---
tag: FETMR0M0M8
kind: concept
name: Support of a module
msc: [13C05, 13B30]
sources:
  - url: https://en.wikipedia.org/wiki/Support_of_a_module
same_as:
  wikidata: Q7644587
  wikipedia: Support of a module
---

## Summary

The support of a [[module|0NZ8JA25WK]] $M$ over a [[commutative ring|0310REJJY8]] $R$ is the set of [[prime ideals|00MHHZ9SNE]] $\mathfrak p$ at which the [[localization|68E0F8N94E]] $M_{\mathfrak p}$ is nonzero.
It is a subset of the [[spectrum|FB45BHHA2J]] of $R$, and it tells where on that space the module "lives".
For a [[finitely generated module|05QY1VEBYD]], it is the closed set of primes containing the [[annihilator|HB4XDKJTW8]] of $M$.

## Explanations

### L1 human

Think of the prime ideals of $R$ as points and of the module as something spread over these points.
The support is the set of points where the module is not zero.
For the integers, the module $\Z/6\Z$ lives only at the primes $2$ and $3$, since every other prime becomes invertible and kills it.
The [[definition|B0223N93XS]] makes this precise.