NYLL
Module

Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.lean

Nyll.AlgebraicCurves.Geometry.Sheaves.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
12 declarations indexed from sourceView source ↗

Module documentation

Scheme Modules exists is Invertible pullback Section eq zero iff not Mem of is Affine Open

Imports8
Imported by2

Declarations

lemma

exists_notMem_mul_mem

L85 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mWeilCartier.exists_notMem_mul_mem
Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mWeilCartier
section SpreadOut
variable {R : Type*} [CommRing R] (p : Ideal R) [p.IsPrime]
  (S : Type*) [CommRing S] [Algebra R S] [IsLocalization.AtPrime S p]
Lean source · statement
lemma exists_notMem_mul_mem (𝔞 : Ideal R) (h𝔞 : 𝔞.FG) (g : R)
    (h : 𝔞.map (algebraMap R S) = Ideal.span {algebraMap R S g}) :
    ∃ f ∉ p, (∀ a ∈ 𝔞, ∃ r : R, f * a = r * g) ∧ f * g ∈ 𝔞
lemma

map_eq_span_of_away

L144 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mWeilCartier.map_eq_span_of_away
Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mWeilCartier
section SpreadOut
variable {R : Type*} [CommRing R] (p : Ideal R) [p.IsPrime]
  (S : Type*) [CommRing S] [Algebra R S] [IsLocalization.AtPrime S p]
Lean source · statement
lemma map_eq_span_of_away {𝔞 : Ideal R} {f g : R} (h1 : ∀ a ∈ 𝔞, ∃ r : R, f * a = r * g)
    (h2 : f * g ∈ 𝔞) (T : Type*) [CommRing T] [Algebra R T] [IsLocalization.Away f T] :
    𝔞.map (algebraMap R T) = Ideal.span {algebraMap R T g}
lemma

prod_dvd_of_forall_prime_dvd

L167 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mWeilCartier.prod_dvd_of_forall_prime_dvd
Source context 15 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mWeilCartier
section UFD
variable {S : Type*} [CommRing S] [IsDomain S] [UniqueFactorizationMonoid S]
omit [IsDomain S] [UniqueFactorizationMonoid S] in
Lean source · statement
lemma prod_dvd_of_forall_prime_dvd {ι : Type*} (s : Finset ι) (π : ι → S)
    (hπ : ∀ i ∈ s, Prime (π i)) (hne : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → ¬ π i ∣ π j)
    (a : S) (ha : ∀ i ∈ s, π i ∣ a) : (∏ i ∈ s, π i) ∣ a
lemma

exists_eq_span_singleton_of_radical

L188 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mWeilCartier.exists_eq_span_singleton_of_radical
Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mWeilCartier
section UFD
variable {S : Type*} [CommRing S] [IsDomain S] [UniqueFactorizationMonoid S]
Lean source · statement
lemma exists_eq_span_singleton_of_radical (J : Ideal S) (hJ0 : J ≠ ⊥) (hJrad : J.IsRadical)
    (hfin : J.minimalPrimes.Finite)
    (hht : ∀ Q ∈ J.minimalPrimes, ∀ P : Ideal S, P.IsPrime → P ≠ ⊥ → P ≤ Q → P = Q) :
    ∃ γ : S, J = Ideal.span {γ}
lemma

exists_generator_of_notMem

L234 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mWeilCartier.exists_generator_of_notMem
Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mWeilCartier
section Geometry
variable {X : Scheme.{u}}
Lean source · statement
lemma exists_generator_of_notMem (Z : Closeds X) {x : X} (hxZ : x ∉ Z) :
    ∃ (U : X.affineOpens) (f : Γ(X, U)), x ∈ X.basicOpen f ∧
      ∃ g ∈ nonZeroDivisors Γ(X, X.affineBasicOpen f),
        (Scheme.IdealSheafData.vanishingIdeal Z).ideal (X.affineBasicOpen f) = Ideal.span {g}
theorem

isInvertible_vanishingIdeal

L250 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mWeilCartier.isInvertible_vanishingIdeal
Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mWeilCartier
section Geometry
variable {X : Scheme.{u}}
Lean source · statement
theorem isInvertible_vanishingIdeal [IsIntegral X] [IsLocallyNoetherian X]
    (hX : ∀ x : X, UniqueFactorizationMonoid (X.presheaf.stalk x))
    (Z : Closeds X) (hZ : Z ≠ ⊤)
    (hcodim : ∀ y ∈ Z, (∀ y' ∈ Z, y' ⤳ y → y' = y) → ringKrullDim (X.presheaf.stalk y) ≤ 1) :
    (Scheme.IdealSheafData.vanishingIdeal Z).IsInvertible
lemma

pullbackSection_zero

L424 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mCartierCompl.pullbackSection_zero
Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mCartierCompl
variable {X : Scheme.{u}}
Lean source · statement
lemma pullbackSection_zero {X' : Scheme.{u}} (F : X' ⟶ X) (M : X.Modules) :
    Scheme.Modules.pullbackSection F (0 : 𝟙_ X.Modules ⟶ M) = 0
lemma

isSeparated_of_isSeparated_over

L429 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mCartierCompl.isSeparated_of_isSeparated_over
Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mCartierCompl
variable {X : Scheme.{u}}
Lean source · statement
lemma isSeparated_of_isSeparated_over (k : Type u) [Field k] (t : X ⟶ Spec (CommRingCat.of k))
    [IsSeparated t] : X.IsSeparated
lemma

isRegularRing_stalk

L434 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mCartierCompl.isRegularRing_stalk
Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mCartierCompl
variable {X : Scheme.{u}}
Lean source · statement
lemma isRegularRing_stalk (k : Type u) [Field k] (t : X ⟶ Spec (CommRingCat.of k)) [Smooth t]
    (x : X) : IsRegularRing (X.presheaf.stalk x)
lemma

uniqueFactorizationMonoid_stalk

L462 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mCartierCompl.uniqueFactorizationMonoid_stalk
Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mCartierCompl
variable {X : Scheme.{u}}
Lean source · statement
lemma uniqueFactorizationMonoid_stalk (k : Type u) [Field k] (t : X ⟶ Spec (CommRingCat.of k))
    [Smooth t] [IsIntegral X] (x : X) : UniqueFactorizationMonoid (X.presheaf.stalk x)
lemma

ringKrullDim_stalk_le_one_of_notMem

L467 ↗
AlgebraicCurves.Geometry.Sheaves.Proofs.Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen.P2mCartierCompl.ringKrullDim_stalk_le_one_of_notMem
Source context 13 commands

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

section ProofImplementation
namespace AlgebraicCurves.Geometry.Sheaves.Proofs
namespace Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
attribute [-instance]
  SheafOfModules.isIso_ihomModelToIhom
  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
  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
  instTopologicallyFGOfFiniteType
attribute [-simp] PresheafOfModules.InternalHom.IsSheafAux.appAt_toPresheafHom
  SheafOfModules.ihomSectionsEquivFamily_unit
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_apply SheafOfModules.ihomEval_unit_app
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_right
  AlgebraicGeometry.Scheme.Modules.ihomEval_zero_left
  AlgebraicGeometry.Scheme.Modules.homOfFamily_app_apply
  SheafOfModules.unit_ihomSectionsEquivFamily AlgebraicGeometry.Scheme.Modules.familyOfHom_app
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_hom_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictUnitIso_inv_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomOfFamily_app_apply
  AlgebraicGeometry.Scheme.Modules.restrictHomEquivFamily_symm_apply
  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
  AlgebraicGeometry.Scheme.Modules.tensorPow_zero AlgebraicGeometry.Scheme.Modules.tensorPow_succ
  PresheafOfModules.freeεIso_hom_app PresheafOfModules.freeμIso_hom_app
universe u
open _root_.CategoryTheory
open _root_.CategoryTheory.Limits
open _root_.CategoryTheory.MonoidalCategory
open _root_.AlgebraicGeometry
open _root_.TopologicalSpace
namespace P2mCartierCompl
variable {X : Scheme.{u}}
Lean source · statement
lemma ringKrullDim_stalk_le_one_of_notMem [IsIntegral X] [IsLocallyNoetherian X] [X.IsSeparated]
    (hX : ∀ x : X, IsIntegrallyClosed (X.presheaf.stalk x))
    (U : X.Opens) (hU : IsAffineOpen U) (hUne : (U : Set X).Nonempty) {y : X} (hy : y ∉ U)
    (hmax : ∀ y' : X, y' ∉ U → y' ⤳ y → y' = y) :
    ringKrullDim (X.presheaf.stalk y) ≤ 1
theorem

exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen

L512 ↗
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
Source context 11 commands

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

section ProofImplementation
namespace AlgebraicGeometry.Scheme.Modules
universe u
open _root_.AlgebraicCurves.Geometry.Sheaves.Proofs in
open Scheme_Modules_exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen in
open _root_.CategoryTheory in
open _root_.CategoryTheory.Limits in
open _root_.CategoryTheory.MonoidalCategory in
open _root_.AlgebraicGeometry in
open _root_.TopologicalSpace in
open P2mCartierCompl in
Lean source · statement
theorem exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen
    (k : Type u) [Field k] {X : Scheme.{u}} (t : X ⟶ Spec (CommRingCat.of k))
    [IsSeparated t] [IsIntegral X] [Smooth t] (U : X.Opens) (hU : IsAffineOpen U) :
    ∃ (L : X.Modules) (_ : Scheme.Modules.IsInvertible L) (θ : 𝟙_ X.Modules ⟶ L),
      ∀ (K : Type u) [Field K] (z : Spec (CommRingCat.of K) ⟶ X),
        Scheme.Modules.pullbackSection z θ = 0 ↔ z (IsLocalRing.closedPoint K) ∉ U

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