NYLL
Module

RelativeGroupLaw_exists_fg_subalgebra_abelianSchemePropertyBundle_isPullback_of_isNoetherianRing.lean

Nyll.AlgebraicCurves.Geometry.AbelianSchemes.Theorems.RelativeGroupLaw_exists_fg_subalgebra_abelianSchemePropertyBundle_isPullback_of_isNoetherianRing
35 declarations indexed from sourceView source ↗

Module documentation

Descending abelian scheme data to a finitely generated base algebra

Imports19
Imported by3

Declarations

theorem

baseChange_mul_fst

L202 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.baseChange_mul_fst
Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
Lean source · full declaration
theorem baseChange_mul_fst {S S' : Type u} [CommRing S] [CommRing S']
    (u : Spec (CommRingCat.of S') ⟶ Spec (CommRingCat.of S))
    {A : Scheme.{u}} {f : A ⟶ Spec (CommRingCat.of S)} (G : RelativeGroupLaw S f)
    {T : Scheme.{u}} (t' : T ⟶ Spec (CommRingCat.of S')) (x y : SchemeHomOver t' (pullback.snd f
      u)) :
    ((G.baseChange u).mul t' x y).1 ≫ pullback.fst f u =
      (G.mul (t' ≫ u)
        ⟨x.1 ≫ pullback.fst f u, by rw [Category.assoc, pullback.condition, ← Category.assoc, x.2]⟩
        ⟨y.1 ≫ pullback.fst f u,
          by rw [Category.assoc, pullback.condition, ← Category.assoc, y.2]⟩).1 := by
  have h := congrArg Subtype.val (RelativeGroupLaw.baseChangePointToBase_mul u G t' x y)
  rw [RelativeGroupLaw.baseChangePointToBase_coe] at h
  exact h
theorem

mul_val_congr

L216 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.mul_val_congr
Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
Lean source · statement
theorem mul_val_congr {S : Type u} [CommRing S] {A : Scheme.{u}} {f : A ⟶ Spec (CommRingCat.of S)}
    (G : RelativeGroupLaw S f) {T : Scheme.{u}} {t₁ t₂ : T ⟶ Spec (CommRingCat.of S)} (h : t₁ = t₂)
    (P₁ Q₁ : SchemeHomOver t₁ f) (P₂ Q₂ : SchemeHomOver t₂ f) (hP : P₁.1 = P₂.1) (hQ : Q₁.1 = Q₂.1)
      :
    (G.mul t₁ P₁ Q₁).1 = (G.mul t₂ P₂ Q₂).1
theorem

inv_comp_eq_of_hom_comp_eq

L232 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.inv_comp_eq_of_hom_comp_eq
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
include he in
Lean source · statement
theorem inv_comp_eq_of_hom_comp_eq : e.inv ≫ fX = fY
def

pointBack

L236 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointBack
Documentation

Transport a point through the inverse of the given scheme isomorphism over the base.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
def pointBack {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of S)} (x : SchemeHomOver t fY) :
  SchemeHomOver t fX
def

pointForth

L241 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointForth
Documentation

Transport a point through the given scheme isomorphism over the base.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
def pointForth {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of S)} (x : SchemeHomOver t fX) :
  SchemeHomOver t fY
theorem

pointBack_coe

L245 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointBack_coe
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
theorem pointBack_coe {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of S)} (x :
  SchemeHomOver t fY) :
    (pointBack e he x).1 = x.1 ≫ e.inv
theorem

pointForth_coe

L249 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointForth_coe
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
theorem pointForth_coe {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of S)} (x :
  SchemeHomOver t fX) :
    (pointForth e he x).1 = x.1 ≫ e.hom
theorem

pointBack_pointForth

L253 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointBack_pointForth
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
theorem pointBack_pointForth {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of S)} (x :
  SchemeHomOver t fX) :
    pointBack e he (pointForth e he x) = x
theorem

pointForth_pointBack

L258 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointForth_pointBack
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
theorem pointForth_pointBack {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of S)} (x :
  SchemeHomOver t fY) :
    pointForth e he (pointBack e he x) = x
def

transportIso

L264 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.transportIso
Documentation

The relative group law transported through the scheme isomorphism.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · full declaration
def transportIso (G : RelativeGroupLaw S fX) : RelativeGroupLaw S fY where
  mul t x y := pointForth e he (G.mul t (pointBack e he x) (pointBack e he y))
  one t := pointForth e he (G.one t)
  inv t x := pointForth e he (G.inv t (pointBack e he x))
  mul_assoc t x y z := by simp only [pointBack_pointForth, G.mul_assoc]
  one_mul t x := by simp only [pointBack_pointForth, G.one_mul, pointForth_pointBack]
  mul_one t x := by simp only [pointBack_pointForth, G.mul_one, pointForth_pointBack]
  inv_mul_cancel t x := by simp only [pointBack_pointForth, G.inv_mul_cancel]
  mul_natural t t' ψ hψ x y := by
    apply Subtype.ext
    have hb : ∀ z : SchemeHomOver t fY, pointBack e he (schemeHomOverComp ψ hψ z) =
        schemeHomOverComp ψ hψ (pointBack e he z) := fun z => Subtype.ext (Category.assoc _ _ _)
    rw [hb, hb, ← G.mul_natural]
    simp only [_root_.GoodReductionJacobian.schemeHomOverComp_coe, pointForth_coe, Category.assoc]
theorem

transportIso_mul_coe

L279 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.transportIso_mul_coe
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
theorem transportIso_mul_coe (G : RelativeGroupLaw S fX) {T : Scheme.{u}} (t : T ⟶
  Spec (CommRingCat.of S))
    (x y : SchemeHomOver t fY) :
    ((transportIso e he G).mul t x y).1 = (G.mul t (pointBack e he x) (pointBack e he y)).1 ≫ e.hom
theorem

transportIso_hom

L285 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.transportIso_hom
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · full declaration
theorem transportIso_hom (G : RelativeGroupLaw S fX) {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of
  S))
    (x y : SchemeHomOver t fX) :
    (G.mul t x y).1 ≫ e.hom =
      ((transportIso e he G).mul t ⟨x.1 ≫ e.hom, by rw [Category.assoc, he, x.2]⟩
        ⟨y.1 ≫ e.hom, by rw [Category.assoc, he, y.2]⟩).1 := by
  rw [transportIso_mul_coe]
  congr 1
  apply mul_val_congr G rfl
  · simp
  · simp
theorem

transportIso_isCommutative

L297 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.transportIso_isCommutative
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Transport
variable {S : Type u} [CommRing S] {X Y : Scheme.{u}} {fX : X ⟶ Spec (CommRingCat.of S)}
  {fY : Y ⟶ Spec (CommRingCat.of S)} (e : X ≅ Y) (he : e.hom ≫ fY = fX)
Lean source · statement
theorem transportIso_isCommutative {G : RelativeGroupLaw S fX} (hG : G.IsCommutative) :
    (transportIso e he G).IsCommutative
theorem

geometricallyConnected_of_bundle

L304 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.geometricallyConnected_of_bundle
Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
Lean source · statement
theorem geometricallyConnected_of_bundle {R : Type u} [CommRing R] {X : Scheme.{u}}
    {f : X ⟶ Spec (CommRingCat.of R)} (L : RelativeGroupLaw R f) (hX : AbelianSchemePropertyBundle
      R f) :
    GeometricallyConnected f
theorem

bundle_of_geometricallyConnected

L328 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.bundle_of_geometricallyConnected
Source context 20 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
Lean source · statement
theorem bundle_of_geometricallyConnected {R : Type u} [CommRing R] {X : Scheme.{u}}
    {f : X ⟶ Spec (CommRingCat.of R)} [Smooth f] [IsProper f] [GeometricallyConnected f] (L :
      RelativeGroupLaw R f) :
    AbelianSchemePropertyBundle R f
abbrev

bT

L339 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.bT
Documentation

The spectrum morphism induced by the inclusion of an intermediate subalgebra.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
Lean source · statement
abbrev bT (T : Subalgebra A₀ A) : (Spec (CommRingCat.of ↥T) : Scheme.{u}) ⟶ Spec (CommRingCat.of
  A₀)
abbrev

uA

L344 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.uA
Documentation

The spectrum morphism from the ambient algebra to an intermediate subalgebra.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
Lean source · statement
abbrev uA (T : Subalgebra A₀ A) : (Spec (CommRingCat.of A) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥T)
abbrev

uT

L349 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.uT
Documentation

The spectrum morphism induced by an inclusion between intermediate subalgebras.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
Lean source · statement
abbrev uT {T₁ T₂ : Subalgebra A₀ A} (h : T₁ ≤ T₂) :
    (Spec (CommRingCat.of ↥T₂) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥T₁)
theorem

uA_uT

L353 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.uA_uT
Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
Lean source · statement
theorem uA_uT {T₁ T₂ : Subalgebra A₀ A} (h : T₁ ≤ T₂) : uA T₂ ≫ uT h = uA T₁
def

flat

L360 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.flat
Documentation

Regards a subalgebra over an intermediate algebra as a subalgebra over the original base.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
def flat : Subalgebra A₀ A
theorem

le_flat

L362 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.le_flat
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem le_flat : T ≤ flat T T'
theorem

flat_fg

L364 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.flat_fg
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem flat_fg (hT : T.FG) (hT' : T'.FG) : (flat T T').FG
def

flatHom

L372 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.flatHom
Documentation

The ring homomorphism identifying a subalgebra with its restriction of scalars.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · full declaration
def flatHom : ↥T' →+* ↥(flat T T') where
  toFun x := ⟨x.1, x.2⟩
  map_one' := rfl
  map_mul' _ _ := rfl
  map_zero' := rfl
  map_add' _ _ := rfl
abbrev

uF

L380 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.uF
Documentation

The spectrum morphism associated with the chosen flat intermediate algebra map.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
abbrev uF : (Spec (CommRingCat.of ↥(flat T T')) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥T')
theorem

uF_comp

L383 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.uF_comp
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem uF_comp : uF T T' ≫ Spec.map (CommRingCat.ofHom (algebraMap ↥T ↥T')) = uT (le_flat T T')
theorem

uA_uF

L388 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.uA_uF
Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem uA_uF : uA (flat T T') ≫ uF T T' = Spec.map (CommRingCat.ofHom (algebraMap ↥T' A))
theorem

isNoetherianRing_stage

L394 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.isNoetherianRing_stage
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
theorem isNoetherianRing_stage [IsNoetherianRing A₀] (hT : T.FG) : IsNoetherianRing ↥T
theorem

isPullback_lift_of_fac

L398 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.isPullback_lift_of_fac
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · full declaration
theorem isPullback_lift_of_fac {X Y C B B' : Scheme.{u}} {π : X ⟶ Y} {f : X ⟶ C} {fY : Y ⟶ B} {b :
  C ⟶ B}
    (hπ : IsPullback π f fY b) (v : B' ⟶ B) (c : C ⟶ B') (hc : c ≫ v = b) :
    IsPullback (pullback.lift π (f ≫ c)
      (by rw [Category.assoc, hc, hπ.w])) f (pullback.snd fY v) c := by
  refine IsPullback.of_right (h₁₂ := pullback.fst fY v) (h₂₂ := v) ?_ (pullback.lift_snd _ _ _)
    (IsPullback.of_hasPullback fY v)
  rw [pullback.lift_fst, hc]
  exact hπ
theorem

prop_snd_of_isPullback

L408 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.prop_snd_of_isPullback
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
theorem prop_snd_of_isPullback {X Y C B : Scheme.{u}} {π : X ⟶ Y} {f : X ⟶ C} {fY : Y ⟶ B} {b : C ⟶
  B}
    (hπ : IsPullback π f fY b) (P : MorphismProperty Scheme.{u}) [P.RespectsIso] (h : P f) :
    P (pullback.snd fY b)
theorem

isPullback_paste

L415 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.isPullback_paste
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
theorem isPullback_paste {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S''
  ⟶ S)
    (h : b ≫ a = c) :
    IsPullback (pullback.fst (pullback.snd f a) b ≫ pullback.fst f a) (pullback.snd (pullback.snd f
      a) b) f c
def

rebaseIso

L427 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.rebaseIso
Documentation

The canonical isomorphism between an iterated base change and the base change along the composite.

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
def rebaseIso {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S'' ⟶ S) (h :
  b ≫ a = c) :
    pullback (pullback.snd f a) b ≅ pullback f c
theorem

rebaseIso_hom_snd

L432 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.rebaseIso_hom_snd
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
theorem rebaseIso_hom_snd {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b :
  S'' ⟶ S') (c : S'' ⟶ S)
    (h : b ≫ a = c) : (rebaseIso f a b c h).hom ≫ pullback.snd f c = pullback.snd _ _
theorem

prop_snd_of_fac

L437 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.prop_snd_of_fac
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
theorem prop_snd_of_fac {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S''
  ⟶ S)
    (h : b ≫ a = c) (P : MorphismProperty Scheme.{u}) [P.IsStableUnderBaseChange] [P.RespectsIso]
    (hP : P (pullback.snd f a)) : P (pullback.snd f c)
theorem

prop_snd_uT_of_prop_snd_algebraMap

L447 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.prop_snd_uT_of_prop_snd_algebraMap
Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent
attribute [-instance] IsDirectLimit.Module.instDirectLimitCoeLinearMapIdOfOfNonempty
  AlgebraicGeometry.SubalgebraStages.compactSpace_pullback
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_pullback
  AlgebraicGeometry.SubalgebraStages.compactSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiSeparatedSpace_obj
  AlgebraicGeometry.SubalgebraStages.quasiCompact_snd
  AlgebraicGeometry.SubalgebraStages.isAffineHom_leg
  AlgebraicGeometry.SubalgebraStages.isAffineHom_trans
  AlgebraicGeometry.SubalgebraStages.isAffineHom_diagram_map
  AlgebraicGeometry.SubalgebraStages.quasiSeparated_snd
  AlgebraicGeometry.SubalgebraStages.isCofiltered_op AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU AlgebraicGeometry.ChowDatum.hp_proper
  AlgebraicGeometry.ProjSpace.algebraAway AlgebraicGeometry.ProjSpace.instIsProperProdOverπ
  AlgebraicGeometry.ChowDatumProj.hp_isoU AlgebraicGeometry.ProjSpace.isProper_π
  AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections 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
attribute [-instance] 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
  AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I ProjSpaceCech.GradedModule.H.module
  ProjSpaceCech.GradedModule.H.addCommGroup ProjSpaceCech.GradedModule.sec.instAdd
  ProjSpaceCech.GradedModule.sec.instNeg ProjSpaceCech.GradedModule.acg
attribute [-instance] ProjSpaceCech.GradedModule.Frac.setoid ProjSpaceCech.GradedModule.modR
  ProjSpaceCech.GradedModule.sec.instModule ProjSpaceCech.GradedModule.sec.instAddCommGroup
  ProjSpaceCech.GradedModule.sec.instZero ProjSpaceCech.GradedModule.Presentation.fJ
  ProjSpaceCech.GradedModule.sec.instSMul ProjSpaceCech.Twist.H.module
  ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] IsDirectLimit.Module.linearEquiv_symm_apply IsDirectLimit.linearEquiv_symm_apply
  IsDirectLimit.lift_of IsDirectLimit.Module.linearEquiv_apply IsDirectLimit.Module.lift_of
  IsDirectLimit.Equiv_apply AlgebraicGeometry.SubalgebraStages.specCone_π_app
  AlgebraicGeometry.SubalgebraStages.specLeg_specHom
  AlgebraicGeometry.SubalgebraStages.specLeg_specTrans AlgebraicGeometry.SubalgebraStages.cone_pt
  AlgebraicGeometry.SubalgebraStages.trans_fst AlgebraicGeometry.SubalgebraStages.diagram_obj
  AlgebraicGeometry.SubalgebraStages.leg_snd AlgebraicGeometry.SubalgebraStages.diagram_map
  AlgebraicGeometry.SubalgebraStages.specTrans_refl AlgebraicGeometry.SubalgebraStages.trans_snd
  AlgebraicGeometry.SubalgebraStages.specTrans_specHom
  AlgebraicGeometry.SubalgebraStages.specCone_pt AlgebraicGeometry.SubalgebraStages.cone_π_app
  AlgebraicGeometry.SubalgebraStages.specDiagram_map
  AlgebraicGeometry.SubalgebraStages.specDiagram_obj AlgebraicGeometry.SubalgebraStages.leg_fst
  AlgebraicGeometry.SubalgebraStages.leg_trans AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq 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
attribute [-simp] 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.OModulePresheaf.prod_obj AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
noncomputable section
namespace GoodReductionJacobian
namespace RelativeGroupLaw
namespace FgAbelianDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T T'}
Lean source · statement
theorem prop_snd_uT_of_prop_snd_algebraMap {Y : Scheme.{u}} (fY : Y ⟶ Spec (CommRingCat.of ↥T))
    (P : MorphismProperty Scheme.{u}) [P.IsStableUnderBaseChange] [P.RespectsIso]
    (hP : P (pullback.snd fY (Spec.map (CommRingCat.ofHom (algebraMap ↥T ↥T'))))) :
    P (pullback.snd fY (uT (le_flat T T')))
theorem

exists_fg_subalgebra_abelianSchemePropertyBundle_isPullback_of_isNoetherianRing

L510 ↗
GoodReductionJacobian.RelativeGroupLaw.exists_fg_subalgebra_abelianSchemePropertyBundle_isPullback_of_isNoetherianRing
Source context 12 commands

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

section ProofImplementation
namespace GoodReductionJacobian.RelativeGroupLaw
universe u
open CategoryTheory CategoryTheory.Limits AlgebraicGeometry NeronModelInfra GoodReductionJacobian
attribute [local simp]
  AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointBack_coe
attribute [local simp]
  AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointForth_coe
attribute [local simp]
  GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointBack_pointForth
attribute [local simp]
  GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.pointForth_pointBack
attribute [local simp]
  GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.transportIso_mul_coe
attribute [local simp]
  AbelianNoetherianDescent.GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent.rebaseIso_hom_snd
open _root_.AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.AbelianNoetherianDescent in
open GoodReductionJacobian.RelativeGroupLaw.FgAbelianDescent in
Lean source · full declaration
theorem exists_fg_subalgebra_abelianSchemePropertyBundle_isPullback_of_isNoetherianRing
    {A₀ : Type u} [CommRing A₀] [IsNoetherianRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
    {X : Scheme.{u}} {f : X ⟶ Spec (CommRingCat.of A)} (L : RelativeGroupLaw A f)
    (hX : AbelianSchemePropertyBundle A f) (s : Finset A) :
    ∃ (T : Subalgebra A₀ A), T.FG ∧ (↑s : Set A) ⊆ T ∧
      ∃ (X₀ : Scheme.{u}) (f₀ : X₀ ⟶ Spec (CommRingCat.of ↥T)) (L₀ : RelativeGroupLaw ↥T f₀) (π : X
        ⟶ X₀)
        (hπ : IsPullback π f f₀ (Spec.map (CommRingCat.ofHom (algebraMap ↥T A)))),
        AbelianSchemePropertyBundle ↥T f₀ ∧ GeometricallyConnected f₀ ∧ (L.IsCommutative →
          L₀.IsCommutative) ∧
        ∀ {T' : Scheme.{u}} (t : T' ⟶ Spec (CommRingCat.of A)) (x y : SchemeHomOver t f),
          (L.mul t x y).1 ≫ π =
            (L₀.mul (t ≫ Spec.map (CommRingCat.ofHom (algebraMap ↥T A)))
              ⟨x.1 ≫ π, by rw [Category.assoc, hπ.w, ← Category.assoc, x.2]⟩
              ⟨y.1 ≫ π, by rw [Category.assoc, hπ.w, ← Category.assoc, y.2]⟩).1 := by
  classical
  have : MorphismProperty.RespectsIso (@Smooth : MorphismProperty Scheme.{u}) :=
    MorphismProperty.IsStableUnderBaseChange.respectsIso
  have : MorphismProperty.RespectsIso (@GeometricallyConnected : MorphismProperty Scheme.{u}) :=
    MorphismProperty.IsStableUnderBaseChange.respectsIso
  have : Smooth f := hX.smooth
  have : IsProper f := hX.proper
  have hGC : GeometricallyConnected f := geometricallyConnected_of_bundle L hX
  obtain ⟨T₁, hT₁, hs₁, Y₁, f₁, π₁, hlfp₁, hqc₁, hqs₁, hπ₁⟩ :=
    AlgebraicGeometry.exists_fg_subalgebra_isPullback_of_locallyOfFinitePresentation (A₀ := A₀) (A
      := A) f s
  have := hlfp₁
  have := hqc₁
  have := hqs₁
  have : Smooth (pullback.snd f₁ (uA T₁)) := prop_snd_of_isPullback hπ₁ @Smooth inferInstance
  obtain ⟨U₂, hU₂, -, hS₂⟩ :=
    AlgebraicGeometry.Smooth.exists_fg_subalgebra_of_smooth_pullback_snd (A₀ := ↥T₁) (A := A) f₁ ∅
  let T₂ : Subalgebra A₀ A := flat T₁ U₂
  have hT₂ : T₂.FG := flat_fg T₁ U₂ hT₁ hU₂
  have h₁₂ : T₁ ≤ T₂ := le_flat T₁ U₂
  let Y₂ : Scheme.{u} := pullback f₁ (uT h₁₂)
  let f₂ : Y₂ ⟶ Spec (CommRingCat.of ↥T₂) := pullback.snd f₁ (uT h₁₂)
  have hS₂' : Smooth f₂ := prop_snd_uT_of_prop_snd_algebraMap f₁ @Smooth hS₂
  let π₂ : X ⟶ Y₂ := pullback.lift π₁ (f ≫ uA T₂) (by rw [Category.assoc, uA_uT, hπ₁.w])
  have hπ₂ : IsPullback π₂ f f₂ (uA T₂) := isPullback_lift_of_fac hπ₁ (uT h₁₂) (uA T₂) (uA_uT h₁₂)
  have : IsSeparated (pullback.snd f₂ (uA T₂)) := prop_snd_of_isPullback hπ₂ @IsSeparated
    inferInstance
  obtain ⟨U₃, hU₃, -, hSep₃⟩ :=
    AlgebraicGeometry.IsSeparated.exists_fg_subalgebra_of_isSeparated_pullback_snd (A₀ := ↥T₂) (A
      := A) f₂ ∅
  let T₃ : Subalgebra A₀ A := flat T₂ U₃
  have hT₃ : T₃.FG := flat_fg T₂ U₃ hT₂ hU₃
  have h₂₃ : T₂ ≤ T₃ := le_flat T₂ U₃
  let Y₃ : Scheme.{u} := pullback f₂ (uT h₂₃)
  let f₃ : Y₃ ⟶ Spec (CommRingCat.of ↥T₃) := pullback.snd f₂ (uT h₂₃)
  have hSep₃' : IsSeparated f₃ := prop_snd_uT_of_prop_snd_algebraMap f₂ @IsSeparated hSep₃
  let π₃ : X ⟶ Y₃ := pullback.lift π₂ (f ≫ uA T₃) (by rw [Category.assoc, uA_uT, hπ₂.w])
  have hπ₃ : IsPullback π₃ f f₃ (uA T₃) := isPullback_lift_of_fac hπ₂ (uT h₂₃) (uA T₃) (uA_uT h₂₃)
  have : IsNoetherianRing ↥T₃ := isNoetherianRing_stage hT₃
  have : IsProper (pullback.snd f₃ (uA T₃)) := prop_snd_of_isPullback hπ₃ @IsProper inferInstance
  obtain ⟨U₄, hU₄, -, hP₄⟩ :=
    AlgebraicGeometry.IsProper.exists_fg_subalgebra_of_isProper_pullback_snd (A₀ := ↥T₃) (A := A)
      f₃ ∅
  let T₄ : Subalgebra A₀ A := flat T₃ U₄
  have hT₄ : T₄.FG := flat_fg T₃ U₄ hT₃ hU₄
  have h₃₄ : T₃ ≤ T₄ := le_flat T₃ U₄
  let Y₄ : Scheme.{u} := pullback f₃ (uT h₃₄)
  let f₄ : Y₄ ⟶ Spec (CommRingCat.of ↥T₄) := pullback.snd f₃ (uT h₃₄)
  have hP₄' : IsProper f₄ := prop_snd_uT_of_prop_snd_algebraMap f₃ @IsProper hP₄
  let π₄ : X ⟶ Y₄ := pullback.lift π₃ (f ≫ uA T₄) (by rw [Category.assoc, uA_uT, hπ₃.w])
  have hπ₄ : IsPullback π₄ f f₄ (uA T₄) := isPullback_lift_of_fac hπ₃ (uT h₃₄) (uA T₄) (uA_uT h₃₄)
  have : IsNoetherianRing ↥T₄ := isNoetherianRing_stage hT₄
  have : GeometricallyConnected (pullback.snd f₄ (uA T₄)) :=
    prop_snd_of_isPullback hπ₄ @GeometricallyConnected inferInstance
  obtain ⟨U₅, hU₅, -, hG₅⟩ :=
    GeometricallyConnected.exists_fg_subalgebra_of_geometricallyConnected_pullback_snd
      (A₀ := ↥T₄) (A := A) f₄ ∅
  let T₅ : Subalgebra A₀ A := flat T₄ U₅
  have hT₅ : T₅.FG := flat_fg T₄ U₅ hT₄ hU₅
  have h₄₅ : T₄ ≤ T₅ := le_flat T₄ U₅
  let Y₅ : Scheme.{u} := pullback f₄ (uT h₄₅)
  let f₅ : Y₅ ⟶ Spec (CommRingCat.of ↥T₅) := pullback.snd f₄ (uT h₄₅)
  have hG₅' : GeometricallyConnected f₅ := prop_snd_uT_of_prop_snd_algebraMap f₄
    @GeometricallyConnected hG₅
  let π₅ : X ⟶ Y₅ := pullback.lift π₄ (f ≫ uA T₅) (by rw [Category.assoc, uA_uT, hπ₄.w])
  have hπ₅ : IsPullback π₅ f f₅ (uA T₅) := isPullback_lift_of_fac hπ₄ (uT h₄₅) (uA T₅) (uA_uT h₄₅)
  let e₅ : X ≅ pullback f₅ (uA T₅) := hπ₅.isoPullback
  have he₅ : e₅.hom ≫ pullback.snd f₅ (uA T₅) = f := hπ₅.isoPullback_hom_snd
  let L' : RelativeGroupLaw A (pullback.snd f₅ (uA T₅)) := transportIso e₅ he₅ L
  obtain ⟨U₆, hU₆, -, L₆, c, hciso, hc_snd, hc_fst, hcmul, hccomm⟩ :=
    exists_fg_subalgebra_relativeGroupLaw_pullback_snd_of_locallyOfFinitePresentation
      (A₀ := ↥T₅) (A := A) f₅ L' ∅
  have := hciso
  let T : Subalgebra A₀ A := flat T₅ U₆
  have hT : T.FG := flat_fg T₅ U₆ hT₅ hU₆
  have h₅T : T₅ ≤ T := le_flat T₅ U₆
  have hsT : (↑s : Set A) ⊆ T := hs₁.trans (h₁₂.trans (h₂₃.trans (h₃₄.trans (h₄₅.trans h₅T))))
  let b₆ : (Spec (CommRingCat.of ↥U₆) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥T₅) :=
    Spec.map (CommRingCat.ofHom (algebraMap ↥T₅ ↥U₆))
  let g₆ : pullback f₅ b₆ ⟶ Spec (CommRingCat.of ↥U₆) := pullback.snd f₅ b₆
  let a₆ : (Spec (CommRingCat.of A) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥U₆) :=
    Spec.map (CommRingCat.ofHom (algebraMap ↥U₆ A))
  let X₀ : Scheme.{u} := pullback g₆ (uF T₅ U₆)
  let f₀ : X₀ ⟶ Spec (CommRingCat.of ↥T) := pullback.snd g₆ (uF T₅ U₆)
  let L₀ : RelativeGroupLaw ↥T f₀ := L₆.baseChange (uF T₅ U₆)
  have : Smooth f₀ := inferInstance
  have : IsProper f₀ := inferInstance
  have : GeometricallyConnected f₀ := inferInstance
  have hA₀ : AbelianSchemePropertyBundle ↥T f₀ := bundle_of_geometricallyConnected L₀
  have hfac : uA T ≫ uF T₅ U₆ = a₆ := uA_uF T₅ U₆
  let κ : pullback g₆ a₆ ⟶ X₀ :=
    pullback.lift (pullback.fst g₆ a₆) (pullback.snd g₆ a₆ ≫ uA T)
      (by rw [Category.assoc, hfac, pullback.condition])
  have hκ_fst : κ ≫ pullback.fst g₆ (uF T₅ U₆) = pullback.fst g₆ a₆ := pullback.lift_fst _ _ _
  have hκ_snd : κ ≫ f₀ = pullback.snd g₆ a₆ ≫ uA T := pullback.lift_snd _ _ _
  have hκ : IsPullback κ (pullback.snd g₆ a₆) f₀ (uA T) := by
    refine IsPullback.of_right (h₁₂ := pullback.fst g₆ (uF T₅ U₆)) (h₂₂ := uF T₅ U₆) ?_ hκ_snd
      (IsPullback.of_hasPullback _ _)
    rw [hκ_fst, hfac]
    exact IsPullback.of_hasPullback _ _
  let ψ : X ⟶ pullback g₆ a₆ := e₅.hom ≫ c
  have hψ : ψ ≫ pullback.snd g₆ a₆ = f := by
    change (e₅.hom ≫ c) ≫ _ = f
    rw [Category.assoc, hc_snd, he₅]
  have : IsIso ψ := IsIso.comp_isIso
  have hψκ : IsPullback (ψ ≫ κ) f f₀ (uA T) := by
    have hsq : IsPullback ψ f (pullback.snd g₆ a₆) (𝟙 _) := IsPullback.of_horiz_isIso
      ⟨by rw [Category.comp_id, hψ]⟩
    have := hsq.paste_horiz hκ
    rwa [Category.id_comp] at this
  refine ⟨T, hT, hsT, X₀, f₀, L₀, ψ ≫ κ, hψκ, hA₀, inferInstance,
    fun hL => RelativeGroupLaw.IsCommutative.baseChange (uF T₅ U₆) (hccomm
      (transportIso_isCommutative e₅ he₅ hL)), ?_⟩
  intro T' t x y
  have ha := transportIso_hom e₅ he₅ L t x y
  have hb := hcmul t ⟨x.1 ≫ e₅.hom,
    by rw [Category.assoc, he₅, x.2]⟩ ⟨y.1 ≫ e₅.hom, by rw [Category.assoc, he₅, y.2]⟩
  have hab : (L.mul t x y).1 ≫ ψ =
      ((L₆.baseChange a₆).mul t
        ⟨(x.1 ≫ e₅.hom) ≫ c, by rw [Category.assoc, hc_snd, Category.assoc, he₅, x.2]⟩
        ⟨(y.1 ≫ e₅.hom) ≫ c, by rw [Category.assoc, hc_snd, Category.assoc, he₅, y.2]⟩).1 := by
    change (L.mul t x y).1 ≫ (e₅.hom ≫ c) = _
    rw [← Category.assoc, ha, hb]
  apply pullback.hom_ext
  · rw [Category.assoc, Category.assoc, hκ_fst, ← Category.assoc, hab, baseChange_mul_fst _ L₆ t,
      baseChange_mul_fst _ L₆ (t ≫ uA T)]
    refine mul_val_congr L₆ ?_ _ _ _ _ ?_ ?_
    · rw [Category.assoc, hfac]
    · change ((x.1 ≫ e₅.hom) ≫ c) ≫ pullback.fst g₆ a₆ =
        (x.1 ≫ (e₅.hom ≫ c) ≫ κ) ≫ pullback.fst g₆ (uF T₅ U₆)
      simp only [Category.assoc, hκ_fst]
    · change ((y.1 ≫ e₅.hom) ≫ c) ≫ pullback.fst g₆ a₆ =
        (y.1 ≫ (e₅.hom ≫ c) ≫ κ) ≫ pullback.fst g₆ (uF T₅ U₆)
      simp only [Category.assoc, hκ_fst]
  · have h1 : ((L.mul t x y).1 ≫ ψ ≫ κ) ≫ f₀ = t ≫ uA T := by
      rw [Category.assoc, Category.assoc, hκ_snd, ← Category.assoc, ← Category.assoc,
        Category.assoc _ ψ, hψ,
        (L.mul t x y).2]
    exact h1.trans ((L₀.mul (t ≫ uA T) _ _).2).symm

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