Characteristic 0 and p analogies, and some motivic cohomology

Abstract

The purpose of this survey is to explain some recent results about analogies between characteristic 0 and characteristic $p>0$ geometry, and to discuss an infinitesimal variant of motivic cohomology. More specifically, we review results showing that the big de Rham Witt complex of a field is a complex of 0-cycles (section 2), as well as results showing analogies between char. $p>0$ and char. 0 geometry. Some of those results determine local cohomology invariants of singular varieties (section 3) and others are concerned with congruences modulo $q$-powers for the number of rational points of varieties defined over $\F_q$ (section 1).