The Stacks project

Lemma 30.9.8. Let $i : Z \to X$ be a closed immersion of locally Noetherian schemes. 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, Lemma 29.4.1. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}$. By Morphisms, Lemma 29.4.1 we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. By Modules, Lemma 17.13.3 we see that $\mathcal{G}$ is of finite type. Hence $\mathcal{G}$ is coherent by Lemma 30.9.1. Thus the functor is also essentially surjective as desired. $\square$


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