Cohomology of p-adic Stein spaces

  • Pierre ColmezEmail author
  • Gabriel Dospinescu
  • Wiesława Nizioł


We compute p-adic étale and pro-étale cohomologies of Drinfeld half-spaces. In the pro-étale case, the main input is a comparison theorem for p-adic Stein spaces; the cohomology groups involved here are much bigger than in the case of étale cohomology of algebraic varieties or proper analytic spaces considered in all previous works. In the étale case, the classical p-adic comparison theorems allow us to pass to a computation of integral differential forms cohomologies which can be done because the standard formal models of Drinfeld half-spaces are pro-ordinary and their differential forms are acyclic.

Mathematics Subject Classification

11F85 14F20 14F30 20G25 22Fxx 



This paper owes great deal to the work of Elmar Grosse-Klönne. We are very grateful to him for his patient and detailed explanations of the computations and constructions in his papers. We would like to thank Fabrizio Andreatta, Bruno Chiarellotto, Frédéric Déglise, Ehud de Shalit, Veronika Ertl, Laurent Fargues, Florian Herzig, Luc Illusie, Arthur-César Le Bras, Sophie Morel, Arthur Ogus, and Lue Pan for helpful conversations related to the subject of this paper. We also thank the referee for useful comments. This paper was partly written during our visits to the IAS at Princeton, the Tata Institute in Mumbai, Banach Center in Warsaw (P.C, W.N), BICMR in Beijing (P.C.), Fudan University in Shanghai (W.N.), Princeton University (W.N.), and the Mittag-Leffler Institute (W.N.). We thank these institutions for their hospitality.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Pierre Colmez
    • 1
    Email author
  • Gabriel Dospinescu
    • 2
  • Wiesława Nizioł
    • 2
  1. 1.CNRS, IMJ-PRGSorbonne UniversitéParisFrance
  2. 2.CNRS, UMPAÉcole Normale Supérieure de LyonLyonFrance

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