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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