Documentation

Mathlib.AlgebraicGeometry.Noetherian

Noetherian and Locally Noetherian Schemes #

We introduce the concept of (locally) Noetherian schemes, giving definitions, equivalent conditions, and basic properties.

Main definitions #

Main results #

References #

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class AlgebraicGeometry.IsLocallyNoetherian (X : AlgebraicGeometry.Scheme) :
Prop

A scheme X is locally Noetherian if 𝒪ₓ(U) is Noetherian for all affine U.

Instances
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    theorem AlgebraicGeometry.isNoetherianRing_of_away {R : Type u} [CommRing R] (S : Finset R) (hS : Ideal.span ↑S = ⊤) (hN : ∀ (s : { x : R // x ∈ S }), IsNoetherianRing (Localization.Away ↑s)) :
    IsNoetherianRing R

    Let R be a ring, and f i a finite collection of elements of R generating the unit ideal. If the localization of R at each f i is noetherian, so is R.

    We follow the proof given in [Har77], Proposition II.3.2

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    theorem AlgebraicGeometry.isLocallyNoetherian_of_affine_cover {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens} (hS : ⨆ (i : ι), ↑(S i) = ⊤) (hS' : ∀ (i : ι), IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑(S i)))) :
    AlgebraicGeometry.IsLocallyNoetherian X

    If a scheme X has a cover by affine opens whose sections are Noetherian rings, then X is locally Noetherian.

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    theorem AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens} (hS : ⨆ (i : ι), ↑(S i) = ⊤) :
    AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : ι), IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑(S i)))

    A scheme is locally Noetherian if and only if it is covered by affine opens whose sections are noetherian rings.

    See [Har77], Proposition II.3.2.

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    theorem AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover {X : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover) [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)] :
    AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing ↑((𝒰.obj i).presheaf.obj (Opposite.op ⊤))

    A version of isLocallyNoetherian_iff_of_iSup_eq_top using Scheme.OpenCover.

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    theorem AlgebraicGeometry.isLocallyNoetherian_of_isOpenImmersion {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsLocallyNoetherian Y] :
    AlgebraicGeometry.IsLocallyNoetherian X
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    theorem AlgebraicGeometry.isLocallyNoetherian_iff_openCover {X : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover) :
    AlgebraicGeometry.IsLocallyNoetherian X ↔ ∀ (i : 𝒰.J), AlgebraicGeometry.IsLocallyNoetherian (𝒰.obj i)

    If 𝒰 is an open cover of a scheme X, then X is locally noetherian if and only if 𝒰.obj i are all locally noetherian.

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    instance AlgebraicGeometry.instNoetherianSpaceαTopologicalSpaceCarrierCommRingCatSpecOfIsNoetherianRingCommRing {R : CommRingCat} [IsNoetherianRing ↑R] :
    TopologicalSpace.NoetherianSpace ↑↑(AlgebraicGeometry.Spec R).toPresheafedSpace

    If R is a noetherian ring, Spec R is a noetherian topological space.

    Equations
    • ⋯ = ⋯
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    theorem AlgebraicGeometry.noetherianSpace_of_isAffine {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine X] [IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ⊤))] :
    TopologicalSpace.NoetherianSpace ↑↑X.toPresheafedSpace
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    theorem AlgebraicGeometry.noetherianSpace_of_isAffineOpen {X : AlgebraicGeometry.Scheme} (U : X.Opens) (hU : AlgebraicGeometry.IsAffineOpen U) [IsNoetherianRing ↑(X.presheaf.obj (Opposite.op U))] :
    TopologicalSpace.NoetherianSpace ↑↑(↑U).toPresheafedSpace
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    @[instance 100]
    instance AlgebraicGeometry.instQuasiCompactOfIsLocallyNoetherianOfIsOpenImmersion {X Z : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocallyNoetherian X] {f : Z ⟶ X} [AlgebraicGeometry.IsOpenImmersion f] :
    AlgebraicGeometry.QuasiCompact f

    Any open immersion Z ⟶ X with X locally Noetherian is quasi-compact.

    Stacks: Lemma 01OX

    Equations
    • ⋯ = ⋯
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    @[instance 100]
    instance AlgebraicGeometry.IsLocallyNoetherian.quasiSeparatedSpace {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocallyNoetherian X] :
    QuasiSeparatedSpace ↑↑X.toPresheafedSpace

    A locally Noetherian scheme is quasi-separated.

    Stacks: Lemma 01OY

    Equations
    • ⋯ = ⋯
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    class AlgebraicGeometry.IsNoetherian (X : AlgebraicGeometry.Scheme) extends AlgebraicGeometry.IsLocallyNoetherian X, CompactSpace ↑↑X.toPresheafedSpace :
    Prop

    A scheme X is Noetherian if it is locally Noetherian and compact.

    Instances
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      theorem AlgebraicGeometry.isNoetherian_iff (X : AlgebraicGeometry.Scheme) :
      AlgebraicGeometry.IsNoetherian X ↔ AlgebraicGeometry.IsLocallyNoetherian X ∧ CompactSpace ↑↑X.toPresheafedSpace
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      theorem AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top {X : AlgebraicGeometry.Scheme} {ι : Sort u_1} [Finite ι] {S : ι → ↑X.affineOpens} (hS : ⨆ (i : ι), ↑(S i) = ⊤) :
      AlgebraicGeometry.IsNoetherian X ↔ ∀ (i : ι), IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑(S i)))

      A scheme is Noetherian if and only if it is covered by finitely many affine opens whose sections are noetherian rings.

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      theorem AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover {X : AlgebraicGeometry.Scheme} {𝒰 : X.OpenCover} [Finite 𝒰.J] [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)] :
      AlgebraicGeometry.IsNoetherian X ↔ ∀ (i : 𝒰.J), IsNoetherianRing ↑((𝒰.obj i).presheaf.obj (Opposite.op ⊤))

      A version of isNoetherian_iff_of_finite_iSup_eq_top using Scheme.OpenCover.

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      @[instance 100]
      instance AlgebraicGeometry.IsNoetherian.noetherianSpace {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsNoetherian X] :
      TopologicalSpace.NoetherianSpace ↑↑X.toPresheafedSpace

      A Noetherian scheme has a Noetherian underlying topological space.

      Stacks, Lemma 01OZ

      Equations
      • ⋯ = ⋯
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      @[instance 100]
      instance AlgebraicGeometry.quasiCompact_of_noetherianSpace_source {X Y : AlgebraicGeometry.Scheme} [TopologicalSpace.NoetherianSpace ↑↑X.toPresheafedSpace] (f : X ⟶ Y) :
      AlgebraicGeometry.QuasiCompact f

      Any morphism of schemes f : X ⟶ Y with X Noetherian is quasi-compact.

      Stacks, Lemma 01P0

      Equations
      • ⋯ = ⋯
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      instance AlgebraicGeometry.instIsLocallyNoetherianSpecOfIsNoetherianRingαCommRing {R : CommRingCat} [IsNoetherianRing ↑R] :
      AlgebraicGeometry.IsLocallyNoetherian (AlgebraicGeometry.Spec R)

      If R is a Noetherian ring, Spec R is a locally Noetherian scheme.

      Equations
      • ⋯ = ⋯
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      @[instance 100]
      instance AlgebraicGeometry.instIsNoetherianRingαCommRingOfIsLocallyNoetherianSpec {R : CommRingCat} [AlgebraicGeometry.IsLocallyNoetherian (AlgebraicGeometry.Spec R)] :
      IsNoetherianRing ↑R
      Equations
      • ⋯ = ⋯
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      instance AlgebraicGeometry.instIsNoetherianSpecOfIsNoetherianRingαCommRing {R : CommRingCat} [IsNoetherianRing ↑R] :
      AlgebraicGeometry.IsNoetherian (AlgebraicGeometry.Spec R)

      If R is a Noetherian ring, Spec R is a Noetherian scheme.

      Equations
      • ⋯ = ⋯
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      instance AlgebraicGeometry.instIsNoetherianSpecOfOfIsNoetherianRing {R : Type u_1} [CommRing R] [IsNoetherianRing R] :
      AlgebraicGeometry.IsNoetherian (AlgebraicGeometry.Spec (CommRingCat.of R))
      Equations
      • ⋯ = ⋯
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      theorem AlgebraicGeometry.isNoetherian_Spec {R : CommRingCat} :
      AlgebraicGeometry.IsNoetherian (AlgebraicGeometry.Spec R) ↔ IsNoetherianRing ↑R

      R is a Noetherian ring if and only if Spec R is a Noetherian scheme.

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      theorem AlgebraicGeometry.finite_irreducibleComponents_of_isNoetherian {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsNoetherian X] :
      (irreducibleComponents ↑↑X.toPresheafedSpace).Finite

      A Noetherian scheme has a finite number of irreducible components.

      Stacks, Lemma 0BA8