In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets
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,
But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring.
The definitions extend to formal schemes.
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.
One of the most important structure theorems about Noetherian rings and Noetherian schemes is the dévissage theorem.
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.
Every Noetherian scheme can only have finitely many components.
[2] There are many nice homological properties of Noetherian schemes.
This makes it possible to compute the sheaf cohomology of
using Čech cohomology for the standard open cover.
of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphism
preserve direct limits and coproducts.
Given a locally finite type morphism
with bounded coherent cohomology such that the sheaves
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 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.
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.
A tower of such deformations can be used to construct formal Noetherian schemes.
There is a notion of algebraic geometry over such rings developed by Weil and Alexander Grothendieck.
[6] Given an infinite Galois field extension
(by adjoining all roots of unity), the ring of integers
This breaks the intuition that finite dimensional schemes are necessarily Noetherian.
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.