Localization respects associated primes for Noetherian rings

This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
View more results involving Noetherianness or Read a survey article on applying Noetherianness

Statement

Suppose $A$ is a Noetherian commutative unital ring and $M$ is any $A$-module (not necessarily finitely generated. Let $S$ be a multiplicatively closed subset of $A$.

There is a natural inclusion on spectra:

$Spec(S^{-1}A)\to Spec(A)$

The set of associated primes for $S^{-1}M$ as an $S^{-1}A$-module is the inverse image in $Spec(S^{-1}A)$ of the set of associated primes for $M$ as an $A$-module.

If we identify $Spec(S^{-1}A)$ with its image, a subset of $Spec(A)$, then we can write:

$As{s}_{S^{-1}A}{S}^{-1}M=As{s}_{A}M\cap Spec(S^{-1}A)$

Proof

The key ingredient in the proof is the fact that if $m\in M$, the union of annihilators of all elements of $Sm$, can be realized as the annihilator of a single element $sm$.