NYLL
Module

exists_representsRelSubPic_algEquivZeroCut_of_openCharts_of_bijective_sections.lean

Nyll.JacobianVarieties.Picard.exists_representsRelSubPic_algEquivZeroCut_of_openCharts_of_bijective_sections
4 declarations indexed from sourceView source ↗

Module documentation

Construct the representing scheme from finitely many open charts and bijective base sections.

Imports35
Imported by2

Declarations

theorem

toGlued_toBase

L267 ↗
JacobianVarieties.Picard.Proofs.OpenChartRepresentation.LEG2Core.toGlued_toBase
Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChartRepresentation
attribute [-instance]
  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
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  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
attribute [-instance]
  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
  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
attribute [-instance] ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.thetaBundle_def
  AlgebraicGeometry.RelPicard.picardBundle_def
  AlgebraicGeometry.RelEffCartierDiv.toRelEffDivisor_ofRelEffDivisor
  AlgebraicGeometry.RelEffCartierDiv.toRelEffDivisor_I
  AlgebraicGeometry.mapOnProdOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.mk.sizeOf_spec
  AlgebraicGeometry.RelEffCartierDiv.mk.injEq
  AlgebraicGeometry.mapOnProdOver_snd
  AlgebraicGeometry.mapOnProdOver_fst
  AlgebraicGeometry.mapOnProdOver_snd_assoc
  AlgebraicGeometry.mapOnProdOver_id
  AlgebraicCurve.RelEffDivisor.mk.sizeOf_spec
  AlgebraicCurve.mapOnProd_fst
  AlgebraicCurve.mapOnProd_fst_assoc
  AlgebraicCurve.mapOnProd_snd
  AlgebraicCurve.UnivDivisorPack.mk.injEq
  AlgebraicCurve.RelEffDivisor.mk.injEq
  AlgebraicCurve.UnivDivisorPack.mk.sizeOf_spec
  AlgebraicCurve.mapOnProd_snd_assoc
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj
  PresheafOfModules.exteriorPower_map_ιMulti
attribute [-simp]
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  CoherentBaseChange.FibreH0Family.mk.sizeOf_spec
  CoherentBaseChange.FibreH0Family.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
attribute [-simp]
  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.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
namespace LEG2Core
section glued
variable {R : Type u} [CommRing R] {C : Scheme.{u}} {c : C ⟶ Spec (CommRingCat.of R)}
  {ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c}
  (hG : Presieve.IsSheaf (Scheme.zariskiTopology.over (Spec (CommRingCat.of R)))
    (relSubPicPresheaf c ε (algEquivZeroCut c ε)))
  {ι : Type u} {X : ι → Scheme.{u}}
  (f : ∀ i, uliftYoneda.{u + 1}.obj (X i) ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal)
  (hf : ∀ i, MorphismProperty.presheafULift.{u + 1} @IsOpenImmersion (f i))
  [Presheaf.IsLocallySurjective Scheme.zariskiTopology (Limits.Sigma.desc f)]
Lean source · statement
theorem toGlued_toBase (i : ι) :
    Scheme.LocalRepresentabilityULift.toGlued (F := relSubPicTotalSheaf hG) (f := f) hf i ≫
      (gluedDesignation hG f hf).toBase = (uliftYonedaEquiv (f i)).1
theorem

locallyOfFiniteType_gluedDesignation

L283 ↗
JacobianVarieties.Picard.Proofs.OpenChartRepresentation.LEG2Core.locallyOfFiniteType_gluedDesignation
Source context 22 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChartRepresentation
attribute [-instance]
  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
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  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
attribute [-instance]
  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
  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
attribute [-instance] ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.thetaBundle_def
  AlgebraicGeometry.RelPicard.picardBundle_def
  AlgebraicGeometry.RelEffCartierDiv.toRelEffDivisor_ofRelEffDivisor
  AlgebraicGeometry.RelEffCartierDiv.toRelEffDivisor_I
  AlgebraicGeometry.mapOnProdOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.mk.sizeOf_spec
  AlgebraicGeometry.RelEffCartierDiv.mk.injEq
  AlgebraicGeometry.mapOnProdOver_snd
  AlgebraicGeometry.mapOnProdOver_fst
  AlgebraicGeometry.mapOnProdOver_snd_assoc
  AlgebraicGeometry.mapOnProdOver_id
  AlgebraicCurve.RelEffDivisor.mk.sizeOf_spec
  AlgebraicCurve.mapOnProd_fst
  AlgebraicCurve.mapOnProd_fst_assoc
  AlgebraicCurve.mapOnProd_snd
  AlgebraicCurve.UnivDivisorPack.mk.injEq
  AlgebraicCurve.RelEffDivisor.mk.injEq
  AlgebraicCurve.UnivDivisorPack.mk.sizeOf_spec
  AlgebraicCurve.mapOnProd_snd_assoc
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj
  PresheafOfModules.exteriorPower_map_ιMulti
attribute [-simp]
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  CoherentBaseChange.FibreH0Family.mk.sizeOf_spec
  CoherentBaseChange.FibreH0Family.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
attribute [-simp]
  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.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
namespace LEG2Core
section glued
variable {R : Type u} [CommRing R] {C : Scheme.{u}} {c : C ⟶ Spec (CommRingCat.of R)}
  {ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c}
  (hG : Presieve.IsSheaf (Scheme.zariskiTopology.over (Spec (CommRingCat.of R)))
    (relSubPicPresheaf c ε (algEquivZeroCut c ε)))
  {ι : Type u} {X : ι → Scheme.{u}}
  (f : ∀ i, uliftYoneda.{u + 1}.obj (X i) ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal)
  (hf : ∀ i, MorphismProperty.presheafULift.{u + 1} @IsOpenImmersion (f i))
  [Presheaf.IsLocallySurjective Scheme.zariskiTopology (Limits.Sigma.desc f)]
include hG f hf in
Lean source · statement
theorem locallyOfFiniteType_gluedDesignation
    (hft : ∀ i, LocallyOfFiniteType (uliftYonedaEquiv (f i)).1) :
    LocallyOfFiniteType (gluedDesignation hG f hf).toBase
theorem

quasiCompact_gluedDesignation

L300 ↗
JacobianVarieties.Picard.Proofs.OpenChartRepresentation.LEG2Core.quasiCompact_gluedDesignation
Source context 21 commands

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

section ProofImplementation
namespace JacobianVarieties.Picard.Proofs.OpenChartRepresentation
attribute [-instance]
  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
  PresheafOfModules.PullbackMonoidal.pullback_monoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ
  PresheafOfModules.pushforward_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_η
  PresheafOfModules.free_monoidal
  PresheafOfModules.restrictScalars_laxMonoidal
  PresheafOfModules.PullbackMonoidal.isIso_δ_gS
  PresheafOfModules.pullback_oplaxMonoidal
attribute [-instance]
  instPreservesColimitsOfSizeCompOppositeCommRingCatRingCatForget₂RingHomCarrierCarrierPb
attribute [-instance]
  PresheafOfModules.pullback_monoidal'
  AlgebraicGeometry.SmoothOfRelativeDimension.fiberToSpecResidueField
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_snd
  AlgebraicGeometry.SmoothOfRelativeDimension.pullback_fst
  AlgebraicGeometry.SmoothOfRelativeDimension.smooth_one
  AlgebraicGeometry.SmoothProperCurve.isIntegral_pullback_Spec_field
  AlgebraicGeometry.IsProper.fiberToSpecResidueField
  PresheafOfModules.ExteriorPower.instModulePresheafAb
  AlgebraicGeometry.OModulePresheaf.isScalarTower
  AlgebraicGeometry.Scheme.OrderedAffineCover.instLinearOrder
  AlgebraicGeometry.OModulePresheaf.module
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintype
  AlgebraicGeometry.Scheme.OrderedAffineCover.instFintypeIdx
  AlgebraicGeometry.OModulePresheaf.addCommGroup
  AlgebraicGeometry.Scheme.OrderedAffineCover.instDecidableEqIdx
  AlgebraicGeometry.OModulePresheaf.moduleSections
  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
attribute [-instance]
  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
  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
attribute [-instance] ProjSpaceCech.Twist.cochain.instModule
attribute [-simp]
  AlgebraicGeometry.RelPicard.rigSection_snd
  AlgebraicGeometry.RelPicard.RigidifiedLineBundle.ofInvertible_L
  AlgebraicGeometry.RelPicard.rigSection_snd_assoc
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.injEq
  AlgebraicGeometry.RelPicard.SubPicGroupCondition.mk.sizeOf_spec
  GoodReductionJacobian.relativeGroupLawOfGrpObj_inv
  GoodReductionJacobian.relativeGroupLawOfGrpObj_mul
  GoodReductionJacobian.overHomEquivSchemeHomOver_apply_coe
  GoodReductionJacobian.relativeGroupLawOfGrpObj_one
  GoodReductionJacobian.overHomEquivSchemeHomOver_symm_apply_left
  PresheafOfModules.freeεIso_hom_app
  PresheafOfModules.freeμIso_hom_app
  AlgebraicGeometry.tilde.functorCompPullbackSpecIso_app
  AlgebraicGeometry.RelPicard.LFP.stageHom_val
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_hom_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.relSubPicPresheafRestrictIso_inv_app_coe
  AlgebraicGeometry.RelPicard.BaseChange.restrict_P
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_snd
  AlgebraicGeometry.SmoothProperCurve.sectionBaseChange_coe_fst
  AlgebraicGeometry.RelPicard.thetaBundle_def
  AlgebraicGeometry.RelPicard.picardBundle_def
  AlgebraicGeometry.RelEffCartierDiv.toRelEffDivisor_ofRelEffDivisor
  AlgebraicGeometry.RelEffCartierDiv.toRelEffDivisor_I
  AlgebraicGeometry.mapOnProdOver_fst_assoc
  AlgebraicGeometry.RelEffCartierDiv.mk.sizeOf_spec
  AlgebraicGeometry.RelEffCartierDiv.mk.injEq
  AlgebraicGeometry.mapOnProdOver_snd
  AlgebraicGeometry.mapOnProdOver_fst
  AlgebraicGeometry.mapOnProdOver_snd_assoc
  AlgebraicGeometry.mapOnProdOver_id
  AlgebraicCurve.RelEffDivisor.mk.sizeOf_spec
  AlgebraicCurve.mapOnProd_fst
  AlgebraicCurve.mapOnProd_fst_assoc
  AlgebraicCurve.mapOnProd_snd
  AlgebraicCurve.UnivDivisorPack.mk.injEq
  AlgebraicCurve.RelEffDivisor.mk.injEq
  AlgebraicCurve.UnivDivisorPack.mk.sizeOf_spec
  AlgebraicCurve.mapOnProd_snd_assoc
  AlgebraicGeometry.Scheme.Modules.exteriorPower_obj
  PresheafOfModules.exteriorPower_map_ιMulti
attribute [-simp]
  PresheafOfModules.ExteriorPower.appₗ_apply
  AlgebraicGeometry.RelPicard.algEquivZeroGroupCut_toSubPicCondition
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.injEq
  AlgebraicGeometry.OModulePresheaf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.mk.injEq
  CoherentBaseChange.FibreH0Family.mk.sizeOf_spec
  CoherentBaseChange.FibreH0Family.mk.injEq
  AlgebraicGeometry.OModulePresheaf.prod_obj
  AlgebraicGeometry.OModulePresheaf.restrOpen_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.injEq
  AlgebraicGeometry.OModulePresheaf.pushforward_obj
  AlgebraicGeometry.OModulePresheaf.im_obj
  AlgebraicGeometry.OModulePresheaf.pow_obj
  AlgebraicGeometry.OModulePresheaf.fstHom_app
  AlgebraicGeometry.OModulePresheaf.ker_obj
  AlgebraicGeometry.OModulePresheaf.coker_obj
  AlgebraicGeometry.OModulePresheaf.DevissageStep.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_U
  AlgebraicGeometry.OModulePresheaf.sndHom_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.id_app
  AlgebraicGeometry.OModulePresheaf.AffHom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffHom.comp_app
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.kerMap_coe
  AlgebraicGeometry.OModulePresheaf.AffHom.id_app
  AlgebraicGeometry.OModulePresheaf.Hom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.toAffHom_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.sizeOf_spec
  AlgebraicGeometry.OModulePresheaf.Hom.comp_app
  AlgebraicGeometry.OModulePresheaf.SES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.AffHom.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Hom.appSections_apply
  AlgebraicGeometry.OModulePresheaf.AffSES.mk.injEq
  AlgebraicGeometry.OModulePresheaf.Leray.restrictToPreimage_U
  AlgebraicGeometry.Scheme.OrderedAffineCover.toCoverOf_U
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.sizeOf_spec
  AlgebraicGeometry.Scheme.OrderedAffineCoverOf.mk.injEq
attribute [-simp]
  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.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
namespace LEG2Core
section glued
variable {R : Type u} [CommRing R] {C : Scheme.{u}} {c : C ⟶ Spec (CommRingCat.of R)}
  {ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c}
  (hG : Presieve.IsSheaf (Scheme.zariskiTopology.over (Spec (CommRingCat.of R)))
    (relSubPicPresheaf c ε (algEquivZeroCut c ε)))
  {ι : Type u} {X : ι → Scheme.{u}}
  (f : ∀ i, uliftYoneda.{u + 1}.obj (X i) ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c ε)).overTotal)
  (hf : ∀ i, MorphismProperty.presheafULift.{u + 1} @IsOpenImmersion (f i))
  [Presheaf.IsLocallySurjective Scheme.zariskiTopology (Limits.Sigma.desc f)]
Lean source · statement
theorem quasiCompact_gluedDesignation [Finite ι] [∀ i, CompactSpace (X i)] :
    QuasiCompact (gluedDesignation hG f hf).toBase
theorem

exists_representsRelSubPic_algEquivZeroCut_of_openCharts_of_bijective_sections

L333 ↗
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_openCharts_of_bijective_sections
Documentation

Construct the representing scheme from finitely many open charts and bijective base sections.

Source context 13 commands

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

section ProofImplementation
namespace AlgebraicGeometry.RelPicard
open _root_.JacobianVarieties.Picard.Proofs
open _root_.JacobianVarieties.Picard.Proofs.OpenChartRepresentation
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.AlgebraicGeometry.RelPicard
open _root_.NeronModelInfra
open _root_.GoodReductionJacobian
open _root_.JacobianVarieties.Picard.Proofs.OpenChartRepresentation.LEG2Core in
Lean source · full declaration
theorem exists_representsRelSubPic_algEquivZeroCut_of_openCharts_of_bijective_sections
    (R : Type u) [CommRing R] [IsNoetherianRing R] {C : Scheme.{u}} (c : C ⟶ Spec (CommRingCat.of
      R))
    [IsProper c] [Flat c]
    (𝒱 : C.TwoAffineOpenCover)
    (ε : SchemeHomOver (𝟙 (Spec (CommRingCat.of R))) c)
    (hH0 : ∀ (A : Type u) [CommRing A] [Algebra R A],
      let : Algebra A Γ(Limits.pullback c (Scheme.TwoAffineOpenCover.specMap R A), ⊤) :=
        Scheme.TwoAffineOpenCover.algebraOfHom
          (Limits.pullback.snd c (Scheme.TwoAffineOpenCover.specMap R A)) ⊤
      Function.Bijective (algebraMap A Γ(Limits.pullback c (Scheme.TwoAffineOpenCover.specMap R A),
        ⊤)))
    (hfib : ∀ (k : Type u) [Field k] [IsAlgClosed k]
      (x : Spec (CommRingCat.of k) ⟶ Spec (CommRingCat.of R))
      (L : (pullback c x).Modules), Scheme.Modules.IsInvertible L →
      IsAlgEquivZero (pullback.snd c x) L →
      ∀ s : 𝟙_ (pullback c x).Modules ⟶ L, s ≠ 0 → Nonempty (L ≅ 𝟙_ (pullback c x).Modules))
    {ι : Type u} [Finite ι] (X : ι → Scheme.{u}) [∀ i, CompactSpace (X i)]
    (f : ∀ i, uliftYoneda.{u + 1}.obj (X i) ⟶ (relSubPicPresheaf c ε (algEquivZeroCut c
      ε)).overTotal)
    (hf : ∀ i, MorphismProperty.presheafULift.{u + 1} @IsOpenImmersion (f i))
    (hft : ∀ i, LocallyOfFiniteType (uliftYonedaEquiv (f i)).1)
    (hsurj : Presheaf.IsLocallySurjective Scheme.zariskiTopology (Limits.Sigma.desc f)) :
    ∃ D : RelativePic0Designation R c,
      Nonempty (RepresentsRelSubPic c ε (algEquivZeroCut c ε) D) ∧
        Smooth D.toBase ∧ IsSeparated D.toBase ∧ QuasiCompact D.toBase := by
  have hG := isSheaf_relSubPicPresheaf_algEquivZeroCut_zariski_of_bijective_sections R c ε hH0
  have : Presheaf.IsLocallySurjective Scheme.zariskiTopology (Limits.Sigma.desc f) := hsurj
  obtain ⟨h⟩ := RepresentsRelSubPic.nonempty_gluedDesignation hG f hf
  have : LocallyOfFiniteType (gluedDesignation hG f hf).toBase :=
    locallyOfFiniteType_gluedDesignation hG f hf hft
  exact ⟨gluedDesignation hG f hf, ⟨h⟩,
    smooth_of_representsRelSubPic_algEquivZeroCut_of_twoAffineOpenCover R c 𝒱 ε _ h,
    isSeparated_of_representsRelSubPic_algEquivZeroCut_of_bijective_sections R c 𝒱 ε hH0 hfib _ h,
    quasiCompact_gluedDesignation hG f hf⟩

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