Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology

Table of Contents - April 2012 - Volume 9, Issue 02  

Research Article

Symmetric monoidal structure on non-commutative motives

Denis-Charles Cisinskia1 and Gonçalo Tabuadaa2*

a1 Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse cedex 9, France denis-charles.cisinski@math-toulouse.fr

a2 Department of Mathematics, MIT, Cambridge MA 02139 USA, Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugal tabuada@math.mit.edu

Abstract

In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(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 Toën's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.

(Received August 17 2010)

Key Words

  • Non-commutative motives;
  • Non-commutative algebraic geometry;
  • Non-connective algebraic K-theory;
  • Secondary K-theory;
  • Hochschild homology;
  • Negative cyclic homology;
  • Periodic cyclic homology

Footnotes

* The first named author was partially supported by the ANR (grant No. ANR-07-BLAN-042). The second named author was partially supported by the Estímulo à Investigação Award 2008 - Calouste Gulbenkian Foundation and by the FCT-Portugal grant PTDC/MAT/098317/2008.