Module

OModulePresheaf_isCoherent_cechPushforward_of_isProper.lean

Nyll.AlgebraicCurves.Cohomology.OModulePresheaf_isCoherent_cechPushforward_of_isProper

70 declarations indexed from sourceView source ↗

Module documentation

OModule Presheaf is Coherent cech Pushforward of is Proper

Imports8

Mathlib.AlgebraicGeometry.Morphisms.Proper
Mathlib.AlgebraicGeometry.Noetherian
Mathlib.RingTheory.Finiteness.Basic
Mathlib.LinearAlgebra.Pi
Mathlib.Tactic.FinCases
Mathlib.Tactic.Choose
Nyll.AlgebraicCurves.Cohomology.Cech.OModulePresheafCechPushforward
Nyll.SchemeTheory.Cohomology.Cech.OModulePresheaf_cechFinite_of_isProper

Imported by1

Nyll.AlgebraicCurves.Cohomology.All

abbrev

pW

L115 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.pW

Documentation

The inverse image of an affine chart under the given morphism.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

abbrev pW : V'.Opens

def

qW

L118 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.qW

Documentation

The restricted morphism over the spectrum of the affine chart ring.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

def qW : ((pW p W) : Scheme.{u}) ⟶ Spec (CommRingCat.of Γ(P, (W : P.Opens)))

theorem

qW_def

L121 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.qW_def

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

theorem qW_def : qW p W = (p ∣_ (W : P.Opens)) ≫ W.2.isoSpec.hom

abbrev

Bch

L124 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.Bch

Documentation

The ring of regular sections on the chosen affine chart.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

abbrev Bch : Type u

abbrev

ΓW

L127 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.ΓW

Documentation

Regular sections on an open of the inverse-image chart.

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

abbrev ΓW (O : ((pW p W) : Scheme.{u}).Opens) : Type u

instance

isProper_qW

L129 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isProper_qW

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

instance isProper_qW [IsProper p] : IsProper (qW p W)

theorem

isNoetherian_of_isProper

L132 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isNoetherian_of_isProper

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

theorem isNoetherian_of_isProper [IsNoetherianRing A] [IsProper q] : IsNoetherian P

theorem

isNoetherianRing_chart

L139 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isNoetherianRing_chart

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

theorem isNoetherianRing_chart [IsNoetherianRing A] [IsProper q] : IsNoetherianRing Γ(P, (W :
  P.Opens))

theorem

ΓSpecIso_inv_qW_app_map

L144 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.ΓSpecIso_inv_qW_app_map

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · statement

theorem ΓSpecIso_inv_qW_app_map (O : ((pW p W) : Scheme.{u}).Opens) :
    (Scheme.ΓSpecIso (CommRingCat.of Γ(P, (W : P.Opens)))).inv ≫ (qW p W).app ⊤ ≫
        ((pW p W) : Scheme.{u}).presheaf.map (homOfLE (le_top : O ≤ ⊤)).op =
      p.appLE (W : P.Opens) ((pW p W).ι ''ᵁ O) (Scheme.Opens.ι_image_le _ _)

theorem

algebraMap_qW_eq

L159 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.algebraMap_qW_eq

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · full declaration

theorem algebraMap_qW_eq (O : ((pW p W) : Scheme.{u}).Opens) (b : Γ(P, (W : P.Opens))) :
    (letI := Scheme.TwoAffineOpenCover.algebraOfHom (qW p W) O
     algebraMap Γ(P, (W : P.Opens)) Γ(((pW p W) : Scheme.{u}), O) b) =
      (p.appLE (W : P.Opens) ((pW p W).ι ''ᵁ O) (Scheme.Opens.ι_image_le _ _)).hom b := by
  have h := ΓSpecIso_inv_qW_app_map p W O
  exact DFunLike.congr_fun (congrArg CommRingCat.Hom.hom h) b

theorem

map_algebraMap_qW

L166 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.map_algebraMap_qW

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · full declaration

theorem map_algebraMap_qW {O O' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') (b : Γ(P, (W :
  P.Opens))) :
    (((pW p W) : Scheme.{u}).presheaf.map (homOfLE h).op).hom
        (letI := Scheme.TwoAffineOpenCover.algebraOfHom (qW p W) O'
         algebraMap Γ(P, (W : P.Opens)) Γ(((pW p W) : Scheme.{u}), O') b) =
      (letI := Scheme.TwoAffineOpenCover.algebraOfHom (qW p W) O
       algebraMap Γ(P, (W : P.Opens)) Γ(((pW p W) : Scheme.{u}), O) b) := by
  change (((pW p W) : Scheme.{u}).presheaf.map (homOfLE h).op).hom
      (((Scheme.ΓSpecIso (CommRingCat.of Γ(P, (W : P.Opens)))).inv ≫ (qW p W).appLE ⊤ O'
        le_top).hom b) =
    ((Scheme.ΓSpecIso (CommRingCat.of Γ(P, (W : P.Opens)))).inv ≫ (qW p W).appLE ⊤ O le_top).hom b
  rw [← Scheme.Hom.appLE_map (qW p W) (le_top : O' ≤ (qW p W) ⁻¹ᵁ ⊤) (homOfLE h).op]
  rfl

theorem

appLE_algebraMap_chart

L180 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.appLE_algebraMap_chart

Source context 19 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)

Lean source · full declaration

theorem appLE_algebraMap_chart (O : ((pW p W) : Scheme.{u}).Opens) (a : A) :
    (p.appLE (W : P.Opens) ((pW p W).ι ''ᵁ O) (Scheme.Opens.ι_image_le _ _)).hom
        (letI := Scheme.TwoAffineOpenCover.algebraOfHom q (W : P.Opens)
         algebraMap A Γ(P, (W : P.Opens)) a) =
      (letI := Scheme.TwoAffineOpenCover.algebraOfHom (p ≫ q) ((pW p W).ι ''ᵁ O)
       algebraMap A Γ(V', (pW p W).ι ''ᵁ O) a) := by
  change ((Scheme.ΓSpecIso (CommRingCat.of A)).inv ≫ q.appLE ⊤ (W : P.Opens) le_top ≫
      p.appLE (W : P.Opens) ((pW p W).ι ''ᵁ O) (Scheme.Opens.ι_image_le _ _)).hom a =
    ((Scheme.ΓSpecIso (CommRingCat.of A)).inv ≫ (p ≫ q).appLE ⊤ ((pW p W).ι ''ᵁ O) le_top).hom a
  have hcomp := Scheme.Hom.appLE_comp_appLE p q ⊤ (W : P.Opens)
    ((pW p W).ι ''ᵁ O) le_top (Scheme.Opens.ι_image_le _ _)
  exact congrArg (fun f => ((Scheme.ΓSpecIso (CommRingCat.of A)).inv ≫ f).hom a) hcomp

abbrev

modSec

L198 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.modSec

Documentation

The original sheaf action on sections over the image open.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))

Lean source · statement

abbrev modSec (O : ((pW p W) : Scheme.{u}).Opens) : Module (ΓW p W O) (G.obj ((pW p W).ι ''ᵁ O))

abbrev

modB

L204 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.modB

Documentation

The chart-ring action obtained from the restricted structure morphism.

Source context 22 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec

Lean source · statement

abbrev modB (O : ((pW p W) : Scheme.{u}).Opens) : Module (Bch W) (G.obj ((pW p W).ι ''ᵁ O))

theorem

modB_smul_def

L210 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.modB_smul_def

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · full declaration

theorem modB_smul_def (O : ((pW p W) : Scheme.{u}).Opens) (b : Bch W) (x : G.obj ((pW p W).ι ''ᵁ
  O)) :
    b • x = (letI := Scheme.TwoAffineOpenCover.algebraOfHom (qW p W) O; algebraMap (Bch W) (ΓW p W
      O) b) • x :=
  rfl

theorem

modB_smul_def'

L216 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.modB_smul_def'

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem modB_smul_def' (O : ((pW p W) : Scheme.{u}).Opens) (b : Bch W) (x : G.obj ((pW p W).ι ''ᵁ
  O)) :
    b • x = (p.appLE (W : P.Opens) ((pW p W).ι ''ᵁ O) (Scheme.Opens.ι_image_le _ _)).hom b • x

theorem

isScalarTower_modB

L221 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isScalarTower_modB

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · full declaration

theorem isScalarTower_modB (O : ((pW p W) : Scheme.{u}).Opens) :
    letI := Scheme.TwoAffineOpenCover.algebraOfHom (qW p W) O
    IsScalarTower (Bch W) (ΓW p W O) (G.obj ((pW p W).ι ''ᵁ O)) :=
  let := Scheme.TwoAffineOpenCover.algebraOfHom (qW p W) O
  IsScalarTower.of_algebraMap_smul fun _ _ => rfl

theorem

res_smul_piece

L227 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.res_smul_piece

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem res_smul_piece {O O' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') (a : ΓW p W O')
    (x : G.obj ((pW p W).ι ''ᵁ O')) :
    G.res ((pW p W).ι.image_mono h) (a • x) =
      (((pW p W) : Scheme.{u}).presheaf.map (homOfLE h).op).hom a • G.res ((pW p W).ι.image_mono h)
        x

def

resW

L235 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.resW

Documentation

Restriction of the rebased sheaf as a linear map over the chart ring.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

def resW {O O' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') :
    G.obj ((pW p W).ι ''ᵁ O') →ₗ[Bch W] G.obj ((pW p W).ι ''ᵁ O)

theorem

resW_apply

L243 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.resW_apply

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem resW_apply {O O' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') (x : G.obj ((pW p W).ι ''ᵁ
  O')) :
    resW q p W G h x = G.res ((pW p W).ι.image_mono h) x

theorem

resW_smul

L247 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.resW_smul

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem resW_smul {O O' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') (a : ΓW p W O')
    (x : G.obj ((pW p W).ι ''ᵁ O')) :
    resW q p W G h (a • x) = (((pW p W) : Scheme.{u}).presheaf.map (homOfLE h).op).hom a • resW q p
      W G h x

theorem

resW_refl

L253 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.resW_refl

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem resW_refl (O : ((pW p W) : Scheme.{u}).Opens) : resW q p W G (le_refl O) = LinearMap.id

theorem

resW_comp

L256 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.resW_comp

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem resW_comp {O O' O'' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') (h' : O' ≤ O'') :
    resW q p W G (h.trans h') = resW q p W G h ∘ₗ resW q p W G h'

def

GW

L261 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.GW

Documentation

The original module presheaf restricted to the inverse-image chart.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · full declaration

def GW : OModulePresheaf (qW p W) where
  obj O := G.obj ((pW p W).ι ''ᵁ O)
  addCommGroup _ := G.addCommGroup _
  module O := modB q p W G O
  moduleSections O := modSec q p W G O
  isScalarTower O := isScalarTower_modB q p W G O
  res h := resW q p W G h
  res_smul h a x := resW_smul q p W G h a x
  res_refl O := resW_refl q p W G O
  res_comp h h' := resW_comp q p W G h h'

theorem

GW_obj

L272 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.GW_obj

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

theorem GW_obj (O : ((pW p W) : Scheme.{u}).Opens) : (GW q p W G).obj O = G.obj ((pW p W).ι ''ᵁ O)

theorem

GW_res_apply

L275 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.GW_res_apply

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · full declaration

theorem GW_res_apply {O O' : ((pW p W) : Scheme.{u}).Opens} (h : O ≤ O') (x : (GW q p W G).obj O') :
    (GW q p W G).res h x = G.res ((pW p W).ι.image_mono h) (show G.obj ((pW p W).ι ''ᵁ O') from x)
      := rfl

theorem

GW_smul_def

L279 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.GW_smul_def

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · full declaration

theorem GW_smul_def (O : ((pW p W) : Scheme.{u}).Opens) (b : Bch W) (x : (GW q p W G).obj O) :
    b • x = (show (GW q p W G).obj O from
        (p.appLE (W : P.Opens) ((pW p W).ι ''ᵁ O) (Scheme.Opens.ι_image_le _ _)).hom b •
          (show G.obj ((pW p W).ι ''ᵁ O) from x)) :=
  modB_smul_def' q p W G O b x

abbrev

modA

L286 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.modA

Documentation

The original base-ring action on rebased sections.

Source context 23 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB

Lean source · statement

abbrev modA (O : ((pW p W) : Scheme.{u}).Opens) : Module A ((GW q p W G).obj O)

theorem

algebraMap_smul_GW

L291 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.algebraMap_smul_GW

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB
attribute [local instance] modA

Lean source · full declaration

theorem algebraMap_smul_GW (O : ((pW p W) : Scheme.{u}).Opens) (a : A) (x : (GW q p W G).obj O) :
    (letI := Scheme.TwoAffineOpenCover.algebraOfHom q (W : P.Opens)
     algebraMap A (Bch W) a • x) = a • x := by
  let := Scheme.TwoAffineOpenCover.algebraOfHom q (W : P.Opens)
  let := Scheme.TwoAffineOpenCover.algebraOfHom (p ≫ q) ((pW p W).ι ''ᵁ O)
  rw [GW_smul_def, appLE_algebraMap_chart]
  exact algebraMap_smul Γ(V', (pW p W).ι ''ᵁ O) a (show G.obj ((pW p W).ι ''ᵁ O) from x)

theorem

isScalarTower_GW

L299 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isScalarTower_GW

Source context 24 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Datum
variable (G : OModulePresheaf (p ≫ q))
attribute [local instance] modSec
attribute [local instance] modB
attribute [local instance] modA

Lean source · full declaration

theorem isScalarTower_GW (O : ((pW p W) : Scheme.{u}).Opens) :
    letI := Scheme.TwoAffineOpenCover.algebraOfHom q (W : P.Opens)
    IsScalarTower A (Bch W) ((GW q p W G).obj O) :=
  let := Scheme.TwoAffineOpenCover.algebraOfHom q (W : P.Opens)
  IsScalarTower.of_algebraMap_smul fun a x => algebraMap_smul_GW q p W G O a x

def

KW

L312 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.KW

Documentation

The induced ordered affine cover of the inverse-image chart.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Cover
variable (K' : V'.OrderedAffineCover)

Lean source · full declaration

def KW [IsSeparated q] : Scheme.OrderedAffineCover ((pW p W) : Scheme.{u}) where
  ι := K'.ι
  U i := (pW p W).ι ⁻¹ᵁ K'.U i
  isAffineOpen i := by
    rw [← (pW p W).ι.isAffineOpen_iff_of_isOpenImmersion,
      Scheme.Hom.image_preimage_eq_opensRange_inf,
      Scheme.Opens.opensRange_ι, inf_comm]
    exact Scheme.OrderedAffineCover.fiberAffineOpen p q (K'.isAffineOpen i) W.2
  iSup_eq_top := by
    rw [← Scheme.Hom.preimage_iSup, K'.iSup_eq_top]
    rfl

theorem

KW_ι

L324 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.KW_ι

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Cover
variable (K' : V'.OrderedAffineCover)

Lean source · statement

theorem KW_ι [IsSeparated q] : (KW q p W K').ι = K'.ι

theorem

KW_U

L326 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.KW_U

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Cover
variable (K' : V'.OrderedAffineCover)

Lean source · statement

theorem KW_U [IsSeparated q] (i : K'.ι) : (KW q p W K').U i = (pW p W).ι ⁻¹ᵁ K'.U i

theorem

image_KW_U

L328 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.image_KW_U

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Cover
variable (K' : V'.OrderedAffineCover)

Lean source · statement

theorem image_KW_U [IsSeparated q] (i : K'.ι) :
    (pW p W).ι ''ᵁ (KW q p W K').U i = K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)

theorem

presheaf_res_res

L338 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.presheaf_res_res

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Transfer
variable (G : OModulePresheaf (p ≫ q))

Lean source · statement

theorem presheaf_res_res {O₁ O₂ O₃ : V'.Opens} (h12 : O₁ ≤ O₂) (h23 : O₂ ≤ O₃) (x : Γ(V', O₃)) :
    (V'.presheaf.map (homOfLE h12).op).hom ((V'.presheaf.map (homOfLE h23).op).hom x) =
      (V'.presheaf.map (homOfLE (h12.trans h23)).op).hom x

theorem

ι_image_basicOpen_piece

L343 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.ι_image_basicOpen_piece

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Transfer
variable (G : OModulePresheaf (p ≫ q))

Lean source · full declaration

theorem ι_image_basicOpen_piece (O : ((pW p W) : Scheme.{u}).Opens) (r : ΓW p W O) :
    (pW p W).ι ''ᵁ (((pW p W) : Scheme.{u}).basicOpen r) = V'.basicOpen (show Γ(V', (pW p W).ι ''ᵁ
      O) from r) := by
  ext x
  constructor
  · rintro ⟨y, hy, rfl⟩
    exact Scheme.Opens.mem_basicOpen_toScheme.1 hy
  · intro hx
    have hxW : x ∈ (pW p W : V'.Opens) :=
      Scheme.Opens.ι_image_le _ _ (V'.basicOpen_le (show Γ(V', (pW p W).ι ''ᵁ O) from r) hx)
    exact ⟨⟨x, hxW⟩, Scheme.Opens.mem_basicOpen_toScheme.2 hx, rfl⟩

theorem

isCoherent_GW

L355 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isCoherent_GW

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Transfer
variable (G : OModulePresheaf (p ≫ q))

Lean source · statement

theorem isCoherent_GW (hc : G.IsCoherent) : (GW q p W G).IsCoherent

theorem

isQuasicoherent_GW

L358 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.isQuasicoherent_GW

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Transfer
variable (G : OModulePresheaf (p ≫ q))

Lean source · statement

theorem isQuasicoherent_GW (hqc : G.IsQuasicoherent) : (GW q p W G).IsQuasicoherent

def

σ

L391 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.σ

Documentation

The degree-zero Cech index associated with one cover member.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

def σ (i : K'.ι) : (KW q p W K').Idx 0

theorem

σ_val

L394 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.σ_val

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem σ_val (i : K'.ι) (k : Fin 1) : (σ q p W K' i).1 k = i

theorem

inter_σ

L396 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.inter_σ

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem inter_σ (i : K'.ι) : (KW q p W K').inter (σ q p W K' i) = (KW q p W K').U i

theorem

image_inter_σ

L400 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.image_inter_σ

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem image_inter_σ (i : K'.ι) :
    (pW p W).ι ''ᵁ (KW q p W K').inter (σ q p W K' i) = (K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens))

theorem

image_inter_le

L404 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.image_inter_le

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem image_inter_le {b : ℕ} (s : (KW q p W K').Idx b) (k : Fin (b + 1)) :
    (pW p W).ι ''ᵁ (KW q p W K').inter s ≤ (K'.U (s.1 k) ⊓ p ⁻¹ᵁ (W : P.Opens))

theorem

image_inf

L409 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.image_inf

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem image_inf (O₁ O₂ : ((pW p W) : Scheme.{u}).Opens) :
    (pW p W).ι ''ᵁ (O₁ ⊓ O₂) = (pW p W).ι ''ᵁ O₁ ⊓ (pW p W).ι ''ᵁ O₂

theorem

chW_inf_le_image_inter

L415 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.chW_inf_le_image_inter

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem chW_inf_le_image_inter (s : (KW q p W K').Idx 1) :
    (K'.U (s.1 0) ⊓ p ⁻¹ᵁ (W : P.Opens)) ⊓ (K'.U (s.1 1) ⊓ p ⁻¹ᵁ (W : P.Opens)) ≤ (pW p W).ι ''ᵁ
      (KW q p W K').inter s

theorem

image_inter_le_inf

L424 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.image_inter_le_inf

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem image_inter_le_inf (s : (KW q p W K').Idx 1) :
    (pW p W).ι ''ᵁ (KW q p W K').inter s ≤ (K'.U (s.1 0) ⊓ p ⁻¹ᵁ (W : P.Opens)) ⊓ (K'.U (s.1 1) ⊓ p
      ⁻¹ᵁ (W : P.Opens))

def

e0

L430 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0

Documentation

Degree-zero cochains expressed on the original cover restricted to the chart.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

def e0 (x : (GW q p W G).cochain (KW q p W K') 0) (i : K'.ι) : G.obj ((K'.U i ⊓ p ⁻¹ᵁ (W :
  P.Opens)))

theorem

e0_apply

L435 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_apply

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · full declaration

theorem e0_apply (x : (GW q p W G).cochain (KW q p W K') 0) (i : K'.ι) :
    e0 q p W G K' x i = G.res (image_inter_σ q p W K' i).ge
      (show G.obj ((pW p W).ι ''ᵁ (KW q p W K').inter (σ q p W K' i)) from x (σ q p W K' i)) := rfl

theorem

e0_add

L439 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_add

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_add (x y : (GW q p W G).cochain (KW q p W K') 0) :
    e0 q p W G K' (x + y) = e0 q p W G K' x + e0 q p W G K' y

theorem

e0_zero

L443 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_zero

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_zero : e0 q p W G K' (0 : (GW q p W G).cochain (KW q p W K') 0) = 0

theorem

e0_neg

L446 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_neg

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_neg (x : (GW q p W G).cochain (KW q p W K') 0) :
    e0 q p W G K' (-x) = -e0 q p W G K' x

theorem

e0_sub

L450 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_sub

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_sub (x y : (GW q p W G).cochain (KW q p W K') 0) :
    e0 q p W G K' (x - y) = e0 q p W G K' x - e0 q p W G K' y

def

e0Hom

L455 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom

Documentation

The additive map sending rebased degree-zero cochains to restricted sections.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · full declaration

def e0Hom : (GW q p W G).cochain (KW q p W K') 0 →+ (∀ i : K'.ι, G.obj ((K'.U i ⊓ p ⁻¹ᵁ (W :
  P.Opens)))) where
  toFun := e0 q p W G K'
  map_zero' := e0_zero q p W G K'
  map_add' := e0_add q p W G K'

theorem

e0Hom_apply

L461 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom_apply

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0Hom_apply (x : (GW q p W G).cochain (KW q p W K') 0) :
    e0Hom q p W G K' x = e0 q p W G K' x

theorem

e0_smulB

L464 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_smulB

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_smulB (b : Bch W) (x : (GW q p W G).cochain (KW q p W K') 0) (i : K'.ι) :
    e0 q p W G K' (b • x) i =
      (p.appLE (W : P.Opens) ((K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens))) inf_le_right).hom b • e0 q p W G K' x
        i

theorem

e0_smulA

L473 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_smulA

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · full declaration

theorem e0_smulA (a : A) (x : (GW q p W G).cochain (KW q p W K') 0) (i : K'.ι) :
    e0 q p W G K' (fun s => (letI := modA q p W G ((KW q p W K').inter s); a • x s)) i =
      a • e0 q p W G K' x i :=
  LinearMap.map_smul (G.res _) a _

def

e0inv

L479 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0inv

Documentation

Restricted cover sections regarded as rebased degree-zero cochains.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

def e0inv (y : ∀ i : K'.ι, G.obj ((K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)))) : (GW q p W G).cochain (KW q p W
  K') 0

theorem

e0_e0inv

L484 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_e0inv

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_e0inv (y : ∀ i : K'.ι, G.obj ((K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)))) : e0 q p W G K' (e0inv q
  p W G K' y) = y

theorem

e0_surjective

L490 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.e0_surjective

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem e0_surjective : Function.Surjective (e0 q p W G K')

theorem

res_congr_idx

L493 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.res_congr_idx

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · full declaration

theorem res_congr_idx (x : (GW q p W G).cochain (KW q p W K') 0) {t t' : (KW q p W K').Idx 0} (h :
  t = t')
    {O : V'.Opens} (h1 : O ≤ (pW p W).ι ''ᵁ (KW q p W K').inter t) :
    G.res h1 (show G.obj ((pW p W).ι ''ᵁ (KW q p W K').inter t) from x t) =
      G.res (h ▸ h1) (show G.obj ((pW p W).ι ''ᵁ (KW q p W K').inter t') from x t') := by
  subst h; rfl

theorem

face_zero_eq

L500 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.face_zero_eq

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem face_zero_eq (s : (KW q p W K').Idx 1) : (KW q p W K').face s 0 = σ q p W K' (s.1 1)

theorem

face_one_eq

L506 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.face_one_eq

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem face_one_eq (s : (KW q p W K').Idx 1) : (KW q p W K').face s 1 = σ q p W K' (s.1 0)

theorem

d0_apply

L512 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.d0_apply

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem d0_apply (x : (GW q p W G).cochain (KW q p W K') 0) (s : (KW q p W K').Idx 1) :
    (GW q p W G).d (KW q p W K') 0 x s =
      (GW q p W G).res ((KW q p W K').inter_le_inter_face s 0) (x ((KW q p W K').face s 0)) -
        (GW q p W G).res ((KW q p W K').inter_le_inter_face s 1) (x ((KW q p W K').face s 1))

def

σ₂

L520 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.σ₂

Documentation

The degree-one Cech index associated with two increasing cover indices.

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

def σ₂ {i j : K'.ι} (hij : i < j) : (KW q p W K').Idx 1

theorem

σ₂_fst

L523 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.σ₂_fst

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem σ₂_fst {i j : K'.ι} (hij : i < j) : (σ₂ q p W K' hij).1 0 = i

theorem

σ₂_snd

L524 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.σ₂_snd

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem σ₂_snd {i j : K'.ι} (hij : i < j) : (σ₂ q p W K' hij).1 1 = j

theorem

d0_eq_zero_iff

L526 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.d0_eq_zero_iff

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem d0_eq_zero_iff (x : (GW q p W G).cochain (KW q p W K') 0) :
    (GW q p W G).d (KW q p W K') 0 x = 0 ↔ ∀ i j : K'.ι,
      G.res (U := (K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)) ⊓ (K'.U j ⊓ p ⁻¹ᵁ (W : P.Opens))) inf_le_left (e0
        q p W G K' x i)
        = G.res (U := (K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)) ⊓ (K'.U j ⊓ p ⁻¹ᵁ (W : P.Opens))) inf_le_right
          (e0 q p W G K' x j)

theorem

mem_H0_iff

L577 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.Rebase.mem_H0_iff

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
noncomputable section
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
open _root_.Opposite
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace Rebase
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
variable {A : Type u} [CommRing A] {P V' : Scheme.{u}}
  (q : P ⟶ Spec (CommRingCat.of A)) (p : V' ⟶ P) (W : P.affineOpens)
section Bridge0
variable [IsSeparated q] (G : OModulePresheaf (p ≫ q)) (K' : V'.OrderedAffineCover)

Lean source · statement

theorem mem_H0_iff (x : (GW q p W G).cochain (KW q p W K') 0) :
    x ∈ (GW q p W G).H0 (KW q p W K') ↔ ∀ i j : K'.ι,
      G.res (U := (K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)) ⊓ (K'.U j ⊓ p ⁻¹ᵁ (W : P.Opens))) inf_le_left (e0
        q p W G K' x i)
        = G.res (U := (K'.U i ⊓ p ⁻¹ᵁ (W : P.Opens)) ⊓ (K'.U j ⊓ p ⁻¹ᵁ (W : P.Opens))) inf_le_right
          (e0 q p W G K' x j)

theorem

finite_of_coord

L631 ↗

AlgebraicCurves.Cohomology.Proofs.S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper.AlgebraicGeometry.OModulePresheaf.C2FIN.finite_of_coord

Source context 16 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
attribute [local instance] AlgebraicGeometry.OModulePresheaf.Rebase.isProper_qW
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.σ_val
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom_apply
namespace AlgebraicGeometry
namespace OModulePresheaf
namespace C2FIN
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry

Lean source · statement

theorem finite_of_coord {B : Type u} [CommRing B]
    {ι : Type u} {M : ι → Type u} [∀ i, AddCommGroup (M i)]
    (act : ∀ i, B → M i → M i)
    {C : Type u} [AddCommGroup C] [Module B C]
    (H : Submodule B C) (hfin : Module.Finite B H)
    {N : Type u} [AddCommGroup N] [Module B N]
    (val : N → ∀ i, M i) (val_inj : Function.Injective val)
    (val_add : ∀ x y, val (x + y) = val x + val y)
    (val_smul : ∀ (b : B) (x : N) (i : ι), val (b • x) i = act i b (val x i))
    (e : C →+ (∀ i, M i))
    (e_smul : ∀ (b : B) (x : C) (i : ι), x ∈ H → e (b • x) i = act i b (e x i))
    (lift : ∀ x ∈ H, ∃ n : N, val n = e x)
    (cover : ∀ n : N, ∃ x ∈ H, e x = val n) :
    Module.Finite B N

theorem

isCoherent_cechPushforward_of_isProper

L678 ↗

AlgebraicGeometry.OModulePresheaf.isCoherent_cechPushforward_of_isProper

Source context 21 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Proofs
namespace S_AlgebraicGeometry_OModulePresheaf_isCoherent_cechPushforward_of_isProper
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  DoubleComplex.instModuleE₂I DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
  ProjSpaceCech.GradedModule.Presentation.fJ ProjSpaceCech.GradedModule.sec.instSMul
  ProjSpaceCech.Twist.H.module ProjSpaceCech.Idx.instFintype ProjSpaceCech.Twist.H.addCommGroup
  ProjSpaceCech.Idx.instDecidableEq ProjSpaceCech.Twist.Mon.instDecidableEq
  ProjSpaceCech.Twist.cochain.instAddCommGroup ProjSpaceCech.Twist.cochain.instModule
attribute [-simp] AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app DoubleComplex.Bounded.mk.injEq
  DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq AlgebraicGeometry.ChowDatumProj.mk.sizeOf_spec
  AlgebraicGeometry.ChowDatum.mk.sizeOf_spec AlgebraicGeometry.ChowDatumProj.mk.injEq
  AlgebraicGeometry.ChowDatum.mk.injEq ProjSpaceCech.GradedModule.mk.injEq
  ProjSpaceCech.GradedModule.mk.sizeOf_spec ProjSpaceCech.GradedModule.Frac.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Presentation.mk.injEq ProjSpaceCech.GradedModule.Frac.mk.injEq
  ProjSpaceCech.GradedModule.Presentation.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.shift_toLinearMap ProjSpaceCech.GradedModule.Hom.mk.sizeOf_spec
  ProjSpaceCech.GradedModule.Hom.mk.injEq
universe u
open _root_.CategoryTheory
open _root_.AlgebraicGeometry
attribute [local instance] AlgebraicGeometry.OModulePresheaf.Rebase.isProper_qW
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.σ_val
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom_apply
attribute [local instance] AlgebraicGeometry.OModulePresheaf.Rebase.isProper_qW
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.σ_val
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom_apply
attribute [local instance] AlgebraicGeometry.OModulePresheaf.Rebase.isProper_qW
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.σ_val
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom_apply
attribute [local instance] AlgebraicGeometry.OModulePresheaf.Rebase.isProper_qW
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.σ_val
attribute [local simp] AlgebraicGeometry.OModulePresheaf.Rebase.e0Hom_apply
open _root_.AlgebraicGeometry.OModulePresheaf AlgebraicGeometry.OModulePresheaf in

Lean source · statement

theorem _root_.AlgebraicGeometry.OModulePresheaf.isCoherent_cechPushforward_of_isProper
    {A : Type u} [CommRing A] [IsNoetherianRing A]
    {P : Scheme.{u}} (q : P ⟶ Spec (CommRingCat.of A)) [IsSeparated q] [LocallyOfFiniteType q]
    {V' : Scheme.{u}} (p : V' ⟶ P) [IsProper p]
    (K' : V'.OrderedAffineCover) (G : OModulePresheaf (p ≫ q)) (hc : G.IsCoherent) (hqc :
      G.IsQuasicoherent) :
    (OModulePresheaf.cechPushforward p q K' G).IsCoherent

Module documentation

OModule Presheaf is Coherent cech Pushforward of is Proper

Declarations