NYLL
Module

Scheme_TwoAffineOpenCover_isLocallyConstant_finrank_ker_sub_finrank_coker_cechDiff_baseChange.lean

Nyll.AlgebraicCurves.Cohomology.Cech.Scheme_TwoAffineOpenCover_isLocallyConstant_finrank_ker_sub_finrank_coker_cechDiff_baseChange
2 declarations indexed from sourceView source ↗

Module documentation

Local constancy of the fibre Euler characteristic of two-chart sections of an invertible sheaf.

Imports10
Imported by6

Declarations

theorem

flat_prod

L111 ↗
AlgebraicCurves.Cohomology.Cech.Proofs.TwoAffineEulerCharacteristic.ChiLocallyConstant.flat_prod
Source context 10 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Cech.Proofs
namespace
  TwoAffineEulerCharacteristic
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
attribute [-instance] 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] TwoChartCech.Mumford.dK_apply TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq 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 CategoryTheory CategoryTheory.Limits AlgebraicGeometry TensorProduct
namespace ChiLocallyConstant
Lean source · statement
theorem flat_prod {R : Type*} [CommRing R] (M N : Type*) [AddCommGroup M] [Module R M]
    [AddCommGroup N]
    [Module R N] [Module.Flat R M] [Module.Flat R N] : Module.Flat R (M × N)
theorem

isLocallyConstant_finrank_ker_sub_finrank_coker_cechDiff_baseChange

L147 ↗
AlgebraicGeometry.Scheme.TwoAffineOpenCover.isLocallyConstant_finrank_ker_sub_finrank_coker_cechDiff_baseChange
Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Cech.Proofs
namespace
  TwoAffineEulerCharacteristic
attribute [-instance] AlgebraicGeometry.OModulePresheaf.instSubsingletonObjZero
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.relAltC_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.biC_abGrp
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_modΓ
  AlgebraicGeometry.OModulePresheaf.Leray.ker_relAltd_smul
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_scalarTower
  AlgebraicGeometry.OModulePresheaf.Leray.relAltH_smul
  AlgebraicGeometry.OModulePresheaf.Leray.biC_module
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instDecidableEqIdx
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instLinearOrder
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.instFintypeIdx DoubleComplex.instModuleE₂I
  DoubleComplex.Bounded.modR DoubleComplex.instModuleE₂II DoubleComplex.instAddCommGroupE₂II
  DoubleComplex.Bounded.abGrp DoubleComplex.instAddCommGroupE₂I
  AlgebraicGeometry.ChowDatum.hι_closed AlgebraicGeometry.ChowDatumProj.hιN_closed
  AlgebraicGeometry.ChowDatumProj.hp_proper AlgebraicGeometry.ChowDatum.hp_isoU
  AlgebraicGeometry.ChowDatum.hp_proper AlgebraicGeometry.ProjSpace.algebraAway
  AlgebraicGeometry.ProjSpace.instIsProperProdOverπ AlgebraicGeometry.ChowDatumProj.hp_isoU
  AlgebraicGeometry.ProjSpace.isProper_π AlgebraicGeometry.ProjSpace.finiteType_mvPolynomial
  ProjSpaceCech.GradedModule.H.module ProjSpaceCech.GradedModule.H.addCommGroup
  ProjSpaceCech.GradedModule.sec.instAdd ProjSpaceCech.GradedModule.sec.instNeg
  ProjSpaceCech.GradedModule.acg ProjSpaceCech.GradedModule.Frac.setoid
  ProjSpaceCech.GradedModule.modR ProjSpaceCech.GradedModule.sec.instModule
  ProjSpaceCech.GradedModule.sec.instAddCommGroup ProjSpaceCech.GradedModule.sec.instZero
attribute [-instance] 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] TwoChartCech.Mumford.dK_apply TwoChartCech.Mumford.ι0_apply
  TwoChartCech.Mumford.ι1_apply TwoChartCech.KerCoprod.dK_apply TwoChartCech.KerCoprod.ι1_apply
  TwoChartCech.KerCoprod.ι0_apply AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
  AlgebraicGeometry.Scheme.OrderedAffineCover.restrict_U DoubleComplex.Bounded.mk.injEq
attribute [-simp] DoubleComplex.Bounded.mk.sizeOf_spec DoubleComplex.Convergence.mk.injEq
  DoubleComplex.Convergence.mk.sizeOf_spec DoubleComplex.SubQuot.mk.sizeOf_spec
  DoubleComplex.SubQuot.mk.injEq 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 CategoryTheory CategoryTheory.Limits AlgebraicGeometry TensorProduct
open _root_.AlgebraicCurves.Cohomology.Cech.Proofs in
open TwoAffineEulerCharacteristic
  in
with_weak_namespace _root_.AlgebraicGeometry.Scheme.TwoAffineOpenCover
Lean source · full declaration
theorem isLocallyConstant_finrank_ker_sub_finrank_coker_cechDiff_baseChange
    {A : Type u} [CommRing A] [IsNoetherianRing A] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of
      A))
    [IsProper c] [Flat c] (𝒱 : C.TwoAffineOpenCover) (M : C.Modules) (hM :
      Scheme.Modules.IsInvertible M) :
    IsLocallyConstant fun 𝔭 : PrimeSpectrum A =>
      (Module.finrank 𝔭.asIdeal.ResidueField
          (LinearMap.ker ((𝒱.sectionsOf c M).cechDiff.baseChange 𝔭.asIdeal.ResidueField)) : ℤ) -
        Module.finrank 𝔭.asIdeal.ResidueField
          ((𝔭.asIdeal.ResidueField ⊗[A] (𝒱.sectionsOf c M).M01) ⧸
            LinearMap.range ((𝒱.sectionsOf c M).cechDiff.baseChange 𝔭.asIdeal.ResidueField)) := by
  have flatOf : ∀ (U : C.Opens) (hU : IsAffineOpen U),
      let := Scheme.TwoAffineOpenCover.moduleSectionsOfHom c M U
      Module.Flat A Γ(M, U) := by
    intro U hU
    let := Scheme.TwoAffineOpenCover.algebraOfHom c U
    let := Scheme.TwoAffineOpenCover.moduleSectionsOfHom c M U
    have := Scheme.TwoAffineOpenCover.isScalarTower_sections c M U
    have : Module.Flat A Γ(C, U) := Scheme.TwoAffineOpenCover.flat_sections_of_flat c U hU
    have : Module.Projective Γ(C, U) Γ(M, U) :=
      Scheme.Modules.projective_sections_of_locallyTrivial _ hM.1 ⟨U, hU⟩
    exact Module.Flat.trans A Γ(C, U) Γ(M, U)
  have : Module.Flat A (𝒱.sectionsOf c M).M0 := flatOf _ 𝒱.isAffineOpen_U0
  have : Module.Flat A (𝒱.sectionsOf c M).M1 := flatOf _ 𝒱.isAffineOpen_U1
  have : Module.Flat A (𝒱.sectionsOf c M).M01 := flatOf _ 𝒱.isAffineOpen_inf
  have : Module.Flat A ((𝒱.sectionsOf c M).M0 × (𝒱.sectionsOf c M).M1) :=
    ChiLocallyConstant.flat_prod _ _
  have hcf := OModulePresheaf.cechFinite_ofModules_of_locallyTrivial c M hM.1 𝒱.toOrderedAffineCover
  obtain ⟨hfin0, hfin1⟩ := (OModulePresheaf.cechFinite_toOrderedAffineCover_iff _ _).mp hcf
  have : Module.Finite A (LinearMap.ker (𝒱.sectionsOf c M).cechDiff) := hfin0
  have : Module.Finite A ((𝒱.sectionsOf c M).M01 ⧸ LinearMap.range (𝒱.sectionsOf c M).cechDiff) :=
    hfin1
  exact TwoChartCech.isLocallyConstant_fibreEulerChar (𝒱.sectionsOf c M).cechDiff

Statements and documentation are read from the Lean source at the revision shown. Source entries are not a measure of mathematical completeness.