# Noetherian scheme

In algebraic geometry, a **noetherian scheme** is a scheme that admits a finite covering by open affine subsets , 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 is an open affine subset, then *A* is a noetherian ring. In particular, is a noetherian scheme if and only if *A* is a noetherian ring. Let *X* be a locally noetherian scheme. Then the local rings 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*.