Notes on isocrystals

TL;DR

This paper provides an overview of overconvergent F-isocrystals, key coefficient objects in p-adic cohomology, highlighting their properties and differences from l-adic sheaves, with a focus on their role in varieties over fields of characteristic p.

Contribution

It offers a concise guide to overconvergent F-isocrystals, detailing their features and setting the stage for future work on their relationship with l-adic sheaves.

Findings

Overconvergent F-isocrystals share some features with l-adic sheaves, like purity and weights.

They exhibit unique p-adic properties such as Newton polygons and overconvergence.

The paper discusses the foundational aspects, preparing for a sequel on the relationship with l-adic cohomology.

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.