Noetherian scheme

In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets ${\displaystyle \operatorname {Spec} A_{i}}$, where each ${\displaystyle A_{i}}$ is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether.

It can be shown that, in a locally Noetherian scheme, if  ${\displaystyle \operatorname {Spec} A}$ is an open affine subset, then A is a Noetherian ring; in particular, ${\displaystyle \operatorname {Spec} A}$ is a Noetherian scheme if and only if A is a Noetherian ring. For a locally Noetherian scheme X, the local rings ${\displaystyle {\mathcal {O}}_{X,x}}$ are also Noetherian rings.

A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring.

The definitions extend to formal schemes.