nLab motivic slice tower

Idea

In motivic homotopy theory, the slice filtration is a filtration of $\mathrm{H}(S)$ and of $SH(S)$ which is analogous to the Postnikov filtration for (∞,1)-topoi. It generalizes the coniveau filtration in algebraic K-theory, the fundamental filtration on Witt groups?, and the weight filtration on mixed Tate motives.

If $S$ is smooth over a field, the layers of the slice filtration of a motivic spectrum (called its slices) are modules over the motivic Eilenberg–Mac Lane spectrum $H(\mathbb{Z})$. At least if $S$ is a field of characteristic zero, this is the same thing as an integral motive. The spectral sequences associated to the slice filtration are analogous to the Atiyah-Hirzebruch spectral sequences in that their first page consists of motivic cohomology groups.

Related concepts

• Postnikov tower, slice filtration
• motivic homotopy theory

References

• Vladimir Voevodsky, Open problems in the motivic stable homotopy theory, I, web.

• Vladimir Voevodsky, A possible new approach to the motivic spectral sequence for algebraic K-theory, web.

• Marc Levine, Motivic Postnikov towers, talk notes, 2007, pdf.

In the unstable motivic homotopy category?:

• Pablo Pelaez?, The Unstable Slice Filtration, arXiv:1201.3657.