An algebraic scheme is locally noetherian if it can be covered by a family of open subsets of the form where are noetherian rings. A scheme is noetherian if it is locally noetherian and quasicompact.
If is a commutative noetherian ring, then every localization of the form where is also noetherian. If a scheme is locally noetherian this implies that there is a base of topology of consisting of spectra of noetherian rings. In particular every affine subscheme in has a basis of topology of spectra of noetherian rings.
An affine scheme is a spectrum of a noetherian ring precisely if it is a locally noetherian scheme.
Every affine subscheme of a locally noetherian scheme is the spectrum of a noetherian ring. A scheme is noetherian if and only if it has a finite cover by spectra of noetherian rings.
Last revised on January 11, 2014 at 11:14:11. See the history of this page for a list of all contributions to it.