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

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 $f,g\notin S^2$.

Final version, to appear in Journal of Algebra; minor changes, unabbreviated title, updated references

References in corpus (1)

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

Cited by in corpus (2)

• On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic
• Remarks on the Small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules