The Stacks project

Lemma 31.12.12. Let $X$ be an integral locally Noetherian scheme. Let $j : U \to X$ be an open subscheme with complement $Z$. Assume $\mathcal{O}_{X, z}$ has depth $\geq 2$ for all $z \in Z$. Then $j^*$ and $j_*$ define an equivalence of categories between the category of coherent reflexive $\mathcal{O}_ X$-modules and the category of coherent reflexive $\mathcal{O}_ U$-modules.

Proof. Let $\mathcal{F}$ be a coherent reflexive $\mathcal{O}_ X$-module. For $z \in Z$ the stalk $\mathcal{F}_ z$ has depth $\geq 2$ by Lemma 31.12.11. Thus $\mathcal{F} \to j_*j^*\mathcal{F}$ is an isomorphism by Lemma 31.5.11. Conversely, let $\mathcal{G}$ be a coherent reflexive $\mathcal{O}_ U$-module. It suffices to show that $j_*\mathcal{G}$ is a coherent reflexive $\mathcal{O}_ X$-module. To prove this we may assume $X$ is affine. By Properties, Lemma 28.22.5 there exists a coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\mathcal{G} = j^*\mathcal{F}$. After replacing $\mathcal{F}$ by its reflexive hull, we may assume $\mathcal{F}$ is reflexive (see discussion above and in particular Lemma 31.12.8). By the above $j_*\mathcal{G} = j_*j^*\mathcal{F} = \mathcal{F}$ as desired. $\square$


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