The Stacks project

Lemma 15.23.16. Let $R$ be a Noetherian domain. Let $M$ be a finite reflexive $R$-module. Let $\mathfrak p \subset R$ be a prime ideal.

  1. If $\text{depth}(R_\mathfrak p) \geq 2$, then $\text{depth}(M_\mathfrak p) \geq 2$.

  2. If $R$ is $(S_2)$, then $M$ is $(S_2)$.

Proof. Since formation of reflexive hull $\mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, R), R)$ commutes with localization (Algebra, Lemma 10.10.2) part (1) follows from Lemma 15.23.10. Part (2) is immediate from Lemma 15.23.11. $\square$

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