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

TL;DR

This note explores the conditions under which the integral closure of an unramified regular local ring in a certain extension is Cohen-Macaulay, linking it to local cohomology and Serre's condition.

Contribution

It establishes a precise equivalence between Cohen-Macaulayness, vanishing of a local cohomology module, and Serre's condition for the integral closure in mixed characteristic.

Findings

R is Cohen-Macaulay iff H^{d-1}_n(R)=0.

Vanishing of local cohomology characterizes Cohen-Macaulayness.

Hom_S(R,S) satisfies Serre's (S_3) condition iff R is Cohen-Macaulay.

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 $(S_3)$.