Module

Pic0_exists_schemeHomOver_pushforwardAlong_of_representsRelSubPic.lean

Nyll.JacobianVarieties.Pic0.Pic0_exists_schemeHomOver_pushforwardAlong_of_representsRelSubPic

10 declarations indexed from sourceView source ↗

Module documentation

Representing Picard pushforward by a scheme morphism

Imports21

Imported by1

Nyll.JacobianVarieties.Pic0.All

theorem

finiteAlong_of_curveModel

L144 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.finiteAlong_of_curveModel

Documentation

An integral homomorphism into a curve model function field is finite.

Source context 18 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward

Lean source · statement

theorem finiteAlong_of_curveModel {K : Type u} [Field K] {F : Type v} [Field F] [Algebra K F]
    {F' : Type w} [Field F'] [Algebra K F'] (M' : CurveModel K F')
    (φ : F →ₐ[K] F') (hφ : φ.toRingHom.IsIntegral) : FiniteAlong K φ

theorem

inertiaDegAlong_eq_one

L166 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.inertiaDegAlong_eq_one

Documentation

Residue field degrees are one over an algebraically closed base field.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem inertiaDegAlong_eq_one [IsCurveOver K F] [IsCurveOver K F']
    (φ : F →ₐ[K] F') (hφ : φ.toRingHom.IsIntegral) (w : Place K F') :
    w.inertiaDegAlong φ hφ = 1

theorem

pushforwardAlong_single_one

L177 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.pushforwardAlong_single_one

Documentation

The pushforward of a degree one place is its restricted place.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem pushforwardAlong_single_one [IsCurveOver K F] [IsCurveOver K F']
    (φ : F →ₐ[K] F') (hφ : φ.toRingHom.IsIntegral) (w : Place K F') :
    Divisor.pushforwardAlong φ hφ (Finsupp.single w 1) = Finsupp.single (w.restrictAlong φ hφ) 1

theorem

pointEquivPlace_comp_eq

L184 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.pointEquivPlace_comp_eq

Documentation

The place of the image of a point is the restriction of its place.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem pointEquivPlace_comp_eq (M : CurveModel K F) (M' : CurveModel K F')
    (φ : F →ₐ[K] F') (hφ : φ.toRingHom.IsIntegral)
    (π : M'.C ⟶ M.C)
    (hcl : ∀ y : closedPoints M'.C, ∃ h : π.base y.1 ∈ closedPoints M.C,
        M.placeOfPoint ⟨π.base y.1, h⟩ = (M'.placeOfPoint y).restrictAlong φ hφ)
    (y : {q : Spec (CommRingCat.of K) ⟶ M'.C // q ≫ M'.toBase = 𝟙 _})
    (hy : (y.1 ≫ π) ≫ M.toBase = 𝟙 _) :
    M.pointEquivPlace ⟨y.1 ≫ π, hy⟩ = (M'.pointEquivPlace y).restrictAlong φ hφ

def

retract

L197 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.retract

Documentation

Project divisors to degree zero by subtracting a fixed place with their degree.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

def retract [IsCurveOver K F'] (v₀ : Place K F') :
    Divisor K F' →+ Divisor.degZero (K := K) (F := F')

theorem

retract_coe

L210 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.retract_coe

Documentation

The underlying divisor of the degree zero projection.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem retract_coe [IsCurveOver K F'] (v₀ : Place K F') (D : Divisor K F') :
    (retract v₀ D : Divisor K F') = D - Divisor.degree D • Finsupp.single v₀ 1

theorem

retract_single_one_coe

L214 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.retract_single_one_coe

Documentation

A single place projects to its difference with the chosen base place.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem retract_single_one_coe [IsCurveOver K F'] (v₀ w : Place K F') :
    (retract v₀ (Finsupp.single w 1) : Divisor K F') = Finsupp.single w 1 - Finsupp.single v₀ 1

theorem

retract_of_mem

L222 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.retract_of_mem

Documentation

The divisor projection fixes divisors of degree zero.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem retract_of_mem [IsCurveOver K F'] (v₀ : Place K F') {D : Divisor K F'}
    (hD : D ∈ Divisor.degZero (K := K) (F := F')) : retract v₀ D = ⟨D, hD⟩

theorem

degZero_addMonoidHom_ext

L228 ↗

JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward.degZero_addMonoidHom_ext

Documentation

Homomorphisms on degree zero divisors are determined by point differences.

Source context 20 commands

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

section ProofImplementation
namespace JacobianVarieties.Pic0.Proofs.PicardPushforward
attribute [-instance]
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instNeg
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulInt
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAddCommGroup
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instAdd
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSub
attribute [-instance]
  instModuleCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  instSMulCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTop
attribute [-instance]
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instSMulNat
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.instZero TwoChartCech.Sections.M0_moduleA
  TwoChartCech.Sections.M1_module TwoChartCech.Cover.A01_algebra TwoChartCech.Cover.A0_algebra
  TwoChartCech.Cover.A1_commRing TwoChartCech.Cover.A1_algebra TwoChartCech.Sections.M01_module
  TwoChartCech.Sections.M0_addCommGroup TwoChartCech.Sections.M0_tower
  TwoChartCech.Sections.M01_addCommGroup TwoChartCech.Cover.A0_commRing
  TwoChartCech.Sections.M1_tower TwoChartCech.Sections.M01_moduleA TwoChartCech.Sections.M0_module
  TwoChartCech.Sections.M1_moduleA TwoChartCech.Sections.M1_addCommGroup
  TwoChartCech.Cover.A01_commRing TwoChartCech.Sections.M01_tower
  CoherentBaseChange.TwoTermComplex.C0_module CoherentBaseChange.TwoTermComplex.C0_addCommGroup
  CoherentBaseChange.TwoTermComplex.C1_module CoherentBaseChange.TwoTermComplex.C1_addCommGroup
  CoherentBaseChange.TwoTermComplex.C0_free CoherentBaseChange.TwoTermComplex.C1_finite
  CoherentBaseChange.TwoTermComplex.C0_finite CoherentBaseChange.TwoTermComplex.C1_free
  SheafOfModules.isIso_ihomModelToIhom
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.Scheme.Modules.normModuleFunctor_obj
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj PresheafOfModules.exteriorPower_map_ιMulti
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.sub_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sresTop
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.add_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.IdealSheafData.coe_resLE
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.smul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_ofLE_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.ideal_range
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zero_app
  AlgebraicGeometry.Scheme.IdealSheafData.sres_sres
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.comp_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.neg_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.id_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.coe_mulRight_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.nsmul_app
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.mk.injEq
  AlgebraicGeometry.Scheme.IdealSheafData.IdealHom.zsmul_app
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.injEq
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U1
  AlgebraicGeometry.Scheme.TwoAffineOpenCover.pullback_U0 TwoChartCech.Sections.mk.injEq
  TwoChartCech.Cover.mk.injEq TwoChartCech.GrothendieckComplex.mk.injEq
  TwoChartCech.Sections.mk.sizeOf_spec TwoChartCech.Cover.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r0_apply TwoChartCech.GrothendieckComplex.mk.sizeOf_spec
  TwoChartCech.Cover.lineBundle_r1_apply CoherentBaseChange.TwoTermComplex.mk.sizeOf_spec
  CoherentBaseChange.TwoTermComplex.mk.injEq AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_right
attribute [-simp]
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.pushforwardCongr_inv_app_app_apply
  PresheafOfModules.pushforwardNatTrans_app_app_apply
  PresheafOfModules.pushforwardCongr_hom_app_app_apply TwoChartCech.Mumford.dK_apply
  TwoChartCech.Mumford.ι0_apply TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply
  TwoChartCech.KerCoprod.ι1_apply TwoChartCech.KerCoprod.ι0_apply
  AlgebraicGeometry.Scheme.Modules.toUnitSection_ofUnitSection
  AlgebraicGeometry.Scheme.Modules.pullbackSection_def
  AlgebraicGeometry.Scheme.Modules.ofUnitSection_toUnitSection
  PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply SheafOfModules.unit_ihomSectionsEquivFamily
  AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
namespace P2mAlbanesePushforward
section Places
variable {K : Type u} [Field K] [IsAlgClosed K]
  {F : Type v} [Field F] [Algebra K F] {F' : Type w} [Field F'] [Algebra K F']

Lean source · statement

theorem degZero_addMonoidHom_ext [IsCurveOver K F'] {A : Type*} [AddGroup A] (v₀ : Place K F')
    {f g : Divisor.degZero (K := K) (F := F') →+ A}
    (hfg : ∀ w : Place K F', f (retract v₀ (Finsupp.single w 1)) = g (retract v₀ (Finsupp.single w
      1))) :
    f = g

theorem

exists_schemeHomOver_pushforwardAlong_of_representsRelSubPic

L266 ↗

AlgebraicCurve.Pic0.exists_schemeHomOver_pushforwardAlong_of_representsRelSubPic

Documentation

A finite map of curve function fields induces the representing Picard pushforward morphism.

Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurve.Pic0
open _root_.JacobianVarieties.Pic0.Proofs.PicardPushforward
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.AlgebraicCurve
universe u v w
open _root_.JacobianVarieties.Pic0.Proofs.PicardPushforward.P2mAlbanesePushforward

Lean source · statement

theorem exists_schemeHomOver_pushforwardAlong_of_representsRelSubPic
    (K : Type u) [Field K] [IsAlgClosed K]
    (F : Type v) [Field F] [Algebra K F] [IsCurveOver K F] (M : CurveModel K F)
    (s : {q : Spec (CommRingCat.of K) ⟶ M.C // q ≫ M.toBase = 𝟙 _})
    (D : RelativePic0Designation K M.toBase)
    (h : RepresentsRelSubPic M.toBase s (algEquivZeroCut M.toBase s) D)
    (aj : SchemeHomOver M.toBase D.toBase)
    (haj : ∀ (K' : Type u) [Field K'] (t : Spec (CommRingCat.of K') ⟶ Spec (CommRingCat.of K))
        (x : SchemeHomOver t M.toBase),
      Nonempty ((h.poincare.pullbackAlong
          ⟨x.1 ≫ aj.1, (Category.assoc _ _ _).trans ((congrArg (x.1 ≫ ·) aj.2).trans x.2)⟩).L ≅
        (RelEffCartierDiv.ofPoint M.toBase x.1 x.2).lineBundle ⊗
          (RelEffCartierDiv.ofPoint M.toBase (t ≫ s.1)
            ((Category.assoc _ _ _).trans ((congrArg (t ≫ ·) s.2).trans (Category.comp_id
              t)))).idealModule))
    (F' : Type w) [Field F'] [Algebra K F'] [IsCurveOver K F'] (M' : CurveModel K F')
    (s' : {q : Spec (CommRingCat.of K) ⟶ M'.C // q ≫ M'.toBase = 𝟙 _})
    (D' : RelativePic0Designation K M'.toBase)
    (h' : RepresentsRelSubPic M'.toBase s' (algEquivZeroCut M'.toBase s') D')
    (aj' : SchemeHomOver M'.toBase D'.toBase)
    (haj' : ∀ (K' : Type u) [Field K'] (t : Spec (CommRingCat.of K') ⟶ Spec (CommRingCat.of K))
        (x : SchemeHomOver t M'.toBase),
      Nonempty ((h'.poincare.pullbackAlong
          ⟨x.1 ≫ aj'.1, (Category.assoc _ _ _).trans ((congrArg (x.1 ≫ ·) aj'.2).trans x.2)⟩).L ≅
        (RelEffCartierDiv.ofPoint M'.toBase x.1 x.2).lineBundle ⊗
          (RelEffCartierDiv.ofPoint M'.toBase (t ≫ s'.1)
            ((Category.assoc _ _ _).trans ((congrArg (t ≫ ·) s'.2).trans (Category.comp_id
              t)))).idealModule))
    (φ : F →ₐ[K] F') (hφi : φ.toRingHom.IsIntegral) :
    ∃ N : SchemeHomOver D'.toBase D.toBase,
      (∀ {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of K)) (x y : SchemeHomOver t D'.toBase),
        NeronModelInfra.schemeHomOverComp
            ((RepresentsRelSubPic.relativeGroupLaw (P := algEquivZeroGroupCut M'.toBase s') h').mul
              t x y) N =
          (RepresentsRelSubPic.relativeGroupLaw (P := algEquivZeroGroupCut M.toBase s) h).mul t
            (NeronModelInfra.schemeHomOverComp x N) (NeronModelInfra.schemeHomOverComp y N)) ∧
      ∀ (pts : Pic0 K F ≃ SchemeHomOver (𝟙 (Spec (CommRingCat.of K))) D.toBase)
        (pts' : Pic0 K F' ≃ SchemeHomOver (𝟙 (Spec (CommRingCat.of K))) D'.toBase),
        (∀ x y : Pic0 K F, pts (x + y) =
          (RepresentsRelSubPic.relativeGroupLaw (P := algEquivZeroGroupCut M.toBase s) h).mul _
            (pts x) (pts y)) →
        (∀ x : {q : Spec (CommRingCat.of K) ⟶ M.C // q ≫ M.toBase = 𝟙 _},
          ∃ Dv : Divisor.degZero (K := K) (F := F),
            (Dv : Divisor K F) =
              Finsupp.single (M.pointEquivPlace x) 1 - Finsupp.single (M.pointEquivPlace s) 1 ∧
            (pts (Pic0.mk Dv)).1 = x.1 ≫ aj.1) →
        (∀ x y : Pic0 K F', pts' (x + y) =
          (RepresentsRelSubPic.relativeGroupLaw (P := algEquivZeroGroupCut M'.toBase s') h').mul _
            (pts' x) (pts' y)) →
        (∀ x : {q : Spec (CommRingCat.of K) ⟶ M'.C // q ≫ M'.toBase = 𝟙 _},
          ∃ Dv : Divisor.degZero (K := K) (F := F'),
            (Dv : Divisor K F') =
              Finsupp.single (M'.pointEquivPlace x) 1 - Finsupp.single (M'.pointEquivPlace s') 1 ∧
            (pts' (Pic0.mk Dv)).1 = x.1 ≫ aj'.1) →
        ∀ Dv : Divisor.degZero (K := K) (F := F'),
          (pts (Pic0.mk ⟨Divisor.pushforwardAlong φ hφi Dv,
              Divisor.pushforwardAlong_mem_degZero φ hφi Dv.2⟩)).1 =
            (pts' (Pic0.mk Dv)).1 ≫ N.1

Module documentation

Representing Picard pushforward by a scheme morphism

Declarations