The Motivic Spectral Sequence

In this chapter we explain the Atiyah-Hirzebruch spectral sequence that relates topological K-theory to singular cohomology and try to motivate the search for a motivic version. In the time since [18] appeared, which concerns motivation for such a motivic spectral sequence, many authors have produced results in this direction. We describe the Bloch-Lichtenbaum spectral sequence [8] for the spectrum of a field together with the Friedlander-Suslin and Levine extensions [12, 28] to the global case for a smooth variety over a field. We explain the Goodwillie-Lichtenbaum idea involving tuples of commuting automorphisms and the theorem [19] that uses it to produce a motivic spectral sequence for an affine regular noetherian scheme, unfortunately involving certain non-standard motivic cohomology groups. We present Suslin’s result [41], that, for smooth varieties over a field, these non-standard motivic cohomology groups are isomorphic to the standard groups. We sketch Voevodsky’s approach via the slice filtration [43, 47, 49], much of which remains conjectural. Finally, we sketch Levine’s recent preprint [29], which gives a novel approach that yields a spectral sequence for smooth varieties over a field and makes it extremely clear which formal properties of K-theory are used in the proof. At this point we refer the reader also to [11] where a similar spectral sequence is developed for semi-topological K-theory. The importance of the motivic spectral sequence lies in its applications. Important work of Voevodsky [42, 44, 48] makes motivic cohomology amenable