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Symmetric monoidal structure on non-commutative motives

Published online by Cambridge University Press:  21 November 2011

Denis-Charles Cisinski  and
Gonçalo Tabuada
Denis-Charles Cisinski
Affiliation:
Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse cedex 9, Francedenis-charles.cisinski@math-toulouse.fr
Gonçalo Tabuada
Affiliation:
Department of Mathematics, MIT, Cambridge MA 02139USA, Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugaltabuada@math.mit.edu
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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.

Keywords

Non-commutative motivesNon-commutative algebraic geometryNon-connective algebraic K-theorySecondary K-theoryHochschild homologyNegative cyclic homologyPeriodic cyclic homology
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 2 , April 2012 , pp. 201 - 268
Copyright
Copyright © ISOPP 2011

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