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Mathlib.AlgebraicGeometry.Morphisms.LocalClosure

Local closure of morphism properties #

We define the source local closure of a property P w.r.t. a morphism property W and show it inherits stability properties from P.

source
def AlgebraicGeometry.sourceLocalClosure (W P : CategoryTheory.MorphismProperty Scheme) :
CategoryTheory.MorphismProperty Scheme

The source (Zariski-)local closure of P is satisfied if there exists an open cover of the source on which P is satisfied.

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    noncomputable def AlgebraicGeometry.sourceLocalClosure.cover {W P : CategoryTheory.MorphismProperty Scheme} {X Y : Scheme} {f : X āŸ¶ Y} (hf : sourceLocalClosure W P f) :
    Scheme.Cover (Scheme.precoverage W) X

    A choice of open cover on which P is satisfied if f satisfies the source local closure of P.

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      theorem AlgebraicGeometry.sourceLocalClosure.property_coverMap_comp {W P : CategoryTheory.MorphismProperty Scheme} {X Y : Scheme} {f : X āŸ¶ Y} (hf : sourceLocalClosure W P f) (i : hf.cover.Iā‚€) :
      P (CategoryTheory.CategoryStruct.comp (hf.cover.f i) f)
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      theorem AlgebraicGeometry.sourceLocalClosure.le {W P : CategoryTheory.MorphismProperty Scheme} [W.ContainsIdentities] [W.RespectsIso] :
      P ā‰¤ sourceLocalClosure W P
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      theorem AlgebraicGeometry.sourceLocalClosure.iff_forall_exists {P : CategoryTheory.MorphismProperty Scheme} {X Y : Scheme} [P.RespectsIso] {f : X āŸ¶ Y} :
      sourceLocalClosure IsOpenImmersion P f ā†” āˆ€ (x : ā†„X), āˆƒ (U : X.Opens), x āˆˆ U āˆ§ P (CategoryTheory.CategoryStruct.comp U.Ī¹ f)
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      instance AlgebraicGeometry.sourceLocalClosure.instRespectsLeftSchemeOfIsStableUnderBaseChange {W P Q : CategoryTheory.MorphismProperty Scheme} [W.IsStableUnderBaseChange] [Scheme.IsJointlySurjectivePreserving W] [P.RespectsLeft Q] [Q.IsStableUnderBaseChange] :
      (sourceLocalClosure W P).RespectsLeft Q
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      instance AlgebraicGeometry.sourceLocalClosure.instRespectsRightScheme {W P Q : CategoryTheory.MorphismProperty Scheme} [P.RespectsRight Q] :
      (sourceLocalClosure W P).RespectsRight Q
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      instance AlgebraicGeometry.sourceLocalClosure.instRespectsIsoScheme {W P : CategoryTheory.MorphismProperty Scheme} [W.IsStableUnderBaseChange] [Scheme.IsJointlySurjectivePreserving W] [P.RespectsIso] :
      (sourceLocalClosure W P).RespectsIso
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      instance AlgebraicGeometry.sourceLocalClosure.instIsZariskiLocalAtSourceIsOpenImmersionOfRespectsIsoOfRespectsLeftScheme {P : CategoryTheory.MorphismProperty Scheme} [P.RespectsIso] [P.RespectsLeft IsOpenImmersion] :
      IsZariskiLocalAtSource (sourceLocalClosure IsOpenImmersion P)
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      instance AlgebraicGeometry.sourceLocalClosure.instIsStableUnderBaseChangeScheme {W P : CategoryTheory.MorphismProperty Scheme} [W.IsStableUnderBaseChange] [Scheme.IsJointlySurjectivePreserving W] [P.IsStableUnderBaseChange] :
      (sourceLocalClosure W P).IsStableUnderBaseChange
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      instance AlgebraicGeometry.sourceLocalClosure.instContainsIdentitiesScheme {W P : CategoryTheory.MorphismProperty Scheme} [W.IsStableUnderBaseChange] [W.ContainsIdentities] [P.ContainsIdentities] :
      (sourceLocalClosure W P).ContainsIdentities
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      instance AlgebraicGeometry.sourceLocalClosure.instIsStableUnderCompositionSchemeOfIsStableUnderBaseChange {W P : CategoryTheory.MorphismProperty Scheme} [W.IsStableUnderBaseChange] [Scheme.IsJointlySurjectivePreserving W] [W.IsStableUnderComposition] [P.IsStableUnderBaseChange] [P.IsStableUnderComposition] :
      (sourceLocalClosure W P).IsStableUnderComposition
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      instance AlgebraicGeometry.sourceLocalClosure.instIsMultiplicativeSchemeOfIsStableUnderBaseChange {W P : CategoryTheory.MorphismProperty Scheme} [W.IsStableUnderBaseChange] [Scheme.IsJointlySurjectivePreserving W] [W.IsMultiplicative] [P.IsStableUnderBaseChange] [P.IsMultiplicative] :
      (sourceLocalClosure W P).IsMultiplicative