Higher Chern classes and Steenrod operations in motivic cohomology (Q702908)

The paper is devoted to the construction of higher Chern classes from \(K\)-theory to motivic cohomology, in the spirit of [\textit{H. Gillet}, Adv. Math. 40, 203--289 (1981; Zbl 0478.14010)]. Let \(X\) be a smooth scheme over a field \(k\) and let \(l\) be a prime different from \(\text{char\,}k\). Then there are higher Chern classes from \(K\)-theory to motivic cohomology \[ c_{p,q}: K_p(X)\to H^{2q-p}(X, \mathbb{Z}(q))\to H^{2q-p}(X, \mathbb{Z}/l(q)) \] such that the composite \[ P^i\circ c_{p,q}: K_p(X)\to H^{2q- p+ 2i(l-1)}(X, \mathbb{Z}/l(q+ i(l- 1))) \] coincides with \(\left(\begin{smallmatrix} q-1\\ i\end{smallmatrix}\right)\cdot c_{p,q+ i(l-1)}\) if \(p> 0\). Here \[ P^i: H^n(X, \mathbb{Z}/l(n))\to H^{n+ 2i(l-1)}(X, \mathbb{Z}/l(q+ i(l- 1))) \] is the \(i\)th reduced power operation in motivic cohomology, constructed by V. Voevodsky. The same result holds for the Chern classes defined on the \(K\)-theory with coefficients \(K_p(X, \mathbb{Z}/l)\) if \(p> 2\) or \(p= 2\) and \(l\neq 2\). The paper also contains an application to the computation of the motivic algebra of \(\text{GL}_n\).