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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