In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets Spec A i {\displaystyle \operatorname {Spec} A_{i}} , where each A i {\displaystyle A_{i}} is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether. It can be shown that, in a locally Noetherian scheme, if Spec A {\displaystyle \operatorname {Spec} A} is an open affine subset, then A is a Noetherian ring; in particular, Spec A {\displaystyle \operatorname {Spec} A} is a Noetherian scheme if and only if A is a Noetherian ring. For a locally Noetherian scheme X, the local rings O X , x {\displaystyle {\mathcal {O}}_{X,x}} are also Noetherian rings. A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring. The definitions extend to formal schemes.
Properties and Noetherian hypotheses Having a (locally) Noetherian hypothesis for a statement about schemes generally makes a lot of problems more accessible because they sufficiently rigidify many of its properties. Any Noetherian scheme can only have finitely many irreducible components. Every morphism from a Noetherian scheme X → S {\displaystyle X\to S} is quasi-compact.
Dévissage One of the most important structure theorems about Noetherian rings and Noetherian schemes is the dévissage theorem. This makes it possible to decompose arguments about coherent sheaves into inductive arguments. Given a short exact sequence of coherent sheaves 0 → E ′ → E → E ″ → 0 , {\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0,} proving one of the sheaves has some property is equivalent to proving the other two have the property. In particular, given a fixed coherent sheaf F {\displaystyle {\mathcal {F}}} and a sub-coherent sheaf F ′ {\displaystyle {\mathcal {F}}'} , showing F {\displaystyle {\mathcal {F}}} has some property can be reduced to looking at F ′ {\displaystyle {\mathcal {F}}'} and F / F ′ {\displaystyle {\mathcal {F}}/{\mathcal {F}}'} . Since this process can only be non-trivially applied only a finite number of times, this makes many induction arguments possible.
Homological properties There are many nice homological properties of Noetherian schemes.
Čech and sheaf cohomology Čech cohomology and sheaf cohomology agree on an affine open cover. This makes it possible to compute the sheaf cohomology of P S n {\displaystyle \mathbb {P} _{S}^{n}} using Čech cohomology for the standard open cover.
Compatibility of colimits with cohomology Given a direct system { F α , ϕ α β } α ∈ Λ {\displaystyle \{{\mathcal {F}}_{\alpha },\phi _{\alpha \beta }\}_{\alpha \in \Lambda }} of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphism lim → H i ( X , F α ) → H i ( X , lim → F α ) {\displaystyle \varinjlim H^{i}(X,{\mathcal {F}}_{\alpha })\to H^{i}(X,\varinjlim {\mathcal {F}}_{\alpha })} meaning the functors H i ( X , − ) : Ab ( X ) → Ab {\displaystyle H^{i}(X,-):{\text{Ab}}(X)\to {\text{Ab}}} preserve direct limits and coproducts.
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