The Stacks project

Lemma 69.12.8. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of locally Noetherian algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $i_*$ induces an equivalence between the category of coherent $\mathcal{O}_ X$-modules annihilated by $\mathcal{I}$ and the category of coherent $\mathcal{O}_ Z$-modules.

Proof. The functor is fully faithful by Morphisms of Spaces, Lemma 67.14.1. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}$. By Morphisms of Spaces, Lemma 67.14.1 we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. To check that $\mathcal{G}$ is coherent we can work étale locally (Lemma 69.12.2). Choosing an étale covering by a scheme we conclude that $\mathcal{G}$ is coherent by the case of schemes (Cohomology of Schemes, Lemma 30.9.8). Hence the functor is fully faithful and the proof is done. $\square$

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