Module

IsLFPInj_relPicardPresheaf.lean

Nyll.JacobianVarieties.Picard.IsLFPInj_relPicardPresheaf

73 declarations indexed from sourceView source ↗

Module documentation

Relative Picard varieties and fibre cohomology

Descend line-bundle isomorphisms through finitely generated base algebras to prove the finite- presentation injectivity of the relative Picard presheaf.

Imports13

Imported by3

theorem

exists_sectionsOf_linearEquiv_of_iso

L89 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.exists_sectionsOf_linearEquiv_of_iso

Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover

Lean source · statement

theorem exists_sectionsOf_linearEquiv_of_iso
    {R : Type u} [CommRing R] {X : Scheme.{u}} (𝒱 : X.TwoAffineOpenCover) (c : X ⟶ Spec (.of R))
    {L L' : X.Modules} (ψ : L ≅ L') :
    ∃ (e0 : (𝒱.sectionsOf c L).M0 ≃ₗ[(𝒱.cover c).A0] (𝒱.sectionsOf c L').M0)
      (e1 : (𝒱.sectionsOf c L).M1 ≃ₗ[(𝒱.cover c).A1] (𝒱.sectionsOf c L').M1)
      (e01 : (𝒱.sectionsOf c L).M01 ≃ₗ[(𝒱.cover c).A01] (𝒱.sectionsOf c L').M01),
      (∀ m, e01 ((𝒱.sectionsOf c L).r0 m) = (𝒱.sectionsOf c L').r0 (e0 m)) ∧
      (∀ m, e01 ((𝒱.sectionsOf c L).r1 m) = (𝒱.sectionsOf c L').r1 (e1 m)) ∧
      (∀ m, e0 m = (ψ.hom.app 𝒱.U0).hom m) ∧ (∀ m, e1 m = (ψ.hom.app 𝒱.U1).hom m) ∧
      (∀ m, e01 m = (ψ.hom.app (𝒱.U0 ⊓ 𝒱.U1)).hom m)

def

actR

L138 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.actR

Documentation

The base-changed linear action of a scalar from the chart ring on the tensor module.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

def actR (s : Γ) : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P

theorem

actR_tmul

L141 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.actR_tmul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_tmul (s : Γ) (a : A') (m : P) :
    actR k A' s (a ⊗ₜ[k] m) = a ⊗ₜ[k] (s • m)

theorem

actR_one

L145 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.actR_one

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_one : actR k A' (1 : Γ) = (LinearMap.id : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P)

theorem

actR_mul

L150 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.actR_mul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_mul (s t : Γ) : actR k A' (P := P) (s * t) = actR k A' s ∘ₗ actR k A' t

theorem

actR_add

L156 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.actR_add

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem actR_add (s t : Γ) : actR k A' (P := P) (s + t) = actR k A' s + actR k A' t

theorem

rTensor_actR

L162 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.rTensor_actR

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem rTensor_actR {A'' : Type*} [CommRing A''] [Algebra k A''] (h : A' →ₐ[k] A'') (s : Γ)
    (x : A' ⊗[k] P) :
    h.toLinearMap.rTensor P (actR k A' s x) = actR k A'' s (h.toLinearMap.rTensor P x)

theorem

rTensor_smul'

L170 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.rTensor_smul'

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem rTensor_smul' {A'' : Type*} [CommRing A''] [Algebra k A''] (h : A' →ₐ[k] A'')
    {Q : Type*} [AddCommGroup Q] [Module k Q] (a : A') (x : A' ⊗[k] Q) :
    h.toLinearMap.rTensor Q (a • x) = h a • h.toLinearMap.rTensor Q x

theorem

tmul_eq_smul_one_tmul

L179 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.tmul_eq_smul_one_tmul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section ActR
variable (k : Type*) [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem tmul_eq_smul_one_tmul {Q : Type*} [AddCommGroup Q] [Module k Q] (a : A') (m : Q) :
    a ⊗ₜ[k] m = a • ((1 : A') ⊗ₜ[k] m)

def

Intertwines

L196 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.Intertwines

Documentation

A tensor-module map intertwines the chart-ring actions through the given scalar map.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

def Intertwines (σ : Γ → Γ') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) : Prop

theorem

comp

L199 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.Intertwines.comp

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.comp {σ : Γ → Γ'} {σ' : Γ' → Γ''} {f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q}
    {g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] N} (hf : Intertwines σ f) (hg : Intertwines σ' g) :
    Intertwines (σ' ∘ σ) (g ∘ₗ f)

theorem

id

L205 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.Intertwines.id

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.id : Intertwines (_root_.id : Γ → Γ) (LinearMap.id :
  A' ⊗[k] P →ₗ[A'] A' ⊗[k] P)

theorem

symm

L209 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.Intertwines.symm

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.symm {e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q} {σ : Γ ≃ Γ'}
    (he : Intertwines σ (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)) :
    Intertwines σ.symm (e.symm : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] P)

theorem

baseChange

L219 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.Intertwines.baseChange

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Intertwines
variable {k : Type*} [CommRing k] {Γ Γ' Γ'' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  [CommRing Γ''] [Algebra k Γ'']
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module Γ'' N] [Module k N] [IsScalarTower k Γ'' N]

Lean source · statement

theorem Intertwines.baseChange (σ : Γ → Γ') (r : P →ₗ[k] Q) (hr : ∀ (s : Γ) (m : P), r (s • m) =
  σ s • r m) :
    Intertwines σ (r.baseChange A')

theorem

linearMap_ext_of_span

L235 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.linearMap_ext_of_span

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Ext
variable {k : Type*} [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {N : Type*} [AddCommGroup N] [Module A' N]

Lean source · statement

theorem linearMap_ext_of_span (F G : A' ⊗[k] P →ₗ[A'] N) (T : Γ → N → N)
    (hF : ∀ (s : Γ) (x : A' ⊗[k] P), F (actR k A' s x) = T s (F x))
    (hG : ∀ (s : Γ) (x : A' ⊗[k] P), G (actR k A' s x) = T s (G x))
    (gens : Set P) (hgens : Submodule.span Γ gens = ⊤)
    (h : ∀ m ∈ gens, F ((1 : A') ⊗ₜ[k] m) = G ((1 : A') ⊗ₜ[k] m)) : F = G

theorem

linearMap_square_of_span

L260 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.linearMap_square_of_span

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Ext
variable {k : Type*} [CommRing k] {Γ : Type*} [CommRing Γ] [Algebra k Γ]
  {A' : Type*} [CommRing A'] [Algebra k A']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {N : Type*} [AddCommGroup N] [Module A' N]

Lean source · statement

theorem linearMap_square_of_span
    {Q Q' : Type*} [AddCommGroup Q] [Module A' Q] [AddCommGroup Q'] [Module A' Q']
    (f : A' ⊗[k] P →ₗ[A'] Q) (g : Q' →ₗ[A'] N)
    (r : A' ⊗[k] P →ₗ[A'] Q') (r' : Q →ₗ[A'] N)
    (T : Γ → Q → Q) (T' : Γ → Q' → Q') (T'' : Γ → N → N)
    (hf : ∀ (a : Γ) (x : A' ⊗[k] P), f (actR k A' a x) = T a (f x))
    (hr : ∀ (a : Γ) (x : A' ⊗[k] P), r (actR k A' a x) = T' a (r x))
    (hr' : ∀ (a : Γ) (y : Q), r' (T a y) = T'' a (r' y))
    (hg : ∀ (a : Γ) (y : Q'), g (T' a y) = T'' a (g y))
    (gens : Set P) (hgens : Submodule.span Γ gens = ⊤)
    (hgen : ∀ m ∈ gens, g (r ((1 : A') ⊗ₜ[k] m)) = r' (f ((1 : A') ⊗ₜ[k] m))) :
    g ∘ₗ r = r' ∘ₗ f

def

push

L289 ↗

AlgebraicGeometry.RelPicard.LFP.push

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]

Lean source · statement

def _root_.AlgebraicGeometry.RelPicard.LFP.push (h : A' →ₐ[k] A'') (f :
  A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) :
    A'' ⊗[k] P →ₗ[A''] A'' ⊗[k] Q

theorem

push_tmul

L296 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_tmul

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_tmul (h : A' →ₐ[k] A'') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (a : A'')
  (m : P) :
    push h f (a ⊗ₜ[k] m) = a • h.toLinearMap.rTensor Q (f ((1 : A') ⊗ₜ[k] m))

theorem

push_rTensor

L302 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_rTensor

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_rTensor (h : A' →ₐ[k] A'') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (x : A' ⊗[k] P) :
    push h f (h.toLinearMap.rTensor P x) = h.toLinearMap.rTensor Q (f x)

theorem

push_id

L310 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_id

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_id (h : A' →ₐ[k] A'') :
    push h (LinearMap.id : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P) = LinearMap.id

theorem

push_comp

L318 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_comp

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_comp (h : A' →ₐ[k] A'') (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)
    (g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] N) :
    push h (g ∘ₗ f) = push h g ∘ₗ push h f

theorem

push

L326 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.Intertwines.push

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem Intertwines.push {σ : Γ → Γ'} (h : A' →ₐ[k] A'') {f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q}
    (hf : Intertwines σ f) : Intertwines σ (push h f)

theorem

rTensor_baseChange

L333 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.rTensor_baseChange

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem rTensor_baseChange (h : A' →ₐ[k] A'') {M M' : Type*} [AddCommGroup M] [Module k M]
  [AddCommGroup M']
    [Module k M'] (r : M →ₗ[k] M') (x : A' ⊗[k] M) :
    h.toLinearMap.rTensor M' (r.baseChange A' x) = r.baseChange A''
      (h.toLinearMap.rTensor M x)

theorem

push_square

L344 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_square

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem push_square (h : A' →ₐ[k] A'') {P' : Type*} [AddCommGroup P'] [Module k P'] {Q' : Type*}
  [AddCommGroup Q']
    [Module k Q'] (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P') (g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] Q')
    (r : P →ₗ[k] Q) (r' : P' →ₗ[k] Q') (hsq : ∀ x, g (r.baseChange A' x) = r'.baseChange A' (f x))
    (y : A'' ⊗[k] P) : push h g (r.baseChange A'' y) = r'.baseChange A'' (push h f y)

theorem

mem_span_range_of_algEquiv

L355 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.mem_span_range_of_algEquiv

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem mem_span_range_of_algEquiv {Γ'' : Type*} [CommRing Γ''] [Algebra A' Γ'']
    (e : A' ⊗[k] Γ ≃ₐ[A'] Γ'') (φ : Γ → Γ'') (he : ∀ s, e ((1 : A') ⊗ₜ[k] s) = φ s) (y : Γ'') :
    y ∈ Submodule.span A' (Set.range φ)

def

pushEquiv

L366 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.pushEquiv

Documentation

Base change a tensor-module linear equivalence along the given algebra homomorphism.

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

def pushEquiv (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) :
    A'' ⊗[k] P ≃ₗ[A''] A'' ⊗[k] Q

theorem

pushEquiv_apply

L373 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.pushEquiv_apply

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem pushEquiv_apply (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) (x :
  A'' ⊗[k] P) :
    pushEquiv h e x = push h (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) x

theorem

pushEquiv_symm_apply

L377 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.pushEquiv_symm_apply

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem pushEquiv_symm_apply (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q)
    (x : A'' ⊗[k] Q) :
    (pushEquiv h e).symm x = push h (e.symm : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] P) x

theorem

coe_pushEquiv

L381 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.coe_pushEquiv

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Push
variable {k : Type*} [CommRing k] {Γ Γ' : Type*} [CommRing Γ] [Algebra k Γ] [CommRing Γ']
  [Algebra k Γ']
  {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module k Q] [IsScalarTower k Γ' Q]
  {N : Type*} [AddCommGroup N] [Module k N]
open _root_.AlgebraicGeometry.RelPicard.LFP

Lean source · statement

theorem coe_pushEquiv (h : A' →ₐ[k] A'') (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) :
    (pushEquiv h e : A'' ⊗[k] P →ₗ[A''] A'' ⊗[k] Q) = push h (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)

def

orient

L394 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient

Documentation

Reorder and cancel the algebra tensor factors to identify the chart-base-change module with its tensor module over the new base.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

def orient : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[k] A' ⊗[k] P

theorem

orient_tmul

L399 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_tmul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_tmul (γ : Γ) (a : A') (m : P) :
    orient k Γ A' P ((γ ⊗ₜ[k] a) ⊗ₜ[Γ] m) = a ⊗ₜ[k] (γ • m)

theorem

orient_symm_tmul

L408 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_symm_tmul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_symm_tmul (a : A') (m : P) :
    (orient k Γ A' P).symm (a ⊗ₜ[k] m) = ((1 : Γ) ⊗ₜ[k] a) ⊗ₜ[Γ] m

theorem

orient_smul

L412 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_smul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_smul (γ : Γ) (a : A') (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A' P ((γ ⊗ₜ[k] a) • x) = a • actR k A' γ (orient k Γ A' P x)

theorem

orient_smul_left

L423 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_smul_left

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_smul_left (a : A') (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A' P (((1 : Γ) ⊗ₜ[k] a) • x) = a • orient k Γ A' P x

theorem

orient_smul_right

L427 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_smul_right

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_smul_right (γ : Γ) (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A' P ((γ ⊗ₜ[k] (1 : A')) • x) = actR k A' γ (orient k Γ A' P x)

theorem

orient_symm_smul

L431 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_symm_smul

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_symm_smul (a : A') (y : A' ⊗[k] P) :
    (orient k Γ A' P).symm (a • y) = ((1 : Γ) ⊗ₜ[k] a) • (orient k Γ A' P).symm y

theorem

orient_symm_actR

L435 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_symm_actR

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_symm_actR (γ : Γ) (y : A' ⊗[k] P) :
    (orient k Γ A' P).symm (actR k A' γ y) = (γ ⊗ₜ[k] (1 : A')) • (orient k Γ A' P).symm y

theorem

orient_rTensor_map

L439 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.orient_rTensor_map

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]

Lean source · statement

theorem orient_rTensor_map {A'' : Type*} [CommRing A''] [Algebra k A''] (h : A' →ₐ[k] A'')
    (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    orient k Γ A'' P ((Algebra.TensorProduct.map (AlgHom.id Γ Γ) h).toLinearMap.rTensor P x) =
      h.toLinearMap.rTensor P (orient k Γ A' P x)

def

fromOrient

L457 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fromOrient

Documentation

Express a linear map between chart-base-change modules in the reordered tensor coordinates.

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · full declaration

def fromOrient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q where
  toFun y := orient k Γ A' Q (f ((orient k Γ A' P).symm y))
  map_add' y y' := by simp only [map_add]
  map_smul' a y := by
    rw [orient_symm_smul, map_smul, orient_smul_left, RingHom.id_apply]

theorem

fromOrient_apply

L464 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fromOrient_apply

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrient_apply (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) (y : A' ⊗[k] P) :
    fromOrient f y = orient k Γ A' Q (f ((orient k Γ A' P).symm y))

theorem

fromOrient_orient

L467 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fromOrient_orient

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrient_orient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q)
    (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    fromOrient f (orient k Γ A' P x) = orient k Γ A' Q (f x)

theorem

intertwines_fromOrient

L472 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.intertwines_fromOrient

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem intertwines_fromOrient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    Intertwines (_root_.id : Γ → Γ) (fromOrient f)

def

toOrient

L478 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.toOrient

Documentation

Lift an intertwining map in tensor coordinates to a linear map over the base-changed chart ring.

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · full declaration

def toOrient (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (hf : Intertwines (_root_.id : Γ →
  Γ) f) :
    (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q where
  toFun x := (orient k Γ A' Q).symm (f (orient k Γ A' P x))
  map_add' x x' := by simp only [map_add]
  map_smul' b x := by
    rw [RingHom.id_apply]
    induction b using TensorProduct.inductionOn with
    | tmul γ a =>
      rw [orient_smul, map_smul, hf, LinearEquiv.symm_apply_eq, orient_smul,
        LinearEquiv.apply_symm_apply]
      rfl
    | add b b' hb hb' => simp only [add_smul, map_add, hb, hb']

theorem

toOrient_apply

L492 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.toOrient_apply

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem toOrient_apply (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (hf : Intertwines (_root_.id : Γ → Γ) f)
    (x : (Γ ⊗[k] A') ⊗[Γ] P) :
    toOrient f hf x = (orient k Γ A' Q).symm (f (orient k Γ A' P x))

theorem

fromOrient_toOrient

L496 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fromOrient_toOrient

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrient_toOrient (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (hf : Intertwines (_root_.id : Γ →
  Γ) f) :
    fromOrient (toOrient f hf) = f

theorem

toOrient_fromOrient

L502 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.toOrient_fromOrient

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem toOrient_fromOrient (f : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    toOrient (fromOrient f) (intertwines_fromOrient f) = f

def

fromOrientEquiv

L508 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fromOrientEquiv

Documentation

Transport a chart-base-change linear equivalence to the reordered tensor coordinates.

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

def fromOrientEquiv (e : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) :
    A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q

theorem

coe_fromOrientEquiv

L527 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.coe_fromOrientEquiv

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem coe_fromOrientEquiv (e : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A']
  (Γ ⊗[k] A') ⊗[Γ] Q) :
    (fromOrientEquiv e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) =
      fromOrient (e : (Γ ⊗[k] A') ⊗[Γ] P →ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q)

theorem

fromOrientEquiv_apply

L532 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fromOrientEquiv_apply

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem fromOrientEquiv_apply (e : (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q) (y :
  A' ⊗[k] P) :
    fromOrientEquiv e y = orient k Γ A' Q (e ((orient k Γ A' P).symm y))

def

toOrientEquiv

L538 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.toOrientEquiv

Documentation

Transport an intertwining tensor-coordinate equivalence to the base-changed chart modules.

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

def toOrientEquiv (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q)
    (he : Intertwines (_root_.id : Γ → Γ) (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)) :
    (Γ ⊗[k] A') ⊗[Γ] P ≃ₗ[Γ ⊗[k] A'] (Γ ⊗[k] A') ⊗[Γ] Q

theorem

toOrientEquiv_apply

L559 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.toOrientEquiv_apply

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Orient
variable (k : Type*) [CommRing k] (Γ : Type*) [CommRing Γ] [Algebra k Γ]
  (A' : Type*) [CommRing A'] [Algebra k A']
  (P : Type*) [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
variable {k Γ A' P}
variable {Q : Type*} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem toOrientEquiv_apply (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q)
    (he : Intertwines (_root_.id : Γ → Γ) (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q)) (x :
      (Γ ⊗[k] A') ⊗[Γ] P) :
    toOrientEquiv e he x = (orient k Γ A' Q).symm (e (orient k Γ A' P x))

theorem

exists_fg_le_forall_rTensor_inclusion_eq

L571 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.exists_fg_le_forall_rTensor_inclusion_eq

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section FG
variable {k : Type*} [CommRing k] {A : Type*} [CommRing A] [Algebra k A]
  {N : Type*} [AddCommGroup N] [Module k N]

Lean source · statement

theorem exists_fg_le_forall_rTensor_inclusion_eq (D : Subalgebra k A) (hD : D.FG)
    {X : Type*} (S : Finset X) (t t' : X → ↥D ⊗[k] N)
    (h : ∀ x ∈ S, D.val.toLinearMap.rTensor N (t x) = D.val.toLinearMap.rTensor N (t' x)) :
    ∃ (D' : Subalgebra k A) (hDD' : D ≤ D'), D'.FG ∧
      ∀ x ∈ S, (Subalgebra.inclusion hDD').toLinearMap.rTensor N (t x) =
        (Subalgebra.inclusion hDD').toLinearMap.rTensor N (t' x)

theorem

exists_fg_linearEquiv_of_intertwines

L610 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.exists_fg_linearEquiv_of_intertwines

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section System
variable (k : Type u) [CommRing k] (A : Type u) [CommRing A] [Algebra k A]
  (Γ : Type u) [CommRing Γ] [Algebra k Γ]
variable {k A Γ}
variable {P : Type w} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {Q : Type w} [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]

Lean source · statement

theorem exists_fg_linearEquiv_of_intertwines [Module.FinitePresentation Γ P]
  [Module.FinitePresentation Γ Q]
    (e : A ⊗[k] P ≃ₗ[A] A ⊗[k] Q) (he : Intertwines (_root_.id : Γ → Γ) (e :
      A ⊗[k] P →ₗ[A] A ⊗[k] Q)) :
    ∃ (D : Subalgebra k A) (_ : D.FG) (eD : ↥D ⊗[k] P ≃ₗ[↥D] ↥D ⊗[k] Q),
      Intertwines (_root_.id : Γ → Γ) (eD : ↥D ⊗[k] P →ₗ[↥D] ↥D ⊗[k] Q) ∧
      ∀ x, D.val.toLinearMap.rTensor Q (eD x) = e (D.val.toLinearMap.rTensor P x)

theorem

smul_comm_of_tower

L644 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.smul_comm_of_tower

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']
omit [Algebra k A'] in

Lean source · statement

theorem smul_comm_of_tower (a : A') (t : Γ') (q : Q) : a • t • q = t • a • q

theorem

apply_actR_of_semilinear

L647 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.apply_actR_of_semilinear

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem apply_actR_of_semilinear (φ : Γ → Γ') (β : A' ⊗[k] P ≃ₗ[A'] Q)
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (s : Γ) (x : A' ⊗[k] P) : β (actR k A' s x) = φ s • β x

theorem

symm_apply_smul_of_semilinear

L657 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.symm_apply_smul_of_semilinear

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem symm_apply_smul_of_semilinear (φ : Γ → Γ') (β : A' ⊗[k] P ≃ₗ[A'] Q)
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (s : Γ) (q : Q) : β.symm (φ s • q) = actR k A' s (β.symm q)

def

conjIn

L664 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.conjIn

Documentation

Conjugate the chart-module equivalence into tensor coordinates using the two given comparison equivalences.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

def conjIn (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q') (ψ : Q ≃ₗ[Γ'] Q') :
    A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P'

theorem

conjIn_apply

L668 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.conjIn_apply

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem conjIn_apply (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q') (ψ : Q ≃ₗ[Γ'] Q')
    (x : A' ⊗[k] P) : conjIn β β' ψ x = β'.symm (ψ (β x))

theorem

intertwines_conjIn

L671 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.intertwines_conjIn

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem intertwines_conjIn (φ : Γ → Γ') (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q')
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (hs' : ∀ (s : Γ) (m : P'), β' ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β' ((1 : A') ⊗ₜ[k] m))
    (ψ : Q ≃ₗ[Γ'] Q') :
    Intertwines (_root_.id : Γ → Γ) (conjIn β β' ψ : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P')

def

conjOut

L681 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.conjOut

Documentation

Conjugate an intertwining tensor-coordinate equivalence to a chart-ring linear equivalence.

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

def conjOut (φ : Γ → Γ') (hgen : ∀ y : Γ', y ∈ Submodule.span A' (Set.range φ))
    (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q')
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (hs' : ∀ (s : Γ) (m : P'), β' ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β' ((1 : A') ⊗ₜ[k] m))
    (f : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P') (hf : Intertwines (_root_.id : Γ → Γ) (f :
      A' ⊗[k] P →ₗ[A'] A' ⊗[k] P')) :
    Q ≃ₗ[Γ'] Q'

theorem

conjOut_apply

L711 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.conjOut_apply

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section OneChart
variable {k : Type*} [CommRing k] {A' : Type*} [CommRing A'] [Algebra k A']
  {Γ : Type*} [CommRing Γ] [Algebra k Γ] {Γ' : Type*} [CommRing Γ'] [Algebra A' Γ']
  {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
  {P' : Type*} [AddCommGroup P'] [Module Γ P'] [Module k P'] [IsScalarTower k Γ P']
  {Q : Type*} [AddCommGroup Q] [Module Γ' Q] [Module A' Q] [IsScalarTower A' Γ' Q]
  {Q' : Type*} [AddCommGroup Q'] [Module Γ' Q'] [Module A' Q'] [IsScalarTower A' Γ' Q']

Lean source · statement

theorem conjOut_apply (φ : Γ → Γ') (hgen : ∀ y : Γ', y ∈ Submodule.span A' (Set.range φ))
    (β : A' ⊗[k] P ≃ₗ[A'] Q) (β' : A' ⊗[k] P' ≃ₗ[A'] Q')
    (hs : ∀ (s : Γ) (m : P), β ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β ((1 : A') ⊗ₜ[k] m))
    (hs' : ∀ (s : Γ) (m : P'), β' ((1 : A') ⊗ₜ[k] (s • m)) = φ s • β' ((1 : A') ⊗ₜ[k] m))
    (f : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P') (hf : Intertwines (_root_.id : Γ → Γ) (f :
      A' ⊗[k] P →ₗ[A'] A' ⊗[k] P'))
    (q : Q) : conjOut φ hgen β β' hs hs' f hf q = β' (f (β.symm q))

structure

ChartBC

L730 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.ChartBC

Documentation

Tensor comparison equivalences on both affine charts and their overlap, compatible with restriction maps and scalar actions.

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section TwoChart
open TwoChartCech
variable {k : Type u} [CommRing k] {A' : Type u} [CommRing A'] [Algebra k A']
  {𝒰 : Cover.{u, u} k} {𝒰' : Cover.{u, u} A'}

Lean source · full declaration

structure ChartBC (φ0 : 𝒰.A0 → 𝒰'.A0) (φ1 : 𝒰.A1 → 𝒰'.A1) (φ01 : 𝒰.A01 → 𝒰'.A01)
    (S : Sections.{u, u, w} 𝒰) (T : Sections.{u, u, w} 𝒰') where
  /-- The tensor comparison equivalence on the first affine chart. -/
  β0 : A' ⊗[k] S.M0 ≃ₗ[A'] T.M0
  /-- The tensor comparison equivalence on the second affine chart. -/
  β1 : A' ⊗[k] S.M1 ≃ₗ[A'] T.M1
  /-- The tensor comparison equivalence on the chart overlap. -/
  β01 : A' ⊗[k] S.M01 ≃ₗ[A'] T.M01
  /-- The first-chart restriction commutes with the tensor comparison equivalences. -/
  hr0 : ∀ x, β01 (S.r0.baseChange A' x) = T.r0 (β0 x)
  /-- The second-chart restriction commutes with the tensor comparison equivalences. -/
  hr1 : ∀ x, β01 (S.r1.baseChange A' x) = T.r1 (β1 x)
  /-- The first-chart comparison respects the given chart scalar map on elementary tensors. -/
  hs0 : ∀ (s : 𝒰.A0) (m : S.M0), β0 ((1 : A') ⊗ₜ[k] (s • m)) = φ0 s • β0 ((1 : A') ⊗ₜ[k] m)
  /-- The second-chart comparison respects the given chart scalar map on elementary tensors. -/
  hs1 : ∀ (s : 𝒰.A1) (m : S.M1), β1 ((1 : A') ⊗ₜ[k] (s • m)) = φ1 s • β1 ((1 : A') ⊗ₜ[k] m)
  /-- The overlap comparison respects the given overlap scalar map on elementary tensors. -/
  hs01 : ∀ (s : 𝒰.A01) (m : S.M01), β01 ((1 : A') ⊗ₜ[k] (s • m)) = φ01 s • β01 ((1 : A') ⊗ₜ[k] m)

theorem

exists_intertwines_of_chartIso

L752 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.ChartBC.exists_intertwines_of_chartIso

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section TwoChart
open TwoChartCech
variable {k : Type u} [CommRing k] {A' : Type u} [CommRing A'] [Algebra k A']
  {𝒰 : Cover.{u, u} k} {𝒰' : Cover.{u, u} A'}
variable {φ0 : 𝒰.A0 → 𝒰'.A0} {φ1 : 𝒰.A1 → 𝒰'.A1} {φ01 : 𝒰.A01 → 𝒰'.A01}
  {S S' : Sections.{u, u, w} 𝒰} {T T' : Sections.{u, u, w} 𝒰'}

Lean source · statement

theorem ChartBC.exists_intertwines_of_chartIso (β : ChartBC φ0 φ1 φ01 S T) (β' :
  ChartBC φ0 φ1 φ01 S' T')
    (ψ0 : T.M0 ≃ₗ[𝒰'.A0] T'.M0) (ψ1 : T.M1 ≃ₗ[𝒰'.A1] T'.M1) (ψ01 : T.M01 ≃ₗ[𝒰'.A01] T'.M01)
    (hψ0 : ∀ m, ψ01 (T.r0 m) = T'.r0 (ψ0 m)) (hψ1 : ∀ m, ψ01 (T.r1 m) = T'.r1 (ψ1 m)) :
    ∃ (e0 : A' ⊗[k] S.M0 ≃ₗ[A'] A' ⊗[k] S'.M0) (e1 : A' ⊗[k] S.M1 ≃ₗ[A'] A' ⊗[k] S'.M1)
      (e01 : A' ⊗[k] S.M01 ≃ₗ[A'] A' ⊗[k] S'.M01),
      Intertwines (_root_.id : 𝒰.A0 → 𝒰.A0) (e0 : A' ⊗[k] S.M0 →ₗ[A'] A' ⊗[k] S'.M0) ∧
      Intertwines (_root_.id : 𝒰.A1 → 𝒰.A1) (e1 : A' ⊗[k] S.M1 →ₗ[A'] A' ⊗[k] S'.M1) ∧
      Intertwines (_root_.id : 𝒰.A01 → 𝒰.A01) (e01 : A' ⊗[k] S.M01 →ₗ[A'] A' ⊗[k] S'.M01) ∧
      (∀ x, e01 (S.r0.baseChange A' x) = S'.r0.baseChange A' (e0 x)) ∧
      (∀ x, e01 (S.r1.baseChange A' x) = S'.r1.baseChange A' (e1 x))

theorem

exists_chartIso_of_intertwines

L773 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.ChartBC.exists_chartIso_of_intertwines

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section TwoChart
open TwoChartCech
variable {k : Type u} [CommRing k] {A' : Type u} [CommRing A'] [Algebra k A']
  {𝒰 : Cover.{u, u} k} {𝒰' : Cover.{u, u} A'}
variable {φ0 : 𝒰.A0 → 𝒰'.A0} {φ1 : 𝒰.A1 → 𝒰'.A1} {φ01 : 𝒰.A01 → 𝒰'.A01}
  {S S' : Sections.{u, u, w} 𝒰} {T T' : Sections.{u, u, w} 𝒰'}

Lean source · statement

theorem ChartBC.exists_chartIso_of_intertwines (β : ChartBC φ0 φ1 φ01 S T) (β' :
  ChartBC φ0 φ1 φ01 S' T')
    (hgen0 : ∀ y : 𝒰'.A0, y ∈ Submodule.span A' (Set.range φ0))
    (hgen1 : ∀ y : 𝒰'.A1, y ∈ Submodule.span A' (Set.range φ1))
    (hgen01 : ∀ y : 𝒰'.A01, y ∈ Submodule.span A' (Set.range φ01))
    (f0 : A' ⊗[k] S.M0 ≃ₗ[A'] A' ⊗[k] S'.M0) (f1 : A' ⊗[k] S.M1 ≃ₗ[A'] A' ⊗[k] S'.M1)
    (f01 : A' ⊗[k] S.M01 ≃ₗ[A'] A' ⊗[k] S'.M01)
    (hf0 : Intertwines (_root_.id : 𝒰.A0 → 𝒰.A0) (f0 : A' ⊗[k] S.M0 →ₗ[A'] A' ⊗[k] S'.M0))
    (hf1 : Intertwines (_root_.id : 𝒰.A1 → 𝒰.A1) (f1 : A' ⊗[k] S.M1 →ₗ[A'] A' ⊗[k] S'.M1))
    (hf01 : Intertwines (_root_.id : 𝒰.A01 → 𝒰.A01) (f01 : A' ⊗[k] S.M01 →ₗ[A'] A' ⊗[k] S'.M01))
    (hsq0 : ∀ x, f01 (S.r0.baseChange A' x) = S'.r0.baseChange A' (f0 x))
    (hsq1 : ∀ x, f01 (S.r1.baseChange A' x) = S'.r1.baseChange A' (f1 x)) :
    ∃ (E0 : T.M0 ≃ₗ[𝒰'.A0] T'.M0) (E1 : T.M1 ≃ₗ[𝒰'.A1] T'.M1) (E01 : T.M01 ≃ₗ[𝒰'.A01] T'.M01),
      (∀ m, E01 (T.r0 m) = T'.r0 (E0 m)) ∧ (∀ m, E01 (T.r1 m) = T'.r1 (E1 m))

theorem

push_rel

L805 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_rel

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Descent
open TwoChartCech
variable {k : Type u} [CommRing k]

Lean source · statement

theorem push_rel {A A' A'' : Type*} [CommRing A] [Algebra k A] [CommRing A'] [Algebra k A']
  [CommRing A'']
    [Algebra k A''] (h : A' →ₐ[k] A'') (v' : A' →ₐ[k] A) (v'' : A'' →ₐ[k] A) (hv : v''.comp h = v')
    {P Q : Type*} [AddCommGroup P] [Module k P] [AddCommGroup Q] [Module k Q]
    (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q) (E : A ⊗[k] P →ₗ[A] A ⊗[k] Q)
    (hf : ∀ x, v'.toLinearMap.rTensor Q (f x) = E (v'.toLinearMap.rTensor P x)) (y : A'' ⊗[k] P) :
    v''.toLinearMap.rTensor Q (push h f y) = E (v''.toLinearMap.rTensor P y)

theorem

pushEquiv_square_of_linearMap_square

L821 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.pushEquiv_square_of_linearMap_square

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Descent
open TwoChartCech
variable {k : Type u} [CommRing k]

Lean source · statement

theorem pushEquiv_square_of_linearMap_square
    {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
    {P P' Q Q' : Type*} [AddCommGroup P] [Module k P] [AddCommGroup P'] [Module k P']
    [AddCommGroup Q] [Module k Q] [AddCommGroup Q'] [Module k Q']
    (h : A' →ₐ[k] A'')
    (f : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P') (g : A' ⊗[k] Q ≃ₗ[A'] A' ⊗[k] Q')
    (r : P →ₗ[k] Q) (r' : P' →ₗ[k] Q')
    (hsq : push h g.toLinearMap ∘ₗ r.baseChange A'' =
      r'.baseChange A'' ∘ₗ push h f.toLinearMap) :
    ∀ x : A'' ⊗[k] P,
      pushEquiv h g (r.baseChange A'' x) = r'.baseChange A'' (pushEquiv h f x)

theorem

push_square_of_span_of_rTensor_eq

L836 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.push_square_of_span_of_rTensor_eq

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Descent
open TwoChartCech
variable {k : Type u} [CommRing k]

Lean source · statement

theorem push_square_of_span_of_rTensor_eq
    {Γ : Type*} [CommRing Γ] [Algebra k Γ]
    {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
    {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
    {P' Q Q' : Type*} [AddCommGroup P'] [Module k P']
    [AddCommGroup Q] [Module k Q] [AddCommGroup Q'] [Module k Q']
    (h : A' →ₐ[k] A'')
    (f : A' ⊗[k] P →ₗ[A'] A' ⊗[k] P') (g : A' ⊗[k] Q →ₗ[A'] A' ⊗[k] Q')
    (r : P →ₗ[k] Q) (r' : P' →ₗ[k] Q')
    (T : Γ → P' →ₗ[k] P') (U : Γ → Q →ₗ[k] Q) (V : Γ → Q' →ₗ[k] Q')
    (hf : ∀ (a : Γ) (x : A'' ⊗[k] P),
      push h f (actR k A'' a x) = (T a).baseChange A'' (push h f x))
    (hg : ∀ (a : Γ) (y : A'' ⊗[k] Q),
      push h g ((U a).baseChange A'' y) = (V a).baseChange A'' (push h g y))
    (hr : ∀ (a : Γ) (m : P), r (a • m) = U a (r m))
    (hr' : ∀ (a : Γ) (m : P'), r' (T a m) = V a (r' m))
    (gens : Set P) (hgens : Submodule.span Γ gens = ⊤)
    (hsep : ∀ m ∈ gens,
      h.toLinearMap.rTensor Q' (g (r.baseChange A' ((1 : A') ⊗ₜ[k] m))) =
        h.toLinearMap.rTensor Q' (r'.baseChange A' (f ((1 : A') ⊗ₜ[k] m)))) :
    push h g ∘ₗ r.baseChange A'' = r'.baseChange A'' ∘ₗ push h f

theorem

pushEquiv_square_of_span_of_indexed_eq

L875 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.pushEquiv_square_of_span_of_indexed_eq

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Descent
open TwoChartCech
variable {k : Type u} [CommRing k]

Lean source · statement

theorem pushEquiv_square_of_span_of_indexed_eq
    {Γ : Type*} [CommRing Γ] [Algebra k Γ]
    {A' A'' : Type*} [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
    {P : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
    {P' Q Q' : Type*} [AddCommGroup P'] [Module k P']
    [AddCommGroup Q] [Module k Q] [AddCommGroup Q'] [Module k Q']
    (h : A' →ₐ[k] A'')
    (f : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] P') (g : A' ⊗[k] Q ≃ₗ[A'] A' ⊗[k] Q')
    (r : P →ₗ[k] Q) (r' : P' →ₗ[k] Q')
    (T : Γ → P' →ₗ[k] P') (U : Γ → Q →ₗ[k] Q) (V : Γ → Q' →ₗ[k] Q')
    (hf : ∀ (a : Γ) (x : A'' ⊗[k] P),
      push h f.toLinearMap (actR k A'' a x) = (T a).baseChange A'' (push h f.toLinearMap x))
    (hg : ∀ (a : Γ) (y : A'' ⊗[k] Q),
      push h g.toLinearMap ((U a).baseChange A'' y) =
        (V a).baseChange A'' (push h g.toLinearMap y))
    (hr : ∀ (a : Γ) (m : P), r (a • m) = U a (r m))
    (hr' : ∀ (a : Γ) (m : P'), r' (T a m) = V a (r' m))
    (gens : Set P) (hgens : Submodule.span Γ gens = ⊤)
    {I : Type*} (s : Finset I) (j : P → I) (hj : ∀ m ∈ gens, j m ∈ s)
    (t t' : I → A' ⊗[k] Q')
    (ht : ∀ m : P, t (j m) = g (r.baseChange A' ((1 : A') ⊗ₜ[k] m)))
    (ht' : ∀ m : P, t' (j m) = r'.baseChange A' (f ((1 : A') ⊗ₜ[k] m)))
    (hsep : ∀ i ∈ s, h.toLinearMap.rTensor Q' (t i) = h.toLinearMap.rTensor Q' (t' i)) :
    ∀ x : A'' ⊗[k] P,
      pushEquiv h g (r.baseChange A'' x) = r'.baseChange A'' (pushEquiv h f x)

theorem

exists_pushEquiv_of_commuting_triangle

L910 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.exists_pushEquiv_of_commuting_triangle

Source context 23 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Descent
open TwoChartCech
variable {k : Type u} [CommRing k]

Lean source · statement

theorem exists_pushEquiv_of_commuting_triangle
    {Γ : Type*} [CommRing Γ] [Algebra k Γ]
    {A A' A'' : Type*} [CommRing A] [Algebra k A]
    [CommRing A'] [Algebra k A'] [CommRing A''] [Algebra k A'']
    {P Q : Type*} [AddCommGroup P] [Module Γ P] [Module k P] [IsScalarTower k Γ P]
    [AddCommGroup Q] [Module Γ Q] [Module k Q] [IsScalarTower k Γ Q]
    (h : A' →ₐ[k] A'') (v' : A' →ₐ[k] A) (v'' : A'' →ₐ[k] A)
    (hv : v''.comp h = v')
    (e : A' ⊗[k] P ≃ₗ[A'] A' ⊗[k] Q) (E : A ⊗[k] P →ₗ[A] A ⊗[k] Q)
    (he : Intertwines (_root_.id : Γ → Γ) (e : A' ⊗[k] P →ₗ[A'] A' ⊗[k] Q))
    (hE : ∀ x, v'.toLinearMap.rTensor Q (e x) = E (v'.toLinearMap.rTensor P x)) :
    ∃ p : A'' ⊗[k] P ≃ₗ[A''] A'' ⊗[k] Q,
      Intertwines (_root_.id : Γ → Γ) (p : A'' ⊗[k] P →ₗ[A''] A'' ⊗[k] Q) ∧
      ∀ y, v''.toLinearMap.rTensor Q (p y) = E (v''.toLinearMap.rTensor P y)

theorem

exists_fg_descent

L934 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.exists_fg_descent

Source context 24 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
section Descent
open TwoChartCech
variable {k : Type u} [CommRing k]
variable {A : Type u} [CommRing A] [Algebra k A] {𝒰 : Cover.{u, u} k} {S S' : Sections.{u, u, w} 𝒰}

Lean source · statement

theorem exists_fg_descent
    [Module.FinitePresentation 𝒰.A0 S.M0] [Module.FinitePresentation 𝒰.A1 S.M1]
    [Module.FinitePresentation 𝒰.A01 S.M01] [Module.FinitePresentation 𝒰.A0 S'.M0]
    [Module.FinitePresentation 𝒰.A1 S'.M1] [Module.FinitePresentation 𝒰.A01 S'.M01]
    (e0 : A ⊗[k] S.M0 ≃ₗ[A] A ⊗[k] S'.M0) (e1 : A ⊗[k] S.M1 ≃ₗ[A] A ⊗[k] S'.M1)
    (e01 : A ⊗[k] S.M01 ≃ₗ[A] A ⊗[k] S'.M01)
    (he0 : Intertwines (_root_.id : 𝒰.A0 → 𝒰.A0) (e0 : A ⊗[k] S.M0 →ₗ[A] A ⊗[k] S'.M0))
    (he1 : Intertwines (_root_.id : 𝒰.A1 → 𝒰.A1) (e1 : A ⊗[k] S.M1 →ₗ[A] A ⊗[k] S'.M1))
    (he01 : Intertwines (_root_.id : 𝒰.A01 → 𝒰.A01) (e01 : A ⊗[k] S.M01 →ₗ[A] A ⊗[k] S'.M01))
    (hsq0 : ∀ x, e01 (S.r0.baseChange A x) = S'.r0.baseChange A (e0 x))
    (hsq1 : ∀ x, e01 (S.r1.baseChange A x) = S'.r1.baseChange A (e1 x)) :
    ∃ (D : Subalgebra k A) (_ : D.FG) (f0 : ↥D ⊗[k] S.M0 ≃ₗ[↥D] ↥D ⊗[k] S'.M0)
      (f1 : ↥D ⊗[k] S.M1 ≃ₗ[↥D] ↥D ⊗[k] S'.M1) (f01 : ↥D ⊗[k] S.M01 ≃ₗ[↥D] ↥D ⊗[k] S'.M01),
      Intertwines (_root_.id : 𝒰.A0 → 𝒰.A0) (f0 : ↥D ⊗[k] S.M0 →ₗ[↥D] ↥D ⊗[k] S'.M0) ∧
      Intertwines (_root_.id : 𝒰.A1 → 𝒰.A1) (f1 : ↥D ⊗[k] S.M1 →ₗ[↥D] ↥D ⊗[k] S'.M1) ∧
      Intertwines (_root_.id : 𝒰.A01 → 𝒰.A01) (f01 : ↥D ⊗[k] S.M01 →ₗ[↥D] ↥D ⊗[k] S'.M01) ∧
      (∀ x, f01 (S.r0.baseChange ↥D x) = S'.r0.baseChange ↥D (f0 x)) ∧
      (∀ x, f01 (S.r1.baseChange ↥D x) = S'.r1.baseChange ↥D (f1 x))

theorem

fg_restrictScalars_of_fg

L1069 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.fg_restrictScalars_of_fg

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech

Lean source · statement

theorem fg_restrictScalars_of_fg {R A : Type u} [CommRing R] [CommRing A] [Algebra R A]
    (A₀ : Subalgebra R A) (hA₀ : A₀.FG) (D : Subalgebra ↥A₀ A) (hD : D.FG) :
      (D.restrictScalars R).FG

theorem

exists_fg_le_nonempty_iso_pullbackAlong

L1078 ↗

JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf.AlgebraicGeometry.RelPicard.LFP.exists_fg_le_nonempty_iso_pullbackAlong

Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.IsLFPInj_relPicardPresheaf
attribute [-instance] AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
attribute [-simp]
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicCondition.mk.sizeOf_spec
  AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.sizeOf_spec
  GoodReductionJacobian.AvatarSchemeBridge.mk.injEq
  GoodReductionJacobian.AvatarSchemeBridge.mk.sizeOf_spec
  GoodReductionJacobian.RelativePic0Designation.mk.injEq
  GoodReductionJacobian.schemeHomOverComp_coe NeronModelInfra.schemeHomOverEquivOverHom_apply
  GoodReductionJacobian.RelativeGroupLaw.mk.sizeOf_spec
  NeronModelInfra.schemeHomOverEquivOverHom_symm_apply
  NeronModelInfra.overHomToSchemeHomOver_coe GoodReductionJacobian.RelativeGroupLaw.mk.injEq
  NeronModelInfra.overHomToSchemeHomOver_schemeHomOverToOverHom
  NeronModelInfra.schemeHomOverToOverHom_left
  NeronModelInfra.schemeHomOverToOverHom_overHomToSchemeHomOver
  AlgebraicGeometry.RelPicard.SubPicCondition.onClasses_mk
  AlgebraicGeometry.RelPicard.relSubPicPresheaf_map_coe
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
universe u v w
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.Opposite
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.TensorProduct
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace RelPicard
open _root_.AlgebraicGeometry.RelPicard
namespace LFP
open _root_.AlgebraicGeometry.RelPicard.LFP
open _root_.AlgebraicGeometry.RelPicard
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
open _root_.TwoChartCech

Lean source · statement

theorem exists_fg_le_nonempty_iso_pullbackAlong
    (R : Type u) [CommRing R] {C : Scheme.{u}} (𝒱 : C.TwoAffineOpenCover) (c : C ⟶ Spec
      (CommRingCat.of R))
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (A : Type u) [CommRing A] [Algebra R A] (A₀ : Subalgebra R A) (hA₀ : A₀.FG)
    (M M' : RigidifiedLineBundle c ε (specMap R A₀))
    (h : Nonempty ((M.pullbackAlong (stageHom R A₀.val)).L ≅ (M'.pullbackAlong
      (stageHom R A₀.val)).L)) :
    ∃ (A₁ : Subalgebra R A) (_ : A₁.FG) (hle : A₀ ≤ A₁),
      Nonempty ((M.pullbackAlong (stageHom R (Subalgebra.inclusion hle))).L ≅
        (M'.pullbackAlong (stageHom R (Subalgebra.inclusion hle))).L)

theorem