We study Weil-etale cohomology, introduced
by Lichtenbaum for varieties over finite fields.
In the first half of the paper we give an explicit
description of the base change from Weil-etale
cohomology to etale cohomology. As a consequence,
we get a long exact sequence relating Weil-etale
cohomology to etale cohomology, show that for finite
coefficients the cohomology theories agree, and with
rational coefficients a Weil-etale cohomology group is
the direct sum of two etale cohomology groups.
In the second half of the paper we restrict ourselves
to Weil-etale cohomology of the motivic complex. We
show that for smooth projective varieties over finite
fields, finite generation of Weil-etale cohomology
is equivalent to Weil-etale cohomology being an integral
model of l-adic cohomology, and also equivalent to the
conjunction of Tate's conjecture and (rational) equality
of rational and numerical equivalence.
We give several examples where these conjectures hold,
and express special values of zeta functions in terms
of Weil-etale cohomology.
The first version, posted May 14, 2002, has been updated March 26, 2003,
by the author. The results are the same, but the exposition is slightly
different.
A revised version has been posted April 23, 2004.