Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T06:59:52.992Z Has data issue: false hasContentIssue false
  • English
  • Français

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, [email protected]
Gonçalo Tabuada
Affiliation:
Department of Mathematics, MIT, Cambridge MA 02139USA, Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Barwick, C., On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 (2010), no. 2, 245–320.Google Scholar
2.Berthelot, P., Grothendieck, A., and Illusie, L., Théorie des intersections et théorème de Riemann-Roch (SGA 6), Lectures Notes in Mathematics 225, Springer-Verlag, 1971.Google Scholar
3.Blumberg, A. and Mandell, M., Localization theorems in topological Hochschild homology and topological cyclic homology. Available at arXiv:0802:3938.Google Scholar
4.Bondal, A. and Kapranov, M., Framed triangulated categories (Russian) Mat. Sb. 181 (1990) no. 5, 669–683; translation in Math. USSR-Sb. 70, no. 1, 93–107.Google Scholar
5.Bondal, A. and Van den Bergh, M., Generators and Representability of Functors in commutative and Noncommutative Geometry. Moscow Mathematical Journal 3(1), 1–37 (2003)Google Scholar
6.Borceux, F., Handbook of categorical algebra 2. Encyclopedia of Mathematics and its Applications 51, Cambridge University Press, 1994.Google Scholar
7.Bousfield, A. K. and Kan, D., Homotopy limits, completions, and localizations. Lecture Notes in Mathematics 304, Springer-Verlag, 1972.Google Scholar
8.Cisinski, D.-C., Images directes cohomologiques dans les catégories de modèles. Annales Mathématiques Blaise Pascal 10 (2003), 195–244.CrossRefGoogle Scholar
9.Cisinski, D.-C., Les préfaisceaux comme modèles des type d'homotopie. Astérisque 308, Soc. Math. France, 2006.Google Scholar
10.Cisinski, D.-C., Propriétés universelles et extensions de Kan dérivées. Theory and Applications of Categories 20 (2008), no. 17, 605–649.Google Scholar
11.Cisinski, D.-C., Catégories dérivables. Bull. Soc. Math. France 138 (2010), no. 3, 317–393.CrossRefGoogle Scholar
12.Cisinski, D.-C. and Tabuada, G., Non-connective K-theory via universal invariants. Compositio Mathematica 147 (2011), 1281–1320.Google Scholar
13.Cisinski, D.-C. and Neeman, A., Additivity for derivator K-theory. Adv. Math. 217 (2008), no. 4, 1381–1475.CrossRefGoogle Scholar
14.Day, B., On closed categories of functors. Reports of the midwest category seminar IV. Lecture notes in Math. 137 (1970), 1–38.Google Scholar
15.Dugger, D., Combinatorial model categories have presentations. Adv. Math. 164 (2001), no. 1, 177–201.Google Scholar
16.Dugger, D., Universal homotopy theories. Adv. Math. 164 (2001), no. 1, 144–176.CrossRefGoogle Scholar
17.Drinfeld, V., DG quotients of DG categories. J. Algebra 272 (2004), 643–691.Google Scholar
18.Drinfeld, V., DG categories. University of Chicago Geometric Langlands Seminar. Available at http://www.math.utexas.edu/users/benzvi/GRASP/lectures/Langlands.html.Google Scholar
19.Dwyer, W. G. and Kan, D. M., Equivalences between homotopy theories of diagrams, Algebraic topology and algebraic K-theory, Annals of Math. Studies 113, Princeton University Press, 1987, pp. 180–205.Google Scholar
20.Eilenberg, S. and Moore, J. C., Homology and fibrations I. Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966), 199–236.CrossRefGoogle Scholar
21.Goerss, P. and Jardine, J., Simplicial homotopy theory. Progress in Mathematics 174, 1999.Google Scholar
22.Heller, A., Homotopy theories. Mem. Amer. Math. Soc. 71 (1988), no. 383.Google Scholar
23.Heller, A., Stable homotopy theories and stabilization. J. Pure Appl. Algebra 115 (1997), 113–130.CrossRefGoogle Scholar
24.Hirschhorn, P., Model categories and their localizations. Mathematical Surveys and Monographs 99, American Mathematical Society, 2003.Google Scholar
25.Hovey, M., Model categories. Mathematical Surveys and Monographs 63, American Mathematical Society, 1999.Google Scholar
26.Hovey, M., Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra 165 (2001), 63–127.CrossRefGoogle Scholar
27.Hovey, M., Model category structures on chain complexes of sheaves. Trans. Amer. Math. Soc. 353 (2001), no.6, 2441–2457.CrossRefGoogle Scholar
28.Hovey, M., Shipley, B. and Smith, J., Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.CrossRefGoogle Scholar
29.Grothendieck, A., Les Dérivateurs. Available at http://people.math.jussieu.fr/maltsin/groth/Derivateurs.html.Google Scholar
30.Kassel, C., Cyclic homology, comodules and mixed complexes, J. Algebra 107 (1987), 195–216.CrossRefGoogle Scholar
31.Keller, B., On differential graded categories. International Congress of Mathematicians (Madrid), Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006.Google Scholar
32.Keller, B., On the cyclic homology of exact categories. J. Pure Appl. Algebra 136 (1999), no. 1, 1–56.Google Scholar
33.Keller, B., On the cyclic homology of ringed spaces and schemes. Doc. Math. 3 (1998), 231–259 (electronic).CrossRefGoogle Scholar
34.Kontsevich, M., Non-commutative 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
35.Kontsevich, M., Notes on motives in finite characteristic. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 213-247, Progr. Math. 270, Birkhäuser Boston, Inc., Boston, MA, 2009.Google Scholar
36.Kontsevich, M., Categorification, NC Motives, Geometric Langlands and Lattice Models. Talk at the University of Chicago Geometric Langlands Seminar (2006). Available at http://www.math.utexas.edu/users/benzvi/notes.html.Google Scholar
37.Kontsevich, M. and Soibelmann, Y., Notes on A-infinity algebras, A-infinity categories and non-commutative geometry I. In “Homological Mirror Symmetry: New Developments and Perspectives” (A.Kapustin et al. (Eds.)). Lect. Notes in Physics 757, Springer, Berlin Heidelberg, 2009, 153–219.Google Scholar
38.May, J. P., The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34–73.Google Scholar
39.Loday, J.-L., Cyclic homology. Grundlehren der Mathematischen Wissenschaften 301. Springer-Verlag, Berlin, 1992.Google Scholar
40.Neeman, A., Triangulated categories. Annals of Mathematics Studies 148, Princeton University Press, 2001.Google Scholar
41.Quillen, D., Homotopical algebra. Lecture Notes in Mathematics 43, Springer-Verlag, 1967.Google Scholar
42.Schlichting, M., Negative K-theory of derived categories. Math. Z. 253 (2006), no. 1, 97–134.Google Scholar
43.Tabuada, G., Higher K-theory via universal invariants. Duke Math. J. 145 (2008), no. 1, 121–206.CrossRefGoogle Scholar
44.Tabuada, G., Invariants additifs de dg-catégories. Int. Math. Res. Not. 53 (2005), 3309–3339.CrossRefGoogle Scholar
45.Tabuada, G., Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.Google Scholar
46.Tabuada, G., Generalized spectral categories, topological Hochschild homology, and trace maps. Algebraic and Geometric Topology, 10 (2010), 137–213.CrossRefGoogle Scholar
47.Tabuada, G., Matrix invariants of spectral categories. Int. Math. Res. Not. 13 (2010), 2459–2511.Google Scholar
48.Tabuada, G., Products, multiplicative Chern characters, and finite coefficients via non-commutative motives. Available at arXiv:1101.0731v2.Google Scholar
49.Thomason, R.W. and Trobaugh, T., Higher algebraic K -theory of schemes and of derived categories. In The Grothendieck Festschrift, Volume III. Progress in Math. 88, 247–436. Birkhauser, Boston, Bassel, Berlin, 1990.Google Scholar
50.Toën, B., The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167 (2007), no. 3, 615–667.Google Scholar
51.Toën, B., Lectures on dg-categories. Sedano winter school on K-theory, January 2007. Available at http://www.math.univ-toulouse.fr/toen/note.html.Google Scholar
52.Toën, B., Secondary K-theory I-III. Lecture notes from the workshop and conference on Topological Field Theories, Northwestern University, May 2009. Available at http://www.math.utexas.edu/users/benzvi/GRASP/lectures/NWTFT.html.Google Scholar
53.Toën, B. and Vaquié, M., Moduli of objects in dg-categories. Ann. Sci. de l’ENS 40 (2007), Issue 3, 387–444.Google Scholar
54.Toën, B. and Vezzosi, G., A note on Chern character, loop spaces and derived algebraic geometry. Available at arXiv:0804.1274. To appear in the Abel Symposium, Oslo 2007.Google Scholar
55.Verdier, J. L., Des catégories dérivées des catégories abéliennes, Astérisque 239, Soc. Math. France, 1996.Google Scholar
56.Weibel, C., Cyclic homology for schemes. Proc. Amer. Math. Soc. 124 (1996), no. 6, 1655–1662.Google Scholar