Local closure of morphism properties #
We define the source local closure of a property P
wrt. a morphism property W
and show it
inherits stability properties from P
.
The source (Zariski-)local closure of P
is satisfied if there exists
an open cover of the source on which P
is satisfied.
Equations
- AlgebraicGeometry.sourceLocalClosure W P f = ∃ (𝒰 : AlgebraicGeometry.Scheme.Cover W X), ∀ (i : 𝒰.J), P (CategoryTheory.CategoryStruct.comp (𝒰.map i) f)
Instances For
noncomputable def
AlgebraicGeometry.sourceLocalClosure.cover
{W P : CategoryTheory.MorphismProperty Scheme}
{X Y : Scheme}
{f : X ⟶ Y}
(hf : sourceLocalClosure W P f)
:
Scheme.Cover W X
A choice of open cover on which P
is satisfied if f
satisfies the source local closure
of P
.
Equations
- hf.cover = Exists.choose hf
Instances For
theorem
AlgebraicGeometry.sourceLocalClosure.property_coverMap_comp
{W P : CategoryTheory.MorphismProperty Scheme}
{X Y : Scheme}
{f : X ⟶ Y}
(hf : sourceLocalClosure W P f)
(i : hf.cover.J)
:
P (CategoryTheory.CategoryStruct.comp (hf.cover.map i) f)
theorem
AlgebraicGeometry.sourceLocalClosure.le
{W P : CategoryTheory.MorphismProperty Scheme}
[W.ContainsIdentities]
[W.RespectsIso]
:
theorem
AlgebraicGeometry.sourceLocalClosure.iff_forall_exists
{P : CategoryTheory.MorphismProperty Scheme}
{X Y : Scheme}
[P.RespectsIso]
{f : X ⟶ Y}
:
sourceLocalClosure IsOpenImmersion P f ↔ ∀ (x : ↥X), ∃ (U : X.Opens), x ∈ U ∧ P (CategoryTheory.CategoryStruct.comp U.ι f)
instance
AlgebraicGeometry.sourceLocalClosure.instRespectsRightScheme
{W P Q : CategoryTheory.MorphismProperty Scheme}
[P.RespectsRight Q]
:
(sourceLocalClosure W P).RespectsRight Q
instance
AlgebraicGeometry.sourceLocalClosure.instIsStableUnderCompositionSchemeOfIsStableUnderBaseChange
{W P : CategoryTheory.MorphismProperty Scheme}
[W.IsStableUnderBaseChange]
[Scheme.IsJointlySurjectivePreserving W]
[W.IsStableUnderComposition]
[P.IsStableUnderBaseChange]
[P.IsStableUnderComposition]
: