Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.

*(English)*Zbl 1057.14026In this short research notice, the author shows that two differently defined types of motivic cohomology groups for smooth schemes over any field are in fact isomorphic. On the one hand, E. M. Friedlander and A. Suslin recently gave a construction of a certain motivic cohomology theory that was directly related, through a spectral sequence, to the higher \(K\)-theory of Chow groups [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, No. 6, 773–875 (2002; Zbl 1047.14011)]. Another kind of motivic cohomology was introduced by A. Suslin and the author of the present note in order to approach the so-called Bloch-Kato conjecture [in: The arithmetic and geometry of algebraic cycles. Proceedings of the NATO Advanced Study Institute, Banff, Canada, Jun 7–19, 1998. Math. Phys. Sci. 548, 117–189 (2000; Zbl 1005.19001)]. The proof that these two kinds of motivic cohomology are equivalent, which is the main result of the present note, implies that the Suslin-Voevodsky motivic cohomology groups of a smooth scheme over any field are isomorphic to certain higher Chow groups. More precisely, there is a natural isomorphism
\[
H^{p,q}(X,\mathbb{Z})\cong CH^q(X, 2q- p),
\]
and the same identity holds for arbitrary coefficients on both sides.

Reviewer: Werner Kleinert (Berlin)