Symmetric monoidal structure on Non-commutative motives

Denis-Charles Cisinski (1) , Gonçalo Tabuada (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 MIT - Massachusetts Institute of Technology

In this article we further the study of non-commutative motives. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Mot of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Mot in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMM of non-commutative mixed motives into the base category Mot(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between Toen's secondary K-theory and the Grothendieck ring of KMM ; (5) a description of the Euler characteristic in KMM in terms of Hochschild homology.

Domaines: K-théorie et homologie [math.KT]; Géométrie algébrique [math.AG]

Denis-Charles Cisinski, Gonçalo Tabuada. Symmetric monoidal structure on Non-commutative motives. Journal of K-theory , 2012, 9 (02), pp.201-268. ⟨10.1017/is011011005jkt169⟩. ⟨hal-00461941⟩