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The Stacks project

Lemma 15.80.9. Let $R \to A$ be a finite type ring map. Let $0 \to M' \to M \to M'' \to 0$ be a short exact sequence of $A$-modules.

  1. If $M', M''$ are finitely presented relative to $R$, then so is $M$.

  2. If $M'$ is a finite type $A$-module and $M$ is finitely presented relative to $R$, then $M''$ is finitely presented relative to $R$.

Proof. Follows immediately from Algebra, Lemma 10.5.3. $\square$

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