The Geisser-Levine method revisited and algebraic cycles over a finite field

Abstract

We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated etale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous etale cohomology.