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Let $M$ be a finitely generated $A$-module and $\mathfrak{p}\subset A$ a prime ideal. Is it true that $\operatorname{Supp}_A M_{\mathfrak{p}}$ is the closure of $\operatorname{Supp}_{A_{\mathfrak{p}}} M_{\mathfrak{p}} \subset \operatorname{Spec}A$ ?

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    It is true if $A$ is Noetherian and $M=A$. It follows from the fact that $(A_{\mathfrak{p}})_{\mathfrak{q}}\neq0$ if and only if $\mathfrak{p}\cap\mathfrak{q}$ contains a prime.2011-09-23

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