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


We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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).

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  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, (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

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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.

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

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Efimov, A.I. Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration. Invent. math. (2020).

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