Lemma 17.12.5 (01BZ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.12: Coherent modules
Lemma 17.12.5 (cite)

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Lemma 17.12.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module. Assume $\mathcal{O}_ X$ is a coherent $\mathcal{O}_ X$ -module. Then $\mathcal{F}$ is coherent if and only if it is of finite presentation.

Proof. Omitted. $\square$

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