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 : ↑(AlgebraicGeometry.Scheme.affineOpens Y)) (V : ↑(AlgebraicGeometry.Scheme.affineOpens X)) (e : ↑V ≤ (TopologicalSpace.Opens.map f.val.base).obj ↑U), RingHom.FiniteType (AlgebraicGeometry.Scheme.Hom.appLe f 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_eq :
    @AlgebraicGeometry.LocallyOfFiniteType = AlgebraicGeometry.affineLocally @RingHom.FiniteType
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    instance AlgebraicGeometry.locallyOfFiniteTypeOfIsOpenImmersion {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] :
    AlgebraicGeometry.LocallyOfFiniteType f
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    theorem AlgebraicGeometry.locallyOfFiniteType_stableUnderComposition :
    CategoryTheory.MorphismProperty.StableUnderComposition @AlgebraicGeometry.LocallyOfFiniteType
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    instance AlgebraicGeometry.locallyOfFiniteTypeComp {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)
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    theorem AlgebraicGeometry.locallyOfFiniteTypeOfComp {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 : AlgebraicGeometry.Scheme.OpenCover Y) [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (AlgebraicGeometry.Scheme.OpenCover.obj 𝒰 i)] (𝒰' : (i : (AlgebraicGeometry.Scheme.OpenCover.pullbackCover 𝒰 f).J) → AlgebraicGeometry.Scheme.OpenCover (AlgebraicGeometry.Scheme.OpenCover.obj (AlgebraicGeometry.Scheme.OpenCover.pullbackCover 𝒰 f) i)) [∀ (i : (AlgebraicGeometry.Scheme.OpenCover.pullbackCover 𝒰 f).J) (j : (𝒰' i).J), AlgebraicGeometry.IsAffine (AlgebraicGeometry.Scheme.OpenCover.obj (𝒰' i) j)] :
    AlgebraicGeometry.LocallyOfFiniteType f ↔ ∀ (i : (AlgebraicGeometry.Scheme.OpenCover.pullbackCover 𝒰 f).J) (j : (𝒰' i).J), RingHom.FiniteType (AlgebraicGeometry.Scheme.Γ.map (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.OpenCover.map (𝒰' i) 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 : AlgebraicGeometry.Scheme.OpenCover X) :
    AlgebraicGeometry.LocallyOfFiniteType f ↔ ∀ (i : 𝒰.J), AlgebraicGeometry.LocallyOfFiniteType (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.OpenCover.map 𝒰 i) f)
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    theorem AlgebraicGeometry.LocallyOfFiniteType.openCover_iff {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : AlgebraicGeometry.Scheme.OpenCover Y) :
    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