Notes on isocrystals

Abstract

For varieties over a perfect field of characteristic p, étale cohomology with Qℓ-coefficients is a Weil cohomology theory only when ℓ≠p; the corresponding role for ℓ=p is played by Berthelot's rigid cohomology. In that theory, the coefficient objects analogous to lisse ℓ-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with ℓ-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship between the two cases, via the theory of companions, will be treated in sequel papers.