Lemma 10.40.7 (07Z5)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.40: Supports and annihilators
Lemma 10.40.7 (cite)

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Lemma 10.40.7. Let $R$ be a ring, let $M$ be an $R$-module, and let $m \in M$. Then $\mathfrak p \in V(\text{Ann}(m))$ if and only if $m$ does not map to zero in $M_\mathfrak p$.

Proof. We may replace $M$ by $Rm \subset M$. Then (1) $\text{Ann}(m) = \text{Ann}(M)$ and (2) $m$ does not map to zero in $M_\mathfrak p$ if and only if $\mathfrak p \in \text{Supp}(M)$. The result now follows from Lemma 10.40.5. $\square$

Comments (5)

Comment #4162 by Robin on April 12, 2019 at 17:17

I think we need M to be finite here.

Comment #4165 by Johan on April 13, 2019 at 13:58

Note that the first step of the proof replaces $M$ by $Rm$ which is finite.

Comment #4166 by Robin on April 13, 2019 at 14:03

Ah yes you are right, sorry.

Comment #6987 by Xiaolong Liu on January 25, 2022 at 09:05

Replace ' $x$ ' by ' $m$ ' in the proof.

Comment #7220 by Johan on April 29, 2022 at 11:08

THanks and fixed here.

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