The Stacks project

Lemma 27.22.3. Let $X$ be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of $\mathcal{O}_ X$-modules is the directed colimit of its quasi-coherent $\mathcal{O}_ X$-submodules which are of finite type.

Proof. The colimit is directed because if $\mathcal{G}_1$, $\mathcal{G}_2$ are quasi-coherent subsheaves of finite type, then $\mathcal{G}_1 + \mathcal{G}_2 \subset \mathcal{F}$ is a quasi-coherent subsheaf of finite type. Let $U \subset X$ be any affine open, and let $s \in \Gamma (U, \mathcal{F})$ be any section. Let $\mathcal{G} \subset \mathcal{F}|_ U$ be the subsheaf generated by $s$. Then clearly $\mathcal{G}$ is quasi-coherent and has finite type as an $\mathcal{O}_ U$-module. By Lemma 27.22.2 we see that $\mathcal{G}$ is the restriction of a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{F}$ which has finite type. Since $X$ has a basis for the topology consisting of affine opens we conclude that every local section of $\mathcal{F}$ is locally contained in a quasi-coherent submodule of finite type. Thus we win. $\square$


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