The Overconvergent Site, Descent, and Cohomology with Compact Support

Abstract

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 Poincare duality from the classical case over \(\mathscr {E}_K\).