Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration

Abstract

We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin (in: Algebra, geometry, and physics in the 21st century. Birkhäuser/Springer, Cham, pp 99–129, 2017). In particular, we show that there exists a minimal 10-dimensional \(A_{\infty }\)-algebra over a field of characteristic zero, for which the supertrace of \(\mu _3\) on the second argument is non-zero. As a byproduct, we obtain an example of a homotopically finitely presented DG category (over a field of characteristic zero) that does not have a smooth categorical compactification, giving a negative answer to a question of Toën. This can be interpreted as a lack of resolution of singularities in the noncommutative setup. We also obtain an example of a proper DG category which does not admit a categorical resolution of singularities in the terminology of Kuznetsov and Lunts (Int Math Res Not 2015(13):4536–4625, 2015) (that is, it cannot be embedded into a smooth and proper DG category).

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Cisinski, D.-C., Tabuada, G.: Non-connective K-theory via universal invariants. Compos. Math. 147, 1281–1320 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Connes, A.: Non-commutative Geometry. Academic Press, Cambridge (1994)

    Google Scholar 

  3. 3.

    Deligne, P., Illusie, L.: Relévements modulo \(p^2\) et décomposition du complexe de de Rham. Invent. Math. 89, 247–270 (1987)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Efimov, A.I.: Generalized non-commutative degeneration conjecture. Proc. Steklov Inst. Math. 290(1), 1–10 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Efimov, A.I.: Homotopy finiteness of some DG categories from algebraic geometry. arXiv:1308.0135 (preprint), to appear in JEMS

  7. 7.

    Efimov, A.I., Lunts, V., Orlov, D.O.: Deformation theory of objects in homotopy and derived categories I: general theory. Adv. Math. 222(2), 359–401 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Feigin, B.L., Tsygan, B.L.: Cohomologies of Lie algebras of generalized Jacobi matrices. Funkts. Anal. Prilozh. 17(2), 86–87 (1983)

    MATH  Google Scholar 

  9. 9.

    Feigin, B.L., Tsygan, B.L.: Cohomologies of Lie algebras of generalized Jacobi matrices. Funct. Anal. Appl. 17, 153–155 (1983)

    Article  Google Scholar 

  10. 10.

    Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. Amer. Math. Soc, Providence (1998)

    Google Scholar 

  11. 11.

    Kaledin, D.: Spectral sequences for cyclic homology. In: Algebra, geometry, and physics in the 21st century, 99–129, Progr. Math., 324, Birkhäuser/Springer, Cham (2017)

  12. 12.

    Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Keller, B.: On the cyclic homology category of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Keller, B.: A-infinity algebras, modules and functor categories. Trends in representation theory of algebras and related topics, 67–93, Contemp. Math., 406, Amer. Math. Soc., Providence, RI (2006)

  15. 15.

    Keller, B.: On the cyclic homology of ringed spaces and schemes. Doc. Math. J. DMV 3, 231–259 (1998)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kontsevich, M.: Private communication (2012)

  17. 17.

    Kontsevich, M., Soibelman, Y.: Notes on A-infinity algebras, A-infinity categories and non-commutative geometry, I. In: Homological Mirror Symmetry. Lecture Notes in Physics, vol. 757. Springer, Berlin

  18. 18.

    Kuznetsov, A., Lunts, V.: Categorical resolutions of irrational singularities. Int. Math. Res. Not. 2015(13), 4536–4625 (2015)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Lefèvre-Hasegawa, K.: Sur les \(A_{\infty }\)-catégories. Ph.D. thesis, Université Paris 7, U.F.R. de Mathématiques, http://arXiv.org/abs/math.CT/0310337 (2003)

  20. 20.

    Loday, J.-L.: Cyclic Homology. Grundlehren der Mathematischen Wissenschaften, 301. Springer, Berlin (1992)

    Google Scholar 

  21. 21.

    Lunts, V.: Categorical resolution of singularities. J. Algebra 323(10), 2977–3003 (2010)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mathew, A.: Kaledin’s degeneration theorem and topological Hochschild homology. arXiv:1710.09045 (preprint)

  23. 23.

    Nagata, M.: A generalization of the imbedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ. 3, 89–102 (1963)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Quillen, D.: Higher algebraic K-theory: I. In: Bass, H. (ed.) Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer, Berlin (1973)

    Google Scholar 

  25. 25.

    Rizzardo, A., Van den Bergh, M., Neeman, A.: An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves. Invent. Math. 216(3), 927–1004 (2019)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Rosenberg, J.: Algebraic K-Theory and Its Applications. Graduate Texts in Mathematics, 147. Springer, Berlin (1994)

    Google Scholar 

  27. 27.

    Sheridan, N.: Formulae in noncommutative Hodge theory. J. Homotopy Relat. Struct. 15, 249–299 (2020)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Shklyarov, D.: Hirzebruch–Riemann–Roch-type formula for DG algebras. Proc. Lond. Math. Soc. 106(1), 1–32 (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Tabuada, G.: Théorie homotopique des DG-catégories. Thése de L’Université Paris Diderot—Paris 7

  30. 30.

    Tabuada, G.: Une structure de catégorie de modéles de Quillen sur la catégorie des dg-catégories. C. R. Acad. Sci. Paris Ser. I Math. 340(1), 15–19 (2005)

    Article  Google Scholar 

  31. 31.

    Toën, B.: Private communication (2012)

  32. 32.

    Toën, B.: The homotopy theory of \(dg\)-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Toën, B.: Lectures on DG-Categories. In: Topics in Algebraic and Topological K-Theory. Lecture Notes in Mathematics, vol 2008. Springer, Berlin (2011)

  34. 34.

    Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann. Sci. École Norm. Sup. (4) 40(3), 387–444 (2007)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Tsygan, B.L.: The homology of matrix Lie algebras over rings and the Hochschild homology. Usp. Mat. Nauk 38(2), 217–218 (1983)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Tsygan, B.L.: The homology of matrix Lie algebras over rings and the Hochschild homology. Russ. Math. Surv. 38(2), 198–199 (1983)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Waldhausen, F.: 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

Download references

Acknowledgements

I am grateful to Dmitry Kaledin, Maxim Kontsevich and Bertrand Toën for useful discussions. I am also grateful to an anonymous referee for a number of useful comments and suggestions, and for pointing out a number of inaccuracies in the previous version of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexander I. Efimov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. N 14.641.31.0001.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Efimov, A.I. Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. math. (2020). https://doi.org/10.1007/s00222-020-00980-9

Download citation