Weight structure on noncommutative motives

TL;DR

This paper introduces a weight structure on Kontsevich's category of noncommutative mixed motives, enabling spectral sequences for invariants and establishing a ring isomorphism between related Grothendieck rings.

Contribution

It constructs a non-degenerate weight structure on KMM and derives new spectral sequences and ring isomorphisms for noncommutative motives.

Findings

Established a convergent weight spectral sequence for additive invariants.

Proved a ring isomorphism between Grothendieck rings of KMM and noncommutative Chow motives.

Enhanced understanding of the structure of noncommutative motives.

Abstract

In this note we endow Kontsevich's category KMM of noncommutative mixed motives with a non-degenerate weight structure in the sense of Bondarko. As an application we obtain a convergent weight spectral sequence for every additive invariant (e.g. algebraic K-theory, cyclic homology, topological Hochschild homology, etc.), and a ring isomorphism between the Grothendieck ring of KMM and the Grothendieck ring of the category of noncommutative Chow motives.