Localization respects associated primes for Noetherian rings: Difference between revisions

Revision as of 16:52, 27 February 2008

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:

$S p e c (S^{- 1} A) \to S p e c (A)$

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

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

$A s s_{S^{- 1} A} S^{- 1} M = A s s_{A} M \cap S p e c (S^{- 1} A)$

Revision as of 16:51, 27 February 2008 (view source) Vipul (talk \| contribs) (New page: ==Statement== Suppose <math>A</math> is a Noetherian commutative unital ring and <math>M</math> is any <math>A</math>-module (not necessarily finitely generate...)		Revision as of 16:52, 27 February 2008 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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