### Tate's Conjecture, algebraic cycles and rational K-theory in characteristic p, by Thomas Geisser

We discuss conjectures on motives, algebraic cycles and K-theory of smooth
projective varieties over finite fields.
We give a characterization of Tate's conjecture for varieties over finite
fields in terms of motives and their Frobenius endomorphism and a criterion
in terms of motives for rational and numerical equivalence over finite fields
to agree. This together with Tate's conjecture implies that higher rational
K-groups of smooth projective varieties over finite fields vanish (Parshin's
conjecture). Parshin's conjecture in turn implies a conjecture of Beilinson
and Kahn giving bounds on rational K-groups of fields in finite
characteristic. We go on to derive further corollaries.

A revised version of this preprint was placed here November 27, 1997.

Thomas Geisser <geisser@math.harvard.edu>