  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).
