The Stacks project

Lemma 10.126.5. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime ideal. Let $M$ be an $R$-module.

  1. If $M_{\mathfrak p}$ is a finite $R_{\mathfrak p}$-module then there exists a finite $R$-module $M'$ and a map $M' \to M$ which induces an isomorphism $M'_{\mathfrak p} \to M_{\mathfrak p}$.

  2. If $M_{\mathfrak p}$ is a finitely presented $R_{\mathfrak p}$-module then there exists an $R$-module $M'$ of finite presentation and a map $M' \to M$ which induces an isomorphism $M'_{\mathfrak p} \to M_{\mathfrak p}$.

Proof. This is a special case of Lemma 10.126.4 $\square$

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