Lemma 10.39.7. 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) $x$ 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 10.39.5. $\square$

Comment #4162 by Robin on

I think we need M to be finite here.

Comment #4165 by on

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

Comment #4166 by Robin on

Ah yes you are right, sorry.

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