The homotopy coniveau tower

Abstract

Abstract. We examine the "homotopy coniveau tower" for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with Voevodsky's slice tower for S 1 -spectra, giving a proof of a connectedness conjecture of Voevodsky.The homotopy coniveau tower construction extends to a tower of functors on the Morel-Voevodsky stable homotopy category, and we identify this P 1 -stable homotopy coniveau tower with Voevodsky's slice filtration for P 1 -spectra. We also show that the 0th layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general P 1 -spectrum the structure of a module over motivic cohomology. This recovers and extends recent results of Voevodsky on the 0th layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence.

Motives of Azumaya algebras

“…We also recall some basic results on Azumaya algebras in the appendix A. Specializing to the case in which X is smooth over a field k and A is the pull-back to X of a central simple algebra A over k, the results of [18] translate our computation of the slices of the homotopy coniveau tower to give theorem 1.…”

“…The second author has shown in [18] that the nth layer for K is the Eilenberg-Maclane spectrum for the Tate motive Z(n) [2n]; similarly, the direct sum decomposition Here Y is any smooth irreducible scheme over k. Letting s n and s mot n denote the n layer of the motivic Postnikov tower in SH S 1 (k) and DM ef f (k), respectively, and letting EM : DM ef f (k) → SH S 1 (k) denote the Eilenberg-Maclane functor [27], our main results are Theorem 1. Let A be a central simple algebra over a field k. Then…”

“…We begin our paper in section 1 with a quick review of the motivic Postnikov tower in SH S 1 (k) and DM ef f (k), recalling the basic constructions as well as some of the results from [18] that we will need. In section 2 we recall some of the first author's theory of birational motives 1 as well as pointing out the role these motives play as the Tate twists of slices of an arbitrary t-spectrum.…”

“…We review Voevodsky's construction of the motivic Postnikov tower in SH(k) and SH S 1 (k), as well as the analog of the tower in DM ef f (k). We also give the description of these towers in terms of the homotopy coniveau tower, following [18], and recall the main results of [18] on well-connected theories and on various generalizations of Bloch's higher Chow groups to higher Chow groups with coefficients.…”

Motivic twistedK–theory

“…For closely related papers on spectral sequences computing (nontwisted) K -groups in terms of motivic cohomology groups, refer to Bloch and Lichtenbaum [7], Friedlander and Suslin [16], Grayson [18], Levine [28], Suslin [45] and Voevodsky [50]. The question of strong convergence of these spectral sequences is a tricky problem.…”