The Overconvergent Site, Descent, and Cohomology with Compact Support

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


In this chapter we introduce a version of Le Stum’s overconvergent site for \({\mathscr {E}}_{K}^{\dagger }\)-valued cohomology, and show that \(H^i_\mathrm {rig}(X/{\mathscr {E}}_{K}^{\dagger },\mathscr {E})\) can be computed as the cohomology of this site. This then allows us to prove that cohomological descent holds for both fppf and proper hypercovers, again by adapting the proofs in the classical case. By using de Jong’s theorem on alteration, we may then deduce finite dimensionality of \(H^i_\mathrm {rig}(X/{\mathscr {E}}_{K}^{\dagger },\mathscr {E})\) in general, extending the case of smooth schemes in the previous chapter. We also introduce a version of \({\mathscr {E}}_{K}^{\dagger }\)-valued rigid cohomology with compact support, although can only prove the required finiteness results under strong assumptions on the coefficients. Under these assumption, we also deduce a version of Poincaré duality from the classical case over \(\mathscr {E}_K\).


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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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