The Stacks project

Lemma 27.22.9. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $U \subset X$ be a quasi-compact open such that $\mathcal{F}|_ U$ is of finite presentation. Then there exists a map of $\mathcal{O}_ X$-modules $\varphi : \mathcal{G} \to \mathcal{F}$ with (a) $\mathcal{G}$ of finite presentation, (b) $\varphi $ is surjective, and (c) $\varphi |_ U$ is an isomorphism.

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a directed colimit with each $\mathcal{F}_ i$ of finite presentation, see Lemma 27.22.6. Choose a finite affine open covering $X = \bigcup V_ j$ and choose finitely many sections $s_{jl} \in \mathcal{F}(V_ j)$ generating $\mathcal{F}|_{V_ j}$, see Lemma 27.16.1. By Sheaves, Lemma 6.29.1 we see that $s_{jl}$ is in the image of $\mathcal{F}_ i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{F}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{F}_ i$ be the kernel of the map $\mathcal{F}_ i \to \mathcal{F}$. Since $\mathcal{F}_ U$ is of finite presentation, we see that $\mathcal{K}|_ U$ is of finite type, see Modules, Lemma 17.11.3. Hence we can find a finite type quasi-coherent submodule $\mathcal{K}' \subset \mathcal{K}$ with $\mathcal{K}'|_ U = \mathcal{K}|_ U$, see Lemma 27.22.2. Then $\mathcal{G} = \mathcal{F}_ i/\mathcal{K}'$ with the given map $\mathcal{G} \to \mathcal{F}$ is a solution. $\square$


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