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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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