Noetherian scheme

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. 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. 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 number of times in a non-trivial manner, this makes many induction arguments possible. Every Noetherian scheme can only have finitely many components. Every morphism from a Noetherian scheme is quasi-compact. 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.