Weight structure on Kontsevich's noncommutative mixed motives

Abstract

In this article we endow Kontsevich’s triangulated category $\mathrm{KMM}_k$ of noncommutative mixed motives with a nondegenerate 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 $K_0(\mathrm{KMM}_k)$ 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.