Localization respects associated primes for Noetherian rings

From Commalg

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

There is a natural inclusion on spectra:

The set of associated primes for as an -module is the inverse image in of the set of associated primes for as an -module.

If we identify with its image, a subset of , then we can write:

Proof

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