Noetherian scheme

A scheme admitting a finite open covering by spectra of Noetherian rings (cf. Noetherian ring). An affine Noetherian scheme is precisely the spectrum of a Noetherian ring. The topological space of a Noetherian scheme $X$ is a Noetherian topological 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.

References

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