In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets, 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 is an open affine subset, then A is a noetherian ring. In particular, is a noetherian scheme if and only if A is a noetherian ring. Let X be a locally noetherian scheme. Then the local rings 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.
Properties and Noetherian hypotheses
Having a 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 theorem makes it possible to decompose arguments about coherent sheaves into inductive arguments. It is because given a short exact sequence of coherent sheavesproving one of the sheaves has some property is equivalent to proving the other two have the property. In particular, given a fixed coherent sheaf and a sub-coherent sheaf, showing has some property can be reduced to looking at and. Since this process can only be applied a finite amount of times in a non-trivial manner, 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 is quasi-compact.
Homological properties
There are many nice homological properties of Noetherian schemes.
Cech cohomology and sheaf cohomology agree on an affine open cover. This makes it possible to compute the Sheaf cohomology of using Cech cohomology for the standard open cover.
Given a locally finite type morphism to a Noetherian scheme and a complex of sheaves with bounded coherent cohomology such that the sheaves have proper support over, then the derived pushforward has bounded coherent cohomology over, meaning it is an object in.
Examples
Many of the schemes found in the wild are Noetherian schemes.
Locally of finite type over a Noetherian base
Another class of examples of Noetherian schemes are families of schemes where the base is Noetherian and is of finite type over. This includes many examples, such as the connected components of a Hilbert scheme, i.e. with a fixed Hilbert polynomial. This is important because it implies many moduli spaces encoutered 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.
In particular, infinitesimal deformations of Noetherian schemes are again Noetherian. For example, given a curve, any deformation is also a Noetherian scheme. A tower of such deformations can be used to construct formal Noetherian schemes.
Given an infinite Galois field extension, such as , the ring of integers is a Non-noetherian ring which is dimension. 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, can be an interesting and fruitful subject.
Artinian ring with infinitely many generators
Another example of a non-Noetherian finite-dimensional scheme is an artin ring with infinitely many generators. For example, the ringgives such an example.
