# Noetherian scheme

From Encyclopedia of Mathematics

A scheme admitting a finite open covering by spectra of Noetherian rings. An affine Noetherian scheme is precisely the spectrum of a Noetherian ring. The topological space of a Noetherian scheme $X$ is a Noetherian space, and the local rings $\mathcal O_{X,x}$ are Noetherian. If every point of a scheme has an open affine Noetherian neighbourhood, the scheme is called locally Noetherian. A quasi-compact locally Noetherian scheme is a Noetherian scheme. An example of a Noetherian scheme is a scheme of finite type over a field (an algebraic variety) or over any Noetherian ring.

#### Comments

#### References

[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |

**How to Cite This Entry:**

Noetherian scheme.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Noetherian_scheme&oldid=34135

This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article