Noetherian and Locally Noetherian Schemes

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

Main definitions

• AlgebraicGeometry.IsLocallyNoetherian: A scheme is locally Noetherian if the components of the structure sheaf at each affine open are Noetherian rings.

• AlgebraicGeometry.IsNoetherian: A scheme is Noetherian if it is locally Noetherian and quasi-compact as a topological space.

Main results

• AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover: A scheme is locally Noetherian if and only if it is covered by affine opens whose sections are Noetherian rings.

• AlgebraicGeometry.IsLocallyNoetherian.quasiSeparatedSpace: A locally Noetherian scheme is quasi-separated.

• AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover: A scheme is Noetherian if and only if it is covered by finitely many affine opens whose sections are Noetherian rings.

• AlgebraicGeometry.IsNoetherian.noetherianSpace: A Noetherian scheme is topologically a Noetherian space.

References

• Stacks: Noetherian Schemes
• [Robin Hartshorne, Algebraic Geometry][Har77]

class AlgebraicGeometry.IsLocallyNoetherian (X : Scheme) : Prop

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

• component_noetherian (U : ↑X.affineOpens) : IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑U))

theorem AlgebraicGeometry.isNoetherianRing_of_away {R : Type u} [CommRing R] (S : Finset R) (hS : Ideal.span ↑S = ⊤) (hN : ∀ (s : ↥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

theorem AlgebraicGeometry.isLocallyNoetherian_of_affine_cover {X : 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)))) : IsLocallyNoetherian X

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

theorem AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top {X : Scheme} {ι : Sort u_1} {S : ι → ↑X.affineOpens} (hS : ⨆ (i : ι), ↑(S i) = ⊤) : 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.

theorem AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover {X : Scheme} (𝒰 : X.OpenCover) [∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)] : IsLocallyNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing ↑((𝒰.X i).presheaf.obj (Opposite.op ⊤))

A version of isLocallyNoetherian_iff_of_iSup_eq_top using Scheme.OpenCover.

theorem AlgebraicGeometry.isLocallyNoetherian_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] [IsLocallyNoetherian Y] : IsLocallyNoetherian X

theorem AlgebraicGeometry.isLocallyNoetherian_iff_openCover {X : Scheme} (𝒰 : X.OpenCover) : IsLocallyNoetherian X ↔ ∀ (i : 𝒰.I₀), IsLocallyNoetherian (𝒰.X i)

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

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

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

theorem AlgebraicGeometry.noetherianSpace_of_isAffine {X : Scheme} [IsAffine X] [IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ⊤))] : TopologicalSpace.NoetherianSpace ↥X

theorem AlgebraicGeometry.noetherianSpace_of_isAffineOpen {X : Scheme} (U : X.Opens) (hU : IsAffineOpen U) [IsNoetherianRing ↑(X.presheaf.obj (Opposite.op U))] : TopologicalSpace.NoetherianSpace ↥↑U

@[instance 100]

instance AlgebraicGeometry.instQuasiCompactOfIsLocallyNoetherianOfIsOpenImmersion {X Z : Scheme} [IsLocallyNoetherian X] {f : Z ⟶ X} [IsOpenImmersion f] : QuasiCompact f

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

@[instance 100]

instance AlgebraicGeometry.IsLocallyNoetherian.quasiSeparatedSpace {X : Scheme} [IsLocallyNoetherian X] : QuasiSeparatedSpace ↥X

A locally Noetherian scheme is quasi-separated.

class AlgebraicGeometry.IsNoetherian (X : Scheme) extends AlgebraicGeometry.IsLocallyNoetherian X, CompactSpace ↥X : Prop

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

• component_noetherian (U : ↑X.affineOpens) : IsNoetherianRing ↑(X.presheaf.obj (Opposite.op ↑U))
• isCompact_univ : IsCompact Set.univ

theorem AlgebraicGeometry.isNoetherian_iff (X : Scheme) : IsNoetherian X ↔ IsLocallyNoetherian X ∧ CompactSpace ↥X

theorem AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top {X : Scheme} {ι : Sort u_1} [Finite ι] {S : ι → ↑X.affineOpens} (hS : ⨆ (i : ι), ↑(S i) = ⊤) : 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.

theorem AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover {X : Scheme} {𝒰 : X.OpenCover} [Finite 𝒰.I₀] [∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)] : IsNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing ↑((𝒰.X i).presheaf.obj (Opposite.op ⊤))

A version of isNoetherian_iff_of_finite_iSup_eq_top using Scheme.OpenCover.

@[instance 100]

instance AlgebraicGeometry.IsNoetherian.noetherianSpace {X : Scheme} [IsNoetherian X] : TopologicalSpace.NoetherianSpace ↥X

A Noetherian scheme has a Noetherian underlying topological space.

@[instance 100]

instance AlgebraicGeometry.quasiCompact_of_noetherianSpace_source {X Y : Scheme} [TopologicalSpace.NoetherianSpace ↥X] (f : X ⟶ Y) : QuasiCompact f

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

instance AlgebraicGeometry.instIsLocallyNoetherianSpecOfIsNoetherianRingCarrier {R : CommRingCat} [IsNoetherianRing ↑R] : IsLocallyNoetherian (Spec R)

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

@[instance 100]

instance AlgebraicGeometry.instIsNoetherianRingCarrierOfIsLocallyNoetherianSpec {R : CommRingCat} [IsLocallyNoetherian (Spec R)] : IsNoetherianRing ↑R

instance AlgebraicGeometry.instIsNoetherianSpecOfIsNoetherianRingCarrier {R : CommRingCat} [IsNoetherianRing ↑R] : IsNoetherian (Spec R)

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

instance AlgebraicGeometry.instIsNoetherianSpecOfOfIsNoetherianRing {R : Type u_1} [CommRing R] [IsNoetherianRing R] : IsNoetherian (Spec { carrier := R, commRing := inst✝ })

instance AlgebraicGeometry.instIsNoetherianRingCarrierStalkCommRingCatPresheafOfIsLocallyNoetherian {X : Scheme} [IsLocallyNoetherian X] {x : ↥X} : IsNoetherianRing ↑(X.presheaf.stalk x)

theorem AlgebraicGeometry.isNoetherian_Spec {R : CommRingCat} : IsNoetherian (Spec R) ↔ IsNoetherianRing ↑R

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

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

A Noetherian scheme has a finite number of irreducible components.