Notes on isocrystals

Abstract

For varieties over a perfect field of characteristic p, etale cohomology with Q_l-coefficients is a Weil cohomology theory only when l is not equal to 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 a sequel paper.