---
tag: 3S58E70Z7Z
kind: concept
name: Local property
msc: [13B30, 13C99]
sources:
  - url: https://en.wikipedia.org/wiki/Local_property
---

## Summary

In commutative algebra, a property of [[modules|0NZ8JA25WK]] or rings is local when it holds globally exactly when it holds after [[localization at every prime ideal|X6JD4AG973]].
Local properties let one prove a statement by checking it over [[local rings|MZ02BKPHJA]], where tools such as [[Nakayama's lemma|15JHBYR6KH]] are available.
Being zero, being [[flat|73RYYEVDKA]], and the exactness of a sequence are local; being Noetherian or being an [[integral domain|0QVD7WBFGN]] is not.

## Explanations

### L1 human

A property is local if you can test it one point at a time.
For a ring $R$, the "points" are its [[prime ideals|00MHHZ9SNE]], and testing at a point $\mathfrak p$ means passing to the localization $M_{\mathfrak p}$.
For example, a module is zero exactly when all its localizations are zero.
So to show that a module vanishes, it is enough to show it near every point.
The [[definition|FD3T25A1DX]] states the condition.