Anne Fayolle | Searcharxiv

arXiv2025

Centers of perfectoid purity

We introduce a mixed characteristic analog of log canonical centers in characteristic $0$ and centers of $F$-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure of) normality of the ring. We also show the existence of a special center of perfectoid purity that detects the perfectoid purity of $R$, analogously to the splitting prime of Aberbach and Enescu, and investigate its behavior under \'etale morphisms.

math.AG↗

arXiv2023

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.

math.AG↗