Lemma 15.23.4. Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module. The following are equivalent:
$M$ is reflexive,
$M_\mathfrak p$ is a reflexive $R_\mathfrak p$-module for all primes $\mathfrak p \subset R$, and
$M_\mathfrak m$ is a reflexive $R_\mathfrak m$-module for all maximal ideals $\mathfrak m$ of $R$.
Proof. The localization of $j : M \to \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, R), R)$ at a prime $\mathfrak p$ is the corresponding map for the module $M_\mathfrak p$ over the Noetherian local domain $R_\mathfrak p$. See Algebra, Lemma 10.10.2. Thus the lemma holds by Algebra, Lemma 10.23.1. $\square$
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