Let $R$ be a local integrally closed domain of dimension $2$. Let $M$ be a nonzero finitely generated $R$-module. We know that "$M$ is reflexive" implies "$M$ is maximal Cohen-Macaulay". Is the converse true?
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If $R$ is a local normal domain with $\dim R=2$, then every MCM is reflexive.
First prove that $M$ is torsion-free. This shows that $M_{\mathfrak p}$ is free over $R_{\mathfrak p}$ for any prime $\mathfrak p$ of height $\le 1$.
Next, if $\mathfrak p$ is a prime of height $2$ it's obvious that $M_{\mathfrak p}$ satisfies Serre's condition $(S_2)$.
In the end, use Proposition 1.4.1(b) from Bruns and Herzog.
(Maybe this can help you.)
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2The claim holds for local normal domains of arbitrary dimension. – user26857 Mar 07 '14 at 16:36