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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

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 X in that Galois category satisfies that (f1(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.

pure $F$-regularity, PLT singularities, fundamental groups, splitting primes, Abhyankar's lemma
Mathematical Subject Classification
Primary: 13A35, 14B05, 14H30
Received: 6 May 2020
Revised: 25 February 2022
Accepted: 4 April 2022
Published: 24 March 2023
Javier Carvajal-Rojas
KU Leuven
Axel Stäbler
Mathematisches Institut
Universität Leipzig

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