The Stacks project

Lemma 27.22.6. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. There exist

  1. a directed set $I$ (see Categories, Definition 4.21.1),

  2. a system $(\mathcal{F}_ i, \varphi _{ii'})$ over $I$ in $\textit{Mod}(\mathcal{O}_ X)$ (see Categories, Definition 4.21.2)

  3. morphisms of $\mathcal{O}_ X$-modules $\varphi _ i : \mathcal{F}_ i \to \mathcal{F}$

such that each $\mathcal{F}_ i$ is of finite presentation and such that the morphisms $\varphi _ i$ induce an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i = \mathcal{F}. \]

Proof. This is a direct consequence of Lemma 27.22.5 and Categories, Lemma 4.21.5 (combined with the fact that colimits exist in the category of sheaves of $\mathcal{O}_ X$-modules, see Sheaves, Section 6.29). $\square$


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