The Stacks project

Lemma 15.23.19. Let $R$ be a Noetherian normal domain. Let $M$ be a finite $R$-module. Then the reflexive hull of $M$ is the intersection

\[ M^{**} = \bigcap \nolimits _{\text{height}(\mathfrak p) = 1} M_{\mathfrak p}/(M_\mathfrak p)_{tors} = \bigcap \nolimits _{\text{height}(\mathfrak p) = 1} (M/M_{tors})_\mathfrak p \]

taken in $M \otimes _ R K$.

Proof. Let $\mathfrak p$ be a prime of height $1$. The kernel of $M_\mathfrak p \to M \otimes _ R K$ is the torsion submodule $(M_\mathfrak p)_{tors}$ of $M_\mathfrak p$. Moreover, we have $(M/M_{tors})_\mathfrak p = M_\mathfrak p/(M_\mathfrak p)_{tors}$ and this is a finite free module over the discrete valuation ring $R_\mathfrak p$ (Lemma 15.22.11). Then $M_\mathfrak p/(M_\mathfrak p)_{tors} \to (M_\mathfrak p)^{**} = (M^{**})_\mathfrak p$ is an isomorphism, hence the lemma is a consequence of Lemma 15.23.18. $\square$

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