Module

OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.lean

Nyll.AlgebraicCurves.Cohomology.Sheaves.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq

30 declarations indexed from sourceView source ↗

Module documentation

OModule Presheaf exists polynomial forall euler Char twist tensor Pow eq

Imports17

Imported by6

def

ext

L125 ↗

P2mChiDatum.ext

Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum

Lean source · statement

def _root_.P2mChiDatum.ext (f : ℕ → ℚ) (Q : ℤ → ℚ) : ℤ → ℚ

theorem

ext_natCast

L129 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.ext_natCast

Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum

Lean source · statement

theorem ext_natCast (f : ℕ → ℚ) (Q : ℤ → ℚ) (n : ℕ) : ext f Q n = f n

theorem

ext_neg_succ

L132 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.ext_neg_succ

Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum

Lean source · statement

theorem ext_neg_succ (f : ℕ → ℚ) (Q : ℤ → ℚ) (j : ℕ) :
    ext f Q (-(j + 1 : ℕ)) = f 0 - ∑ i ∈ Finset.range (j + 1), Q (-(i + 1 : ℕ))

theorem

fwdDiff_ext

L137 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.fwdDiff_ext

Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum

Lean source · statement

theorem fwdDiff_ext (f : ℕ → ℚ) (Q : ℤ → ℚ) (hf : ∀ n : ℕ, f (n + 1) - f n = Q n) (m : ℤ) :
    fwdDiff (1 : ℤ) (ext f Q) m = Q m

theorem

poly_of_delta

L158 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.poly_of_delta

Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum

Lean source · statement

theorem poly_of_delta (f : ℕ → ℚ) (q : ℚ[X]) (e : ℕ) (hq : q.degree < e)
    (hf : ∀ n : ℕ, f (n + 1) - f n = q.eval (n : ℚ)) :
    ∃ p : ℚ[X], p.degree < (e + 1 : ℕ) ∧ ∀ n : ℕ, f n = p.eval (n : ℚ)

theorem

isIrreducible_of_isIntegral_subscheme

L194 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.isIrreducible_of_isIntegral_subscheme

Source context 16 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Dim
variable {V : Scheme.{u}}

Lean source · statement

theorem isIrreducible_of_isIntegral_subscheme (Z₀ : Closeds V)
    (hint : IsIntegral (Scheme.IdealSheafData.vanishingIdeal Z₀).subscheme) :
    IsIrreducible (Z₀ : Set V)

theorem

dim_lt_of_lt

L204 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.dim_lt_of_lt

Source context 16 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Dim
variable {V : Scheme.{u}}

Lean source · statement

theorem dim_lt_of_lt (Z₀ Y' : Closeds V) (hint : IsIntegral (Scheme.IdealSheafData.vanishingIdeal
  Z₀).subscheme)
    (hY : Y' < Z₀) (e : ℕ) (hd : topologicalKrullDim Z₀ < (e + 1 : ℕ)) :
    topologicalKrullDim Y' < (e : ℕ)

theorem

dim_mono

L217 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.dim_mono

Source context 16 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Dim
variable {V : Scheme.{u}}

Lean source · statement

theorem dim_mono (Y Y' : Closeds V) (h : Y' ≤ Y) : topologicalKrullDim Y' ≤ topologicalKrullDim Y

theorem

dim_nonneg

L221 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.dim_nonneg

Source context 16 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Dim
variable {V : Scheme.{u}}

Lean source · statement

theorem dim_nonneg (Y : Closeds V) (hY : (Y : Set V).Nonempty) : 0 ≤ topologicalKrullDim Y

def

chi

L234 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.chi

Documentation

The Euler characteristic after twisting by the indicated tensor power of the line bundle.

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Skeleton
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)

Lean source · statement

def chi (G : OModulePresheaf π) (n : ℕ) : ℤ

def

Good

L237 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.Good

Documentation

Coherence, quasicoherence, and support in the specified closed subset.

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Skeleton
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)

Lean source · statement

def Good (G : OModulePresheaf π) (Y : Closeds V) : Prop

def

P

L242 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.P

Documentation

Polynomial dependence of the twisted Euler characteristic, with degree below the given bound.

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Skeleton
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)

Lean source · statement

def P (e : ℕ) (G : OModulePresheaf π) : Prop

def

IH

L246 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.IH

Documentation

Polynomial dependence for sheaves supported on proper closed subsets of the given support.

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Skeleton
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)

Lean source · statement

def IH (Z₀ : Closeds V) : Prop

abbrev

OZ

L252 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.OZ

Documentation

The pushforward of the structure module of the closed subscheme defined by the vanishing ideal.

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Skeleton
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)

Lean source · statement

abbrev OZ (Z₀ : Closeds V) : OModulePresheaf π

theorem

skeleton

L255 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.skeleton

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Skeleton
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)

Lean source · statement

theorem skeleton
    (hDev : (∀ (Z₀ : Closeds V), (Z₀ : Set V).Nonempty →
        IsIntegral (Scheme.IdealSheafData.vanishingIdeal Z₀).subscheme →
        ∀ e : ℕ, topologicalKrullDim Z₀ < (e + 1 : ℕ) → IH π K L Z₀ → P π K L (e + 1) (OZ π Z₀)) →
      ∀ (Y : Closeds V) (G : OModulePresheaf π) (e : ℕ), Good π G Y → topologicalKrullDim Y < (e :
        ℕ) →
        P π K L e G)
    (hTwist : ∀ (Z₀ : Closeds V), (Z₀ : Set V).Nonempty →
        IsIntegral (Scheme.IdealSheafData.vanishingIdeal Z₀).subscheme →
        ∃ (Y₁ : Closeds V) (G₁ : OModulePresheaf π) (Y₂ : Closeds V) (G₂ : OModulePresheaf π),
          Y₁ < Z₀ ∧ Good π G₁ Y₁ ∧ Y₂ < Z₀ ∧ Good π G₂ Y₂ ∧
          ∀ n : ℕ, chi π K L (OZ π Z₀) (n + 1) - chi π K L (OZ π Z₀) n = chi π K L G₁ (n + 1) - chi
            π K L G₂ n) :
    ∀ (Y : Closeds V) (G : OModulePresheaf π) (e : ℕ), Good π G Y → topologicalKrullDim Y < (e : ℕ)
      →
      P π K L e G

theorem

locTriv_tensorPow

L305 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.locTriv_tensorPow

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem locTriv_tensorPow (n : ℕ) : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj (L.tensorPow n) ≅ SheafOfModules.unit
      U.toScheme.ringCatSheaf)

theorem

good_twist

L314 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.good_twist

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem good_twist (G : OModulePresheaf π) (hc : G.IsCoherent) (hq : G.IsQuasicoherent) (n : ℕ) :
    (G.twist (L.tensorPow n)).IsCoherent ∧ (G.twist (L.tensorPow n)).IsQuasicoherent

theorem

cechFinite_twist

L324 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.cechFinite_twist

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem cechFinite_twist [IsProper π] (G : OModulePresheaf π) (hc : G.IsCoherent) (hq :
  G.IsQuasicoherent) (n : ℕ) :
    (G.twist (L.tensorPow n)).CechFinite K

theorem

flat_sections

L331 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.flat_sections

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem flat_sections (n : ℕ) (U : V.affineOpens) :
    Module.Flat Γ(V, U.1) ((OModulePresheaf.ofModules π (L.tensorPow n)).obj U.1)

theorem

chi_add

L339 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.chi_add

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem chi_add [IsProper π] {G₁ G₂ G₃ : OModulePresheaf π} (S : OModulePresheaf.AffSES G₁ G₂ G₃)
    (h₁c : G₁.IsCoherent) (h₁q : G₁.IsQuasicoherent) (h₂c : G₂.IsCoherent) (h₂q :
      G₂.IsQuasicoherent)
    (h₃c : G₃.IsCoherent) (h₃q : G₃.IsQuasicoherent) (n : ℕ) :
    chi π K L G₂ n = chi π K L G₁ n + chi π K L G₃ n

theorem

eulerChar_eq_zero_of_subsingleton

L350 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.eulerChar_eq_zero_of_subsingleton

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))

Lean source · statement

theorem eulerChar_eq_zero_of_subsingleton [IsSeparated π] (G : OModulePresheaf π)
    (hG : ∀ U : V.affineOpens, Subsingleton (G.obj U.1)) : G.eulerChar K = 0

theorem

chi_eq_zero_of_subsingleton

L369 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.chi_eq_zero_of_subsingleton

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))

Lean source · statement

theorem chi_eq_zero_of_subsingleton [IsSeparated π] (G : OModulePresheaf π)
    (hG : ∀ U : V.affineOpens, Subsingleton (G.obj U.1)) (n : ℕ) : chi π K L G n = 0

def

Q

L377 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.Q

Documentation

Polynomial dependence with every degree bound exceeding the dimension of the support.

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))

Lean source · statement

def Q (Y : Closeds V) (G : OModulePresheaf π) : Prop

theorem

Q_zero

L380 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.Q_zero

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))

Lean source · statement

theorem Q_zero [IsSeparated π] (Y : Closeds V) (G : OModulePresheaf π)
    (hG : ∀ U : V.affineOpens, Subsingleton (G.obj U.1)) : Q π K L Y G

theorem

Q_mono

L385 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.Q_mono

Source context 17 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))

Lean source · statement

theorem Q_mono (Y Y' : Closeds V) (G : OModulePresheaf π) (h : Y' ≤ Y) (hQ : Q π K L Y' G) : Q π K
  L Y G

theorem

Q_ext

L390 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.Q_ext

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem Q_ext [IsProper π] (Y : Closeds V) (G₁ G₂ G₃ : OModulePresheaf π)
    (hS : Nonempty (OModulePresheaf.AffSES G₁ G₂ G₃))
    (h₁c : G₁.IsCoherent) (h₁q : G₁.IsQuasicoherent) (h₂c : G₂.IsCoherent) (h₂q :
      G₂.IsQuasicoherent)
    (h₃c : G₃.IsCoherent) (h₃q : G₃.IsQuasicoherent) :
    (Q π K L Y G₁ → Q π K L Y G₃ → Q π K L Y G₂) ∧ (Q π K L Y G₁ → Q π K L Y G₂ → Q π K L Y G₃) ∧
      (Q π K L Y G₂ → Q π K L Y G₃ → Q π K L Y G₁)

theorem

hDev

L414 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.hDev

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem hDev [IsProper π]
    (hstep : ∀ (Z₀ : Closeds V), (Z₀ : Set V).Nonempty →
      IsIntegral (Scheme.IdealSheafData.vanishingIdeal Z₀).subscheme →
      ∀ e : ℕ, topologicalKrullDim Z₀ < (e + 1 : ℕ) → IH π K L Z₀ → P π K L (e + 1) (OZ π Z₀)) :
    ∀ (Y : Closeds V) (G : OModulePresheaf π) (e : ℕ), Good π G Y → topologicalKrullDim Y < (e : ℕ)
      →
      P π K L e G

theorem

hTwist

L434 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.hTwist

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem hTwist [IsProper π] (Z₀ : Closeds V) (hne : (Z₀ : Set V).Nonempty)
    (hint : IsIntegral (Scheme.IdealSheafData.vanishingIdeal Z₀).subscheme) :
    ∃ (Y₁ : Closeds V) (G₁ : OModulePresheaf π) (Y₂ : Closeds V) (G₂ : OModulePresheaf π),
      Y₁ < Z₀ ∧ Good π G₁ Y₁ ∧ Y₂ < Z₀ ∧ Good π G₂ Y₂ ∧
      ∀ n : ℕ, chi π K L (OZ π Z₀) (n + 1) - chi π K L (OZ π Z₀) n = chi π K L G₁ (n + 1) - chi π K
        L G₂ n

theorem

card

L445 ↗

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq.P2mChiDatum.card

Source context 18 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct
namespace P2mChiDatum
section Inputs
variable {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k))
variable (K : V.OrderedAffineCover) (L : V.Modules)
  (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
    Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
include hL in

Lean source · statement

theorem card [IsProper π] (G : OModulePresheaf π)
    (hc : G.IsCoherent) (hq : G.IsQuasicoherent)
    (Y : TopologicalSpace.Closeds V) (hY : G.SupportedIn Y)
    (d : ℕ) (hd : topologicalKrullDim Y ≤ d) :
    ∃ p : Polynomial ℚ, p.natDegree ≤ d ∧
      ∀ n : ℕ, ((G.twist (L.tensorPow n)).eulerChar K : ℚ) = p.eval (n : ℚ)

theorem

exists_polynomial_forall_eulerChar_twist_tensorPow_eq

L465 ↗

AlgebraicGeometry.OModulePresheaf.exists_polynomial_forall_eulerChar_twist_tensorPow_eq

Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs
namespace OModulePresheaf_exists_polynomial_forall_eulerChar_twist_tensorPow_eq
attribute [-instance]
  pullback_monoidal
  isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.Scheme.Modules.preservesBinaryProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback_monoidal
  AlgebraicGeometry.Scheme.Modules.sheafify_isLocalization'
  AlgebraicGeometry.Scheme.Modules.preservesTerminal_opensMap
  AlgebraicGeometry.Scheme.Modules.pullback₀_monoidal
  AlgebraicGeometry.Scheme.Modules.preservesFiniteProducts_opensMap
  AlgebraicGeometry.Scheme.Modules.sheafifyPullback_lifting
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx
  DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR
  DoubleComplex.instModuleE₂II
  DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp
  DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed
  AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper
  AlgebraicGeometry.ChowDatum.hp_isoU
attribute [-instance] 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.Scheme.Modules.tensorSections_zero_right
  AlgebraicGeometry.Scheme.Modules.map_unitSection
  AlgebraicGeometry.Scheme.Modules.tensorSectionsBilin_apply
  AlgebraicGeometry.Scheme.Modules.tensorPowSection_zero
  AlgebraicGeometry.Scheme.Modules.tensorSections_zero_left PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
  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_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.Polynomial
open _root_.TopologicalSpace
open scoped TensorProduct

Lean source · statement

theorem
  _root_.AlgebraicGeometry.OModulePresheaf.exists_polynomial_forall_eulerChar_twist_tensorPow_eq
    {k : Type u} [Field k] {V : Scheme.{u}} (π : V ⟶ Spec (.of k)) [IsProper π]
    (K : V.OrderedAffineCover) (G : OModulePresheaf π)
    (hc : G.IsCoherent) (hq : G.IsQuasicoherent)
    (Y : TopologicalSpace.Closeds V) (hY : G.SupportedIn Y)
    (d : ℕ) (hd : topologicalKrullDim Y ≤ d)
    (L : V.Modules)
    (hL : ∀ x : V, ∃ (U : V.Opens), x ∈ U ∧
      Nonempty ((Scheme.Modules.pullback U.ι).obj L ≅ SheafOfModules.unit U.toScheme.ringCatSheaf))
        :
    ∃ p : Polynomial ℚ, p.natDegree ≤ d ∧
      ∀ n : ℕ, ((G.twist (L.tensorPow n)).eulerChar K : ℚ) = p.eval (n : ℚ)

Module documentation

OModule Presheaf exists polynomial forall euler Char twist tensor Pow eq

Declarations