Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T22:47:14.877Z Has data issue: false hasContentIssue false
  • English
  • Français

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: matilde@caltech.edu)
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: tabuada@math.mit.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[AK02a]André, Y. and Kahn, B., Nilpotence, radicaux et structures monoïdales, Rend. Semin. Mat. Univ. Padova 108 (2002), 107–291 (French).Google Scholar
[AK02b]André, Y. and Kahn, B., Erratum: Nilpotence, radicaux et structures monoïdales, Rend. Semin. Mat. Univ. Padova 108 (2002), 125–128 (French).Google Scholar
[Bei78]Beilinson, A., Coherent sheaves on ℙn and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), 68–69 (Russian).CrossRefGoogle Scholar
[BK89]Bondal, A. and Kapranov, M., Representable functors, Serre functors, and mutations, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1183–1205.Google Scholar
[BK90]Bondal, A. and Kapranov, M., Framed triangulated categories, Mat. Sb. 181 (1990), 669–683 (Russian; translation in Sb. Math. 70 (1991), 93–107).Google Scholar
[BV03]Bondal, A. and Van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 1–36.CrossRefGoogle Scholar
[CT12]Cisinski, D.-C. and Tabuada, G., Symmetric monoidal structure on non-commutative motives, J. K-Theory 9 (2012), 201–268.CrossRefGoogle Scholar
[Dri02]Drinfeld, V., DG categories, Talk in the University of Chicago geometric Langlands seminar, 2002, notes available at www.math.utexas.edu/users/benzvi/GRASP/lectures/Langlands.html.Google Scholar
[Dri04]Drinfeld, V., DG quotients of DG categories, J. Algebra 272 (2004), 643–691.CrossRefGoogle Scholar
[Hov99]Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
[Kal10]Kaledin, D., Motivic structures in noncommutative geometry, in Proceedings of the International Congress of Mathematicians 2010 (Hyderabad, India), vol. II (Hindustan Book Agency, New Delhi, 2010), 461–496.Google Scholar
[Kel06]Keller, B., On differential graded categories, in International Congress of Mathematicians 2006 (Madrid), vol. II (European Mathematical Society, Zürich, 2006), 151–190.Google Scholar
[Kon98]Kontsevich, M., Triangulated categories and geometry, Course at the École Normale Supérieure, Paris, 1998, notes available at www.math.uchicago.edu/mitya/langlands.html.Google Scholar
[Kon05]Kontsevich, M., Noncommutative motives, Talk at the Institute for Advanced Study on the occasion of the 61st birthday of Pierre Deligne, October 2005, video available at http://video.ias.edu/Geometry-and-Arithmetic.Google Scholar
[Kon09]Kontsevich, M., Notes on motives in finite characteristic, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progress in Mathematics, vol. 270 (Birkhäuser, Boston, 2009), 213–247.CrossRefGoogle Scholar
[Kon10]Kontsevich, M., Mixed noncommutative motives, Talk at the FRG Workshop on Homological Mirror Symmetry, University of Miami, 2010, notes available at www-math.mit.edu/auroux/frg/miami10-notes.Google Scholar
[LO10]Lunts, V. and Orlov, D., Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), 853–908.CrossRefGoogle Scholar
[MT11a]Marcolli, M. and Tabuada, G., Noncommutative motives, numerical equivalence, and semi-simplicity, Amer. J. Math., to appear, available at arXiv:1105.2950.Google Scholar
[MT11b]Marcolli, M. and Tabuada, G., Noncommutative numerical motives, Tannakian structures, and motivic Galois groups, Preprint (2011), arXiv:1110.2438.Google Scholar
[MT11c]Marcolli, M. and Tabuada, G., Unconditional motivic Galois groups and Voevodsky’s nilpotence conjecture in the noncommutative world, Preprint (2011), arXiv:1112.5422.Google Scholar
[Nee01]Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, 2001).CrossRefGoogle Scholar
[Shk07]Shklyarov, D., On Serre duality for compact homologically smooth DG algebras, Preprint (2007), arXiv:math/0702590.Google Scholar
[Tab05]Tabuada, G., Invariants additifs de dg-catégories, Int. Math. Res. Not. 53 (2005), 3309–3339.Google Scholar
[Tab10]Tabuada, G., A guided tour through the garden of noncommutative motives, Extended notes of a survey talk on noncommutative motives given at the 3era Escuela de Inverno Luis Santaló-CIMPA: Topics in Noncommutative Geometry, Buenos Aires, July 26 to August 6, 2010, Clay Mathematics Proceedings, vol. 16, to appear, available at arXiv:1108.3787.Google Scholar
[Tab11]Tabuada, G., Chow motives versus noncommutative motives, J. Noncommut. Geom., to appear, arXiv:1103.0200.Google Scholar