Tamely ramified forms of closed polydisks and laces
type: Item
instance of: scholarly article
title: Tamely ramified forms of closed polydisks and laces
zbMATH Open document ID: 1485.14040
author: Marc Chapuis
publication date: 2021-06-22T00:00:00Z
review text: This paper deals with the description of some forms of Berkovich spaces, including polydiscs and annuli. For the case of polydiscs, the study started with the following result of \textit{A. Ducros} [Math. Z. 273, No. 1--2, 331--353 (2013; Zbl 1264.14035)]: Let \(k\) be a complete non-archimedean valued field, \(X\) a \(k\)-analytic space (in the sense of Berkovich) and \(L\) a finite, separable, tamely ramified extension of \(k\). Then, if \(X_L\) is an open \(L\)-polydisc, \(X\) is an open \(k\)-polydisc. Briefly, every tamely ramified form of an open polydisc is trivial. The proof of this theorem relies on \textit{M. Temkin}'s graded reduction theory (see [Isr. J. Math. 140, 1--27 (2004; Zbl 1066.32025)]) and this strategy led \textit{T. Schmidt} to adapt Hilbert theorem 90 to this context and prove a similar result for closed discs (see [Ann. Inst. Fourier 65, No. 3, 1301--1347 (2015; Zbl 1329.14058)]). In this paper, the author extends the graded Hilbert theorem 90 to the multidimensional case on the assumption that the Galois action is residually affine. Consequently, the following fact is shown: every tamely ramified form of a closed polydisc on which the Galois action is residually affine is trivial. It can be noted that, for every trivial form of a polydisc \(X\), the Galois action on \(X\) is always residually affine. For the case of annuli, the author suggests studying the more general notion of \textit{lace}, which is equivalent in dimension one. To a lace \(X\), one can associate a free \(\mathbb{Z}\)-module called \textit{lattice} and, for every form of \(X\), there is a natural Galois action on the lattice of \(X\). If this action is trivial, the author shows that the form is a lace and describes it. The article ends by an example of a non-trivial tamely ramified form of an annulus.
reviewed by: Dorian Berger
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 16W70
zbMATH DE Number: 7362104
zbMATH Keywords: Berkovich spaces
zbMATH Keywords: tame ramification
zbMATH Keywords: polydisc
zbMATH Keywords: annuli
MaRDI profile type: Publication
full work available at URL: aif.3352
OpenAlex ID: W2744121868
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DOI: 10.5802/AIF.3352
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 7362104