Tame fundamental groups of pure pairs and Abhyankar's lemma

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 $fY\to X$ in that Galois category satisfies that $(f^{-1}(P){)}_{\text{red}}$ 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 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.

arXiv comment45 pages, comments are welcome, typos fixed, shortened down, some parts were rewritten to improve exposition, Proposition 4.8 was removed as it was flawed v3: Major revision, to apper in ANT

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Cited by in corpus (1)

• On tame ramification and centers of $F$-purity