Noetherian scheme

In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets $\operatorname{Spec} A_i$, $A_i$ noetherian rings. 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 quasi-compact. As with noetherian rings, the concept is named after Emmy Noether.

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

References

• Robin Hartshorne, Algebraic geometry.