On tame ramification and centers of $F$-purity

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.