Motivic cohomology and algebraic cycles a categorical approach, by Marc Levine

For each scheme S, we construct a triangulated tensor category DM(S), functorially in S, which we propose as a candidate for the derived category of the conjectural category of mixed motives over S. The resulting cohomology theory has all the properties of a Bloch-Ogus cohomology theory, including cycle classes and Chern classes for higher K-theory. For S a field of characteristic zero, or a smooth curve over a field of characteristic zero, the motivic cohomology agrees with Bloch's higher Chow groups; the same is true in characteristic p>0 if one uses Q-coefficients. In particular, the motivic cohomology agrees rationally with the weight-graded pieces of algebraic K-theory, for S smooth and of dimension at most one over a field. In addition, each reasonable graded cohomology theory Gamma(*) on the category of smooth, quasi-projective schemes over a fixed base S gives rise to a realization functor Re_Gamma for DM(S); for example, we have the Betti, e'tale and Hodge realizations of DM(S).

Marc Levine <>