The slice filtration and mixed Tate motives

Abstract

Using the 'slice filtration', defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology andétale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their E 2 -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.

Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky versus Hanamura

“…For rational coefficients (ii) was proved in Appendix B of [HuK06]: it suffices to note that if there exists an étale finite morphism…”

Motives of Azumaya algebras

“…Birational motives have been introduced and studied by Kahn-Sujatha [16] and Huber-Kahn [14]. In this section we re-examine their theory, emphasizing the relation to the slices in the motivic Postnikov tower.…”

“…Results of Huber-Kahn [14] give a computation of the sheaf H 0 of the slices of M (X) for X any smooth projective variety and show that H n vanishes for n > 0. For the motive of a Severi-Brauer variety X = SB(A), we are able to show (in case A has prime degree ℓ over k) that the negative cohomology vanishes as well.…”

Chow Groups, Chow Cohomology, and Linear Varieties

“…Following a referee's suggestion, I added some comments on the relation between higher Chow groups and Voevodsky's motivic homology groups. I also added some papers to the bibliography which refer to the preprint version of this paper, and which develop the ideas further [7,10,14,17,23]. …”