nLab
noetherian scheme

An algebraic scheme is locally noetherian if it can be covered by a family $U_{\alpha }$ of open subsets of the form $U_{\alpha }=\mathrm{Spec}{R}_{\alpha }$ where $R_{\alpha }$ are noetherian rings. A scheme is noetherian if it is locally noetherian and quasicompact.

Observation

If $R$ is a commutative noetherian ring, every localization of the form $R[f^{-1}]$ where $f\in R$ is also noetherian. If a scheme $X$ is locally noetherian this implies that there is a base of topology of $X$ consisting of spectra of noetherian rings. In particular every affine subscheme in $X$ has a basis of topology of spectra of noetherian rings.

Proposition

An affine scheme is a spectrum of a noetherian ring iff it is a locally noetherian scheme.

Corollary

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.

• Springer eom: Noetherian scheme

category: algebraic geometry