The Stacks project

Lemma 56.6.1. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is a categorically compact object of $\mathit{QCoh}(\mathcal{O}_ X)$ if and only if $\mathcal{F}$ is of finite presentation.

Proof. See Categories, Definition 4.26.1 for our notion of categorically compact objects in a category. If $\mathcal{F}$ is of finite presentation then it is categorically compact by Modules, Lemma 17.22.8. Conversely, any quasi-coherent module $\mathcal{F}$ can be written as a filtered colimit $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ of finitely presented (hence quasi-coherent) $\mathcal{O}_ X$-modules, see Properties, Lemma 28.22.7. If $\mathcal{F}$ is categorically compact, then we find some $i$ and a morphism $\mathcal{F} \to \mathcal{F}_ i$ which is a right inverse to the given map $\mathcal{F}_ i \to \mathcal{F}$. We conclude that $\mathcal{F}$ is a direct summand of a finitely presented module, and hence finitely presented itself. $\square$

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