The isomorphism between motivic cohomology and \(K\)-groups for equi-characteristic regular local rings (Q426726)

\textit{V. Voevodsky}, \textit{A. Suslin} and \textit{E. M. Friedlander} [Cycles, transfers, and motivic homology theories. Ann. Math. Stud. 143 (2000; Zbl 1021.14006)] defined the motivic cohomology groups \(C H^r_{\mathrm {Zar}} (X, n)\) for smooth noetherian schemes \(X\) over a field by using equi-dimensional cycle groups, and \textit{E. M. Friedlander} and \textit{A. Suslin} [Ann. Sci. Éc. Norm. Supér. (4) 35, No. 6, 773--875 (2002; Zbl 1047.14011)] showed that \(C H^r_{\mathrm {Zar}} (X, n) = C H^r (X, n)\) (= the higher Chow groups of \(X)\) when \(X\) is smooth quasi-projective. For such an \(X\), \textit{S. Bloch} [Adv. Math. 61, 267--304 (1986; Zbl 0608.14004)] had proved previously that \(\bigoplus_{r \geq 0} C H^r (X, n)\) coincides with \(K_n (X) \otimes_{\mathbb Z} {\mathbb Q}\).NEWLINENEWLINEHere the author considers the motivic cohomology groups \(C H^r_{\mathrm {Zar}}(X, n)\) for regular schemes and he shows that, after tensoring with \({\mathbb Q},\) they become isomorphic to \(K_n (X)\) for the spectrum of an arbitrary regular ring containing a field. More precisely, if \(R\) is such a ring, a certain cycle class map establishes an isomorphism \(K_n (R)^{(r)} \otimes {\mathbb Q}\;{\buildrel\sim\over \to}\;C H^r_{\mathrm{Zar}} (R, n) \otimes {\mathbb Q}\) for any \(n, r \geq 0,\) the superscript \((\ldotp)^{(r)}\) denoting the eigenspace of the Adams operator \(\Psi^k\;:\;K_n(R) \otimes {\mathbb Q} \to K_n (R) \otimes {\mathbb Q}\) with the eigenvalue \(k^r\) for \(k = 2, 3, \ldots\)