---
tag: ND6XEY68P4
kind: concept
name: Noetherian scheme
msc: [14A15, "13E05"]
specializes: [5A35TYZD02]
sources:
  - url: https://en.wikipedia.org/wiki/Noetherian_scheme
same_as:
  wikidata: Q3475675
  wikipedia: Noetherian scheme
---

## Summary

A Noetherian scheme is a [[scheme|5A35TYZD02]] that is covered by finitely many affine pieces, each the spectrum of a [[Noetherian ring|0670T59HGE]].
It is the finiteness condition under which most of the standard theory of schemes works well.
All varieties over a field and all schemes of finite type over $\Z$ are Noetherian.

## Explanations

### L1 human

A Noetherian ring is one where every ideal has finitely many generators, so every system of polynomial equations is equivalent to a finite one.
A Noetherian scheme is built from finitely many pieces of this kind.
The finiteness has geometric consequences, for example the scheme splits into finitely many irreducible components.
The spectrum of a polynomial ring in infinitely many variables is a typical scheme that is not Noetherian.