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

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