Tame fundamental groups of pure pairs andAbhyankar’s lemma

Let (R, m, k) be a strictly local normal k-domain of positive characteristic and P be 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 (f^-1 (P) ) ₑ₄₃ 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 \'etale-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.