Abhyankar's perfectoid lemma
type: Item
instance of: scholarly article
title: Abhyankar's perfectoid lemma
zbMATH Open document ID: 1419.14030
author: Yves André
published in: Publications Mathématiques
publication date: 2018-08-07T00:00:00Z
review text: The article under review provides an extension of the almost purity theorem of \textit{G. Faltings} [J. Am. Math. Soc. 1, No. 1, 255--299 (1988; Zbl 0764.14012)]. Let $A$ be a perfectoid algebra over a perfectoid field $K$. In the form given by \textit{P. Scholze} [Publ. Math., Inst. Hautes Étud. Sci. 116, 245--313 (2012; Zbl 1263.14022)] and \textit{K. S. Kedlaya} and \textit{R. Liu} [Relative $p$-adic Hodge theory: foundations. Paris: Société Mathématique de France (SMF) (2015; Zbl 1370.14025)], the later theorem claims that each finite étale $A$-algebra $B$ is perfectoid and its set $B^\circ$ of power-bounded elements is almost finite étale over $A^\circ$ in the basic setting $(K^\circ, K^{\circ\circ})$. Here, ``almost'' is to be understood is the sense of Faltings' almost algebra (see [\textit{O. Gabber} and \textit{L. Ramero}, Almost ring theory. Berlin: Springer (2003; Zbl 1045.13002)] for a general reference), roughly meaning that we work up to $K^{\circ\circ}$-torsion. The author proves a similar result but allowing ramification. The basic setting here is $(K^\circ [T^{1/p^\infty}], T^{1/p^\infty} K^{\circ\circ} [T^{1/p^\infty}])$. Assume that $A$ has residue characteristic $p$ and contains a sequence of compatible $p^n$ roots of some $g \in A^\circ$, so as to see $A^\circ$ as a $K^\circ [T^{1/p^\infty}]$-algebra. Then, for each finite étale $A[1/g]$-algebra $B'$, there exists a perfectoid sub-algebra $B$ of $B'$ with $B[1/g] = B'$, $B^\circ$ is almost isomorphic to the integral closure of $g^{-1/p^\infty} A^\circ$ in $B'$ and, for each $m$, $B^\circ/p^m$ is almost finite étale over $A^\circ/p^m$. (The author notes that it is even plausible that $B^\circ$ be almost finite étale over $A^\circ$.) In the sense that it makes ramification vanish, this result is reminiscent of Abhyankar's lemma with some major differences however in the assumptions, in particular the fact that no tameness is required. Let us describe the contents of the paper. The first two sectio…
reviewed by: Jérôme Poineau
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 13J07
Mathematics Subject Classification ID: 13B40
zbMATH DE Number: 6914158
zbMATH Keywords: Abhyankar's lemma
zbMATH Keywords: perfectoid spaces
zbMATH Keywords: Riemann extension theorem
zbMATH Keywords: almost purity
zbMATH Keywords: almost algebra
MaRDI profile type: Publication
full work available at URL: s10240-017-0096-x
OpenAlex ID: W2963015601
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DOI: 10.1007/S10240-017-0096-X
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description: scientific article; zbMATH DE number 6914158