Lemma 10.90.4 (05CX)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.90: Coherent rings
Lemma 10.90.4 (cite)

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Lemma 10.90.4. Let $R$ be a ring. If $R$ is coherent, then a module is coherent if and only if it is finitely presented.

Proof. It is clear that a coherent module is finitely presented (over any ring). Conversely, if $R$ is coherent, then $R^{\oplus n}$ is coherent and so is the cokernel of any map $R^{\oplus m} \to R^{\oplus n}$ , see Lemma 10.90.3. $\square$

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