Every tamely ramified form of an open polydisc is trivial
type: Item
instance of: scholarly article
title: Every tamely ramified form of an open polydisc is trivial
zbMATH Open document ID: 1264.14035
author: Antoine Ducros
published in: Mathematische Zeitschrift
publication date: 2013-02-06T00:00:00Z
review text: Let \(k\) be a complete non-Archimedean valued field and \(X\) be a \(k\)-analytic space. The main result of the paper is the following: if there exists a finite and tamely ramified extension \(L\) of \(k\) such that \(X \hat{\otimes}_{k} L\) is an open polydisc over \(L\), then \(X\) itself is an open polydisc over \(k\). Let us point out that the radii of the polydiscs may be arbitrary elements of \(\mathbb{R}_{>0}\) and do not need to belong to the value group \(|k^*|\). The paper is written in the setting of Berkovich spaces. In the introduction, the author explains his strategy in a simpler case: \(L/k\) is unramified and \(X \hat{\otimes}_{k} L\) is the unit polydisc. In this case, the result asserts that \(X\) is isomorphic to the unit polydisc over \(k\). The proof heavily relies on reduction arguments: we have \(\widetilde{\mathcal{O}(X) \hat{\otimes}_{k} L} = \widetilde{\mathcal{O}(X \hat{\otimes}_{k} L)} \simeq \tilde{L}[\![t_{1},\dots,t_{n}]\!]\). Using the \(t_{i}\)'s, one may construct a morphism from \(X\) to the unit polydisc over \(k\) that becomes an isomorphism over \(L\). The conclusion now follows from a descent argument. In the general case (tamely ramified extensions and arbitrary radii), reduction may behave badly and has to be replaced by \textit{M. Temkin}'s graded reduction (see [Isr. J. Math. 140, 1--27 (2004; Zbl 1066.32025)]). Otherwise, the overall strategy remains the same. In fact, Ducros claims that tamely ramified extensions are the analogues of unramified extensions in the graded setting. The author also adapts several classical notions and results from commutative algebra to the context of graded commutative algebra: Galois theory, étale algebras, etc. The paper ends by an example to show that the result may fail when wild ramification occurs.
reviewed by: Jérôme Poineau
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 6134684
zbMATH Keywords: Berkovich space
zbMATH Keywords: polydisc
zbMATH Keywords: tame ramification
zbMATH Keywords: graded commutative algebra
MaRDI profile type: Publication
full work available at URL: s00209-012-1007-y
OpenAlex ID: W2157023275
cites work: Q3998951
cites work: Étale cohomology for non-Archimedean analytic spaces
cites work: Non-archimedean analytification of algebraic spaces
cites work: On local properties of non-Archimedean analytic spaces. II
DOI: 10.1007/S00209-012-1007-Y
description: scientific article; zbMATH DE number 6134684