First Definitions and Basic Properties

  • Christopher LazdaEmail author
  • Ambrus Pál
Part of the Algebra and Applications book series (AA, volume 21)


In this chapter we introduce a refinement of rigid cohomology for varieties over Laurent series fields \(k(\!(t)\!)\) in characteristic p, taking values in vector spaces over the bounded Robba ring \(\mathscr {E}_K^\dagger \). We achieve this by considering a more general kind of ‘frame’ \((X,Y,\mathfrak {P})\) than in the classical theory, obtained by compactifying our varieties X as schemes over \(k[\![t ]\!]\) rather than just over \(k(\!(t)\!)\). With this definition in place we may transport almost all of Berthelot’s original constructions and results word for word into our setting, showing that these \(H^i_\mathrm {rig}(X/\mathscr {E}_K^\dagger )\) cohomology groups are well defined (i.e. independent of the choice of such a frame). We also introduce categories of coefficients \(F\text {-}{\mathrm {Isoc}}^\dagger (X/\mathscr {E}_K^\dagger )\) for this cohomology theory.


Rigid Cohomology Rigid Analytic Varieties Overconvergent Isocrystals Space Ad Cofinal System 
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  1. 1.
    Berkovich, V.: Spectral theory and analytic geometry over non-Archimedean fields. In: Mathematical Survey and Monographs, vol. 33. American Mathematical Society, Providence (1990)Google Scholar
  2. 2.
    Berthelot, P.: Géométrie rigide et cohomologie des variétés algébriques de caractéristique \(p\). Mem. Soc. Math. France 23, 7–32 (1986)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berthelot, P.: Cohomologie rigide et cohomologie ridige à supports propres, première partie. Preprint (1996)Google Scholar
  4. 4.
    Berthelot, P.: Finitude et pureté cohomologique en cohomologie rigide. Invent. Math. 128(2), 329–377 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Frensel, J., van der Put, M.: Rigid analytic geometry and its applications. In: Progress in Mathematics, vol. 218. Birkhäuser (2004)Google Scholar
  6. 6.
    Fujiwara, K., Kato, F.: Foundations of rigid geometry I. preprint (2013). arXiv:math/1308.4734v1, to appear in EMS Monographs in Mathematics
  7. 7.
    Huber, R.: A generalization of formal schemes and rigid analytic varieties. Math. Z. 217(4), 513–551 (1994)Google Scholar
  8. 8.
    Huber, R.: Étale cohomology of rigid analytic varieties and adic spaces. Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig (1996)Google Scholar
  9. 9.
    Le Stum, B.: Rigid Cohomology, Cambridge Tracts in Mathematics, vol. 172. Cambridge University Press (2007)Google Scholar
  10. 10.
    Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Techniques de “platification” d’un module. Invent. Math. 13, 1–89 (1971)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Università Degli Studi di PadovaPaduaItaly
  2. 2.Imperial College LondonLondonUK

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