Eliminating tame ramification : generalizations of Abhyankar’s lemma

Abstract

A basic version of Abhyankar's Lemma states that for two finite extensions L and F of a local field K, if L|K is tamely ramified and if the ramification index of L|K divides the ramification index of F |K, then the compositum L.F is an unramified extension of F . In this paper, we generalize the result to valued fields with value groups of rational rank 1, and show that the latter condition is necessary. Replacing the condition on the ramification indices by the condition that the value group of L be contained in that of F , we generalize the result further in order to give a necessary and sufficient condition for the elimination of tame ramification of an arbitrary extension F |K by a suitable algebraic extension of the base field K. In addition, we derive more precise ramification theoretical statements and give several examples.

A ruled residue theorem for function fields of conics

“…The condition that p = char kv implies that Kv|k(X)v is a separable extension. It now follows from [5,Theorem 3(2)] that…”

Extensions of absolute values on two subfields

“…There are also variants of Abhyankar's lemma in other settings, such as [16, Lemme 3.6 in Exp. 10] when the residue field extension may be inseparable, [28,Lemma 15.105.4] for formally smooth madic topologies, [13] for transcendental extensions and henselizations, and [2] for perfectoid spaces. Proposition 1.4 has the desirable feature of being sufficiently general to apply to questions in many different branches of math, while also having a concise and easily understood statement and a short self-contained proof.…”