Finiteness of rigid cohomology with coefficients

Abstract

We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology Hrigi(X,E) and rigid cohomology with compact supports Hc,rigi(X,E) are finite-dimensional vector spaces over an appropriate p-adic field. We also establish Poincaré duality and the Künneth formula with coefficients. The arguments use a pushforward construction in relative dimension 1, based on a relative version of Crew's [Cr] conjecture on the quasi-unipotence of certain p-adic differential equations