Documentation

Mathlib.AlgebraicGeometry.Morphisms.FiniteType

Morphisms of finite type #

A morphism of schemes f : X ⟶ Y is locally of finite type if for each affine U ⊆ Y and V ⊆ f ⁻¹' U, The induced map Γ(Y, U) ⟶ Γ(X, V) is of finite type.

A morphism of schemes is of finite type if it is both locally of finite type and quasi-compact.

We show that these properties are local, and are stable under compositions.

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theorem AlgebraicGeometry.locallyOfFiniteType_iff {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) :
AlgebraicGeometry.LocallyOfFiniteType f ↔ ∀ (U : ↑Y.affineOpens) (V : ↑X.affineOpens) (e : ↑V ≤ (TopologicalSpace.Opens.map f.val.base).obj ↑U), RingHom.FiniteType (AlgebraicGeometry.Scheme.Hom.appLE f (↑U) (↑V) e)
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class AlgebraicGeometry.LocallyOfFiniteType {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) :
Prop

A morphism of schemes f : X ⟶ Y is locally of finite type if for each affine U ⊆ Y and V ⊆ f ⁻¹' U, The induced map Γ(Y, U) ⟶ Γ(X, V) is of finite type.

Instances
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    theorem AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.LocallyOfFiniteType f] (U : ↑Y.affineOpens) (V : ↑X.affineOpens) (e : ↑V ≤ (TopologicalSpace.Opens.map f.val.base).obj ↑U) :
    RingHom.FiniteType (AlgebraicGeometry.Scheme.Hom.appLE f (↑U) (↑V) e)
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    theorem AlgebraicGeometry.locallyOfFiniteType_eq :
    @AlgebraicGeometry.LocallyOfFiniteType = AlgebraicGeometry.affineLocally @RingHom.FiniteType
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    @[instance 900]
    instance AlgebraicGeometry.locallyOfFiniteType_of_isOpenImmersion {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] :
    AlgebraicGeometry.LocallyOfFiniteType f
    Equations
    • ⋯ = ⋯
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    instance AlgebraicGeometry.locallyOfFiniteType_isStableUnderComposition :
    CategoryTheory.MorphismProperty.IsStableUnderComposition @AlgebraicGeometry.LocallyOfFiniteType
    Equations
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    instance AlgebraicGeometry.locallyOfFiniteType_comp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [hf : AlgebraicGeometry.LocallyOfFiniteType f] [hg : AlgebraicGeometry.LocallyOfFiniteType g] :
    AlgebraicGeometry.LocallyOfFiniteType (CategoryTheory.CategoryStruct.comp f g)
    Equations
    • ⋯ = ⋯
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    theorem AlgebraicGeometry.locallyOfFiniteType_of_comp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [hf : AlgebraicGeometry.LocallyOfFiniteType (CategoryTheory.CategoryStruct.comp f g)] :
    AlgebraicGeometry.LocallyOfFiniteType f
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    theorem AlgebraicGeometry.LocallyOfFiniteType.affine_openCover_iff {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : Y.OpenCover) [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)] (𝒰' : (i : (𝒰.pullbackCover f).J) → ((𝒰.pullbackCover f).obj i).OpenCover) [∀ (i : (𝒰.pullbackCover f).J) (j : (𝒰' i).J), AlgebraicGeometry.IsAffine ((𝒰' i).obj j)] :
    AlgebraicGeometry.LocallyOfFiniteType f ↔ ∀ (i : (𝒰.pullbackCover f).J) (j : (𝒰' i).J), RingHom.FiniteType (AlgebraicGeometry.Scheme.Γ.map (CategoryTheory.CategoryStruct.comp ((𝒰' i).map j) CategoryTheory.Limits.pullback.snd).op)
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    theorem AlgebraicGeometry.LocallyOfFiniteType.source_openCover_iff {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : X.OpenCover) :
    AlgebraicGeometry.LocallyOfFiniteType f ↔ ∀ (i : 𝒰.J), AlgebraicGeometry.LocallyOfFiniteType (CategoryTheory.CategoryStruct.comp (𝒰.map i) f)
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    theorem AlgebraicGeometry.LocallyOfFiniteType.openCover_iff {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : Y.OpenCover) :
    AlgebraicGeometry.LocallyOfFiniteType f ↔ ∀ (i : 𝒰.J), AlgebraicGeometry.LocallyOfFiniteType CategoryTheory.Limits.pullback.snd
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    theorem AlgebraicGeometry.locallyOfFiniteType_respectsIso :
    CategoryTheory.MorphismProperty.RespectsIso @AlgebraicGeometry.LocallyOfFiniteType