nLab noetherian scheme

Definition

Definition

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

Remark

If $R$ is a commutative noetherian ring, then 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.

Properties

Proposition

An affine scheme is a spectrum of a noetherian ring precisely if 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.

References

• Springer eom: Noetherian scheme

category: algebraic geometry