Fundamental groups of $F$-regular singularities via $F$-signature

TL;DR

This paper establishes a connection between the $F$-signature and the finiteness of the local etale fundamental group in strongly $F$-regular singularities, extending known results from characteristic zero to positive characteristic.

Contribution

It introduces new transformation rules for the $F$-signature under finite etale-in-codimension-one extensions and applies these to fundamental group finiteness and purity of the branch locus.

Findings

The local etale fundamental group is finite and bounded by the reciprocal of the $F$-signature.

Transformation rules for the $F$-signature under certain extensions are developed.

Purity of the branch locus is established for rings with mild singularities.

Abstract

We prove that the local etale fundamental group of a strongly $F$-regular singularity is finite (and likewise for the \'etale fundamental group of the complement of a codimension $\geq 2$ set), analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In fact our result is effective, we show that the reciprocal of the $F$-signature of the singularity gives a bound on the size of this fundamental group. To prove these results and their corollaries, we develop new transformation rules for the $F$-signature under finite etale-in-codimension-one extensions. As another consequence of these transformation rules, we also obtain purity of the branch locus over rings with mild singularities (particularly if the $F$-signature is $> 1/2$). Finally, we generalize our $F$-signature transformation rules to the context of pairs and not-necessarily etale-in-codimension-one extensions, obtaining an analog of another result of Xu.