Generalized non-commutative degeneration conjecture

Article

Abstract

We propose a generalization of the Kontsevich–Soibelman conjecture on the degeneration of the Hochschild-to-cyclic spectral sequence for a smooth compact differential graded category. Our conjecture states identical vanishing of a certain map between bi-additive invariants of arbitrary small differential graded categories over a field of characteristic zero. We show that this generalized conjecture follows from the Kontsevich–Soibelman conjecture and the so-called conjecture on smooth categorical compactification.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D.-C. Cisinski and G. Tabuada, “Non-connective K-theory via universal invariants,” Compos. Math. 147, 1281–1320 (2011).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994).Google Scholar
  3. 3.
    P. Deligne, “ThéorÈme de Lefschetz et critÈres de dégénérescence de suites spectrales,” Publ. Math. Inst. Hautes étud. Sci. 35, 107–126 ( 1968).Google Scholar
  4. 4.
    P. Deligne and L. Illusie, “RelÈvements modulo p2 et décomposition du complexe de de Rham,” Invent. Math. 89, 247–270 (1987).MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. Drinfeld, “DG quotients of DG categories,” J. Algebra 272 (2), 643–691 (2004).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. I. Efimov, “Homotopy finiteness of some DG categories from algebraic geometry,” arXiv: 1308.0135 [math.AG].Google Scholar
  7. 7.
    B. L. Feigin and B. L. Tsygan, “Cohomologies of Lie algebras of generalized Jacobi matrices,” Funkts. Anal. Prilozh. 17 (2), 86–87 (1983)Google Scholar
  8. 7.
    B. L. Feigin Funct. Anal. Appl. 17, 153–155 (1983)].MATHCrossRefGoogle Scholar
  9. 8.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry (J. Wiley & Sons, New York, 1978).Google Scholar
  10. 9.
    H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II,” Ann. Math., Ser. 2, 79, 109–203, 205–326 (1964).Google Scholar
  11. 10.
    W. V. D. Hodge, The Theory and Applications of Harmonic Integrals (Cambridge Univ. Press, Cambridge, 1941).Google Scholar
  12. 11.
    D. Kaledin, “Non-commutative Hodge-to-de Rham degeneration via the method of Deligne–Illusie,” Pure Appl. Math. Q. 4 (3), 785–875 (2008).MATHMathSciNetCrossRefGoogle Scholar
  13. 12.
    B. Keller, “Deriving DG categories,” Ann. Sci. éc. Norm. Supér., Sér. 4, 27 (1), 63–102 (1994).MATHGoogle Scholar
  14. 13.
    B. Keller, “On the cyclic homology of ringed spaces and schemes,” Doc. Math., J. DMV 3, 231–259 (1998).Google Scholar
  15. 14.
    B. Keller, “Invariance and localization for cyclic homology of DG algebras,” J. Pure Appl. Algebra 123 (1–3), 223–273 (1998).MathSciNetGoogle Scholar
  16. 15.
    B. Keller, “On differential graded categories,” in Proc. Int. Congr. Math., Madrid, 2006 (Eur. Math. Soc., ZÜrich, 2006), Vol. 2, pp. 151–190.Google Scholar
  17. 16.
    M. Kontsevich and Y. Soibelman, “Notes on A8-algebras, A8-categories and non-commutative geometry,” in Homological Mirror Symmetry: New Developments and Perspectives, Ed. by A. Kapustin, M. Kreuzer, and K.-G. Schlesinger (Springer, Berlin, 2009)Google Scholar
  18. 16.
    M. Kontsevich and Y. Soibelman, Lect. Notes Phys. 757, pp. 153–219.Google Scholar
  19. 17.
    J.-L. Loday, Cyclic Homology (Springer, Berlin, 1992), Grundl. Math. Wiss. 301.Google Scholar
  20. 18.
    V. A. Lunts and O. M. SchnÜrer, “Smoothness of equivariant derived categories,” Proc. London Math. Soc., Ser. 3, 108 (5), 1226–1276 (2014).Google Scholar
  21. 19.
    M. Nagata, “A generalization of the imbedding problem of an abstract variety in a complete variety,” J. Math. Kyoto Univ. 3, 89–102 (1963).MathSciNetGoogle Scholar
  22. 20.
    D. Shklyarov, “Hirzebruch–Riemann–Roch-type formula for DG algebras,” Proc. London Math. Soc., Ser. 3, 106 (1), 1–32 (2013).MATHGoogle Scholar
  23. 21.
    G. Tabuada, “Invariants additifs de dg-catégories,” Int. Math. Res. Not. 2005 (53), 3309–3339 (2005).CrossRefGoogle Scholar
  24. 22.
    B. Toën and M. Vaquié, “Moduli of objects in dg-categories,” Ann. Sci. éc. Norm. Supér., Sér. 4, 40 (3), 387–444 (2007).Google Scholar
  25. 23.
    B. L. Tsygan, “The homology of matrix Lie algebras over rings and the Hochschild homology,” Usp. Mat. Nauk 38 (2), 217–218 (1983)MATHGoogle Scholar
  26. 23.
    B. L. Tsygan, Russ. Math. Surv. 38 (2), 198–199 (1983)].MATHCrossRefGoogle Scholar
  27. 24.
    F. Waldhausen, “Algebraic K-theory of spaces,” in Algebraic and Geometric Topology: Proc. Conf., New Brunswick, NJ, 1983 (Springer, Berlin, 1985), Lect. Notes Math. 1126, pp. 318–419.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations