Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. (Q2782260)

In 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, \textit{E. M. Friedlander} and \textit{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 \textit{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 NEWLINE\[NEWLINEH^{p,q}(X,\mathbb{Z})\cong CH^q(X, 2q- p),NEWLINE\]NEWLINE and the same identity holds for arbitrary coefficients on both sides.