A note concerning the vanishing of local cohomology for roots in mixed characteristic

Abstract

The goal of this note is to record the following curious fact: let $(S,\n)$ be an unramified regular local ring of mixed characteristic $p>0$ and dimension $d$. Let $L$ denote the quotient field of $S$ and $K=L(\omega)$ with $\omega^p\in L$. Let $R$ denote the integral closure of $S$ in $K$. Then $R$ is Cohen-Macaulay if and only if $\mathrm{H}^{d-1}_{\n}(R)=0$, i.e., the obstruction to the Cohen-Macaulayness of $R$ lies in a single local cohomology module. Furthermore, this is equivalent to the dual module $\Hom_S(R,S)$ satisfying Serre's condition $(S_3)$.