The Stacks project

Lemma 69.12.7. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and $\mathcal{G}$ the quasi-coherent $\mathcal{O}_ Z$-module such that $i_*\mathcal{G} = \mathcal{F}$, see Morphisms of Spaces, Definition 67.15.4. Then $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module.

Proof. The statement of the lemma makes sense as a coherent module is in particular of finite type. Moreover, as $Z \to X$ is a closed immersion it is locally of finite type and hence $Z$ is locally Noetherian, see Morphisms of Spaces, Lemmas 67.23.7 and 67.23.5. Finally, as $\mathcal{G}$ is of finite type it is a coherent $\mathcal{O}_ Z$-module by Lemma 69.12.2 $\square$

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