pow_mul_u_mem_lSpace_of_le

L4858 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.pow_mul_u_mem_lSpace_of_le

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in
namespace PoleDivisorPackage
variable (P : PoleDivisorPackage K F)

Lean source · statement

theorem pow_mul_u_mem_lSpace_of_le {j m : ℕ} (hjm : j + P.c ≤ m) (i : Fin P.n) :
    P.x ^ j * P.u i ∈ LSpace (m • P.B)

theorem

ell_nsmul_poleDivisor_ge

L4868 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.ell_nsmul_poleDivisor_ge

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in
namespace PoleDivisorPackage
variable (P : PoleDivisorPackage K F)

Lean source · statement

theorem ell_nsmul_poleDivisor_ge [IsCurveOver K F]
    [FiniteDimensional K (LSpace (0 : Divisor K F))] {m : ℕ} (hm : P.c ≤ m) :
    (P.n : ℤ) * (m - P.c + 1) ≤ ell (m • P.B)

theorem

degree_nsmul_sub_ell_le

L4894 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.degree_nsmul_sub_ell_le

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in
namespace PoleDivisorPackage
variable (P : PoleDivisorPackage K F)

Lean source · statement

theorem degree_nsmul_sub_ell_le [IsCurveOver K F]
    [FiniteDimensional K (LSpace (0 : Divisor K F))] {m : ℕ} (hm : P.c ≤ m) :
    Divisor.degree (m • P.B) - ell (m • P.B) ≤ (P.n : ℤ) * (P.c - 1)

theorem

ell_nsmul_sub_pos

L4905 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.ell_nsmul_sub_pos

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem PoleDivisorPackage.ell_nsmul_sub_pos [IsCurveOver K F]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (P : PoleDivisorPackage K F) {A : Divisor K F} (hA : 0 ≤ A) {m : ℕ} (hm : P.c ≤ m)
    (hbig : Divisor.degree A < (P.n : ℤ) * (m - P.c + 1)) :
    0 < ell (m • P.B - A)

theorem

degree_sub_ell_le

L4923 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.degree_sub_ell_le

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem PoleDivisorPackage.degree_sub_ell_le [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (P : PoleDivisorPackage K F) (D : Divisor K F) :
    Divisor.degree D - ell D ≤ (P.n : ℤ) * (P.c - 1)

theorem

riemannGenusBounded_of_poleDivisorPackage

L4968 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.riemannGenusBounded_of_poleDivisorPackage

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem riemannGenusBounded_of_poleDivisorPackage [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (P : PoleDivisorPackage K F) :
    RiemannGenusBounded K F

theorem

stichtenothGenusExists_of_poleDivisorPackage

L4974 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.stichtenothGenusExists_of_poleDivisorPackage

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem stichtenothGenusExists_of_poleDivisorPackage [IsCurveOver K F] [Nonempty (Place K F)]
    [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (P : PoleDivisorPackage K F) :
    StichtenothGenusExists K F

theorem

stichtenothGenusExists_of_hasPoleDivisorPackage

L4980 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.stichtenothGenusExists_of_hasPoleDivisorPackage

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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]
variable (K F) in

Lean source · statement

theorem stichtenothGenusExists_of_hasPoleDivisorPackage [IsCurveOver K F]
    [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (h : HasPoleDivisorPackage K F) :
    StichtenothGenusExists K F

structure

TranscendenceTower

L5013 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower

Documentation

A transcendental coordinate with a distinguished degree-one pole place and regularity elsewhere.

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]

Lean source · full declaration

structure TranscendenceTower (K E F : Type*) [Field K] [Field E] [Field F]
    [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F] where
  /-- The transcendental coordinate in the intermediate field. -/
  x : E
  /-- The powers of the coordinate are linearly independent over the base field. -/
  hx_indep : LinearIndependent K (fun j : ℕ => x ^ j)
  /-- The distinguished pole place of the coordinate. -/
  v : Place K E
  /-- The distinguished place has degree one. -/
  hvdeg : v.deg = 1
  /-- The coordinate has a simple pole at the distinguished place. -/
  hxv : v.ord x = -1
  /-- The coordinate is regular at every other place. -/
  hxreg : ∀ u : Place K E, u ≠ v → 0 ≤ u.ord x

theorem

x_ne_zero

L5034 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.x_ne_zero

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F] in

Lean source · statement

theorem x_ne_zero : T.x ≠ 0

abbrev

xF

L5039 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.xF

Documentation

The transcendental coordinate mapped into the extension field.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F] in

Lean source · statement

abbrev xF : F

def

poleDivisor

L5042 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.poleDivisor

Documentation

The pullback to the extension field of the distinguished pole place.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)

Lean source · statement

def poleDivisor : Divisor K F

theorem

poleDivisor_apply

L5045 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.poleDivisor_apply

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] in

Lean source · statement

theorem poleDivisor_apply (w : Place K F) :
    T.poleDivisor w = (w.ramificationIndex E : ℤ) * (Finsupp.single T.v 1) (w.restrict E)

theorem

poleDivisor_nonneg

L5050 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.poleDivisor_nonneg

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] in

Lean source · statement

theorem poleDivisor_nonneg : 0 ≤ T.poleDivisor

theorem

xF_mem_lSpace_poleDivisor

L5058 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.xF_mem_lSpace_poleDivisor

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] in

Lean source · statement

theorem xF_mem_lSpace_poleDivisor : T.xF ∈ LSpace T.poleDivisor

theorem

degree_poleDivisor_eq_finrank

L5068 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.degree_poleDivisor_eq_finrank

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)

Lean source · statement

theorem degree_poleDivisor_eq_finrank :
    Divisor.degree T.poleDivisor = (Module.finrank E F : ℤ)

theorem

mem_lSpace_nsmul_poleDivisor_of_regular_outside

L5074 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.mem_lSpace_nsmul_poleDivisor_of_regular_outside

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] in

Lean source · statement

theorem mem_lSpace_nsmul_poleDivisor_of_regular_outside {f : F}
    (hreg : ∀ w : Place K F, w.restrict E ≠ T.v → 0 ≤ w.ord f)
    {c : ℕ} (hc : ∀ w ∈ T.v.fiber F, -(c : ℤ) ≤ w.ord f) :
    f ∈ LSpace (c • T.poleDivisor)

theorem

exists_forall_mem_lSpace_nsmul_poleDivisor

L5094 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.exists_forall_mem_lSpace_nsmul_poleDivisor

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
namespace TranscendenceTower
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (T : TranscendenceTower K E F)
omit [FiniteDimensional E F] in

Lean source · statement

theorem exists_forall_mem_lSpace_nsmul_poleDivisor {n : ℕ} (u : Fin n → F)
    (hreg : ∀ i, ∀ w : Place K F, w.restrict E ≠ T.v → 0 ≤ w.ord (u i)) :
    ∃ c : ℕ, ∀ i, u i ∈ LSpace (c • T.poleDivisor)

structure

IntegralBasisInLSpace

L5118 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.IntegralBasisInLSpace

Documentation

A linearly independent extension-field family lying in a common bounded Riemann-Roch space.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]

Lean source · full declaration

structure IntegralBasisInLSpace (T : TranscendenceTower K E F) where
  /-- The common multiple of the pole divisor bounding the family. -/
  c : ℕ
  /-- The family indexed by the extension-field dimension. -/
  u : Fin (Module.finrank E F) → F
  /-- The family is linearly independent over the intermediate field. -/
  hu_indep : LinearIndependent E u
  /-- Each family member lies in the specified bounded Riemann-Roch space. -/
  hu_mem : ∀ i, u i ∈ LSpace (c • T.poleDivisor)

def

HasIntegralBasisInLSpace

L5130 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.HasIntegralBasisInLSpace

Documentation

Existence of a bounded linearly independent family for the specified transcendence tower.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in

Lean source · statement

def HasIntegralBasisInLSpace (T : TranscendenceTower K E F) : Prop

def

ofRegularOutside

L5134 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.IntegralBasisInLSpace.ofRegularOutside

Documentation

Bound a family regular outside the distinguished place in a common Riemann-Roch space.

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in

Lean source · full declaration

def IntegralBasisInLSpace.ofRegularOutside (T : TranscendenceTower K E F)
    (u : Fin (Module.finrank E F) → F) (hu_indep : LinearIndependent E u)
    (hreg : ∀ i, ∀ w : Place K F, w.restrict E ≠ T.v → 0 ≤ w.ord (u i)) :
    IntegralBasisInLSpace T where
  c := (T.exists_forall_mem_lSpace_nsmul_poleDivisor u hreg).choose
  u := u
  hu_indep := hu_indep
  hu_mem := (T.exists_forall_mem_lSpace_nsmul_poleDivisor u hreg).choose_spec

def

HasIntegralBasisRegularOutside

L5147 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.HasIntegralBasisRegularOutside

Documentation

Existence of a linearly independent family regular away from places over the distinguished pole.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in

Lean source · statement

def HasIntegralBasisRegularOutside (T : TranscendenceTower K E F) : Prop

theorem

hasIntegralBasisInLSpace_of_regularOutside

L5152 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasIntegralBasisInLSpace_of_regularOutside

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in
omit [FiniteDimensional E F] in

Lean source · statement

theorem hasIntegralBasisInLSpace_of_regularOutside (T : TranscendenceTower K E F)
    (h : HasIntegralBasisRegularOutside K E F T) :
    HasIntegralBasisInLSpace K E F T

theorem

linearIndependent_pow_mul

L5159 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.linearIndependent_pow_mul

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in
omit [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F] in

Lean source · full declaration

theorem linearIndependent_pow_mul {x : E} {n : ℕ} {u : Fin n → F}
    (hx : LinearIndependent K (fun j : ℕ => x ^ j)) (hu : LinearIndependent E u) :
    LinearIndependent K (fun p : ℕ × Fin n => (algebraMap E F x) ^ p.1 * u p.2) := by
  have heq : (fun p : ℕ × Fin n => (algebraMap E F x) ^ p.1 * u p.2)
      = (fun p : ℕ × Fin n => (x ^ p.1) • u p.2) := by
    funext p
    rw [Algebra.smul_def, map_pow]
  rw [heq]
  exact linearIndependent_smul (R := K) (S := E) (A := F) hx hu

def

ofTranscendenceTower

L5172 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.PoleDivisorPackage.ofTranscendenceTower

Documentation

Construct the pole divisor data from a transcendence tower and its bounded independent family.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in

Lean source · full declaration

def PoleDivisorPackage.ofTranscendenceTower (T : TranscendenceTower K E F)
    (IB : IntegralBasisInLSpace T) :
    PoleDivisorPackage K F where
  x := T.xF
  B := T.poleDivisor
  hB_eff := T.poleDivisor_nonneg
  hx_mem := T.xF_mem_lSpace_poleDivisor
  n := Module.finrank E F
  hn_pos := Module.finrank_pos
  degB_eq := T.degree_poleDivisor_eq_finrank
  c := IB.c
  u := IB.u
  hu_mem := IB.hu_mem
  lin_indep := linearIndependent_pow_mul T.hx_indep IB.hu_indep

theorem

hasPoleDivisorPackage_of_transcendenceTower

L5187 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasPoleDivisorPackage_of_transcendenceTower

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in

Lean source · statement

theorem hasPoleDivisorPackage_of_transcendenceTower
    (T : TranscendenceTower K E F) (IB : IntegralBasisInLSpace T) :
    HasPoleDivisorPackage K F

theorem

hasPoleDivisorPackage_of_hasIntegralBasisInLSpace

L5192 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasPoleDivisorPackage_of_hasIntegralBasisInLSpace

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in

Lean source · statement

theorem hasPoleDivisorPackage_of_hasIntegralBasisInLSpace
    (T : TranscendenceTower K E F) (h : HasIntegralBasisInLSpace K E F T) :
    HasPoleDivisorPackage K F

theorem

hasPoleDivisorPackage_of_hasIntegralBasisRegularOutside

L5197 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasPoleDivisorPackage_of_hasIntegralBasisRegularOutside

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.IsDedekindDomain
open _root_.WithZero
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
variable (K E F) in

Lean source · statement

theorem hasPoleDivisorPackage_of_hasIntegralBasisRegularOutside
    (T : TranscendenceTower K E F) (h : HasIntegralBasisRegularOutside K E F T) :
    HasPoleDivisorPackage K F

theorem

mem_of_isIntegral

L5228 ↗

AlgebraicCurve.Place.mem_of_isIntegral

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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] (w : Place K F)

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_of_isIntegral {f : F} (hf : _root_.IsIntegral
    w.toValuationSubring f) :
    f ∈ w.toValuationSubring

theorem

mem_of_isIntegral_of_algebraMap_mem

L5235 ↗

AlgebraicCurve.Place.mem_of_isIntegral_of_algebraMap_mem

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
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] (w : Place K F)
open _root_.AlgebraicCurve.Place

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_of_isIntegral_of_algebraMap_mem {A : Type*}
    [CommRing A] [Algebra A F]
    (hA : ∀ a : A, algebraMap A F a ∈ w.toValuationSubring)
    {f : F} (hf : _root_.IsIntegral A f) : f ∈ w.toValuationSubring

theorem

mem_of_isIntegral_of_restrict

L5256 ↗

AlgebraicCurve.Place.mem_of_isIntegral_of_restrict

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Bridge
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable {A : Type*} [CommRing A] [Algebra A E] [Algebra A F] [IsScalarTower A E F]
namespace Place
open scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.mem_of_isIntegral_of_restrict [Algebra.IsIntegral E F]
    (w : Place K F)
    {u : Place K E} (hw : w.restrict E = u)
    (hA : ∀ a : A, algebraMap A E a ∈ u.toValuationSubring)
    {f : F} (hf : _root_.IsIntegral A f) : f ∈ w.toValuationSubring

theorem

ord_nonneg_of_isIntegral_of_restrict

L5269 ↗

AlgebraicCurve.Place.ord_nonneg_of_isIntegral_of_restrict

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Bridge
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable {A : Type*} [CommRing A] [Algebra A E] [Algebra A F] [IsScalarTower A E F]
open _root_.AlgebraicCurve
namespace Place
open scoped _root_.AlgebraicCurve.Place in

Lean source · statement

theorem _root_.AlgebraicCurve.Place.ord_nonneg_of_isIntegral_of_restrict
    [Algebra.IsIntegral E F] (w : Place K F)
    {u : Place K E} (hw : w.restrict E = u)
    (hA : ∀ a : A, algebraMap A E a ∈ u.toValuationSubring)
    {f : F} (hf : _root_.IsIntegral A f) : 0 ≤ w.ord f

def

RegularOutside

L5280 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.RegularOutside

Documentation

Regularity of the chosen subalgebra away from the distinguished pole place.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Bridge
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable {A : Type*} [CommRing A] [Algebra A E] [Algebra A F] [IsScalarTower A E F]
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (A) in

Lean source · statement

def TranscendenceTower.RegularOutside (T : TranscendenceTower K E F) : Prop

theorem

ord_nonneg_of_isIntegral_of_regularOutside

L5283 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.ord_nonneg_of_isIntegral_of_regularOutside

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Bridge
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable {A : Type*} [CommRing A] [Algebra A E] [Algebra A F] [IsScalarTower A E F]
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable (A) in

Lean source · statement

theorem TranscendenceTower.ord_nonneg_of_isIntegral_of_regularOutside
    [Algebra.IsIntegral E F] (T : TranscendenceTower K E F) (hA : T.RegularOutside A)
    {f : F} (hf : _root_.IsIntegral A f) :
    ∀ w : Place K F, w.restrict E ≠ T.v → 0 ≤ w.ord f

def

HasRegularFractionSubring

L5299 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.HasRegularFractionSubring

Documentation

Existence of a fraction subalgebra regular away from the distinguished pole place.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in

Lean source · statement

def HasRegularFractionSubring (T : TranscendenceTower K E F) : Prop

theorem

linearIndependent_reindex_basis

L5304 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.linearIndependent_reindex_basis

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [FiniteDimensional E F] [Algebra.IsSeparable E F] in

Lean source · full declaration

theorem linearIndependent_reindex_basis {s : Finset F} (b : Basis s E F) :
    LinearIndependent E (fun i : Fin (Module.finrank E F) =>
      b ((Fintype.equivFinOfCardEq (Module.finrank_eq_card_basis b).symm).symm i)) :=
  b.linearIndependent.comp _ (Equiv.injective _)

theorem

hasIntegralBasisRegularOutside_of_isFractionRing

L5310 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasIntegralBasisRegularOutside_of_isFractionRing

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [HasPrincipalDivisors K F] in

Lean source · statement

theorem hasIntegralBasisRegularOutside_of_isFractionRing
    {A : Type*} [CommRing A] [IsDomain A] [Algebra A E] [IsFractionRing A E]
    [Algebra A F] [IsScalarTower A E F]
    (T : TranscendenceTower K E F) (hA : T.RegularOutside A) :
    HasIntegralBasisRegularOutside K E F T

theorem

hasIntegralBasisRegularOutside_of_hasRegularFractionSubring

L5322 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasIntegralBasisRegularOutside_of_hasRegularFractionSubring

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [HasPrincipalDivisors K F] in

Lean source · statement

theorem hasIntegralBasisRegularOutside_of_hasRegularFractionSubring
    (T : TranscendenceTower K E F) (h : HasRegularFractionSubring K E F T) :
    HasIntegralBasisRegularOutside K E F T

theorem

hasPoleDivisorPackage_of_hasRegularFractionSubring

L5330 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasPoleDivisorPackage_of_hasRegularFractionSubring

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in

Lean source · statement

theorem hasPoleDivisorPackage_of_hasRegularFractionSubring
    (T : TranscendenceTower K E F) (h : HasRegularFractionSubring K E F T) :
    HasPoleDivisorPackage K F

theorem

stichtenothGenusExists_of_hasRegularFractionSubring

L5337 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.stichtenothGenusExists_of_hasRegularFractionSubring

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [HasPrincipalDivisors K F] in

Lean source · statement

theorem stichtenothGenusExists_of_hasRegularFractionSubring [IsCurveOver K F]
    [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (T : TranscendenceTower K E F) (h : HasRegularFractionSubring K E F T) :
    StichtenothGenusExists K F

theorem

adjoin_x_regularOutside

L5346 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.TranscendenceTower.adjoin_x_regularOutside

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F] in

Lean source · statement

theorem TranscendenceTower.adjoin_x_regularOutside (T : TranscendenceTower K E F) :
    ∀ u : Place K E, u ≠ T.v →
      ∀ a ∈ Algebra.adjoin K {T.x}, a ∈ u.toValuationSubring

theorem

hasRegularFractionSubring_of_adjoin_x_isFractionRing

L5362 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.hasRegularFractionSubring_of_adjoin_x_isFractionRing

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F] in

Lean source · statement

theorem hasRegularFractionSubring_of_adjoin_x_isFractionRing
    (T : TranscendenceTower K E F)
    (hfrac : IsFractionRing (Algebra.adjoin K {T.x}) E) :
    HasRegularFractionSubring K E F T

theorem

stichtenothGenusExists_of_adjoin_x_isFractionRing

L5369 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.stichtenothGenusExists_of_adjoin_x_isFractionRing

Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Headline
variable {K E F : Type*} [Field K] [Field E] [Field F]
  [Algebra K E] [Algebra K F] [Algebra E F] [IsScalarTower K E F]
variable [FiniteDimensional E F] [Algebra.IsSeparable E F] [HasPrincipalDivisors K F]
variable (K E F) in
omit [HasPrincipalDivisors K F] in

Lean source · statement

theorem stichtenothGenusExists_of_adjoin_x_isFractionRing [IsCurveOver K F]
    [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
    (T : TranscendenceTower K E F)
    (hfrac : IsFractionRing (Algebra.adjoin K {T.x}) E) :
    StichtenothGenusExists K F

theorem

linearIndependent_pow_of_transcendental

L5404 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.linearIndependent_pow_of_transcendental

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.Polynomial
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
section Transcendental
variable {K A : Type*} [CommRing K] [CommRing A] [Algebra K A]

Lean source · full declaration

theorem linearIndependent_pow_of_transcendental {x : A} (hx : Transcendental K x) :
    LinearIndependent K (fun j : ℕ => x ^ j) := by
  have hinj : Function.Injective (Polynomial.aeval x : K[X] →ₐ[K] A) :=
    transcendental_iff_injective.mp hx
  have hXj : LinearIndependent K (fun j : ℕ => (X : K[X]) ^ j) := by
    have hb := (Polynomial.basisMonomials K).linearIndependent
    simp only [Polynomial.coe_basisMonomials] at hb
    convert hb using 2 with j
    exact (Polynomial.monomial_one_right_eq_X_pow j).symm
  have heq : (fun j : ℕ => x ^ j)
      = (fun j : ℕ => (Polynomial.aeval x : K[X] →ₐ[K] A) (X ^ j)) := by
    funext j; simp
  rw [heq]
  exact hXj.map' (Polynomial.aeval x).toLinearMap
    (LinearMap.ker_eq_bot_of_injective hinj)

theorem

algebraMap_polynomial_mem_of_ne_placeInfty

L5429 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.algebraMap_polynomial_mem_of_ne_placeInfty

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]

Lean source · statement

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

theorem

ord_X_nonneg_of_ne_placeInfty

L5436 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.ord_X_nonneg_of_ne_placeInfty

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]

Lean source · statement

theorem ord_X_nonneg_of_ne_placeInfty
    {u : Place K (RatFunc K)} (hu : u ≠ placeInfty K) :
    0 ≤ u.ord (RatFunc.X : RatFunc K)

def

transcendenceTower

L5446 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.transcendenceTower

Documentation

The transcendence tower given by the rational function coordinate and its place at infinity.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]

Lean source · full declaration

def transcendenceTower : TranscendenceTower K (RatFunc K) F where
  x := RatFunc.X
  hx_indep := linearIndependent_pow_of_transcendental RatFunc.transcendental_X
  v := placeInfty K
  hvdeg := deg_placeInfty K
  hxv := ord_placeInfty_X K
  hxreg := fun _ hu => ord_X_nonneg_of_ne_placeInfty K hu

theorem

aeval_X_eq_algebraMap

L5455 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.aeval_X_eq_algebraMap

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem aeval_X_eq_algebraMap (q : K[X]) :
    (Polynomial.aeval (R := K) (RatFunc.X : RatFunc K)) q
      = algebraMap K[X] (RatFunc K) q

theorem

algebraMap_polynomial_mem_adjoin_X

L5464 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.algebraMap_polynomial_mem_adjoin_X

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem algebraMap_polynomial_mem_adjoin_X (q : K[X]) :
    algebraMap K[X] (RatFunc K) q ∈ Algebra.adjoin K {(RatFunc.X : RatFunc K)}

theorem

isFractionRing_adjoin_X

L5470 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.isFractionRing_adjoin_X

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem isFractionRing_adjoin_X :
    IsFractionRing (Algebra.adjoin K {(RatFunc.X : RatFunc K)}) (RatFunc K)

theorem

stichtenothGenusExists_of_ratFunc_tower

L5486 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.stichtenothGenusExists_of_ratFunc_tower

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [HasPrincipalDivisors K F]
variable [IsCurveOver K F] [Nonempty (Place K F)]
  [FiniteDimensional K (LSpace (0 : Divisor K F))]
omit [DecidableEq (RatFunc K)] [HasPrincipalDivisors K F] in

Lean source · statement

theorem stichtenothGenusExists_of_ratFunc_tower :
    StichtenothGenusExists K F

theorem

not_isField_integralClosureAt'

L5524 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.not_isField_integralClosureAt'

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.Polynomial
open _root_.IsDedekindDomain
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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
section LyingOver
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
attribute [local instance 0] Place.valuationSubringAlgebra
omit [DecidableEq (RatFunc K)] [Algebra K F] [IsScalarTower K (RatFunc K) F]
  [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F] in

Lean source · statement

theorem not_isField_integralClosureAt' (v : Place K (RatFunc K)) :
    ¬IsField (Place.integralClosureAt F v)

theorem

exists_restrict_eq

L5553 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.exists_restrict_eq

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.Polynomial
open _root_.IsDedekindDomain
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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
section LyingOver
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
attribute [local instance 0] Place.valuationSubringAlgebra
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem exists_restrict_eq (v : Place K (RatFunc K)) :
    ∃ w : Place K F, w.restrict (RatFunc K) = v

theorem

nonempty_place_of_ratFunc_tower

L5585 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.nonempty_place_of_ratFunc_tower

Source context 27 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.Polynomial
open _root_.IsDedekindDomain
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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
section LyingOver
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
attribute [local instance 0] Place.valuationSubringAlgebra
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem nonempty_place_of_ratFunc_tower : Nonempty (Place K F)

theorem

finiteDimensional_lSpace_zero_of_constantsAreBase

L5608 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.finiteDimensional_lSpace_zero_of_constantsAreBase

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.Polynomial
open _root_.IsDedekindDomain
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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
omit [DecidableEq (RatFunc K)] [Algebra (RatFunc K) F] [IsScalarTower K (RatFunc K) F] in

Lean source · statement

theorem finiteDimensional_lSpace_zero_of_constantsAreBase (hC : ConstantsAreBase K F) :
    FiniteDimensional K (LSpace (0 : Divisor K F))

theorem

stichtenothGenusExists

L5634 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField.stichtenothGenusExists

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.Module
open _root_.Polynomial
open _root_.IsDedekindDomain
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] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsCurveOver K F]
omit [DecidableEq (RatFunc K)] in

Lean source · statement

theorem stichtenothGenusExists (hC : ConstantsAreBase K F) :
    StichtenothGenusExists K F

theorem

p0n20_rr_constantsAreBase_of_isAlgClosed

L5662 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n20_rr_constantsAreBase_of_isAlgClosed

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Mp72a102T2
section
section
noncomputable section
open _root_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
open _root_.IsLocalRing
section AlgClosedEngine
variable (K F : Type*) [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [DecidableEq (RatFunc K)]
variable [Algebra (RatFunc K) F] [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsAlgClosed K] [IsCurveOver K F]
omit [DecidableEq (RatFunc K)] [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n20_rr_constantsAreBase_of_isAlgClosed : ConstantsAreBase K F

theorem

p0n25_wkc_kaehlerResidueTermKFam_single_of_ne

L5699 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_kaehlerResidueTermKFam_single_of_ne

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalProbe
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
open _root_.AlgebraicCurves.Cohomology.RiemannRoch.Proofs in
open scoped _root_.Classical FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Classical in
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_kaehlerResidueTermKFam_single_of_ne
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK) (ω : Ω[F⁄K])
    {v : Place K F} (g : F) {w : Place K F} (hw : w ≠ v) :
    kaehlerResidueTermKFam Rfam ω (Pi.single v g) w = 0

theorem

p0n25_wkc_localResidueData_res_mul_uniformizer_inv

L5707 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_localResidueData_res_mul_uniformizer_inv

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalProbe
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_localResidueData_res_mul_uniformizer_inv
    {v : Place K F} (R : v.LocalResidueData) (c : v.toValuationSubring) :
    R.res ((c : F) * v.uniformizer⁻¹) = IsLocalRing.residue _ c

theorem

p0n25_wkc_weilOfKaehlerK_single

L5723 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_weilOfKaehlerK_single

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalProbe
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]
omit [HasCanonicalLocalResidueKStar K F] in
open _root_.AlgebraicCurves.Cohomology.RiemannRoch.Proofs in
open scoped _root_.Classical FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.Classical in

Lean source · statement

theorem p0n25_wkc_weilOfKaehlerK_single
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) (v : Place K F) (g : F) :
    weilOfKaehlerK Rfam hω (adeleSingle v g)
      = Algebra.trace K v.ResidueField ((Rfam v).res (g * v.differentialCoeff ω))

theorem

p0n25_wkc_weilOfKaehlerK_simplePoleProbe

L5734 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_weilOfKaehlerK_simplePoleProbe

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalProbe
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]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_weilOfKaehlerK_simplePoleProbe
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) (v : Place K F) (c : v.toValuationSubring) :
    weilOfKaehlerK Rfam hω (simplePoleProbe hω v c)
      = Algebra.trace K v.ResidueField (IsLocalRing.residue _ c)

theorem

p0n25_wkc_weilOfKaehlerK_ne_zero

L5752 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_weilOfKaehlerK_ne_zero

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalNonvanishingMaximality
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]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_weilOfKaehlerK_ne_zero [HasSeparableResidue K F] [Nonempty (Place K F)]
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) : weilOfKaehlerK Rfam hω ≠ 0

theorem

p0n25_wkc_trace_eq_zero_of_weilOfKaehlerK_mem

L5762 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_trace_eq_zero_of_weilOfKaehlerK_mem

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalNonvanishingMaximality
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]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_trace_eq_zero_of_weilOfKaehlerK_mem
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) {D : Divisor K F}
    (hD : weilOfKaehlerK Rfam hω ∈ omegaSpace D)
    {v : Place K F} (hDv : canonicalDivisorOf hω v < D v) :
    (Algebra.trace K v.ResidueField : _ →ₗ[K] K) = 0

theorem

p0n25_wkc_weilOfKaehlerK_omegaSpace_le_canonical

L5774 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_weilOfKaehlerK_omegaSpace_le_canonical

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalNonvanishingMaximality
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]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_weilOfKaehlerK_omegaSpace_le_canonical [HasSeparableResidue K F]
    (Rfam : ∀ v : Place K F, v.CanonicalLocalResidueDataK)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0) {D : Divisor K F}
    (hD : weilOfKaehlerK Rfam hω ∈ omegaSpace D) :
    D ≤ canonicalDivisorOf hω

theorem

p0n25_wkc_exists_weilMax_of_residueTheoremK

L5793 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_exists_weilMax_of_residueTheoremK

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section CanonicalSupply
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]
omit [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_exists_weilMax_of_residueTheoremK
    [HasSeparableResidue K F] [Nonempty (Place K F)] [HasPrincipalDivisors K F]
    (hRTK : ResidueTheoremK K F) {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    ∃ φ : Module.Dual K (adeleSpace K F), φ ≠ 0 ∧
      φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω

theorem

p0n25_wkc_finrank_omegaSpace_eq_ell_of_genusReached_of_weilMax

L5814 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_finrank_omegaSpace_eq_ell_of_genusReached_of_weilMax

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorIdentification
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
variable [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
omit [HasCanonicalLocalResidueKStar K F] [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_finrank_omegaSpace_eq_ell_of_genusReached_of_weilMax
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    {ω : Ω[F⁄K]} (hω : ω ≠ 0)
    {φ : Module.Dual K (adeleSpace K F)} (hne : φ ≠ 0)
    (hmem : φ ∈ omegaSpace (canonicalDivisorOf hω))
    (hmax : ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (D : Divisor K F) :
    Module.finrank K (omegaSpace (K := K) (F := F) D) = ell (canonicalDivisorOf hω - D)

theorem

p0n25_wkc_ell_canonicalDivisor_eq_gamma_of_weilMax

L5828 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_ell_canonicalDivisor_eq_gamma_of_weilMax

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorIdentification
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
variable [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
omit [HasCanonicalLocalResidueKStar K F] [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_ell_canonicalDivisor_eq_gamma_of_weilMax
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (hC : ConstantsAreBase K F) {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    (ell (canonicalDivisorOf hω) : ℤ) = γ

theorem

p0n25_wkc_degree_canonicalDivisor_eq_of_weilMax

L5846 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_degree_canonicalDivisor_eq_of_weilMax

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorIdentification
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
variable [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
omit [HasCanonicalLocalResidueKStar K F] [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_degree_canonicalDivisor_eq_of_weilMax
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (hC : ConstantsAreBase K F) {ω : Ω[F⁄K]} (hω : ω ≠ 0) :
    Divisor.degree (canonicalDivisorOf hω) = 2 * γ - 2

theorem

p0n25_wkc_stichtenothGenus_eq_genus_of_weilMax

L5865 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_stichtenothGenus_eq_genus_of_weilMax

Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorIdentification
variable {K F : Type*} [Field K] [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
variable [Nonempty (Place K F)] [FiniteDimensional K (LSpace (0 : Divisor K F))]
omit [HasCanonicalLocalResidueKStar K F] [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_stichtenothGenus_eq_genus_of_weilMax
    {γ : ℤ} {D₀ : Divisor K F} (h : RiemannGenusReachedAt γ D₀)
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (hC : ConstantsAreBase K F) :
    γ = (genus K F : ℤ)

theorem

p0n25_wkc_riemannGenusReached_of_weilMax

L5894 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_riemannGenusReached_of_weilMax

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorAssembly
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F] [Algebra (RatFunc
  K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
omit [DecidableEq (RatFunc K)] [HasCanonicalLocalResidueKStar K F] [∀ (v : Place K F),
  v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_riemannGenusReached_of_weilMax
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (hC : ConstantsAreBase K F) :
    RiemannGenusReached K F (genus K F)

theorem

p0n25_wkc_riemannIndexFormula_of_weilMax

L5908 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_riemannIndexFormula_of_weilMax

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorAssembly
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F] [Algebra (RatFunc
  K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
omit [DecidableEq (RatFunc K)] [IsCurveOver K F] [HasCanonicalLocalResidueKStar K F]
  [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_riemannIndexFormula_of_weilMax
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (hC : ConstantsAreBase K F) :
    RiemannIndexFormula K F

theorem

p0n25_wkc_weilOmegaEllAgrees_of_weilMax

L5921 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_weilOmegaEllAgrees_of_weilMax

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorAssembly
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F] [Algebra (RatFunc
  K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
omit [DecidableEq (RatFunc K)] [HasCanonicalLocalResidueKStar K F] [Algebra (RatFunc K) F]
  [IsScalarTower K (RatFunc K) F] [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc
  K) F] [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_weilOmegaEllAgrees_of_weilMax [Nonempty (Place K F)]
    (hRI : RiemannIndexFormula K F)
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω) :
    WeilOmegaEllAgrees K F

theorem

p0n25_wkc_functionFieldRiemannRoch_of_weilMax_of_constantsAreBase

L5938 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_functionFieldRiemannRoch_of_weilMax_of_constantsAreBase

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorAssembly
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F] [Algebra (RatFunc
  K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
omit [DecidableEq (RatFunc K)] [IsCurveOver K F] [HasCanonicalLocalResidueKStar K F]
  [∀ (v : Place K F), v.DCoordGenerates] in

Lean source · statement

theorem p0n25_wkc_functionFieldRiemannRoch_of_weilMax_of_constantsAreBase
    (hsup : ∀ {ω : Ω[F⁄K]} (hω : ω ≠ 0), ∃ φ : Module.Dual K (adeleSpace K F),
      φ ≠ 0 ∧ φ ∈ omegaSpace (canonicalDivisorOf hω) ∧
      ∀ D : Divisor K F, φ ∈ omegaSpace D → D ≤ canonicalDivisorOf hω)
    (hC : ConstantsAreBase K F) :
    FunctionFieldRiemannRoch K F

theorem

p0n25_wkc_functionFieldRiemannRoch_of_residueTheoremK

L5955 ↗

AlgebraicCurves.Cohomology.RiemannRoch.Proofs.FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.p0n25_wkc_functionFieldRiemannRoch_of_residueTheoremK

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs
namespace FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.AlgebraicCurve.RationalFunctionField
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Lg37
open FunctionFieldRiemannRoch_of_residueTheoremK_of_isAlgClosed.ModularCurve.Mp72a102T3
open FunctionFieldRiemannRoch_of_residueTheoremK_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_.AlgebraicCurve
namespace ModularCurve
local notation "Qbar" => AlgebraicClosure ℚ
section MirrorAssembly
variable (K : Type*) [Field K] [DecidableEq (RatFunc K)]
variable (F : Type*) [Field F] [Algebra K F] [HasCanonicalLocalResidueKStar K F] [Algebra (RatFunc
  K) F]
  [IsScalarTower K (RatFunc K) F]
variable [FiniteDimensional (RatFunc K) F] [Algebra.IsSeparable (RatFunc K) F]
variable [IsCurveOver K F] [HasCanonicalDivisor (K := K) (F := F)]
variable [∀ v : Place K F, v.DCoordGenerates]
omit [DecidableEq (RatFunc K)] [IsCurveOver K F] [HasCanonicalLocalResidueKStar K F] in

Lean source · statement

theorem p0n25_wkc_functionFieldRiemannRoch_of_residueTheoremK
    [HasSeparableResidue K F]
    (hRTK : ResidueTheoremK K F) (hC : ConstantsAreBase K F) :
    FunctionFieldRiemannRoch K F

theorem