Noetherian scheme

In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets $\operatorname {Spec} A_{i}$ , $A_{i}$ noetherian rings. 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 quasi-compact. As with noetherian rings, the concept is named after Emmy Noether.

It can be shown that, in a locally noetherian scheme, if $\operatorname {Spec} A$ is an open affine subset, then A is a noetherian ring. In particular, $\operatorname {Spec} A$ is a noetherian scheme if and only if A is a noetherian ring. Let X be a locally noetherian scheme. Then the local rings ${\mathcal {O}}_{X,x}$ are 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.

Examples

Many of the schemes found in the wild are Noetherian schemes.

Quasi-projective varieties

In particular, quasi-projective varieties are Noetherian schemes. This class includes algebraic curves, elliptic curves, abelian varieties, calabi-yau schemes, shimura varieties, K3 surfaces, and cubic surfaces. Basically all of the objects from classical algebraic geometry fit into this class of examples.

Infinitesimal deformations of Noetherian schemes

In particular, infinitesimal deformations of Noetherian schemes are again Noetherian. For example, given a curve $C/{\text{Spec}}(\mathbb {F} _{q})$ , any deformation ${\mathcal {C}}/{\text{Spec}}(\mathbb {F} _{q^{n}})$ is also a Noetherian scheme. A tower of such deformations can be used to construct formal Noetherian schemes.

Non-examples

Rings of integers over infinite extensions

Given an infinite Galois field extension $K/L$ , such as $\mathbb {Q} (\zeta _{\infty })/\mathbb {Q}$ (by adjoining all roots of unity), the ring of integers ${\mathcal {O}}_{K}$ is a Non-noetherian ring which is dimension $1$ . This breaks the intuition that finite dimensional schemes are necessarily Noetherian. Also, this example provides motivation for why studying schemes over a non-Noetherian base; that is, schemes ${\text{Sch}}/{\text{Spec}}({\mathcal {O}}_{E})$ , can be an interesting and fruitful subject.

Artinian ring with infinitely many generators

Another example of a non-Noetherian finite-dimensional scheme (in fact zero dimensional) is an artin ring with infinitely many generators. For example, the ring

${\frac {\mathbb {Q} [x_{1},x_{2},x_{3},\ldots ]}{(x_{1},x_{2}^{2},x_{3}^{3},\ldots )}}$

gives such an example.

References

Robin Hartshorne, Algebraic geometry.
Noetherian scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Noetherian_scheme&oldid=34135

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