The Stacks project

Lemma 69.15.1. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Every quasi-coherent $\mathcal{O}_ X$-module is the filtered colimit of its coherent submodules.

Proof. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are coherent $\mathcal{O}_ X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another coherent $\mathcal{O}_ X$-submodule which contains both of them (see Lemmas 69.12.3 and 69.12.4). In this way we see that the system is directed. Hence it now suffices to show that $\mathcal{F}$ can be written as a filtered colimit of coherent modules, as then we can take the images of these modules in $\mathcal{F}$ to conclude there are enough of them.

Let $U$ be an affine scheme and $U \to X$ a surjective ├ętale morphism. Set $R = U \times _ X U$ so that $X = U/R$ as usual. By Properties of Spaces, Proposition 66.32.1 we see that $\mathit{QCoh}(\mathcal{O}_ X) = \mathit{QCoh}(U, R, s, t, c)$. Hence we reduce to showing the corresponding thing for $\mathit{QCoh}(U, R, s, t, c)$. Thus the result follows from the more general Groupoids, Lemma 39.15.4. $\square$

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