Tame fundamental groups of pure pairs and Abhyankar's lemma

TL;DR

This paper proves that under certain conditions, tamely ramified covers over a local normal domain have a prime divisor as the ramification locus, generalizing Abhyankar's theorem and exploring implications for the fundamental group.

Contribution

It establishes that Galois covers in a tamely ramified category over purely F-regular pairs have prime divisor ramification, extending classical results to a broader setting.

Findings

Tamely ramified covers have prime divisor ramification locus.

Generalization of Abhyankar's theorem to purely F-regular pairs.

Characteristic zero analog via reduction methods.

Abstract

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. 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 $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ 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 \'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.