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arXiv · 2103.12023

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

Prashanth Sridhar
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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 $f,g\notin S^2$.

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Prashanth Sridhar. 2021-03-22. Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic. https://doi.org/10.1016/j.jalgebra.2021.05.002

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