The Stacks project

Lemma 102.18.1. Let $\mathcal{X}$ be a Noetherian algebraic stack. Every quasi-coherent $\mathcal{O}_\mathcal {X}$-module is the filtered colimit of its coherent submodules.

Proof. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_\mathcal {X}$-module. If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are coherent $\mathcal{O}_\mathcal {X}$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another coherent $\mathcal{O}_\mathcal {X}$-submodule which contains both of them, see Lemma 102.17.7. 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 $U \to \mathcal{X}$ a surjective smooth morphism (Properties of Stacks, Lemma 99.6.2). Set $R = U \times _\mathcal {X} U$ so that $\mathcal{X} = [U/R]$ as in Algebraic Stacks, Lemma 93.16.2. By Lemma 102.17.8 we have $\mathit{QCoh}(\mathcal{O}_ X) = \mathit{QCoh}(U, R, s, t, c)$ and $\textit{Coh}(\mathcal{O}_ X) = \textit{Coh}(U, R, s, t, c)$. In this way we reduce to the problem of proving the corresponding thing for $\mathit{QCoh}(U, R, s, t, c)$. This is Groupoids in Spaces, Lemma 77.13.4; we check its assumptions in the next paragraph.

We urge the reader to skip the rest of the proof. The affine scheme $U$ is Noetherian; this follows from our definition of $\mathcal{X}$ being locally Noetherian, see Properties of Stacks, Definition 99.7.2 and Remark 99.7.3. The projection morphisms $s, t : R \to U$ are smooth (see reference given above) and quasi-separated and quasi-compact (Morphisms of Stacks, Lemma 100.7.8). In particular, $R$ is a quasi-compact and quasi-separated algebraic space smooth over $U$ and hence Noetherian (Morphisms of Spaces, Lemma 66.28.6). $\square$

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