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 #

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

Instances

    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

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

    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.

    A version of isLocallyNoetherian_iff_of_iSup_eq_top using Scheme.OpenCover.

    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.

    @[instance 100]

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

    Stacks Tag 01OX

    @[instance 100]

    A locally Noetherian scheme is quasi-separated.

    Stacks Tag 01OY

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

    Instances

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

      theorem AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover {X : Scheme} {𝒰 : X.OpenCover} [Finite 𝒰.J] [∀ (i : 𝒰.J), IsAffine (𝒰.obj i)] :
      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.

      @[instance 100]

      A Noetherian scheme has a Noetherian underlying topological space.

      Stacks Tag 01OZ

      @[instance 100]

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

      Stacks Tag 01P0

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

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

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

      A Noetherian scheme has a finite number of irreducible components.

      Stacks Tag 0BA8