On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic

TL;DR

This paper proves the existence of birational maximal Cohen-Macaulay modules over certain biradical extensions in mixed characteristic, expanding understanding of module theory in algebraic geometry.

Contribution

It demonstrates that the integral closure of an unramified regular local ring in a specific biradical extension admits a birational maximal Cohen-Macaulay module, a novel result in mixed characteristic.

Findings

Existence of birational maximal Cohen-Macaulay modules over the extension

Extension R is not necessarily Cohen-Macaulay

Extension constructed via adjoining p-th roots of elements in S^p

Abstract

Let $S$ be an unramified regular local ring of mixed characteristic $p\geq 3$ and $S^p$ the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/pS$. Let $R$ be the integral closure of $S$ in a biradical extension of degree $p^2$ of its quotient field obtained by adjoining $p$-th roots of sufficiently general square free elements $f,g\in S^p$. We show that $R$ admits a birational maximal Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay.