On the Cohomology of the Affine Space

  • Pierre ColmezEmail author
  • Wiesława Nizioł
Conference paper
Part of the Simons Symposia book series (SISY)


We compute the p-adic geometric pro-étale cohomology of the rigid analytic affine space (in any dimension). This cohomology is non-zero, contrary to the étale cohomology, and can be described by means of differential forms.



We would like to thank the referee for a careful reading of the manuscript and useful suggestions for improving the exposition.


  1. 1.
    V. Berkovich, On the comparison theorem for étale cohomology of non-Archimedean analytic spaces, Israel J. Math. 92 (1995), 45–59.MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. Colmez, G. Dospinescu, W. NizioŁ, Cohomology of \(p\)-adic Stein spaces. Invent. Math. 219 (2020), 873–985.Google Scholar
  3. 3.
    P. Colmez, W.  NiziołŁ, Syntomic complexes and \(p\)-adic nearby cycles, Invent. Math. 208 (2017), 1–108.Google Scholar
  4. 4.
    J.-M. Fontaine, W. Messing, \(p\)-adic periods and \(p\)-adic étale cohomology, Current Trends in Arithmetical Algebraic Geometry (K. Ribet, ed.), Contemporary Math., vol. 67, AMS, Providence, 1987, 179–207.Google Scholar
  5. 5.
    O. Hyodo, K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, Astérisque 223 (1994), 221–268.MathSciNetzbMATHGoogle Scholar
  6. 6.
    A.-C. Le Bras, Espaces de Banach-Colmez et faisceaux cohérents sur la courbe de Fargues-Fontaine, Duke Math J. 167 (2018) 3455–3532.MathSciNetCrossRefGoogle Scholar
  7. 7.
    T. Tsuji, \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233–411.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.CNRS, IMJ-PRG, Sorbonne UniversitéParisFrance
  2. 2.CNRS, UMPA, École Normale Supérieure de LyonLyonFrance

Personalised recommendations