Lemma 10.126.4. Let $R$ be a ring. Let $S \subset R$ be a multiplicative subset. Let $M$ be an $R$-module.

If $S^{-1}M$ is a finite $S^{-1}R$-module then there exists a finite $R$-module $M'$ and a map $M' \to M$ which induces an isomorphism $S^{-1}M' \to S^{-1}M$.

If $S^{-1}M$ is a finitely presented $S^{-1}R$-module then there exists an $R$-module $M'$ of finite presentation and a map $M' \to M$ which induces an isomorphism $S^{-1}M' \to S^{-1}M$.

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