NYLL
Module

Ell_sub_ell_le_degree_sub_degree.lean

Nyll.AlgebraicCurves.Cohomology.RiemannRoch.Ell_sub_ell_le_degree_sub_degree
29 declarations indexed from sourceView source ↗

Module documentation

Ell sub ell le degree sub degree

Imports4
Imported by2

Declarations

theorem

ord_nonneg_of_mem

L33 ↗
AlgebraicCurve.Place.ord_nonneg_of_mem
Source context 9 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
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_nonneg_of_mem {f : F} (hf : f ∈
  v.toValuationSubring) : 0 ≤ v.ord f
theorem

mem_of_ord_nonneg

L49 ↗
AlgebraicCurve.Place.mem_of_ord_nonneg
Source context 10 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
Lean source · statement
theorem _root_.AlgebraicCurve.Place.mem_of_ord_nonneg {f : F} (hf : f ≠ 0) (h : 0 ≤ v.ord
  f) :
    f ∈ v.toValuationSubring
theorem

mem_iff_ord_nonneg

L58 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.Place.mem_iff_ord_nonneg
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
namespace Place
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
variable {K F : Type*} [Field K] [Field F] [Algebra K F] (v : Place K F)
open _root_.AlgebraicCurve.Place
open _root_.AlgebraicCurve.Place
Lean source · statement
theorem mem_iff_ord_nonneg {f : F} (hf : f ≠ 0) :
    f ∈ v.toValuationSubring ↔ 0 ≤ v.ord f
theorem

degree_eq_sum_support

L72 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.Divisor.degree_eq_sum_support
Source context 7 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
open scoped _root_.AlgebraicCurve.Divisor in
Lean source · statement
theorem Divisor.degree_eq_sum_support (D : Divisor K F) :
    Divisor.degree D = ∑ v ∈ D.support, D v * (v.deg : ℤ)
theorem

degree_nonneg_of_nonneg

L78 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.Divisor.degree_nonneg_of_nonneg
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {D : 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

L82 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.lSpace_eq_bot_of_degree_neg
Source context 7 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {D : 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

L98 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.ell_eq_zero_of_degree_neg
Source context 7 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
namespace AlgebraicCurve
open _root_.AlgebraicCurve
open _root_.AlgebraicCurve
variable {K F : Type*} [Field K] [Field F] [Algebra K F]
variable {D : 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

mk_mem_maximalIdeal_iff

L122 ↗
AlgebraicCurve.Place.mk_mem_maximalIdeal_iff
Source context 15 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

ell_le_ell_sub_single_add_deg

L152 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.ell_le_ell_sub_single_add_deg
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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_le_ell_sub_single_add_deg [IsCurveOver K F] (D : Divisor K F) (P : Place K F) :
    ell D ≤ ell (D - Finsupp.single P 1) + P.deg
theorem

one_le_deg

L244 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.one_le_deg
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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
theorem

ell_le_degree_add_ellZero

L248 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.ell_le_degree_add_ellZero
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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_le_degree_add_ellZero [IsCurveOver K F] {D : Divisor K F}
    (hD : ∀ v, 0 ≤ D v) :
    (ell D : ℤ) ≤ Divisor.degree D + ell (0 : Divisor K F)
theorem

ell_le_degree_add_one_of_effective

L285 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.ell_le_degree_add_one_of_effective
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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_le_degree_add_one_of_effective [IsCurveOver K F]
    (hC : ConstantsAreBase K F) {D : Divisor K F} (hD : ∀ v, 0 ≤ D v) :
    (ell D : ℤ) ≤ Divisor.degree D + 1
def

lSpaceShiftEquiv

L294 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.lSpaceShiftEquiv
Documentation

Multiplication by the inverse of g identifies the spaces for divisors differing by its divisor.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

ell_le_degree_add_one

L328 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.ell_le_degree_add_one
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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_le_degree_add_one [IsCurveOver K F] (hC : ConstantsAreBase K F)
    {D : Divisor K F} (hD : 0 ≤ Divisor.degree D) :
    (ell D : ℤ) ≤ Divisor.degree D + 1
def

nestedComapMapMkQEquiv

L366 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.Submodule.nestedComapMapMkQEquiv
Documentation

Identify the image of B in the quotient of C by A with the induced quotient of B.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L394 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.Submodule.finrank_quotient_chain
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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)
theorem

finrank_quotient_chain'

L413 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.Submodule.finrank_quotient_chain'
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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)
    [hCA : Module.Finite K (C ⧸ A.comap C.subtype)] :
    Module.Finite K (C ⧸ B.comap C.subtype) ∧
      finrank K (C ⧸ B.comap C.subtype)
        = finrank K (C ⧸ A.comap C.subtype) - finrank K (B ⧸ A.comap B.subtype)
def

adeleBddQuotSingleEquivResidueField

L439 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.adeleBddQuotSingleEquivResidueField
Documentation

The quotient obtained by lowering the bound at one place is linearly equivalent to its residue field.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L550 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.finrank_adeleBdd_quotient_single
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L560 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.finrank_adeleBdd_quotient
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L621 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.module_finite_adeleBdd_quotient
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L625 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.finrank_adeleBdd_quotient_eq
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L631 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.lSpaceQuotientToAdeleBddQuotient
Documentation

The map between divisor-bounded quotients induced by the diagonal embedding into adeles.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

L642 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.lSpaceQuotientToAdeleBddQuotient_injective
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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_port

L655 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.ell_sub_ell_le_degree_sub_degree_port
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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_port [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

L674 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.adeleBdd_inf_sup_globalSub
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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

map_diagonal_lSpace_inf_adeleBdd

L679 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.map_diagonal_lSpace_inf_adeleBdd
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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 map_diagonal_lSpace_inf_adeleBdd {D₁ D₂ : Divisor K F} (hD : D₁ ≤ D₂) :
    (LSpace D₂).map (diagonalHom K F) ⊓ adeleBdd D₁ = (LSpace D₁).map (diagonalHom K F)
theorem

gate_finrank_adeleBdd_quotient_single_point

L684 ↗
AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve.gate_finrank_adeleBdd_quotient_single_point
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
noncomputable section
open _root_.Module
open _root_.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 gate_finrank_adeleBdd_quotient_single_point [IsCurveOver K F] (P : Place K F) :
    (finrank K (adeleBdd (Finsupp.single P 1 : Divisor K F)
        ⧸ (adeleBdd (0 : Divisor K F)).comap (adeleBdd (Finsupp.single P 1)).subtype) : ℤ)
      = (P.deg : ℤ)
theorem

ell_sub_ell_le_degree_sub_degree

L700 ↗
AlgebraicCurve.ell_sub_ell_le_degree_sub_degree
Source context 3 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree
open _root_.AlgebraicCurve
  _root_.AlgebraicCurves.Cohomology.RiemannRoch.Proofs.EllSubEllLeDegreeSubDegree.AlgebraicCurve in
Lean source · statement
theorem _root_.AlgebraicCurve.ell_sub_ell_le_degree_sub_degree {K F : Type*} [Field K] [Field F]
  [Algebra K F] [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

Statements and documentation are read from the Lean source at the revision shown. Source entries are not a measure of mathematical completeness.