We prove that the localétale fundamental group of a strongly F -regular singularity is finite (and likewise for theétale fundamental group of the complement of a codimension ≥ 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 finiteétale-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-necessarilý etale-in-codimension-one extensions, obtaining an analog of another result of Xu.

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Abstract. We prove that a strongly F -regular scheme X admits a finite, generically Galois, andétale-in-codimension-one cover X − → X such that theétale fundamental groups of X and X reg agree. Equivalently, every finiteétale cover of X reg extends to a finiteétale cover of X. This is analogous to a result for complex klt varieties by Greb, Kebekus and Peternell.