Noetherian scheme on arvau

Wikipedia / Noetherian scheme

In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname{Spec} 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 is an open affine subset, then A is a Noetherian scheme if and only if A is a Noetherian ring. For a locally Noetherian scheme X, the local rings 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 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. 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 using Čech cohomology for the standard open cover.

Compatibility of colimits with cohomology

Given a direct system of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphismmeaning the functorspreserve direct limits and coproducts.

Derived direct image

Given a locally finite type with bounded coherent cohomology.

Examples

Most schemes of interest are Noetherian schemes.

Locally of finite type over a Noetherian base

Another class of examples of Noetherian schemes is of finite type. This includes many examples, such as the connected componentss of a Hilbert scheme, i.e. with a fixed Hilbert polynomial. This is important because it implies many moduli spaces encountered in the wild are Noetherian, such as the Moduli of algebraic curves and Moduli of stable vector bundles. Also, this property can be used to show many schemes considered in algebraic geometry are in fact Noetherian.

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

, any deformation is also a Noetherian scheme. A tower of such deformations can be used to construct formal Noetherian schemes.

Non-examples

Schemes over Adelic bases

One of the natural rings which are non-Noetherian are the ring of adeles for an algebraic number field. In order to deal with such rings, a topology is considered, giving topological rings. There is a notion of algebraic geometry over such rings developed by André Weil and Alexander Grothendieck.

Rings of integers over infinite extensions

Given an infinite Galois field extension (by adjoining all roots of unity), the ring of integers, can be an interesting and fruitful subject.

One special casepg 93. The induced morphismforms the universal covering.

Polynomial ring with infinitely many generators

Another example of a non-Noetherian finite-dimensional scheme (in fact zero-dimensional) is given by the following quotient of a polynomial ring with infinitely many generators.