Lemma 17.11.5 (01BR)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.11: Modules of finite presentation
Lemma 17.11.5 (cite)

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Lemma 17.11.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $R = \Gamma (X, \mathcal{O}_ X)$ . Let $M$ be an $R$ -module. The $\mathcal{O}_ X$ -module $\mathcal{F}_ M$ associated to $M$ is a directed colimit of finitely presented $\mathcal{O}_ X$ -modules.

Proof. This follows immediately from Lemma 17.10.5 and the fact that any module is a directed colimit of finitely presented modules, see Algebra, Lemma 10.11.3. $\square$

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