Weight one motivic cohomology and \(K\)-theory (Q2731700)

In the quest for motivic cohomology, Grayson made a proposal that uses a functorial filtration \(W^t(X)\) of the \(K\)-theory spectrum \(K(X)\) of a nonsingular \(X\) to set up the desired Atiyah-Hirzebruch spectral sequence [\textit{D. R. Grayson}, ``Weight filtrations via commuting automorphisms'', \(K\)-Theory 9, No. 2, 139-172 (1995; Zbl 0826.19003)]. The author has been collecting evidence in favor of a related proposal. NEWLINENEWLINENEWLINEIn [\textit{M. E. Walker}, ``Adams operations for bivariant \(K\)-theory and a filtration using projective lines'', \(K\)-Theory 21, No. 2, 101-140 (2000; Zbl 0977.19003)] he already gave his construction and some of its properties. In [\textit{M. E. Walker}, ``The primitive topology of a scheme'', J. Algebra 201, No. 2, 656-685 (1998; Zbl 0928.14003)] he laid some of the necessary ground work. NEWLINENEWLINENEWLINEIn the present paper the main result is that the weight one part is correct: If \(X=\text{Spec}(R)\) is a smooth affine variety over an infinite field, then \(\pi_0W^{1/2}(X)=\text{Pic}(R)\), \(\pi_1W^{1/2}(X)=R^\times\) and the other \(\pi_nW^{1/2}(X)\) vanish. This agrees with the predictions of motivic cohomology.