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First Definitions and Basic Properties

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

Abstract

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.

Keywords

Rigid Cohomology Rigid Analytic Varieties Overconvergent Isocrystals Space Ad Cofinal System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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