ÉTALE MOTIVIC COHOMOLOGY AND ALGEBRAIC CYCLES

We consider étale motivic or Lichtenbaum cohomology and its relation to algebraic cycles. We give an geometric interpretation of Lichtenbaum cohomology and use it to show that the usual integral cycle maps extend to maps on integral Lichtenbaum cohomology. We also show that Lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. For example, we formulate integral étale versions of the Hodge and the Tate conjecture, and show that these are equivalent to the usual rational conjectures.