Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic

TL;DR

This paper investigates the Cohen-Macaulay properties of certain integral closures in biquadratic extensions over unramified regular local rings in mixed characteristic two, establishing conditions for the existence of small Cohen-Macaulay modules.

Contribution

It characterizes when the integral closure admits a birational small Cohen-Macaulay module in mixed characteristic two, based on the elements defining the extension.

Findings

R admits a birational small Cohen-Macaulay module when at least one of f,g is in S^2

R is not necessarily Cohen-Macaulay if f,g are in or not in S^2

The Cohen-Macaulayness of R depends on the position of f,g relative to S^2

Abstract

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a birational small Cohen-Macaulay module. It is noted that $R$ is not automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$.