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 $\mathcal{E}$ over $X$, the rigid cohomology $H_{\mathrm{rig}}^i(X,\mathcal{E})$ and rigid cohomology with compact supports $H_{c,\mathrm{rig}}^i(X,\mathcal{E})$ 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