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Galois descent of semi-affinoid spaces

  • Lorenzo Fantini
  • Daniele Turchetti
Article
  • 113 Downloads

Abstract

We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a K-analytic space X, provided that \(X\otimes _KL\) is semi-affinoid for some finite tamely ramified extension L of K. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.

Notes

Acknowledgements

We warmly thank Federico Bambozzi, Marc Chapuis, Antoine Ducros, Florent Martin, Michel Matignon, Ariane Mézard, Johannes Nicaise, and Jêrôme Poineau for interesting discussions about and beyond the content of this paper, and an anonymous referee for his comments and corrections. We are especially grateful to Marc Chapuis for keeping us updated on the status of his related work [9]. During the preparation of this work, the first author’s research was supported by the European Research Council (Starting Grant project “Nonarcomp” no.307856) and by the Agence Nationale de la Recherche (project “Défigéo”), while the second author’s research was supported by the Max Planck Institute for Mathematics and the European Research Council (Starting Grant project “TOSSIBERG” no.637027). Finally, each of us warmly thanks the other author, without whom this paper would have been finished much earlier (but with a lot more mistakes).

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Institut Mathématique de JussieuUniversité Pierre et Marie CurieParisFrance
  2. 2.Laboratoire de Mathématiques Nicolas OresmeUniversité de Caen Basse-NormandieCaenFrance

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