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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