## The spectral sequence relating algebraic $$K$$-theory to motivic cohomology.(English)Zbl 1047.14011

Suppose that $$X$$ is a smooth variety over a field $$F$$. The main result of this paper asserts that there is a spectral sequence $E_{2}^{p,q} = H^{p-q}(X,\mathbb{Z}(-q)) = \text{CH}^{-q}(X,-p-q) \Rightarrow K_{p-q}(X)$ from the motivic cohomology of $$X$$ to the algebraic $$K$$-theory of $$X$$. This result, as stated and proved in this paper, extends and depends on a theorem of Bloch and Lichtenbaum, which asserts the existence of the same spectral sequence in the case where $$X$$ is the field $$F$$.
The motivic cohomology spectral sequence is now a fundamental construction for algebraic $$K$$-theory. In particular, a torsion coefficients analogue is used to show that the Bloch-Kato conjecture identifying Milnor K-theory with Galois cohomology (both with torsion coefficients) implies the Lichtenbaum-Quillen conjecture. The Lichtenbaum-Quillen conjecture says, roughly, that $$K$$-theory with torsion coefficients can be recovered from étale cohomology in all but finitely many degrees.
The method of proof of the main result in the paper under review is to show that the objects in a tower of cofibrations arising from multirelative $$K$$-theory spectra have an adequate theory of transfers and satisfy the homotopy property. Techniques of V. Voevodsky [in: Motives, polylogarithms and Hodge theory, I: Motives and polylogarithms. Int. Press. Lect. Ser. 3, 3–34 (2002; see the following review Zbl 1047.14012)] are then used to show that the fibre sequences making up the Bloch-Lichtenbaum spectral sequence [S. Bloch and S. Lichtenbaum, “A spectral sequence for motivic cohomology” (preprint 1995)] extend to semi-local rings, and then the desired spectral sequence is recovered from a Zariski (i.e., Brown-Gersten) descent argument.
Since the appearance of this paper, A. Suslin [Proc. Steklov Inst. Math. 241, 202–237 (2003)] has removed the dependence on the Bloch-Lichtenbaum result by showing that a variant of motivic cohomology developed by D. Grayson [$$K$$-Theory 9, 139–172 (1995; Zbl 0826.19003)] – which has a cohomological spectral sequence converging to $$K$$-theory for smooth semi-local schemes – coincides with motivic cohomology.
Most recently, M. Levine [“$$K$$-theory and motivic cohomology of schemes” (preprint 1999) and “The homotopy coniveau filtration” (preprint 2003)] has shown that the multirelative $$K$$-theory tower is a special case of Voevodsky’s slice filtration, which is a construction that exists quite generally in the motivic stable category (loc. cit.). The layers of the slice filtration for $$K$$-theory are shifted motivic cohomology spectra, and so one obtains another description of the motivic cohomology spectral sequence.

### MSC:

 14F42 Motivic cohomology; motivic homotopy theory 14C25 Algebraic cycles 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 19E08 $$K$$-theory of schemes

### Citations:

Zbl 0826.19003; Zbl 1047.14012
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### References:

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