Lemma 70.9.8. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U \subset X be a quasi-compact open. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Let \mathcal{G} \subset \mathcal{F}|_ U be a quasi-coherent \mathcal{O}_ U-submodule which is of finite type. Then there exists a quasi-coherent submodule \mathcal{G}' \subset \mathcal{F} which is of finite type such that \mathcal{G}'|_ U = \mathcal{G}.
Proof. Denote j : U \to X the inclusion morphism. As X is quasi-separated and U quasi-compact, the morphism j is quasi-compact. Hence j_*\mathcal{G} \subset j_*\mathcal{F}|_ U are quasi-coherent modules on X (Morphisms of Spaces, Lemma 67.11.2). Let \mathcal{H} = \mathop{\mathrm{Ker}}(j_*\mathcal{G} \oplus \mathcal{F} \to j_*\mathcal{F}|_ U). Then \mathcal{H}|_ U = \mathcal{G}. By Lemma 70.9.2 we can find a finite type quasi-coherent submodule \mathcal{H}' \subset \mathcal{H} such that \mathcal{H}'|_ U = \mathcal{H}|_ U = \mathcal{G}. Set \mathcal{G}' = \mathop{\mathrm{Im}}(\mathcal{H}' \to \mathcal{F}) to conclude. \square
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