The Stacks project

Lemma 70.9.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then we can write $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ i$ where each $\mathcal{F}_ i$ is an $\mathcal{O}_ X$-module of finite presentation and all transition maps $\mathcal{F}_ i \to \mathcal{F}_{i'}$ surjective.

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ as a filtered colimit of finitely presented $\mathcal{O}_ X$-modules (Lemma 70.9.1). We claim that $\mathcal{G}_ i \to \mathcal{F}$ is surjective for some $i$. Namely, choose an ├ętale surjection $U \to X$ where $U$ is an affine scheme. Choose finitely many sections $s_ k \in \mathcal{F}(U)$ generating $\mathcal{F}|_ U$. Since $U$ is affine we see that $s_ k$ is in the image of $\mathcal{G}_ i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_ i$ be the kernel of the map $\mathcal{G}_ i \to \mathcal{F}$. Write $\mathcal{K} = \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma 70.9.2). Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i/\mathcal{K}_ a$ is a solution to the problem posed by the lemma. $\square$

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