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Kontsevich’s noncommutative numerical motives

Part of: Categories with structure Homological algebra Higher algebraic $K$-theory Grothendieck groups and $K_0$ Categories and geometry

Published online by Cambridge University Press:  12 October 2012

Matilde Marcolli  and
Gonçalo Tabuada
Matilde Marcolli
Affiliation:
Department of Mathematics, California Institute of Technology, 253-37 Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA (email: [email protected])
Gonçalo Tabuada
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugal (email: [email protected])
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Abstract

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In this article we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada, Noncommutative motives, numerical equivalence, and semisimplicity, Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that NCnum(k)F is abelian semi-simple as conjectured by Kontsevich.

Keywords

noncommutative algebraic geometrynoncommutative motives

MSC classification

Secondary: 18D20: Enriched categories (over closed or monoidal categories) 18F30: Grothendieck groups 18G55: Homotopical algebra 19A49: $K_0$ of other rings 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1811 - 1820
Copyright
Copyright © Foundation Compositio Mathematica 2012

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