Lemma 31.12.5. Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The following are equivalent

$\mathcal{F}$ is reflexive,

$\mathcal{F}_ x$ is a reflexive $\mathcal{O}_{X, x}$-module for all $x \in X$,

$\mathcal{F}_ x$ is a reflexive $\mathcal{O}_{X, x}$-module for all closed points $x \in X$.

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