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.

It can be shown that, in a locally noetherian scheme, for every open affine subset $U=\operatorname {Spec} A$, A is a noetherian ring. In particular, $\operatorname {Spec} A$ is a noetherian scheme if and only if A is a noetherian ring.

A noetherian scheme is a noetherian topological space. But the converse is false in general.

References

• Hartshorne, Algebraic geometry.