NYLL
Module

RelativeGroupLaw_exists_fg_subalgebra_relativeGroupLaw_pullback_snd_of_locallyOfFinitePresentation.lean

Nyll.AlgebraicCurves.Geometry.AbelianSchemes.Theorems.RelativeGroupLaw_exists_fg_subalgebra_relativeGroupLaw_pullback_snd_of_locallyOfFinitePresentation
165 declarations indexed from sourceView source ↗

Module documentation

Relative Group Law exists fg subalgebra relative Group Law pullback snd of locally Of

Finite Presentation

Imports21
Imported by2

Declarations

abbrev

W₂

L103 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.W₂
Documentation

The fibre square of the scheme over its base.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
Lean source · statement
abbrev W₂ : Scheme.{u}
abbrev

w₂

L106 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.w₂
Documentation

The structure morphism of the fibre square.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
Lean source · statement
abbrev w₂ : W₂ f ⟶ S
abbrev

W₃

L109 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.W₃
Documentation

The triple fibre product, grouped as a fibre square and a third factor.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
Lean source · statement
abbrev W₃ : Scheme.{u}
abbrev

w₃

L112 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.w₃
Documentation

The structure morphism of the triple fibre product.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
Lean source · statement
abbrev w₃ : W₃ f ⟶ S
abbrev

Yb

L117 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Yb
Documentation

The scheme obtained by base change along b.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
abbrev Yb : Scheme.{u}
abbrev

fb

L120 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.fb
Documentation

The structure morphism of the base-changed scheme.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
abbrev fb : Yb f b ⟶ B
abbrev

Q₂

L123 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Q₂
Documentation

The base change of the fibre square.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
abbrev Q₂ : Scheme.{u}
abbrev

Q₃

L126 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Q₃
Documentation

The base change of the triple fibre product.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
abbrev Q₃ : Scheme.{u}
def

p₁

L129 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁
Documentation

The first projection from the base-changed fibre square.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
def p₁ : Q₂ f b ⟶ Yb f b
def

p₂

L134 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₂
Documentation

The second projection from the base-changed fibre square.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
def p₂ : Q₂ f b ⟶ Yb f b
theorem

p₁_fst

L138 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁_fst
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
theorem p₁_fst : p₁ f b ≫ pullback.fst f b = pullback.fst _ _ ≫
  pullback.fst f f
theorem

p₁_snd

L140 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁_snd
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
theorem p₁_snd : p₁ f b ≫ pullback.snd f b = pullback.snd _ _
theorem

p₂_fst

L142 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₂_fst
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
theorem p₂_fst : p₂ f b ≫ pullback.fst f b = pullback.fst _ _ ≫
  pullback.snd f f
theorem

p₂_snd

L144 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₂_snd
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
Lean source · statement
theorem p₂_snd : p₂ f b ≫ pullback.snd f b = pullback.snd _ _
theorem

pair_aux₁

L149 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_aux₁
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_aux₁ {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b = y ≫ fb f b) :
    (x ≫ pullback.fst f b) ≫ f = (y ≫ pullback.fst f b) ≫ f
theorem

pair_aux₂

L158 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_aux₂
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_aux₂ {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b = y ≫ fb f b) :
    pullback.lift (x ≫ pullback.fst f b) (y ≫ pullback.fst f b) (pair_aux₁ x y h) ≫ w₂ f = (x ≫
      pullback.snd f b) ≫ b
def

pair

L164 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair
Documentation

Pairs two morphisms over the same base into the base-changed fibre square.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
def pair {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b = y ≫ fb f b) : T ⟶ Q₂ f b
theorem

pair_fst_fst

L168 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_fst_fst
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_fst_fst {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b =
  y ≫ fb f b) :
    pair x y h ≫ pullback.fst _ _ ≫ pullback.fst f f = x ≫ pullback.fst f b
theorem

pair_fst_snd

L173 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_fst_snd
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_fst_snd {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b =
  y ≫ fb f b) :
    pair x y h ≫ pullback.fst _ _ ≫ pullback.snd f f = y ≫ pullback.fst f b
theorem

pair_snd

L178 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_snd
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_snd {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b = y ≫
  fb f b) :
    pair x y h ≫ pullback.snd _ _ = x ≫ pullback.snd f b
theorem

pair_p₁

L183 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_p₁
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_p₁ {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b = y ≫
  fb f b) :
    pair x y h ≫ p₁ f b = x
theorem

pair_p₂

L190 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_p₂
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem pair_p₂ {T : Scheme.{u}} (x y : T ⟶ Yb f b) (h : x ≫ fb f b = y ≫
  fb f b) :
    pair x y h ≫ p₂ f b = y
theorem

hom_ext₂

L198 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.hom_ext₂
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
Lean source · statement
theorem hom_ext₂ {T : Scheme.{u}} {k k' : T ⟶ Q₂ f b} (h₁ : k ≫ p₁ f b = k' ≫ p₁ f b) (h₂ : k ≫ p₂
  f b = k' ≫ p₂ f b) :
    k = k'
theorem

p₁_fb

L209 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁_fb
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
Lean source · statement
theorem p₁_fb : p₁ f b ≫ fb f b = pullback.snd _ _
theorem

p₂_fb

L210 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₂_fb
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
Lean source · statement
theorem p₂_fb : p₂ f b ≫ fb f b = pullback.snd _ _
theorem

p₁_fb_eq_p₂_fb

L211 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁_fb_eq_p₂_fb
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
Lean source · statement
theorem p₁_fb_eq_p₂_fb : p₁ f b ≫ fb f b = p₂ f b ≫ fb f b
theorem

pair_p₁_p₂

L213 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_p₁_p₂
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
Lean source · statement
theorem pair_p₁_p₂ : pair (p₁ f b) (p₂ f b) (p₁_fb_eq_p₂_fb f b) = 𝟙 _
theorem

comp_pair

L218 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.comp_pair
Source context 23 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
Lean source · full declaration
theorem comp_pair {T T' : Scheme.{u}} (k : T' ⟶ T) (x y : T ⟶ Yb f b) (h : x ≫ fb f b =
  y ≫ fb f b) :
    k ≫ pair x y h = pair (k ≫ x) (k ≫ y) (by rw [Category.assoc, Category.assoc, h]) :=
  hom_ext₂ (by rw [Category.assoc, pair_p₁, pair_p₁]) (by rw [Category.assoc, pair_p₂, pair_p₂])
def

q₁₂

L226 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂
Documentation

The projection onto the first two factors of the base-changed triple product.

Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
def q₁₂ : Q₃ f b ⟶ Q₂ f b
def

q₃

L231 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₃
Documentation

The projection onto the third factor of the base-changed triple product.

Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
def q₃ : Q₃ f b ⟶ Yb f b
theorem

q₁₂_fst

L235 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂_fst
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₁₂_fst : q₁₂ f b ≫ pullback.fst (w₂ f) b = pullback.fst _ _ ≫
  pullback.fst _ _
theorem

q₁₂_snd

L237 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂_snd
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₁₂_snd : q₁₂ f b ≫ pullback.snd (w₂ f) b = pullback.snd _ _
theorem

q₃_fst

L239 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₃_fst
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₃_fst : q₃ f b ≫ pullback.fst f b = pullback.fst _ _ ≫
  pullback.snd _ _
theorem

q₃_snd

L241 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₃_snd
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₃_snd : q₃ f b ≫ pullback.snd f b = pullback.snd _ _
theorem

q₁₂_p₁_fb

L244 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂_p₁_fb
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₁₂_p₁_fb : (q₁₂ f b ≫ p₁ f b) ≫ fb f b = pullback.snd _ _
theorem

q₁₂_p₂_fb

L246 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂_p₂_fb
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₁₂_p₂_fb : (q₁₂ f b ≫ p₂ f b) ≫ fb f b = pullback.snd _ _
theorem

q₃_fb

L248 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₃_fb
Source context 24 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · statement
theorem q₃_fb : q₃ f b ≫ fb f b = pullback.snd _ _
theorem

triple_aux₁

L252 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_aux₁
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_aux₁ {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb f b = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) : (pair x y hxy ≫ pullback.fst _ _) ≫ w₂ f = (z ≫ pullback.fst
      f b) ≫ f
theorem

triple_aux₂

L260 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_aux₂
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_aux₂ {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb f b = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) :
    pullback.lift (pair x y hxy ≫ pullback.fst _ _) (z ≫ pullback.fst f b) (triple_aux₁ x y z hxy
      hxz) ≫ w₃ f =
      (x ≫ pullback.snd f b) ≫ b
def

triple

L268 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple
Documentation

Combines three morphisms over the same base into the base-changed triple product.

Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
def triple {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb f b = y ≫ fb f b) (hxz : x ≫ fb f b
  = z ≫ fb f b) :
    T ⟶ Q₃ f b
theorem

triple_fst_fst

L275 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_fst_fst
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_fst_fst {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb
  f b = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) :
    triple x y z hxy hxz ≫ pullback.fst _ _ ≫ pullback.fst _ _ = pair x y hxy ≫ pullback.fst _ _
theorem

triple_fst_snd

L282 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_fst_snd
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_fst_snd {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb
  f b = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) :
    triple x y z hxy hxz ≫ pullback.fst _ _ ≫ pullback.snd _ _ = z ≫ pullback.fst f b
theorem

triple_snd

L288 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_snd
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_snd {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb f b
  = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) :
    triple x y z hxy hxz ≫ pullback.snd _ _ = x ≫ pullback.snd f b
theorem

triple_q₁₂

L294 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_q₁₂
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_q₁₂ {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb f b
  = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) : triple x y z hxy hxz ≫ q₁₂ f b = pair x y hxy
theorem

triple_q₃

L301 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_q₃
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem triple_q₃ {T : Scheme.{u}} (x y z : T ⟶ Yb f b) (hxy : x ≫ fb f b =
  y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) : triple x y z hxy hxz ≫ q₃ f b = z
theorem

hom_ext₃

L309 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.hom_ext₃
Source context 25 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · statement
theorem hom_ext₃ {T : Scheme.{u}} {k k' : T ⟶ Q₃ f b} (h₁ : k ≫ q₁₂ f b = k' ≫ q₁₂ f b) (h₂ : k ≫
  q₃ f b = k' ≫ q₃ f b) :
    k = k'
theorem

triple_universal

L320 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_universal
Source context 26 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
Lean source · full declaration
theorem triple_universal :
    triple (q₁₂ f b ≫ p₁ f b) (q₁₂ f b ≫ p₂ f b) (q₃ f b)
      (by rw [q₁₂_p₁_fb, q₁₂_p₂_fb]) (by rw [q₁₂_p₁_fb, q₃_fb]) = 𝟙 _ :=
  hom_ext₃
    (by rw [triple_q₁₂, Category.id_comp]; exact hom_ext₂ (by rw [pair_p₁]) (by rw [pair_p₂]))
    (by rw [triple_q₃, Category.id_comp])
theorem

comp_triple

L329 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.comp_triple
Source context 27 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
Lean source · full declaration
theorem comp_triple {T T' : Scheme.{u}} (k : T' ⟶ T) (x y z : T ⟶ Yb f b) (hxy : x ≫ fb
  f b = y ≫ fb f b)
    (hxz : x ≫ fb f b = z ≫ fb f b) :
    k ≫ triple x y z hxy hxz = triple (k ≫ x) (k ≫ y) (k ≫ z)
      (by rw [Category.assoc, Category.assoc, hxy])
      (by rw [Category.assoc, Category.assoc, hxz]) :=
  hom_ext₃ (by rw [Category.assoc, triple_q₁₂, triple_q₁₂, comp_pair])
    (by rw [Category.assoc, triple_q₃, triple_q₃])
def

tW

L344 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW
Documentation

The morphism between two base changes induced by a morphism of their bases.

Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
def tW : pullback w b₂ ⟶ pullback w b₁
theorem

tW_fst

L348 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_fst
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_fst : tW w b₁ b₂ u hu ≫ pullback.fst w b₁ = pullback.fst w b₂
theorem

tW_snd

L350 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_snd
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_snd : tW w b₁ b₂ u hu ≫ pullback.snd w b₁ = pullback.snd w b₂ ≫
  u
theorem

eq_of_comp_tW_eq

L353 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.eq_of_comp_tW_eq
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem eq_of_comp_tW_eq {T : Scheme.{u}} {k k' : T ⟶ pullback w b₂} (h₁ : k ≫ tW w b₁ b₂ u hu = k'
  ≫ tW w b₁ b₂ u hu)
    (h₂ : k ≫ pullback.snd w b₂ = k' ≫ pullback.snd w b₂) : k = k'
theorem

tW_p₁

L359 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_p₁
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_p₁ : tW (w₂ f) b₁ b₂ u hu ≫ p₁ f b₁ = p₁ f b₂ ≫ tW f b₁ b₂ u hu
theorem

tW_p₂

L364 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_p₂
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_p₂ : tW (w₂ f) b₁ b₂ u hu ≫ p₂ f b₁ = p₂ f b₂ ≫ tW f b₁ b₂ u hu
theorem

tW_q₁₂

L369 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_q₁₂
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_q₁₂ : tW (w₃ f) b₁ b₂ u hu ≫ q₁₂ f b₁ = q₁₂ f b₂ ≫ tW (w₂ f) b₁ b₂ u hu
theorem

tW_q₃

L374 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_q₃
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_q₃ : tW (w₃ f) b₁ b₂ u hu ≫ q₃ f b₁ = q₃ f b₂ ≫ tW f b₁ b₂ u hu
theorem

tW_fb

L379 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_fb
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · statement
theorem tW_fb : tW f b₁ b₂ u hu ≫ fb f b₁ = fb f b₂ ≫ u
theorem

pair_tW

L381 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_tW
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · full declaration
theorem pair_tW {T : Scheme.{u}} (x y : T ⟶ Yb f b₂) (h : x ≫ fb f b₂ = y ≫ fb f b₂) :
    pair (x ≫ tW f b₁ b₂ u hu) (y ≫ tW f b₁ b₂ u hu)
      (by rw [Category.assoc, Category.assoc, tW_fb, reassoc_of% h]) =
      pair x y h ≫ tW (w₂ f) b₁ b₂ u hu :=
  hom_ext₂ (by rw [pair_p₁, Category.assoc, tW_p₁, pair_p₁_assoc])
    (by rw [pair_p₂, Category.assoc, tW_p₂, pair_p₂_assoc])
theorem

triple_tW

L388 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_tW
Source context 29 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Presentations
variable {X S : Scheme.{u}} (f : X ⟶ S)
variable {B : Scheme.{u}} (b : B ⟶ S)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
variable (f b)
variable {f b}
section Trans
variable {W B₁ B₂ : Scheme.{u}} (w : W ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B₁) (hu : u ≫ b₁
  = b₂)
Lean source · full declaration
theorem triple_tW {T : Scheme.{u}} (x y z : T ⟶ Yb f b₂) (hxy : x ≫ fb f b₂ = y ≫ fb f b₂)
    (hxz : x ≫ fb f b₂ = z ≫ fb f b₂) :
    triple (x ≫ tW f b₁ b₂ u hu) (y ≫ tW f b₁ b₂ u hu) (z ≫ tW f b₁ b₂ u hu)
        (by rw [Category.assoc, Category.assoc, tW_fb, reassoc_of% hxy])
        (by rw [Category.assoc, Category.assoc, tW_fb, reassoc_of% hxz]) =
      triple x y z hxy hxz ≫ tW (w₃ f) b₁ b₂ u hu :=
  hom_ext₃ (by rw [triple_q₁₂, pair_tW, Category.assoc, tW_q₁₂, triple_q₁₂_assoc])
    (by rw [triple_q₃, Category.assoc, tW_q₃, triple_q₃_assoc])
structure

Dat

L406 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat
Documentation

Multiplication, unit, and inversion morphisms over the base.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
Lean source · full declaration
structure Dat where
  /-- The multiplication morphism. -/
  m : Q₂ f b ⟶ Yb f b
  /-- The unit section. -/
  e : B ⟶ Yb f b
  /-- The inversion morphism. -/
  ι : Yb f b ⟶ Yb f b
  /-- Multiplication is a morphism over the base. -/
  hm : m ≫ fb f b = pullback.snd _ _
  /-- The unit is a section of the structure morphism. -/
  he : e ≫ fb f b = 𝟙 _
  /-- Inversion is a morphism over the base. -/
  hι : ι ≫ fb f b = fb f b
theorem

q₁₂_m_fb

L424 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.q₁₂_m_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem q₁₂_m_fb : (q₁₂ f b ≫ D.m) ≫ fb f b = q₃ f b ≫ fb f b
theorem

p₂_m₂₃_fb

L427 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.p₂_m₂₃_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · full declaration
theorem p₂_m₂₃_fb : (q₁₂ f b ≫ p₁ f b) ≫ fb f b =
    (pair (q₁₂ f b ≫ p₂ f b) (q₃ f b) (by rw [q₁₂_p₂_fb, q₃_fb]) ≫ D.m) ≫ fb f b := by
  rw [q₁₂_p₁_fb, Category.assoc, D.hm, pair_snd, q₁₂_p₂_fb]
theorem

fb_e_fb

L431 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.fb_e_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem fb_e_fb : (fb f b ≫ D.e) ≫ fb f b = 𝟙 _ ≫ fb f b
theorem

ι_fb

L434 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.ι_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem ι_fb : D.ι ≫ fb f b = 𝟙 _ ≫ fb f b
def

assocL

L437 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assocL
Documentation

The multiplication composite representing the left-associated product.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
def assocL : Q₃ f b ⟶ Yb f b
def

assocR

L440 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assocR
Documentation

The multiplication composite representing the right-associated product.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
def assocR : Q₃ f b ⟶ Yb f b
def

oneMulL

L445 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.oneMulL
Documentation

The multiplication composite with the unit in the first factor.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
def oneMulL : Yb f b ⟶ Yb f b
def

invMulL

L448 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invMulL
Documentation

The multiplication composite with the inverse in the first factor.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
def invMulL : Yb f b ⟶ Yb f b
def

unitC

L451 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.unitC
Documentation

The constant unit endomorphism induced by the structure morphism.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
def unitC : Yb f b ⟶ Yb f b
def

commL

L454 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.commL
Documentation

The multiplication morphism with its two factors interchanged.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
def commL : Q₂ f b ⟶ Yb f b
theorem

assocL_fb

L456 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assocL_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem assocL_fb : D.assocL ≫ fb f b = pullback.snd _ _
theorem

assocR_fb

L458 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assocR_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem assocR_fb : D.assocR ≫ fb f b = pullback.snd _ _
theorem

oneMulL_fb

L460 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.oneMulL_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem oneMulL_fb : D.oneMulL ≫ fb f b = fb f b
theorem

invMulL_fb

L462 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invMulL_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem invMulL_fb : D.invMulL ≫ fb f b = fb f b
theorem

unitC_fb

L464 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.unitC_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem unitC_fb : D.unitC ≫ fb f b = fb f b
theorem

commL_fb

L466 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.commL_fb
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · statement
theorem commL_fb : D.commL ≫ fb f b = pullback.snd _ _
structure

Laws

L470 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.Laws
Documentation

Associativity, left-unit, and left-inverse equations for the structural morphisms.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
Lean source · full declaration
structure Laws : Prop where
  /-- The multiplication morphisms satisfy associativity. -/
  assoc : D.assocL = D.assocR
  /-- Multiplication by the unit on the left is the identity. -/
  one_mul : D.oneMulL = 𝟙 _
  /-- Multiplication by the inverse on the left equals the constant unit morphism. -/
  inv_mul : D.invMulL = D.unitC
theorem

res_m_aux

L480 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_m_aux
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_m_aux : (tW (w₂ f) b b₂ u hu ≫ D.m ≫ pullback.fst f b) ≫ f = pullback.snd (w₂ f) b₂ ≫
  b₂
theorem

res_e_aux

L484 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_e_aux
Source context 23 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
include hu in
Lean source · statement
theorem res_e_aux : (u ≫ D.e ≫ pullback.fst f b) ≫ f = 𝟙 _ ≫ b₂
theorem

res_ι_aux

L487 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_ι_aux
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_ι_aux : (tW f b b₂ u hu ≫ D.ι ≫ pullback.fst f b) ≫ f = fb f b₂ ≫ b₂
def

res

L491 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res
Documentation

Base changes the multiplication, unit, and inversion morphisms.

Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · full declaration
def res : Dat f b₂ where
  m := pullback.lift (tW (w₂ f) b b₂ u hu ≫ D.m ≫ pullback.fst f b) (pullback.snd _ _) (D.res_m_aux
    b₂ u hu)
  e := pullback.lift (u ≫ D.e ≫ pullback.fst f b) (𝟙 _) (D.res_e_aux b₂ u hu)
  ι := pullback.lift (tW f b b₂ u hu ≫ D.ι ≫ pullback.fst f b) (fb f b₂) (D.res_ι_aux b₂ u hu)
  hm := pullback.lift_snd _ _ _
  he := pullback.lift_snd _ _ _
  hι := pullback.lift_snd _ _ _
theorem

res_m_fst

L500 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_m_fst
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_m_fst : (D.res b₂ u hu).m ≫ pullback.fst f b₂ = tW (w₂ f) b b₂ u hu ≫ D.m ≫
  pullback.fst f b
theorem

res_e_fst

L503 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_e_fst
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_e_fst : (D.res b₂ u hu).e ≫ pullback.fst f b₂ = u ≫ D.e ≫ pullback.fst f b
theorem

res_ι_fst

L505 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_ι_fst
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_ι_fst : (D.res b₂ u hu).ι ≫ pullback.fst f b₂ = tW f b b₂ u hu ≫ D.ι ≫
  pullback.fst f b
theorem

res_m_tW

L509 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_m_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_m_tW : (D.res b₂ u hu).m ≫ tW f b b₂ u hu = tW (w₂ f) b b₂ u hu ≫ D.m
theorem

res_e_tW

L515 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_e_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_e_tW : (D.res b₂ u hu).e ≫ tW f b b₂ u hu = u ≫ D.e
theorem

res_ι_tW

L522 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_ι_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem res_ι_tW : (D.res b₂ u hu).ι ≫ tW f b b₂ u hu = tW f b b₂ u hu ≫ D.ι
theorem

assocL_tW

L529 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assocL_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem assocL_tW : (D.res b₂ u hu).assocL ≫ tW f b b₂ u hu = tW (w₃ f) b b₂ u hu ≫ D.assocL
theorem

assocR_tW

L536 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assocR_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem assocR_tW : (D.res b₂ u hu).assocR ≫ tW f b b₂ u hu = tW (w₃ f) b b₂ u hu ≫ D.assocR
theorem

oneMulL_tW

L547 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.oneMulL_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem oneMulL_tW : (D.res b₂ u hu).oneMulL ≫ tW f b b₂ u hu = tW f b b₂ u hu ≫ D.oneMulL
theorem

invMulL_tW

L554 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invMulL_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem invMulL_tW : (D.res b₂ u hu).invMulL ≫ tW f b b₂ u hu = tW f b b₂ u hu ≫ D.invMulL
theorem

unitC_tW

L561 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.unitC_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem unitC_tW : (D.res b₂ u hu).unitC ≫ tW f b b₂ u hu = tW f b b₂ u hu ≫ D.unitC
theorem

commL_tW

L564 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.commL_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem commL_tW : (D.res b₂ u hu).commL ≫ tW f b b₂ u hu = tW (w₂ f) b b₂ u hu ≫ D.commL
theorem

m_tW

L571 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.m_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem m_tW : (D.res b₂ u hu).m ≫ tW f b b₂ u hu = tW (w₂ f) b b₂ u hu ≫ D.m
theorem

eq_of_tW

L573 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.eq_of_tW
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem eq_of_tW {T : Scheme.{u}} {k k' : T ⟶ Yb f b₂} (h₁ : k ≫ tW f b b₂ u hu = k' ≫ tW f b b₂ u
  hu)
    (h₂ : k ≫ fb f b₂ = k' ≫ fb f b₂) : k = k'
theorem

assoc_res

L578 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assoc_res
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem assoc_res (h : D.assocL = D.assocR) : (D.res b₂ u hu).assocL = (D.res b₂ u hu).assocR
theorem

one_mul_res

L581 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.one_mul_res
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem one_mul_res (h : D.oneMulL = 𝟙 _) : (D.res b₂ u hu).oneMulL = 𝟙 _
theorem

inv_mul_res

L585 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.inv_mul_res
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem inv_mul_res (h : D.invMulL = D.unitC) : (D.res b₂ u hu).invMulL = (D.res b₂ u hu).unitC
theorem

res

L588 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.Laws.res
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem Laws.res (hD : D.Laws) : (D.res b₂ u hu).Laws
theorem

comm_res

L591 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.comm_res
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Data
variable {X S : Scheme.{u}} (f : X ⟶ S) {B : Scheme.{u}} (b : B ⟶ S)
namespace Dat
variable {f b} (D : Dat f b)
variable {B₂ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
Lean source · statement
theorem comm_res (hD : D.commL = D.m) : (D.res b₂ u hu).commL = (D.res b₂ u hu).m
theorem

pair_congr

L601 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_congr
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Congr
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
Lean source · statement
theorem pair_congr {T : Scheme.{u}} {x y x' y' : T ⟶ Yb f b} (h : x ≫ fb f b = y ≫ fb f b) (h' : x'
  ≫ fb f b = y' ≫ fb f b)
    (hx : x = x') (hy : y = y') : pair x y h = pair x' y' h'
abbrev

qb

L614 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.qb
Documentation

The structure morphism of the base-changed fibre square.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
abbrev qb : Q₂ f b ⟶ Spec (CommRingCat.of R')
def

P₁

L617 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.P₁
Documentation

The first fibre-square projection regarded as a morphism over the base.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
def P₁ : SchemeHomOver (qb f b) (fb f b)
def

P₂

L620 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.P₂
Documentation

The second fibre-square projection regarded as a morphism over the base.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
def P₂ : SchemeHomOver (qb f b) (fb f b)
def

Pid

L623 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Pid
Documentation

The identity morphism regarded as a morphism over the base.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
def Pid : SchemeHomOver (fb f b) (fb f b)
theorem

P₁_val

L625 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.P₁_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
theorem P₁_val : (P₁ f b).1 = p₁ f b
theorem

P₂_val

L626 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.P₂_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
theorem P₂_val : (P₂ f b).1 = p₂ f b
theorem

Pid_val

L627 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Pid_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
Lean source · statement
theorem Pid_val : (Pid f b).1 = 𝟙 _
def

ofLaw

L632 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.ofLaw
Documentation

Extracts multiplication, unit, and inversion morphisms from a relative group law.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · full declaration
def ofLaw : Dat f b where
  m := (L.mul (qb f b) (P₁ f b) (P₂ f b)).1
  e := (L.one (𝟙 _)).1
  ι := (L.inv (fb f b) (Pid f b)).1
  hm := (L.mul (qb f b) (P₁ f b) (P₂ f b)).2
  he := (L.one (𝟙 _)).2
  hι := (L.inv (fb f b) (Pid f b)).2
theorem

mul_fb_eq

L640 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.mul_fb_eq
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · statement
theorem mul_fb_eq {T : Scheme.{u}} {t : T ⟶ Spec (CommRingCat.of R')} (x y : SchemeHomOver t (fb f
  b)) :
    x.1 ≫ fb f b = y.1 ≫ fb f b
theorem

mul_val

L644 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.mul_val
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · statement
theorem mul_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x y : SchemeHomOver t (fb f
  b)) :
    (L.mul t x y).1 = pair x.1 y.1 (mul_fb_eq x y) ≫ (ofLaw L).m
theorem

one_val

L657 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.one_val
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · statement
theorem one_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) : (L.one t).1 = t ≫ (ofLaw L).e
theorem

inv_val

L663 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.inv_val
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · statement
theorem inv_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x : SchemeHomOver t (fb f b)) :
    (L.inv t x).1 = x.1 ≫ (ofLaw L).ι
theorem

pair_P₁_P₂

L677 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_P₁_P₂
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · statement
theorem pair_P₁_P₂ : pair (P₁ f b).1 (P₂ f b).1 (mul_fb_eq (P₁ f b) (P₂ f b)) = 𝟙 _
theorem

ofLaw_laws

L680 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.ofLaw_laws
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · full declaration
theorem ofLaw_laws : (ofLaw L).Laws where
  assoc := by
    let t₃ : Q₃ f b ⟶ Spec (CommRingCat.of R') := pullback.snd (w₃ f) b
    let Z₁ : SchemeHomOver t₃ (fb f b) := ⟨q₁₂ f b ≫ p₁ f b, q₁₂_p₁_fb f b⟩
    let Z₂ : SchemeHomOver t₃ (fb f b) := ⟨q₁₂ f b ≫ p₂ f b, q₁₂_p₂_fb f b⟩
    let Z₃ : SchemeHomOver t₃ (fb f b) := ⟨q₃ f b, q₃_fb f b⟩
    have h := congrArg Subtype.val (L.mul_assoc t₃ Z₁ Z₂ Z₃)
    simp only [mul_val] at h
    have h12 : pair Z₁.1 Z₂.1 (mul_fb_eq Z₁ Z₂) = q₁₂ f b := by
      rw [← Category.comp_id (q₁₂ f b), ← pair_p₁_p₂ f b, comp_pair]
    rw [Dat.assocL, Dat.assocR]
    refine Eq.trans ?_ h
    congr 1
    exact pair_congr _ _ (by rw [h12]) rfl
  one_mul := by
    have h := congrArg Subtype.val (L.one_mul (fb f b) (Pid f b))
    simp only [mul_val, one_val] at h
    exact h
  inv_mul := by
    have h := congrArg Subtype.val (L.inv_mul_cancel (fb f b) (Pid f b))
    simp only [mul_val, inv_val, one_val] at h
    rw [Dat.invMulL, Dat.unitC]
    refine Eq.trans ?_ h
    congr 1
theorem

ofLaw_comm

L705 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.ofLaw_comm
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfLaw
variable {X S : Scheme.{u}} (f : X ⟶ S) {R' : Type u} [CommRing R'] (b : Spec (CommRingCat.of R') ⟶
  S)
variable {f b} (L : RelativeGroupLaw R' (fb f b))
Lean source · statement
theorem ofLaw_comm (hL : L.IsCommutative) : (ofLaw L).commL = (ofLaw L).m
def

mulPt

L719 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.mulPt
Documentation

Multiplies two points over the same test scheme using the multiplication morphism.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
def Dat.mulPt {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x y : SchemeHomOver t (fb f b)) :
    SchemeHomOver t (fb f b)
def

onePt

L724 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.onePt
Documentation

The unit point over a test scheme.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
def Dat.onePt {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) : SchemeHomOver t (fb f b)
def

invPt

L728 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invPt
Documentation

Inverts a point over a test scheme using the inversion morphism.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
def Dat.invPt {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x : SchemeHomOver t (fb f b)) :
    SchemeHomOver t (fb f b)
theorem

mulPt_val

L732 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.mulPt_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
theorem Dat.mulPt_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x y :
  SchemeHomOver t (fb f b)) :
    (D.mulPt t x y).1 = pair x.1 y.1 (mul_fb_eq x y) ≫ D.m
theorem

onePt_val

L736 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.onePt_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
theorem Dat.onePt_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) : (D.onePt
  t).1 = t ≫ D.e
theorem

invPt_val

L739 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invPt_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
theorem Dat.invPt_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x :
  SchemeHomOver t (fb f b)) :
    (D.invPt t x).1 = x.1 ≫ D.ι
theorem

mulPt_assoc

L743 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.mulPt_assoc
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
include hD in
Lean source · statement
theorem Dat.mulPt_assoc {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x y z : SchemeHomOver
  t (fb f b)) :
    D.mulPt t (D.mulPt t x y) z = D.mulPt t x (D.mulPt t y z)
theorem

onePt_mulPt

L758 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.onePt_mulPt
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
include hD in
Lean source · statement
theorem Dat.onePt_mulPt {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x : SchemeHomOver t
  (fb f b)) :
    D.mulPt t (D.onePt t) x = x
theorem

invPt_mulPt

L769 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invPt_mulPt
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
include hD in
Lean source · statement
theorem Dat.invPt_mulPt {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x : SchemeHomOver t
  (fb f b)) :
    D.mulPt t (D.invPt t x) x = D.onePt t
theorem

mulPt_onePt

L780 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.mulPt_onePt
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
include hD in
Lean source · statement
theorem Dat.mulPt_onePt {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x : SchemeHomOver t
  (fb f b)) :
    D.mulPt t x (D.onePt t) = x
def

toLaw

L791 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.toLaw
Documentation

Constructs the relative group law from structural morphisms satisfying the group equations.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · full declaration
def Dat.toLaw : RelativeGroupLaw R' (fb f b) where
  mul t x y := D.mulPt t x y
  one t := D.onePt t
  inv t x := D.invPt t x
  mul_assoc t x y z := D.mulPt_assoc hD t x y z
  one_mul t x := D.onePt_mulPt hD t x
  mul_one t x := D.mulPt_onePt hD t x
  inv_mul_cancel t x := D.invPt_mulPt hD t x
  mul_natural t t' ψ hψ x y := by
    apply Subtype.ext
    simp only [schemeHomOverComp_coe, Dat.mulPt_val, comp_pair_assoc]
theorem

toLaw_mul_val

L803 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.toLaw_mul_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
theorem Dat.toLaw_mul_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) (x y :
  SchemeHomOver t (fb f b)) :
    ((D.toLaw hD).mul t x y).1 = pair x.1 y.1 (mul_fb_eq x y) ≫ D.m
theorem

toLaw_one_val

L808 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.toLaw_one_val
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
theorem Dat.toLaw_one_val {T : Scheme.{u}} (t : T ⟶ Spec (CommRingCat.of R')) :
    ((D.toLaw hD).one t).1 = t ≫ D.e
theorem

toLaw_comm

L812 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.toLaw_comm
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section OfDat
variable {X S : Scheme.{u}} {f : X ⟶ S} {R' : Type u} [CommRing R'] {b : Spec (CommRingCat.of R') ⟶
  S}
  (D : Dat f b) (hD : D.Laws)
Lean source · statement
theorem Dat.toLaw_comm (hc : D.commL = D.m) : (D.toLaw hD).IsCommutative
theorem

isPullback_paste

L831 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.isPullback_paste
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Rebase
variable {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S'' ⟶ S) (h : b ≫ a
  = c)
include h in
Lean source · statement
theorem isPullback_paste :
    IsPullback (pullback.fst (pullback.snd f a) b ≫ pullback.fst f a) (pullback.snd (pullback.snd f
      a) b) f c
def

rebaseIso

L838 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.rebaseIso
Documentation

Identifies an iterated base change with the base change along the composite.

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Rebase
variable {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S'' ⟶ S) (h : b ≫ a
  = c)
Lean source · statement
def rebaseIso : pullback (pullback.snd f a) b ≅ pullback f c
theorem

rebaseIso_hom_snd

L841 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.rebaseIso_hom_snd
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Rebase
variable {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S'' ⟶ S) (h : b ≫ a
  = c)
Lean source · statement
theorem rebaseIso_hom_snd : (rebaseIso f a b c h).hom ≫ pullback.snd f c =
  pullback.snd _ _
theorem

rebaseIso_hom_fst

L845 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.rebaseIso_hom_fst
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Rebase
variable {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S'' ⟶ S) (h : b ≫ a
  = c)
Lean source · statement
theorem rebaseIso_hom_fst :
    (rebaseIso f a b c h).hom ≫ pullback.fst f c = pullback.fst _ _ ≫ pullback.fst f a
theorem

fst_eq_rebaseIso_hom_tW

L849 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.fst_eq_rebaseIso_hom_tW
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Rebase
variable {X S S' S'' : Scheme.{u}} (f : X ⟶ S) (a : S' ⟶ S) (b : S'' ⟶ S') (c : S'' ⟶ S) (h : b ≫ a
  = c)
Lean source · statement
theorem fst_eq_rebaseIso_hom_tW : pullback.fst (pullback.snd f a) b = (rebaseIso f a b c h).hom ≫
  tW f a c b h
theorem

tW_comp

L862 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_comp
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section TransComp
variable {X S B₁ B₂ B₃ : Scheme.{u}} (w : X ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (b₃ : B₃ ⟶ S)
Lean source · full declaration
theorem tW_comp (u : B₂ ⟶ B₁) (hu : u ≫ b₁ = b₂) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃) :
    tW w b₂ b₃ u' hu' ≫ tW w b₁ b₂ u hu = tW w b₁ b₃ (u' ≫ u)
      (by rw [Category.assoc, hu, hu']) := by
  apply pullback.hom_ext
  · rw [Category.assoc, tW_fst, tW_fst, tW_fst]
  · rw [Category.assoc, tW_snd, tW_snd_assoc, tW_snd]
theorem

tW_congr

L869 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_congr
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section TransComp
variable {X S B₁ B₂ B₃ : Scheme.{u}} (w : X ⟶ S) (b₁ : B₁ ⟶ S) (b₂ : B₂ ⟶ S) (b₃ : B₃ ⟶ S)
Lean source · statement
theorem tW_congr {u u' : B₂ ⟶ B₁} (hu : u ≫ b₁ = b₂) (hu' : u' ≫ b₁ = b₂) (e : u = u') :
    tW w b₁ b₂ u hu = tW w b₁ b₂ u' hu'
theorem

eq_of

L880 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.eq_of
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
Lean source · statement
theorem eq_of (D D' : Dat f b) (hm : D.m = D'.m) (he : D.e = D'.e) (hι : D.ι = D'.ι) : D = D'
theorem

res_res

L886 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_res
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
variable (D : Dat f b) {B₂ B₃ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
  (b₃ : B₃ ⟶ S) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃)
Lean source · full declaration
theorem res_res : (D.res b₂ u hu).res b₃ u' hu' = D.res b₃ (u' ≫ u)
  (by rw [Category.assoc, hu, hu']) := by
  refine eq_of _ _ ?_ ?_ ?_
  · apply pullback.hom_ext
    · rw [res_m_fst, res_m_fst, res_m_fst, ← tW_comp (w₂ f) b b₂ b₃ u hu u' hu', Category.assoc]
    · rw [(Dat.res _ b₃ u' hu').hm, (Dat.res _ b₃ _ _).hm]
  · apply pullback.hom_ext
    · rw [res_e_fst, res_e_fst, res_e_fst, Category.assoc]
    · rw [(Dat.res _ b₃ u' hu').he, (Dat.res _ b₃ _ _).he]
  · apply pullback.hom_ext
    · rw [res_ι_fst, res_ι_fst, res_ι_fst, ← tW_comp f b b₂ b₃ u hu u' hu', Category.assoc]
    · rw [(Dat.res _ b₃ u' hu').hι, (Dat.res _ b₃ _ _).hι]
theorem

res_congr

L899 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.res_congr
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
variable (D : Dat f b) {B₂ B₃ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
  (b₃ : B₃ ⟶ S) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃)
Lean source · statement
theorem res_congr {u₁ u₂ : B₂ ⟶ B} (h₁ : u₁ ≫ b = b₂) (h₂ : u₂ ≫ b = b₂) (e : u₁ = u₂) : D.res b₂
  u₁ h₁ = D.res b₂ u₂ h₂
theorem

assoc_res_iff

L903 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.assoc_res_iff
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
variable (D : Dat f b) {B₂ B₃ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
  (b₃ : B₃ ⟶ S) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃)
Lean source · statement
theorem assoc_res_iff : (D.res b₂ u hu).assocL = (D.res b₂ u hu).assocR ↔
    tW (w₃ f) b b₂ u hu ≫ D.assocL = tW (w₃ f) b b₂ u hu ≫ D.assocR
theorem

one_mul_res_iff

L908 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.one_mul_res_iff
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
variable (D : Dat f b) {B₂ B₃ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
  (b₃ : B₃ ⟶ S) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃)
Lean source · statement
theorem one_mul_res_iff : (D.res b₂ u hu).oneMulL = 𝟙 _ ↔ tW f b b₂ u hu ≫ D.oneMulL = tW f b b₂ u
  hu ≫ 𝟙 _
theorem

inv_mul_res_iff

L914 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.inv_mul_res_iff
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
variable (D : Dat f b) {B₂ B₃ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
  (b₃ : B₃ ⟶ S) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃)
Lean source · statement
theorem inv_mul_res_iff : (D.res b₂ u hu).invMulL = (D.res b₂ u hu).unitC ↔
    tW f b b₂ u hu ≫ D.invMulL = tW f b b₂ u hu ≫ D.unitC
theorem

comm_res_iff

L919 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.comm_res_iff
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
namespace Dat
variable {X S : Scheme.{u}} {f : X ⟶ S} {B : Scheme.{u}} {b : B ⟶ S}
variable (D : Dat f b) {B₂ B₃ : Scheme.{u}} (b₂ : B₂ ⟶ S) (u : B₂ ⟶ B) (hu : u ≫ b = b₂)
  (b₃ : B₃ ⟶ S) (u' : B₃ ⟶ B₂) (hu' : u' ≫ b₂ = b₃)
Lean source · statement
theorem comm_res_iff : (D.res b₂ u hu).commL = (D.res b₂ u hu).m ↔
    tW (w₂ f) b b₂ u hu ≫ D.commL = tW (w₂ f) b b₂ u hu ≫ D.m
abbrev

bT

L932 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.bT
Documentation

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

Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
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

bA

L938 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.bA
Documentation

The spectrum morphism induced by the ambient algebra map.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
Lean source · statement
abbrev bA : (Spec (CommRingCat.of A) : Scheme.{u}) ⟶ Spec (CommRingCat.of A₀)
abbrev

uA

L942 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uA
Documentation

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

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
Lean source · statement
abbrev uA (T : Subalgebra A₀ A) : (Spec (CommRingCat.of A) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥T)
abbrev

uT

L947 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uT
Documentation

The spectrum morphism induced by an inclusion between intermediate subalgebras.

Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
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_bT

L951 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uA_bT
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
Lean source · statement
theorem uA_bT (T : Subalgebra A₀ A) : uA T ≫ bT T = bA A₀ A
theorem

uT_bT

L955 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uT_bT
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
Lean source · statement
theorem uT_bT {T₁ T₂ : Subalgebra A₀ A} (h : T₁ ≤ T₂) : uT h ≫ bT T₁ = bT T₂
theorem

uA_uT

L959 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uA_uT
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
Lean source · statement
theorem uA_uT {T₁ T₂ : Subalgebra A₀ A} (h : T₁ ≤ T₂) : uA T₂ ≫ uT h = uA T₁
theorem

tW_snd_val

L963 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_snd_val
Source context 20 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
Lean source · statement
theorem tW_snd_val {W : Scheme.{u}} (w : W ⟶ Spec (CommRingCat.of A₀)) (T : Subalgebra A₀ A) :
    tW w (bT T) (bA A₀ A) (uA T) (uA_bT T) ≫ pullback.snd w (bT T) =
      pullback.snd w (bA A₀ A) ≫ Spec.map (CommRingCat.ofHom T.val.toRingHom)
def

flat

L971 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.flat
Documentation

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

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
def flat : Subalgebra A₀ A
theorem

le_flat

L973 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.le_flat
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem le_flat : T ≤ flat T T'
theorem

flat_fg

L975 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.flat_fg
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
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
theorem

subset_flat

L982 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.subset_flat
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem subset_flat {s : Set A} (h : s ⊆ T') : s ⊆ flat T T'
theorem

flat_mono

L984 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.flat_mono
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem flat_mono {T'' : Subalgebra ↥T A} (h : T' ≤ T'') : flat T T' ≤ flat T T''
def

flatHom

L988 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.flatHom
Documentation

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

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
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

L996 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uF
Documentation

The spectrum morphism induced by inclusion after restricting the subalgebra scalars.

Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
abbrev uF {T₂ : Subalgebra A₀ A} (h : flat T T' ≤ T₂) :
    (Spec (CommRingCat.of ↥T₂) : Scheme.{u}) ⟶ Spec (CommRingCat.of ↥T')
theorem

uF_comp

L1000 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.uF_comp
Source context 21 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
Lean source · statement
theorem uF_comp {T₂ : Subalgebra A₀ A} (h : flat T T' ≤ T₂) (h' : T ≤ T₂) :
    uF T T' h ≫ Spec.map (CommRingCat.ofHom (algebraMap ↥T ↥T')) = uT h'
theorem

exists_finset_of_fg

L1005 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.exists_finset_of_fg
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
omit [Algebra A₀ A] in
Lean source · statement
theorem exists_finset_of_fg {R : Type u} [CommRing R] [Algebra R A] {U : Subalgebra R A} (hU :
  U.FG) :
    ∃ t : Finset A, (↑t : Set A) ⊆ U ∧ ∀ U' : Subalgebra R A, (↑t : Set A) ⊆ U' → U ≤ U'
theorem

exists_tW_comp_eq

L1016 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.exists_tW_comp_eq
Source context 22 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Stages
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
variable (A₀ A) in
variable (T : Subalgebra A₀ A) (T' : Subalgebra ↥T A)
variable {T} {X : Scheme.{u}} (f : X ⟶ Spec (CommRingCat.of A₀)) [LocallyOfFinitePresentation f]
  {W : Scheme.{u}} (w : W ⟶ Spec (CommRingCat.of A₀)) [QuasiCompact w]
Lean source · statement
theorem exists_tW_comp_eq (a b : pullback w (bT T) ⟶ Yb f (bT T))
    (ha : a ≫ fb f (bT T) = pullback.snd _ _) (hb : b ≫ fb f (bT T) = pullback.snd _ _)
    (hab : tW w (bT T) (bA A₀ A) (uA T) (uA_bT T) ≫ a = tW w (bT T) (bA A₀ A) (uA T) (uA_bT T) ≫ b)
      (s : Finset A) :
    ∃ (T' : Subalgebra ↥T A), T'.FG ∧ (↑s : Set A) ⊆ T' ∧
      ∀ (T₂ : Subalgebra A₀ A) (_ : flat T T' ≤ T₂) (h : T ≤ T₂),
        tW w (bT T) (bT T₂) (uT h) (uT_bT h) ≫ a = tW w (bT T) (bT T₂) (uT h) (uT_bT h) ≫ b
theorem

baseChange_mul_fst

L1054 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.baseChange_mul_fst
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Final
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
  {X : Scheme.{u}} (f : X ⟶ Spec (CommRingCat.of A₀))
  [QuasiCompact f] [QuasiSeparated f] [LocallyOfFinitePresentation f]
  (L : RelativeGroupLaw A (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A)))))
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)} (K : RelativeGroupLaw S f)
    {T : Scheme.{u}} (t' : T ⟶ Spec (CommRingCat.of S')) (x y : SchemeHomOver t' (pullback.snd f
      u)) :
    ((K.baseChange u).mul t' x y).1 ≫ pullback.fst f u =
      (K.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 K t' x y)
  rw [RelativeGroupLaw.baseChangePointToBase_coe] at h
  exact h
theorem

exists_relativeGroupLaw

L1068 ↗
AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent.GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.exists_relativeGroupLaw
Source context 19 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
attribute [-instance] 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
attribute [-simp] 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
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 GroupLawDescent
open _root_.GoodReductionJacobian.RelativeGroupLaw
open _root_.GoodReductionJacobian
section Final
variable {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
  {X : Scheme.{u}} (f : X ⟶ Spec (CommRingCat.of A₀))
  [QuasiCompact f] [QuasiSeparated f] [LocallyOfFinitePresentation f]
  (L : RelativeGroupLaw A (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A)))))
Lean source · full declaration
theorem exists_relativeGroupLaw (s : Finset A) :
    ∃ (T : Subalgebra A₀ A), T.FG ∧ (↑s : Set A) ⊆ T ∧
      ∃ (Lₜ : RelativeGroupLaw ↥T (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀
        ↥T)))))
        (c : pullback f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A))) ⟶
          pullback (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀ ↥T))))
            (Spec.map (CommRingCat.ofHom (algebraMap ↥T A))))
        (_ : IsIso c)
        (hc : c ≫ pullback.snd _ _ = pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀
          A))))
        (_ : c ≫ pullback.fst _ _ ≫ pullback.fst _ _ =
          pullback.fst f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A)))),
        (∀ {T' : Scheme.{u}} (t : T' ⟶ Spec (CommRingCat.of A))
            (x y : SchemeHomOver t (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀
              A))))),
          (L.mul t x y).1 ≫ c =
            ((Lₜ.baseChange (Spec.map (CommRingCat.ofHom (algebraMap ↥T A)))).mul t
              ⟨x.1 ≫ c, by rw [Category.assoc, hc, x.2]⟩ ⟨y.1 ≫ c, by rw [Category.assoc, hc,
                y.2]⟩).1) ∧
        (L.IsCommutative → Lₜ.IsCommutative) := by
  classical
  have hDR : (ofLaw L).Laws := ofLaw_laws L
  obtain ⟨T₁, hT₁, hs₁, m₁, hm₁, hq₁⟩ :=
    exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation
      (A₀ := A₀) (A := A)
      (w₂ f) f (ofLaw L).m (ofLaw L).hm s
  obtain ⟨t₁, ht₁, ht₁'⟩ := exists_finset_of_fg hT₁
  obtain ⟨T₂, hT₂, hs₂, e₂, he₂, hq₂⟩ :=
    exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation
      (A₀ := A₀) (A := A)
      (𝟙 _) f (pullback.snd (𝟙 _) (bA A₀ A) ≫ (ofLaw L).e)
      (by rw [Category.assoc, (ofLaw L).he, Category.comp_id]) (s ∪ t₁)
  obtain ⟨t₂, ht₂, ht₂'⟩ := exists_finset_of_fg hT₂
  obtain ⟨T₃, hT₃, hs₃, ι₃, hι₃, hq₃⟩ :=
    exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation
      (A₀ := A₀) (A := A)
      f f (ofLaw L).ι (ofLaw L).hι (s ∪ t₂)
  rw [Finset.coe_union, Set.union_subset_iff] at hs₂ hs₃
  have h₁₂ : T₁ ≤ T₂ := ht₁' T₂ hs₂.2
  have h₂₃ : T₂ ≤ T₃ := ht₂' T₃ hs₃.2
  have h₁₃ : T₁ ≤ T₃ := h₁₂.trans h₂₃
  let s₀ : (Spec (CommRingCat.of ↥T₃) : Scheme.{u}) ⟶ pullback (𝟙 (Spec (CommRingCat.of A₀))) (bT
    T₃) :=
    pullback.lift (bT T₃) (𝟙 _) (by simp)
  have hs₀_fst : s₀ ≫ pullback.fst _ _ = bT T₃ := pullback.lift_fst _ _ _
  have hs₀_snd : s₀ ≫ pullback.snd _ _ = 𝟙 _ := pullback.lift_snd _ _ _
  let t₂ := tW (w₂ f) (bT T₁) (bT T₃) (uT h₁₃) (uT_bT h₁₃)
  let t₀ := tW (𝟙 (Spec (CommRingCat.of A₀))) (bT T₂) (bT T₃) (uT h₂₃) (uT_bT h₂₃)
  have pf_m : (t₂ ≫ m₁ ≫ pullback.fst f (bT T₁)) ≫ f = pullback.snd (w₂ f) (bT T₃) ≫ bT T₃ := by
    rw [Category.assoc, Category.assoc, pullback.condition, reassoc_of% hm₁, tW_snd_assoc, uT_bT]
  have pf_e : ((s₀ ≫ t₀) ≫ e₂ ≫ pullback.fst f (bT T₂)) ≫ f = 𝟙 _ ≫ bT T₃ := by
    rw [Category.assoc, Category.assoc, Category.assoc, pullback.condition, reassoc_of% he₂,
      tW_snd_assoc,
      reassoc_of% hs₀_snd, uT_bT, Category.id_comp]
  let Dj : Dat f (bT T₃) :=
    { m := pullback.lift (t₂ ≫ m₁ ≫ pullback.fst f (bT T₁)) (pullback.snd _ _) pf_m
      e := pullback.lift ((s₀ ≫ t₀) ≫ e₂ ≫ pullback.fst f (bT T₂)) (𝟙 _) pf_e
      ι := ι₃
      hm := pullback.lift_snd _ _ _
      he := pullback.lift_snd _ _ _
      hι := hι₃ }
  have hDjm : Dj.m ≫ pullback.fst _ _ = t₂ ≫ m₁ ≫ pullback.fst f (bT T₁) := pullback.lift_fst _ _ _
  have hDje : Dj.e ≫ pullback.fst _ _ = (s₀ ≫ t₀) ≫ e₂ ≫ pullback.fst f (bT T₂) :=
    pullback.lift_fst _ _ _
  have hres : ofLaw L = Dj.res (bA A₀ A) (uA T₃) (uA_bT T₃) := by
    refine Dat.eq_of _ _ ?_ ?_ ?_
    · apply pullback.hom_ext
      · rw [Dat.res_m_fst, hDjm, ← Category.assoc, tW_comp, tW_congr (w₂ f) _ _ _ (uA_bT T₁) (uA_uT
          h₁₃),
          ← Category.assoc, hq₁ (tW (w₂ f) (bT T₁) (bA A₀ A) (uA T₁) (uA_bT T₁)) (tW f (bT T₁) (bA
            A₀ A) (uA T₁) (uA_bT T₁))
            (tW_fst _ _ _ _ _) (tW_snd_val _ _) (tW_fst _ _ _ _ _) (tW_snd_val _ _),
              Category.assoc, tW_fst]
      · rw [(ofLaw L).hm, (Dj.res _ _ _).hm]
    · apply pullback.hom_ext
      · rw [Dat.res_e_fst, hDje]
        let sR : (Spec (CommRingCat.of A) : Scheme.{u}) ⟶ pullback (𝟙 (Spec (CommRingCat.of A₀)))
          (bA A₀ A) :=
          pullback.lift (bA A₀ A) (𝟙 _) (by simp)
        have hsR : sR ≫ pullback.snd _ _ = 𝟙 _ := pullback.lift_snd _ _ _
        have hsR_fst : sR ≫ pullback.fst _ _ = bA A₀ A := pullback.lift_fst _ _ _
        have hκ := hq₂ (tW (𝟙 _) (bT T₂) (bA A₀ A) (uA T₂) (uA_bT T₂)) (tW f (bT T₂) (bA A₀ A) (uA
          T₂) (uA_bT T₂))
          (tW_fst _ _ _ _ _) (tW_snd_val _ _) (tW_fst _ _ _ _ _) (tW_snd_val _ _)
        have h1 : (ofLaw L).e ≫ pullback.fst f (bA A₀ A) =
            sR ≫ (pullback.snd (𝟙 _) (bA A₀ A) ≫ (ofLaw L).e) ≫ pullback.fst f (bA A₀ A) := by
          rw [Category.assoc, reassoc_of% hsR]
        have h2 : (pullback.snd (𝟙 _) (bA A₀ A) ≫ (ofLaw L).e) ≫ pullback.fst f (bA A₀ A) =
            tW (𝟙 _) (bT T₂) (bA A₀ A) (uA T₂) (uA_bT T₂) ≫ e₂ ≫ pullback.fst f (bT T₂) := by
          rw [← tW_fst f (bT T₂) (bA A₀ A) (uA T₂) (uA_bT T₂), ← Category.assoc, ← hκ,
            Category.assoc]
        rw [h1, h2]
        simp only [← Category.assoc]
        congr 2
        apply pullback.hom_ext
        · rw [Category.assoc, tW_fst, hsR_fst, Category.assoc, Category.assoc, tW_fst, hs₀_fst,
            uA_bT]
        · rw [Category.assoc, tW_snd, reassoc_of% hsR, Category.assoc, Category.assoc, tW_snd,
            reassoc_of% hs₀_snd,
            uA_uT]
      · rw [(ofLaw L).he, (Dj.res _ _ _).he]
    · apply pullback.hom_ext
      · rw [Dat.res_ι_fst]
        change _ = tW f (bT T₃) (bA A₀ A) (uA T₃) (uA_bT T₃) ≫ ι₃ ≫ _
        rw [← Category.assoc (tW _ _ _ _ _),
          hq₃ (tW f (bT T₃) (bA A₀ A) (uA T₃) (uA_bT T₃)) (tW f (bT T₃) (bA A₀ A) (uA T₃) (uA_bT
            T₃))
            (tW_fst _ _ _ _ _) (tW_snd_val _ _) (tW_fst _ _ _ _ _) (tW_snd_val _ _),
              Category.assoc, tW_fst]
      · rw [(ofLaw L).hι, (Dj.res _ _ _).hι]
  have hLR : (Dj.res (bA A₀ A) (uA T₃) (uA_bT T₃)).Laws := hres ▸ hDR
  obtain ⟨U₁, hU₁, -, hk₁⟩ := exists_tW_comp_eq f (w₃ f) Dj.assocL Dj.assocR Dj.assocL_fb
    Dj.assocR_fb
    ((Dj.assoc_res_iff _ _ _).mp hLR.assoc) ∅
  obtain ⟨u₁, hu₁, hu₁'⟩ := exists_finset_of_fg hU₁
  obtain ⟨U₂, hU₂, hsU₂, hk₂⟩ := exists_tW_comp_eq f f Dj.oneMulL (𝟙 _) Dj.oneMulL_fb
    (Category.id_comp _)
    ((Dj.one_mul_res_iff _ _ _).mp hLR.one_mul) u₁
  obtain ⟨u₂, hu₂, hu₂'⟩ := exists_finset_of_fg hU₂
  obtain ⟨U₃, hU₃, hsU₃, hk₃⟩ := exists_tW_comp_eq f f Dj.invMulL Dj.unitC Dj.invMulL_fb Dj.unitC_fb
    ((Dj.inv_mul_res_iff _ _ _).mp hLR.inv_mul) u₂
  obtain ⟨u₃, hu₃, hu₃'⟩ := exists_finset_of_fg hU₃
  have hk₄ : ∃ (U₄ : Subalgebra ↥T₃ A), U₄.FG ∧ (↑u₃ : Set A) ⊆ U₄ ∧ (L.IsCommutative →
      ∀ (T₂ : Subalgebra A₀ A) (_ : flat T₃ U₄ ≤ T₂) (h : T₃ ≤ T₂),
        tW (w₂ f) (bT T₃) (bT T₂) (uT h) (uT_bT h) ≫ Dj.commL = tW (w₂ f) (bT T₃) (bT T₂) (uT h)
          (uT_bT h) ≫ Dj.m) := by
    by_cases hL : L.IsCommutative
    · have hcm : (Dj.res (bA A₀ A) (uA T₃) (uA_bT T₃)).commL = (Dj.res (bA A₀ A) (uA T₃) (uA_bT
        T₃)).m := by
        rw [← hres]; exact ofLaw_comm L hL
      obtain ⟨U₄, hU₄, hsU₄, hk⟩ := exists_tW_comp_eq f (w₂ f) Dj.commL Dj.m Dj.commL_fb Dj.hm
        ((Dj.comm_res_iff _ _ _).mp hcm) u₃
      exact ⟨U₄, hU₄, hsU₄, fun _ => hk⟩
    · exact ⟨U₃, hU₃, hu₃, fun h => absurd @h hL⟩
  obtain ⟨U₄, hU₄, hsU₄, hk₄⟩ := hk₄
  have hU₁₂ : U₁ ≤ U₂ := hu₁' U₂ hsU₂
  have hU₂₃ : U₂ ≤ U₃ := hu₂' U₃ hsU₃
  have hU₃₄ : U₃ ≤ U₄ := hu₃' U₄ hsU₄
  let T : Subalgebra A₀ A := flat T₃ U₄
  have hT : T.FG := flat_fg T₃ U₄ hT₃ hU₄
  have h₃T : T₃ ≤ T := le_flat T₃ U₄
  have hsT : (↑s : Set A) ⊆ T := hs₃.1.trans h₃T
  have hle₁ : flat T₃ U₁ ≤ T := flat_mono T₃ U₁ (hU₁₂.trans (hU₂₃.trans hU₃₄))
  have hle₂ : flat T₃ U₂ ≤ T := flat_mono T₃ U₂ (hU₂₃.trans hU₃₄)
  have hle₃ : flat T₃ U₃ ≤ T := flat_mono T₃ U₃ hU₃₄
  let Dk : Dat f (bT T) := Dj.res (bT T) (uT h₃T) (uT_bT h₃T)
  have hA : Dk.assocL = Dk.assocR := (Dj.assoc_res_iff _ _ _).mpr (hk₁ T hle₁ h₃T)
  have hB : Dk.oneMulL = 𝟙 _ := (Dj.one_mul_res_iff _ _ _).mpr (hk₂ T hle₂ h₃T)
  have hC : Dk.invMulL = Dk.unitC := (Dj.inv_mul_res_iff _ _ _).mpr (hk₃ T hle₃ h₃T)
  have hCm : L.IsCommutative → Dk.commL = Dk.m := fun hL => (Dj.comm_res_iff _ _ _).mpr (hk₄ hL T
    le_rfl h₃T)
  have hDk : Dk.Laws := ⟨hA, hB, hC⟩
  have hDRk : ofLaw L = Dk.res (bA A₀ A) (uA T) (uA_bT T) := by
    rw [hres, Dat.res_res]
    exact Dat.res_congr _ _ _ _ (uA_uT h₃T).symm
  let r := rebaseIso f (bT T) (uA T) (bA A₀ A) (uA_bT T)
  let κ := tW f (bT T) (bA A₀ A) (uA T) (uA_bT T)
  have hc : r.inv ≫ pullback.snd _ _ = pullback.snd f (bA A₀ A) := by
    rw [Iso.inv_comp_eq, rebaseIso_hom_snd]
  have hcfst : r.inv ≫ pullback.fst _ _ ≫ pullback.fst _ _ = pullback.fst f (bA A₀ A) := by
    rw [Iso.inv_comp_eq, rebaseIso_hom_fst]
  have hcκ : r.inv ≫ pullback.fst _ _ = κ := by
    apply pullback.hom_ext
    · rw [Category.assoc, hcfst, tW_fst]
    · rw [Category.assoc, pullback.condition, reassoc_of% hc, tW_snd]
  refine ⟨T, hT, hsT, Dk.toLaw hDk, r.inv, inferInstance, hc, hcfst, ?_, fun hL => Dk.toLaw_comm
    hDk (hCm hL)⟩
  intro T' t x y
  apply pullback.hom_ext
  · rw [Category.assoc, hcκ, mul_val L t x y, hDRk, Category.assoc, Dat.res_m_tW, ← Category.assoc,
      ← pair_tW,
      baseChange_mul_fst, Dat.toLaw_mul_val]
    congr 1
    exact pair_congr _ _
      (by change x.1 ≫ κ = (x.1 ≫ r.inv) ≫ pullback.fst _ _; rw [Category.assoc, hcκ])
      (by change y.1 ≫ κ = (y.1 ≫ r.inv) ≫ pullback.fst _ _; rw [Category.assoc, hcκ])
  · have hl : ((L.mul t x y).1 ≫ r.inv) ≫ pullback.snd _ _ = t := by
      rw [Category.assoc, hc]; exact (L.mul t x y).2
    exact hl.trans ((((Dk.toLaw hDk).baseChange (uA T)).mul t _ _).2).symm
theorem

exists_fg_subalgebra_relativeGroupLaw_pullback_snd_of_locallyOfFinitePresentation

L1365 ↗
GoodReductionJacobian.RelativeGroupLaw.exists_fg_subalgebra_relativeGroupLaw_pullback_snd_of_locallyOfFinitePresentation
Source context 42 commands

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

section ProofImplementation
open _root_.AlgebraicGeometry (
  exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation)
namespace GoodReductionJacobian.RelativeGroupLaw
open _root_.AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.AlgebraicGeometry
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
universe u
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₁_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₂_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.p₂_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_fst_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_fst_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_p₁
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.pair_p₂
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₁₂_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₃_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.q₃_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_fst_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_fst_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_q₁₂
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.triple_q₃
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_fst
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.tW_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.P₁_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.P₂_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Pid_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.mulPt_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.onePt_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.invPt_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.toLaw_mul_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.Dat.toLaw_one_val
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.rebaseIso_hom_snd
attribute [local simp] GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent.rebaseIso_hom_fst
open _root_.AlgebraicCurves.Geometry.AbelianSchemes.Theorems.Proofs.RelativeGroupLawFiniteDescent in
open GoodReductionJacobian.RelativeGroupLaw.GroupLawDescent in
Lean source · full declaration
theorem exists_fg_subalgebra_relativeGroupLaw_pullback_snd_of_locallyOfFinitePresentation
    {A₀ : Type u} [CommRing A₀] {A : Type u} [CommRing A] [Algebra A₀ A]
    {X : Scheme.{u}} (f : X ⟶ Spec (CommRingCat.of A₀))
    [QuasiCompact f] [QuasiSeparated f] [LocallyOfFinitePresentation f]
    (L : RelativeGroupLaw A (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A)))))
    (s : Finset A) :
    ∃ (T : Subalgebra A₀ A), T.FG ∧ (↑s : Set A) ⊆ T ∧
      ∃ (Lₜ : RelativeGroupLaw ↥T (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀
        ↥T)))))
        (c : pullback f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A))) ⟶
          pullback (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀ ↥T))))
            (Spec.map (CommRingCat.ofHom (algebraMap ↥T A))))
        (_ : IsIso c)
        (hc : c ≫ pullback.snd _ _ = pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀
          A))))
        (_ : c ≫ pullback.fst _ _ ≫ pullback.fst _ _ =
          pullback.fst f (Spec.map (CommRingCat.ofHom (algebraMap A₀ A)))),
        (∀ {T' : Scheme.{u}} (t : T' ⟶ Spec (CommRingCat.of A))
            (x y : SchemeHomOver t (pullback.snd f (Spec.map (CommRingCat.ofHom (algebraMap A₀
              A))))),
          (L.mul t x y).1 ≫ c =
            ((Lₜ.baseChange (Spec.map (CommRingCat.ofHom (algebraMap ↥T A)))).mul t
              ⟨x.1 ≫ c, by rw [Category.assoc, hc, x.2]⟩ ⟨y.1 ≫ c, by rw [Category.assoc, hc,
                y.2]⟩).1) ∧
        (L.IsCommutative → Lₜ.IsCommutative) :=
  exists_relativeGroupLaw f L s

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