Covers of rational double points in mixed characteristic

Abstract

We further the classification of rational surface singularities. Suppose $(S,\mathfrak{n},k)$ is a strictly Henselian regular local ring of mixed characteristic $(0,p>5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R,\mathfrak{m},k)$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0,p>5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.

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