Slice Filtration Research Articles

Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub)

We clarify the expected properties of the slice filtration on triangulated motives from the point of view of the generalized Hodge conjecture. In the appendix, J. Ayoub proves unconditionally that the slice filtration does not respect geometric motives. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The homotopy coniveau tower

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.

The slice filtration and mixed Tate motives

Using the ‘slice filtration’, defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology and etale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their -terms then have a simple description. In particular this yields spectral sequences converging to the motivic cohomology of a split connected reductive group. We also describe in detail the multiplicative structure of the motive of a split torus.