Lemma 31.12.10. Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{F}$ is reflexive,

2. for each $x \in X$ one of the following happens

1. $\mathcal{F}_ x$ is a reflexive $\mathcal{O}_{X, x}$-module, or

2. $\text{depth}(\mathcal{F}_ x) \geq 2$.

Proof. Omitted. See More on Algebra, Lemma 15.23.15. $\square$

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