Lemma 15.81.9 (0671)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.81: Relatively finitely presented modules
Lemma 15.81.9 (cite)

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Lemma 15.81.9. 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.

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