NYLL
Module

Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.lean

Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
71 declarations indexed from sourceView source ↗

Module documentation

Divisors representing inverse ideal modules

The inverse module of an invertible ideal admits a function-field presentation. The distinguished inverse-module section is nonzero, and its orders compare the representing divisor with local generators of the ideal.

Imports19
Imported by2

Declarations

abbrev

rM

L119 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM
Documentation

Restriction of module sections between the specified open subsets.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
abbrev rM {V W : X.Opens} (h : V ≤ W) : Γ(M, W) → Γ(M, V)
abbrev

rO

L123 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO
Documentation

Restriction of functions between the specified open subsets.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
abbrev rO {V W : X.Opens} (h : V ≤ W) : Γ(X, W) → Γ(X, V)
theorem

rM_rM

L126 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_rM
Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
theorem rM_rM {V₁ V₂ V₃ : X.Opens} (h₂₁ : V₂ ≤ V₁) (h₃₂ : V₃ ≤ V₂) (y : Γ(M, V₁)) :
    rM M h₃₂ (rM M h₂₁ y) = rM M (h₃₂.trans h₂₁) y
theorem

rO_rO

L131 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_rO
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
omit M in
Lean source · statement
theorem rO_rO {V₁ V₂ V₃ : X.Opens} (h₂₁ : V₂ ≤ V₁) (h₃₂ : V₃ ≤ V₂) (g : Γ(X, V₁)) :
    rO h₃₂ (rO h₂₁ g) = rO (h₃₂.trans h₂₁) g
theorem

rM_self

L136 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_self
Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
theorem rM_self {V : X.Opens} (h : V ≤ V) (y : Γ(M, V)) : rM M h y = y
theorem

rO_self

L140 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_self
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
omit M in
Lean source · statement
theorem rO_self {V : X.Opens} (h : V ≤ V) (g : Γ(X, V)) : rO h g = g
theorem

rM_smul

L144 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_smul
Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
theorem rM_smul {V W : X.Opens} (h : V ≤ W) (g : Γ(X, W)) (y : Γ(M, W)) :
    rM M h (g • y) = rO h g • rM M h y
theorem

rM_add

L147 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_add
Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
theorem rM_add {V W : X.Opens} (h : V ≤ W) (y y' : Γ(M, W)) :
    rM M h (y + y') = rM M h y + rM M h y'
theorem

rM_zero

L150 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_zero
Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
theorem rM_zero {V W : X.Opens} (h : V ≤ W) : rM M h (0 : Γ(M, W)) = 0
theorem

rM_sub

L152 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_sub
Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
Lean source · statement
theorem rM_sub {V W : X.Opens} (h : V ≤ W) (y y' : Γ(M, W)) :
    rM M h (y - y') = rM M h y - rM M h y'
theorem

rO_one

L155 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_one
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
omit M in
Lean source · statement
theorem rO_one {V W : X.Opens} (h : V ≤ W) : rO h (1 : Γ(X, W)) = 1
theorem

rO_mul

L157 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_mul
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
omit M in
Lean source · statement
theorem rO_mul {V W : X.Opens} (h : V ≤ W) (a b : Γ(X, W)) : rO h (a * b) = rO h a * rO h b
theorem

rO_zero

L160 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_zero
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
omit M in
Lean source · statement
theorem rO_zero {V W : X.Opens} (h : V ≤ W) : rO h (0 : Γ(X, W)) = 0
theorem

hat_eq

L166 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.hat_eq
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem hat_eq {W : X.Opens} (hWV : W ≤ V) : V.ι ''ᵁ V.ι ⁻¹ᵁ W = W
theorem

le_hat

L169 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.le_hat
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem le_hat {W : X.Opens} (hWV : W ≤ V) : W ≤ V.ι ''ᵁ V.ι ⁻¹ᵁ W
theorem

hat_le

L170 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.hat_le
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem hat_le {W : X.Opens} (hWV : W ≤ V) : V.ι ''ᵁ V.ι ⁻¹ᵁ W ≤ W
def

eApp

L173 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp
Documentation

The map on section modules induced by the chosen local trivialisation.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
def eApp (W : X.Opens) (m : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) : Γ(X, V.ι ''ᵁ V.ι ⁻¹ᵁ W)
def

eInv

L177 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eInv
Documentation

The inverse map on sections induced by the chosen local trivialisation.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
def eInv (W : X.Opens) (a : Γ(X, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W)
theorem

eInv_eApp

L180 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eInv_eApp
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eInv_eApp (W : X.Opens) (m : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) : eInv M V e W (eApp M V e W m) = m
theorem

eApp_eInv

L186 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp_eInv
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eApp_eInv (W : X.Opens) (a : Γ(X, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) : eApp M V e W (eInv M V e W a) = a
theorem

eApp_zero

L192 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp_zero
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eApp_zero (W : X.Opens) : eApp M V e W 0 = 0
theorem

eApp_add

L194 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp_add
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eApp_add (W : X.Opens) (m m' : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) :
    eApp M V e W (m + m') = eApp M V e W m + eApp M V e W m'
theorem

eApp_injective

L198 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp_injective
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eApp_injective (W : X.Opens) : Function.Injective (eApp M V e W)
theorem

restrict_smul_eq

L201 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.restrict_smul_eq
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem restrict_smul_eq (W' : V.toScheme.Opens) (b : Γ(V.toScheme, W')) (m : Γ(M.restrict V.ι, W'))
    :
    (HSMul.hSMul (α := Γ(V.toScheme, W')) (β := Γ(M.restrict V.ι, W')) b m : Γ(M, V.ι ''ᵁ W'))
      = HSMul.hSMul (α := Γ(X, V.ι ''ᵁ W')) (β := Γ(M, V.ι ''ᵁ W')) ((Scheme.Hom.appIso V.ι W').inv
          b) m
theorem

eApp_smul

L207 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp_smul
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eApp_smul (W : X.Opens) (a : Γ(X, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) (m : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) :
    eApp M V e W (a • m) = a * eApp M V e W m
theorem

eApp_res

L218 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eApp_res
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eApp_res {W₁ W₂ : X.Opens} (h : W₂ ≤ W₁) (m : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W₁)) :
    eApp M V e W₂ (rM M (Scheme.Hom.image_mono V.ι (Scheme.Hom.preimage_mono V.ι h)) m)
      = rO (Scheme.Hom.image_mono V.ι (Scheme.Hom.preimage_mono V.ι h)) (eApp M V e W₁ m)
def

ψ

L231 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ
Documentation

The section coordinate map on an open subset contained in the trivialising open.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
def ψ {W : X.Opens} (hWV : W ≤ V) (m : Γ(M, W)) : Γ(X, W)
def

ψinv

L235 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψinv
Documentation

The inverse section coordinate map on an open subset contained in the trivialising open.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
def ψinv {W : X.Opens} (hWV : W ≤ V) (a : Γ(X, W)) : Γ(M, W)
theorem

rM_roundtrip

L238 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_roundtrip
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem rM_roundtrip {W : X.Opens} (hWV : W ≤ V) (m : Γ(M, W)) : rM M (le_hat V hWV) (rM M (hat_le V
    hWV) m) = m
theorem

rM_roundtrip'

L241 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_roundtrip'
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem rM_roundtrip' {W : X.Opens} (hWV : W ≤ V) (m : Γ(M, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) :
    rM M (hat_le V hWV) (rM M (le_hat V hWV) m) = m
theorem

rO_roundtrip

L245 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_roundtrip
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
omit M e in
Lean source · statement
theorem rO_roundtrip {W : X.Opens} (hWV : W ≤ V) (a : Γ(X, W)) : rO (le_hat V hWV) (rO (hat_le V
    hWV) a) = a
theorem

rO_roundtrip'

L249 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_roundtrip'
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
omit M e in
Lean source · statement
theorem rO_roundtrip' {W : X.Opens} (hWV : W ≤ V) (a : Γ(X, V.ι ''ᵁ V.ι ⁻¹ᵁ W)) :
    rO (hat_le V hWV) (rO (le_hat V hWV) a) = a
theorem

ψ_ψinv

L253 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_ψinv
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_ψinv {W : X.Opens} (hWV : W ≤ V) (a : Γ(X, W)) : ψ M V e hWV (ψinv M V e hWV a) = a
theorem

ψinv_ψ

L257 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψinv_ψ
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψinv_ψ {W : X.Opens} (hWV : W ≤ V) (m : Γ(M, W)) : ψinv M V e hWV (ψ M V e hWV m) = m
theorem

ψ_injective

L261 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_injective
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_injective {W : X.Opens} (hWV : W ≤ V) : Function.Injective (ψ M V e hWV)
theorem

ψ_zero

L264 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_zero
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_zero {W : X.Opens} (hWV : W ≤ V) : ψ M V e hWV 0 = 0
theorem

ψ_add

L269 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_add
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_add {W : X.Opens} (hWV : W ≤ V) (m m' : Γ(M, W)) :
    ψ M V e hWV (m + m') = ψ M V e hWV m + ψ M V e hWV m'
theorem

ψ_smul

L275 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_smul
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_smul {W : X.Opens} (hWV : W ≤ V) (a : Γ(X, W)) (m : Γ(M, W)) :
    ψ M V e hWV (a • m) = a * ψ M V e hWV m
theorem

ψ_res

L284 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_res
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_res {W₁ W₂ : X.Opens} (h₁ : W₁ ≤ V) (h : W₂ ≤ W₁) (m : Γ(M, W₁)) :
    ψ M V e (h.trans h₁) (rM M h m) = rO h (ψ M V e h₁ m)
def

frame

L294 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.frame
Documentation

The local frame corresponding to the unit coordinate in the chosen trivialisation.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
def frame : Γ(M, V)
theorem

ψ_frame

L296 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_frame
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_frame : ψ M V e le_rfl (frame M V e) = 1
theorem

ψ_rM_frame

L298 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_rM_frame
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem ψ_rM_frame {W : X.Opens} (hWV : W ≤ V) : ψ M V e hWV (rM M hWV (frame M V e)) = 1
theorem

eq_smul_frame

L303 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.eq_smul_frame
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Transport
variable (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf)
Lean source · statement
theorem eq_smul_frame {W : X.Opens} (hWV : W ≤ V) (m : Γ(M, W)) :
    m = ψ M V e hWV m • rM M hWV (frame M V e)
theorem

nonempty_inf

L315 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.nonempty_inf
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem nonempty_inf (U W : X.Opens) [hU : Nonempty U] [hW : Nonempty W] : Nonempty (U ⊓ W :
    X.Opens)
abbrev

alg

L323 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg
Documentation

The image of a regular function in the function field.

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
abbrev alg (U : X.Opens) [Nonempty U] (a : Γ(X, U)) : X.functionField
theorem

alg_rO

L326 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg_rO
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem alg_rO {U W : X.Opens} [Nonempty U] [Nonempty W] (h : W ≤ U) (a : Γ(X, U)) :
    alg W (rO h a) = alg U a
theorem

alg_injective

L332 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg_injective
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem alg_injective (U : X.Opens) [Nonempty U] : Function.Injective (alg (X := X) U)
theorem

rO_injective

L335 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rO_injective
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem rO_injective {U W : X.Opens} [Nonempty U] [Nonempty W] (h : W ≤ U) :
    Function.Injective (rO (X := X) h)
theorem

alg_one

L341 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg_one
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem alg_one (U : X.Opens) [Nonempty U] : alg U (1 : Γ(X, U)) = 1
theorem

alg_zero

L343 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg_zero
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem alg_zero (U : X.Opens) [Nonempty U] : alg U (0 : Γ(X, U)) = 0
theorem

alg_add

L345 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg_add
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem alg_add (U : X.Opens) [Nonempty U] (a b : Γ(X, U)) : alg U (a + b) = alg U a + alg U b
theorem

alg_mul

L348 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.alg_mul
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Integral
variable [IsIntegral X]
omit M in
Lean source · statement
theorem alg_mul (U : X.Opens) [Nonempty U] (a b : Γ(X, U)) : alg U (a * b) = alg U a * alg U b
theorem

adicValuation_alg_le_one

L361 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.adicValuation_alg_le_one
Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Places
open AlgebraicCurve WithZero
variable {K : Type u} [Field K] (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X]
    [SmoothOfRelativeDimension 1 x]
omit M in
Lean source · full declaration
theorem adicValuation_alg_le_one (W : X.Opens) (y : X) (hyW : y ∈ W) (hy : IsClosed ({y} : Set X))
    (v : letI := (baseToFunctionField x).toAlgebra; Place K X.functionField)
    (hv : letI := (baseToFunctionField x).toAlgebra
      (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring)
    (a : Γ(X, W)) :
    letI := (baseToFunctionField x).toAlgebra
    haveI : Nonempty W := ⟨⟨y, hyW⟩⟩
    v.adicValuation (alg W a) ≤ 1 := by
  let := (baseToFunctionField x).toAlgebra
  have : Nonempty W := ⟨⟨y, hyW⟩⟩
  obtain ⟨W', hW'aff, hyW', hW'W⟩ :=
    (Opens.isBasis_iff_nbhd.mp X.isBasis_affineOpens) hyW
  have : Nonempty W' := ⟨⟨y, hyW'⟩⟩
  rw [← alg_rO hW'W a]
  apply (Place.mem_iff_adicValuation_le_one v).mp
  have hmem : alg W' (rO hW'W a) ∈ (algebraMap Γ(X, W') X.functionField).range := ⟨rO hW'W a, rfl⟩
  rw [range_algebraMap_functionField_eq_iInf_of_isAffineOpen x W' hW'aff] at hmem
  have h2 := (Subring.mem_iInf.mp hmem) v
  have h3 := (Subring.mem_iInf.mp h2) ⟨y, hyW', hy, hv⟩
  exact h3
def

IsGeneratorAt

L383 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.IsGeneratorAt
Documentation

The condition that a section locally generates the module near the specified point.

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Places
open AlgebraicCurve WithZero
variable {K : Type u} [Field K] (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X]
    [SmoothOfRelativeDimension 1 x]
Lean source · statement
def IsGeneratorAt (U : X.Opens) (s : Γ(M, U)) (y : X) : Prop
theorem

isGeneratorAt_frame

L386 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.isGeneratorAt_frame
Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Places
open AlgebraicCurve WithZero
variable {K : Type u} [Field K] (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X]
    [SmoothOfRelativeDimension 1 x]
omit x [IsIntegral X] in
Lean source · statement
theorem isGeneratorAt_frame (V : X.Opens) (e : M.restrict V.ι ≅ SheafOfModules.unit
    V.toScheme.ringCatSheaf)
    (y : X) : IsGeneratorAt M V (frame M V e) y
theorem

isGeneratorAt_rM

L391 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.isGeneratorAt_rM
Source context 25 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
variable {X : Scheme.{u}} (M : X.Modules)
section Places
open AlgebraicCurve WithZero
variable {K : Type u} [Field K] (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X]
    [SmoothOfRelativeDimension 1 x]
omit x [IsIntegral X] in
Lean source · statement
theorem isGeneratorAt_rM {U W : X.Opens} (h : W ≤ U) (s : Γ(M, U)) (y : X) (hs : IsGeneratorAt M U s
    y) :
    IsGeneratorAt M W (rM M h s) y
def

fn

L426 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn
Documentation

The regular function obtained from a section of the ideal module via its inclusion in the unit module.

Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
def fn (U : X.Opens) (m : Γ(I.module, U)) : Γ(X, U)
theorem

fn_add

L429 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_add
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_add (U : X.Opens) (m m' : Γ(I.module, U)) : fn I U (m + m') = fn I U m + fn I U m'
theorem

fn_zero

L432 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_zero
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_zero (U : X.Opens) : fn I U (0 : Γ(I.module, U)) = 0
theorem

fn_sub

L435 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_sub
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_sub (U : X.Opens) (m m' : Γ(I.module, U)) : fn I U (m - m') = fn I U m - fn I U m'
theorem

fn_smul

L438 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_smul
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_smul (U : X.Opens) (a : Γ(X, U)) (m : Γ(I.module, U)) : fn I U (a • m) = a * fn I U m
theorem

fn_rM

L442 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_rM
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_rM {U W : X.Opens} (h : W ≤ U) (m : Γ(I.module, U)) :
    fn I W (rM I.module h m) = rO h (fn I U m)
theorem

fn_injective_affine

L452 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_injective_affine
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_injective_affine (U : X.Opens) (hU : IsAffineOpen U) : Function.Injective (fn I U)
theorem

range_fn_affine

L456 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.range_fn_affine
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem range_fn_affine (U : X.Opens) (hU : IsAffineOpen U) :
    Set.range (fn I U) = (I.ideal ⟨U, hU⟩ : Set Γ(X, U))
theorem

fn_injective

L476 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.fn_injective
Source context 26 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
Lean source · statement
theorem fn_injective (U : X.Opens) : Function.Injective (fn I U)
theorem

frame_ne_zero

L508 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.frame_ne_zero
Source context 29 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
section Curve
variable {K : Type u} [Field K] (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X]
    [SmoothOfRelativeDimension 1 x]
omit x [IsIntegral X] [SmoothOfRelativeDimension 1 x] in
Lean source · statement
theorem frame_ne_zero (V : X.Opens) [Nonempty V]
    (e : I.module.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf) : frame I.module V e ≠
        0
theorem

adicValuation_alg_eq_of_span_eq

L516 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.PresMod.adicValuation_alg_eq_of_span_eq
Source context 28 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace IdealSheafData
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
namespace PresMod
open _root_.AlgebraicGeometry.Scheme.IdealSheafData
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open RatSec
open _root_.AlgebraicCurve
open _root_.WithZero
variable {X : Scheme.{u}} (I : X.IdealSheafData)
section Curve
variable {K : Type u} [Field K] (x : X ⟶ Spec (CommRingCat.of K)) [IsIntegral X]
    [SmoothOfRelativeDimension 1 x]
Lean source · full declaration
theorem adicValuation_alg_eq_of_span_eq (W : X.Opens) (y : X) (hyW : y ∈ W) (hy : IsClosed ({y} :
    Set X))
    (v : letI := (baseToFunctionField x).toAlgebra; Place K X.functionField)
    (hv : letI := (baseToFunctionField x).toAlgebra
      (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring)
    (g g' : Γ(X, W)) (hgg' : Ideal.span {g} = Ideal.span {g'}) :
    letI := (baseToFunctionField x).toAlgebra
    haveI : Nonempty W := ⟨⟨y, hyW⟩⟩
    v.adicValuation (alg W g) = v.adicValuation (alg W g') := by
  let := (baseToFunctionField x).toAlgebra
  have : Nonempty W := ⟨⟨y, hyW⟩⟩
  have h1 : g ∈ Ideal.span {g'} := hgg' ▸ Ideal.mem_span_singleton_self g
  have h2 : g' ∈ Ideal.span {g} := hgg'.symm ▸ Ideal.mem_span_singleton_self g'
  obtain ⟨a, ha⟩ := Ideal.mem_span_singleton'.mp h1
  obtain ⟨b, hb⟩ := Ideal.mem_span_singleton'.mp h2
  apply le_antisymm
  · rw [← ha, alg_mul, Valuation.map_mul]
    calc v.adicValuation (alg W a) * v.adicValuation (alg W g')
        ≤ 1 * v.adicValuation (alg W g') :=
          mul_le_mul_left (adicValuation_alg_le_one x W y hyW hy v hv a) _
      _ = v.adicValuation (alg W g') := one_mul _
  · rw [← hb, alg_mul, Valuation.map_mul]
    calc v.adicValuation (alg W b) * v.adicValuation (alg W g)
        ≤ 1 * v.adicValuation (alg W g) :=
          mul_le_mul_left (adicValuation_alg_le_one x W y hyW hy v hv b) _
      _ = v.adicValuation (alg W g) := one_mul _
theorem

rM_section_one

L565 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.rM_section_one
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
section Coeff
variable {X : Scheme.{u}} (M : X.Modules) (V : X.Opens)
  (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf) (σ : 𝟙_ X.Modules ⟶ M)
Lean source · statement
theorem rM_section_one {U W : X.Opens} (h : W ≤ U) :
    rM M h (σ.app U (toUnitSection U 1)) = σ.app W (toUnitSection W 1)
theorem

ψ_section_eq_coeff

L578 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.Modules.RatSec.ψ_section_eq_coeff
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace Scheme
open _root_.AlgebraicGeometry.Scheme
namespace Modules
open _root_.AlgebraicGeometry.Scheme.Modules
namespace RatSec
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq
open _root_.AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry
open _root_.CategoryTheory.MonoidalCategory
section Coeff
variable {X : Scheme.{u}} (M : X.Modules) (V : X.Opens)
  (e : M.restrict V.ι ≅ SheafOfModules.unit V.toScheme.ringCatSheaf) (σ : 𝟙_ X.Modules ⟶ M)
Lean source · statement
theorem ψ_section_eq_coeff :
    ψ M V e le_rfl (σ.app V (toUnitSection V 1)) = coeff σ V e.hom
theorem

exists_divisor_range_invModule_eq_lSpaceOn'

L641 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_divisor_range_invModule_eq_lSpaceOn'
Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
open scoped _root_.AlgebraicGeometry _root_.AlgebraicGeometry.Scheme
  _root_.AlgebraicGeometry.Scheme.IdealSheafData
  _root_.AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible in
with_weak_namespace _root_.AlgebraicCurves.Geometry.Sheaves.Proofs
with_weak_namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
with_weak_namespace AlgebraicGeometry.Scheme
open _root_.AlgebraicGeometry.Scheme.Modules hiding restrict_smul_eq in
open Modules Modules.RatSec IdealSheafData.PresMod AlgebraicCurve
    _root_.CategoryTheory.MonoidalCategory in
with_weak_namespace _root_.AlgebraicCurves.Geometry.Sheaves.Proofs
with_weak_namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
Lean source · full declaration
theorem
    AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_divisor_range_invModule_eq_lSpaceOn'
    {K : Type u} [Field K] {X : Scheme.{u}} (x : X ⟶ Spec (CommRingCat.of K))
    [IsIntegral X] [IsSeparated x] [QuasiCompact x] [SmoothOfRelativeDimension 1 x]
    {I : X.IdealSheafData} (hI : I.IsInvertible) :
    letI := (AlgebraicCurve.baseToFunctionField x).toAlgebra
    ∃ (D : AlgebraicCurve.Divisor K X.functionField)
      (φ : ∀ U : X.Opens, Γ(I.invModule, U) →+ (X.functionField : Type u)),
      ((∀ (U V : X.Opens) (h : V ≤ U), Nonempty V →
          ∀ m : Γ(I.invModule, U), φ V (I.invModule.presheaf.map (homOfLE h).op m) = φ U m) ∧
      (∀ (U : X.Opens) [Nonempty U] (a : Γ(X, U)) (m : Γ(I.invModule, U)),
          φ U (a • m) = algebraMap Γ(X, U) X.functionField a * φ U m) ∧
      (∀ U : X.Opens, Nonempty U → Function.Injective (φ U)) ∧
      (∀ U : X.Opens, IsAffineOpen U → Nonempty U →
          Set.range (φ U) = (AlgebraicCurve.lSpaceOn (AlgebraicCurve.placesOf x U) D : Set
              X.functionField)) ∧
      (∀ (U : X.Opens) (s : Γ(I.invModule, U)) (y : X), y ∈ U → IsClosed ({y} : Set X) →
          (∀ (W : X.Opens) (h : W ≤ U), y ∈ W → ∀ m : Γ(I.invModule, W), ∃ a : Γ(X, W), m = a •
              I.invModule.presheaf.map (homOfLE h).op s) →
          ∀ v : AlgebraicCurve.Place K X.functionField,
            (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring
                →
            WithZero.exp (D v) = v.adicValuation (φ U s))) ∧
      φ ⊤ (I.invModuleSection.app ⊤ (Scheme.Modules.toUnitSection ⊤ 1)) ≠ 0 ∧
      (∀ (U : X.Opens) (hU : IsAffineOpen U) (g : Γ(X, U)) (y : X) (hyU : y ∈ U), IsClosed ({y} :
          Set X) →
          I.ideal ⟨U, hU⟩ = Ideal.span {g} →
          ∀ v : AlgebraicCurve.Place K X.functionField,
            (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring
                →
            haveI : Nonempty U := ⟨⟨y, hyU⟩⟩
            D v + v.ord (φ ⊤ (I.invModuleSection.app ⊤ (Scheme.Modules.toUnitSection ⊤ 1))) =
              v.ord (algebraMap Γ(X, U) X.functionField g)) := by
  classical
  let := (AlgebraicCurve.baseToFunctionField x).toAlgebra
  have hM : Scheme.Modules.IsInvertible I.invModule := hI.isInvertible_invModule
  obtain ⟨D, φ, h1, h2, h3, h4, h5⟩ :=
    Scheme.Modules.IsInvertible.exists_divisor_range_eq_lSpaceOn x I.invModule hM
  set σ : 𝟙_ X.Modules ⟶ I.invModule := I.invModuleSection with hσ
  have : Nonempty (⊤ : X.Opens) := ⟨⟨genericPoint X, trivial⟩⟩
  have hZ : Scheme.Modules.zeroSchemeIdeal σ = I := hI.zeroSchemeIdeal_invModuleSection
  have hideal : ∀ (W : X.affineOpens) (τ : I.invModule.restrict W.1.ι ≅ 𝟙_ (W.1 :
      Scheme.{u}).Modules),
      I.ideal W = Ideal.span {ψ I.invModule W.1 τ le_rfl (σ.app W.1 (toUnitSection W.1 1))} := by
    intro W τ
    rw [ψ_section_eq_coeff I.invModule W.1 τ σ]
    conv_lhs => rw [← hZ]
    exact ((Scheme.Modules.IsInvertible.coeffIdeal_le_and_ideal_zeroSchemeIdeal_eq hM σ).2 W τ).1
  have key : ∀ (U : X.Opens) (hU : IsAffineOpen U) (g : Γ(X, U)) (y : X) (hyU : y ∈ U), IsClosed
      ({y} : Set X) →
      I.ideal ⟨U, hU⟩ = Ideal.span {g} →
      ∀ v : AlgebraicCurve.Place K X.functionField,
        (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring →
        haveI : Nonempty U := ⟨⟨y, hyU⟩⟩
        WithZero.exp (D v) * v.adicValuation (alg U g) =
          v.adicValuation (φ ⊤ (σ.app ⊤ (toUnitSection ⊤ 1))) := by
    intro U hU g y hyU hy hg v hv
    have : Nonempty U := ⟨⟨y, hyU⟩⟩
    obtain ⟨W, hyW, hWU, ⟨τ⟩⟩ := Scheme.Modules.IsInvertible.exists_restrict_iso hM (V := U) hyU
    have : Nonempty W.1 := ⟨⟨y, hyW⟩⟩
    let s : Γ(I.invModule, W.1) := frame I.invModule W.1 τ
    have hs : IsGeneratorAt I.invModule W.1 s y := isGeneratorAt_frame I.invModule W.1 τ y
    let a : Γ(X, W.1) := ψ I.invModule W.1 τ le_rfl (σ.app W.1 (toUnitSection W.1 1))
    have hσW : σ.app W.1 (toUnitSection W.1 1) = a • s := by
      have e1 := eq_smul_frame I.invModule W.1 τ le_rfl (σ.app W.1 (toUnitSection W.1 1))
      rw [rM_self] at e1
      exact e1
    have hIW1 : I.ideal W = Ideal.span {a} := hideal W τ
    have hIW2 : I.ideal W = Ideal.span {rO hWU g} := by
      rw [← I.map_ideal (U := W) (V := ⟨U, hU⟩) hWU, hg, Ideal.map_span, Set.image_singleton]
      rfl
    have hval : v.adicValuation (alg W.1 a) = v.adicValuation (alg U g) := by
      rw [adicValuation_alg_eq_of_span_eq x W.1 y hyW hy v hv a (rO hWU g) (hIW1.symm.trans hIW2),
          alg_rO]
    have k5 := h5 W.1 s y hyW hy (fun W' h hyW' m => hs W' h hyW' m) v hv
    have k1 : φ ⊤ (σ.app ⊤ (toUnitSection ⊤ 1)) = φ W.1 (σ.app W.1 (toUnitSection W.1 1)) := by
      rw [← rM_section_one I.invModule σ (le_top : W.1 ≤ ⊤)]
      exact (h1 ⊤ W.1 le_top inferInstance _).symm
    rw [k1, hσW, h2, Valuation.map_mul, ← k5, ← hval]
    change _ = v.adicValuation (alg W.1 a) * _
    rw [mul_comm]
  have hσ1 : σ.app ⊤ (toUnitSection ⊤ 1) ≠ 0 := by
    intro h0
    obtain ⟨U, f, hηf, g, hg, hIg⟩ := hI (genericPoint X)
    obtain ⟨W, hηW, hWU, ⟨τ⟩⟩ :=
      Scheme.Modules.IsInvertible.exists_restrict_iso hM (V := X.affineBasicOpen f) hηf
    have : Nonempty W.1 := ⟨⟨genericPoint X, hηW⟩⟩
    have : Nonempty (X.affineBasicOpen f : X.Opens) := ⟨⟨genericPoint X, hηf⟩⟩
    have hW0 : σ.app W.1 (toUnitSection W.1 1) = 0 := by
      rw [← rM_section_one I.invModule σ (le_top : W.1 ≤ ⊤), h0, rM_zero]
    have hIW1 := hideal W τ
    rw [hW0] at hIW1
    erw [ψ_zero] at hIW1
    have hIW2 : I.ideal W = Ideal.span {rO hWU g} := by
      rw [← I.map_ideal (U := W) (V := X.affineBasicOpen f) hWU, hIg, Ideal.map_span,
          Set.image_singleton]
      rfl
    have hg0 : rO hWU g = 0 := by
      have hmem : rO hWU g ∈ I.ideal W := hIW2 ▸ Ideal.mem_span_singleton_self _
      rw [hIW1] at hmem
      simpa using hmem
    have hg0' : g = 0 := rO_injective hWU (by rw [hg0, rO_zero])
    exact nonZeroDivisors.ne_zero hg hg0'
  have hf : φ ⊤ (σ.app ⊤ (toUnitSection ⊤ 1)) ≠ 0 := by
    intro h0
    exact hσ1 (h3 ⊤ inferInstance (by rw [h0, map_zero]))
  refine ⟨D, φ, ⟨h1, h2, h3, h4, h5⟩, hf, ?_⟩
  intro U hU g y hyU hy hg v hv
  have : Nonempty U := ⟨⟨y, hyU⟩⟩
  have k := key U hU g y hyU hy hg v hv
  have hf' : v.adicValuation (φ ⊤ (σ.app ⊤ (toUnitSection ⊤ 1))) ≠ 0 := (Valuation.ne_zero_iff
      _).mpr hf
  have hg' : v.adicValuation (alg U g) ≠ 0 := by
    intro h0
    rw [h0, mul_zero] at k
    exact hf' k.symm
  have hlog := congrArg WithZero.log k
  rw [WithZero.log_mul WithZero.exp_ne_zero hg', WithZero.log_exp] at hlog
  simp only [Place.ord]
  change D v + -WithZero.log (v.adicValuation (φ ⊤ (σ.app ⊤ (toUnitSection ⊤ 1)))) = -WithZero.log
      (v.adicValuation (alg U g))
  omega
theorem

exists_divisor_range_invModule_eq_lSpaceOn

L773 ↗
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_divisor_range_invModule_eq_lSpaceOn
Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
attribute [-instance]
  AlgebraicCurve.Place.instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.Place.instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring
  AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply
  SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero
  AlgebraicGeometry.Scheme.Modules.tensorPow_succ
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.Opposite
open _root_.TopologicalSpace
with_weak_namespace _root_.AlgebraicCurves.Geometry.Sheaves.Proofs
open _root_.AlgebraicGeometry
    Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn.AlgebraicGeometry
    _root_.CategoryTheory.MonoidalCategory in
with_weak_namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
with_weak_namespace _root_.AlgebraicCurves.Geometry.Sheaves.Proofs
with_weak_namespace Scheme_IdealSheafData_IsInvertible_exists_divisor_range_invModule_eq_lSpaceOn
with_weak_namespace AlgebraicGeometry.Scheme.IdealSheafData
open IsInvertible in
with_weak_namespace _root_.AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible
Lean source · full declaration
theorem exists_divisor_range_invModule_eq_lSpaceOn
    {K : Type u} [Field K] {X : Scheme.{u}} (x : X ⟶ Spec (CommRingCat.of K))
    [IsIntegral X] [IsSeparated x] [QuasiCompact x] [SmoothOfRelativeDimension 1 x]
    {I : X.IdealSheafData} (hI : I.IsInvertible) :
    letI := (AlgebraicCurve.baseToFunctionField x).toAlgebra
    ∃ (D : AlgebraicCurve.Divisor K X.functionField)
      (φ : ∀ U : X.Opens, Γ(I.invModule, U) →+ (X.functionField : Type u)),
      ((∀ (U V : X.Opens) (h : V ≤ U), Nonempty V →
          ∀ m : Γ(I.invModule, U), φ V (I.invModule.presheaf.map (homOfLE h).op m) = φ U m) ∧
      (∀ (U : X.Opens) [Nonempty U] (a : Γ(X, U)) (m : Γ(I.invModule, U)),
          φ U (a • m) = algebraMap Γ(X, U) X.functionField a * φ U m) ∧
      (∀ U : X.Opens, Nonempty U → Function.Injective (φ U)) ∧
      (∀ U : X.Opens, IsAffineOpen U → Nonempty U →
          Set.range (φ U) = (AlgebraicCurve.lSpaceOn (AlgebraicCurve.placesOf x U) D : Set
              X.functionField)) ∧
      (∀ (U : X.Opens) (s : Γ(I.invModule, U)) (y : X), y ∈ U → IsClosed ({y} : Set X) →
          (∀ (W : X.Opens) (h : W ≤ U), y ∈ W → ∀ m : Γ(I.invModule, W), ∃ a : Γ(X, W), m = a •
              I.invModule.presheaf.map (homOfLE h).op s) →
          ∀ v : AlgebraicCurve.Place K X.functionField,
            (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring
                →
            WithZero.exp (D v) = v.adicValuation (φ U s))) ∧
      φ ⊤ (I.invModuleSection.app ⊤ (Scheme.Modules.toUnitSection ⊤ 1)) ≠ 0 ∧
      (∀ (U : X.Opens) (hU : IsAffineOpen U) (g : Γ(X, U)) (y : X) (hyU : y ∈ U), IsClosed ({y} :
          Set X) →
          I.ideal ⟨U, hU⟩ = Ideal.span {g} →
          ∀ v : AlgebraicCurve.Place K X.functionField,
            (algebraMap (X.presheaf.stalk y) X.functionField).range = v.toValuationSubring.toSubring
                →
            haveI : Nonempty U := ⟨⟨y, hyU⟩⟩
            D v + v.ord (φ ⊤ (I.invModuleSection.app ⊤ (Scheme.Modules.toUnitSection ⊤ 1))) =
              v.ord (algebraMap Γ(X, U) X.functionField g)) :=
  exists_divisor_range_invModule_eq_lSpaceOn' x hI

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