Lemma 31.12.2. 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 $U \subset X$ affine open $\mathcal{F}(U)$ is a reflexive $\mathcal{O}(U)$-module.

Proof. Omitted. $\square$

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