Homology, Homotopy and Applications

Volume 14 (2012)

Number 2

Weight structure on Kontsevich’s noncommutative mixed motives

Pages: 129 – 142

DOI: https://dx.doi.org/10.4310/HHA.2012.v14.n2.a8


Gonçalo Tabuada (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.; Departamento de Matematica, FCT-UNL, Quinta da Torre, Caparica, Portugal)


In this article we endow Kontsevich’s triangulated category $KMM_k$ of noncommutative mixed motives with a non-degenerate weight structure in the sense of Bondarko. As an application we obtain: (1) a convergent weight spectral sequence for every additive invariant (e.g., algebraic $K$-theory, cyclic homology, topological Hochschild homology, etc.); (2) a ring isomorphism between $K_0(KMM_k)$ and the Grothendieck ring of the category of noncommutative Chow motives; (3) a precise relationship between Voevodsky’s (virtual) mixed motives and Kontsevich’s noncommutative (virtual) mixed motives.


weight structure, weight spectral sequence, Grothendieck ring, Picard group, Voevodsky motive, Kontsevich noncommutative motive

2010 Mathematics Subject Classification

14A22, 18D20, 18G40, 19L10

Published 4 December 2012