The Stacks project

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

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

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

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

Proof. By Modules, Lemma 17.22.4 we see that (1) and (2) are equivalent. Since every point of $X$ specializes to a closed point (Properties, Lemma 28.5.9) we see that (2) and (3) are equivalent. $\square$

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