Noetherian scheme

It can be shown that, in a locally Noetherian scheme, if ${\displaystyle \mathrm{Spec}A}$ is an open affine subset, then A is a Noetherian ring; in particular, ${\displaystyle \mathrm{Spec}A}$ is a Noetherian scheme if and only if A is a Noetherian ring. For a locally Noetherian scheme X, the local rings ${\displaystyle {\mathcal{O}}_{X,x}}$ 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.

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

${\displaystyle 0\to {\mathcal{E}}^{\prime }\to \mathcal{E}\to {\mathcal{E}}^{″}\to 0,}$

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 ${\displaystyle \mathcal{F}}$ and a sub-coherent sheaf ${\displaystyle {\mathcal{F}}^{\prime }}$, showing ${\displaystyle \mathcal{F}}$ has some property can be reduced to looking at ${\displaystyle {\mathcal{F}}^{\prime }}$ and ${\displaystyle \mathcal{F}/{\mathcal{F}}^{\prime }}$. Since this process can only be non-trivially applied only a finite number of times, this makes many induction arguments possible.

Number of irreducible components

Every Noetherian scheme can only have finitely many components.

Morphisms from Noetherian schemes are quasi-compact

Every morphism from a Noetherian scheme ${\displaystyle X\to S}$ is quasi-compact.

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 ${\displaystyle {\mathbb{P}}_S^n}$ using Čech cohomology for the standard open cover.

Compatibility of colimits with cohomology

Given a direct system ${\displaystyle \{{\mathcal{F}}_{\alpha },{\varphi }_{\alpha \beta }{\}}_{\alpha \in \mathrm{\Lambda }}}$ of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphism

${\displaystyle \underrightarrow{\lim }{H}^i(X,{\mathcal{F}}_{\alpha })\to H^i(X,\underrightarrow{\lim }{\mathcal{F}}_{\alpha })}$

meaning the functors

${\displaystyle H^i(X,-):\text{Ab}(X)\to \text{Ab}}$

preserve direct limits and coproducts.

Derived direct image

Given a locally finite type morphism ${\displaystyle f:X\to S}$ to a Noetherian scheme ${\displaystyle S}$ and a complex of sheaves ${\displaystyle {\mathcal{E}}^{\bullet }\in D_{Coh}^b(X)}$ with bounded coherent cohomology such that the sheaves ${\displaystyle H^i({\mathcal{E}}^{\bullet })}$ have proper support over ${\displaystyle S}$, then the derived pushforward ${\displaystyle \mathbf{R}{f}_{\ast }({\mathcal{E}}^{\bullet })}$ has bounded coherent cohomology over ${\displaystyle S}$, meaning it is an object in ${\displaystyle D_{Coh}^b(S)}$.