On tame ramification and centers of $F$-purity

TL;DR

This paper introduces a generalized notion of tame ramification for finite covers, connecting it to the concept of centers of F-purity, and explores their behavior under finite covers with a focus on transitivity in towers.

Contribution

It extends classical tame ramification notions to higher dimensions and relates them to centers of F-purity, providing new insights into their behavior under finite covers.

Findings

Defines tame ramification for general finite covers.

Connects tame ramification to centers of F-purity.

Demonstrates transitivity property in towers of covers.

Abstract

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of $F$-purity (aka compatibly $F$-split subvariety). As an application, we describe the behavior of centers of $F$-purity under finite covers -- it all comes down to a transitivity property for tame ramification in towers.