Tame fundamental groups of pure pairs and Abhyankar’s lemma

Carvajal-Rojas, Javier; Stäbler, Axel

doi:10.2140/ant.2023.17.309

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Tame fundamental groups of pure pairs and Abhyankar's lemma

Javier Carvajal-Rojas and Axel Stäbler

Vol. 17 (2023), No. 2, 309–358

DOI: 10.2140/ant.2023.17.309

Abstract

Let $(R, 𝔪, k)$ be a strictly local normal $k$ -domain of positive characteristic and $P$ a prime divisor on $X = Spec R$ . We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck–Murre. Assuming that $(X, P)$ is a purely $F$ -regular pair, our main result is that every Galois cover $f : Y \to X$ in that Galois category satisfies that ${(f^{- 1} (P))}_{red}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S. Abhyankar regarding the étale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt–Gabber–Olsson’s methods.

Keywords

pure $F$-regularity, PLT singularities, fundamental groups, splitting primes, Abhyankar's lemma

Mathematical Subject Classification

Primary: 13A35, 14B05, 14H30

Milestones

Received: 6 May 2020

Revised: 25 February 2022

Accepted: 4 April 2022

Published: 24 March 2023

Authors

Javier Carvajal-Rojas
KU Leuven Heverlee Belgium

Axel Stäbler
Mathematisches Institut Universität Leipzig Leipzig Germany

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Vol. 17, No. 2, 2023