The Stacks project

Lemma 15.23.20. Let $A$ be a Noetherian normal domain with fraction field $K$. Let $L$ be a finite extension of $K$. If the integral closure $B$ of $A$ in $L$ is finite over $A$, then $B$ is reflexive as an $A$-module.

Proof. It suffices to show that $B = \bigcap B_\mathfrak p$ where the intersection is over height $1$ primes $\mathfrak p \subset A$, see Lemma 15.23.18. Let $b \in \bigcap B_\mathfrak p$. Let $x^ d + a_1x^{d - 1} + \ldots + a_ d$ be the minimal polynomial of $b$ over $K$. We want to show $a_ i \in A$. By Algebra, Lemma 10.38.6 we see that $a_ i \in A_\mathfrak p$ for all $i$ and all height one primes $\mathfrak p$. Hence we get what we want from Algebra, Lemma 10.157.6 (or the lemma already cited as $A$ is a reflexive module over itself). $\square$

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