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Hochschild (Co)homology and Derived Categories

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Abstract

These are slightly expanded notes of lectures given in April 2019 at the Isfahan school and conference on representations of algebras. We recall the formalism of derived categories and functors and survey invariance results for the Hochschild (co)homology of differential graded algebras and categories.

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Notes

  1. The Hattori–Stallings trace [26, 51] is based on an extension of this idea, cf. [23] for a beautiful application.

References

  1. Amiot, C.: Cluster categories for algebras of global dimension \(2\) and quivers with potential. Ann. de l’institut Fourier 59(6), 2525–2590 (2009)

    Article  MathSciNet  Google Scholar 

  2. Ánh, P.N., Márki, L.: Morita equivalence for rings without identity. Tsukuba J. Math. 11(1), 1–16 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Armenta, M.A., Keller, B.: Derived invariance of the Tamarkin-Tsygan calculus of an algebra. C. R. Math. Acad. Sci. Paris 357(3), 236–240 (2019). MR 3945161

    Article  MathSciNet  Google Scholar 

  4. Baues, H.J.: The double bar and cobar constructions. Composit. Math. 43(3), 331–341 (1981)

    MathSciNet  MATH  Google Scholar 

  5. Bendiffalah, B., Guin, D.: Cohomologie de l’algèbre triangulaire et applications. Algebra Montpellier Announcements 01-2003, 1–5, electronic. (2003)

  6. Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings, 155 pp. (1986)

  7. Cartan, H., Eilenberg, S.: Homological algebra. Princeton University Press, Princeton (1956)

    MATH  Google Scholar 

  8. Chen, X.-W., Li, H., Wang, Z.: Leavitt path algebras, \(B_\infty \)-algebras and Keller’s conjecture for singular Hochschild cohomology, arXiv:2007.06895/math.RT

  9. Cibils, C., Marcos, E., Redondo, M.J., Solotar, A.: Cohomology of split algebras and of trivial extensions, arXiv:math.KT/0102194

  10. Cibils, C.: Tensor Hochschild homology and cohomology, Interactions between ring theory and representations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York, 2000, pp. 35–51. (2000)

  11. Donovan, W., Wemyss, M.: Noncommutative deformations and flops. Duke Math. J. 165(8), 1397–1474 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Dyckerhoff, T.: Compact generators in categories of matrix factorizations. Duke Math. J. 159(2), 223–274 (2011)

    Article  MathSciNet  Google Scholar 

  14. Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260(1), 35–64 (1980)

    Article  MathSciNet  Google Scholar 

  15. Elagin, A., Lunts, V.A., Schnürer, O.M.: Smoothness of derived categories of algebras, arXiv:1810.07626 [math.AG]

  16. Gabriel, P., Roiter, A.V.: Representations of finite-dimensional algebras, Encyclopaedia Math. Sci., vol. 73, Springer–Verlag, 1992

  17. Gert-Martin Greuel and Thuy Huong Pham: Mather-Yau theorem in positive characteristic. J. Algebraic Geom. 26(2), 347–355 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Getzler, E., Jones, J.D.S.: Operads, homotopy algebra, and iterated integrals for double loop spaces, hep-th/9403055

  19. Green, E.L., Solberg, Ø.: Hochschild cohomology rings and triangular rings. In: Happel, D., Zhang, Y.B. (eds.)Proceedings of the Ninth International Conference, Beijing, 2000, vol. II, Beijing Normal University Press, Beijing, pp. 192–200 (2002)

  20. Green, E.L., Marcos, E.N., Snashall, N.: The Hochschild cohomology ring of a one point extension. Comm. Algebra 31(1), 357–379 (2003)

    Article  MathSciNet  Google Scholar 

  21. Guccione, J.A., Guccione, J.J.: Hochschild cohomology of triangular matrix algebras, arXiv:math.KT/0104068

  22. Guccione, J.A., Guccione, J., Redondo, M.J., Villamayor, O.E.: Hochschild and cyclic homology of hypersurfaces. Adv. Math. 95(1), 18–60 (1992)

    Article  MathSciNet  Google Scholar 

  23. Han, Y.: A bimodule approach to the strong no loop conjecture. J. Pure Appl. Algebra 219(6), 2139–2143 (2015)

    Article  MathSciNet  Google Scholar 

  24. Happel, D.: Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126 (1989)

  25. Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62(3), 339–389 (1987)

    Article  MathSciNet  Google Scholar 

  26. Hattori, A.: Rank element of a projective module. Nagoya Math. J. 25, 113–120 (1965)

    Article  MathSciNet  Google Scholar 

  27. Hochschild, G.: On the cohomology groups of an associative algebra. Ann. Math. (2) 46, 58–67 (1945)

    Article  MathSciNet  Google Scholar 

  28. Hua, Z., Keller, B.: Cluster categories and rational curves, arXiv:1810.00749 [math.AG]

  29. Jenny August On the finiteness of the derived equivalence classes of some stable endomorphism rings, arXiv:1801.05687 [math.RT]

  30. Kadeishvili, T.V.: \(A_\infty \)-algebra structure in cohomology and the rational homotopy type, Preprint 37, Forschungsschwerpunkt Geometrie, Universität Heidelberg. Mathematisches Institut 1988, (1988)

  31. Kassel, C.: Cyclic homology, comodules and mixed complexes. J. Algebra 107, 195–216 (1987)

    Article  MathSciNet  Google Scholar 

  32. Keller, B.: Cluster algebras and derived categories. In: Derived categories in algebraic geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 123–183. MR 3050703 (2012)

  33. Keller, B.: Derived categories and tilting, Handbook of Tilting Theory, LMS Lecture Note Series, vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 49–104 (2007)

  34. Keller, B.: Derived invariance of higher structures on the Hochschild complex, preprint, 2003, available at the author’s home page, (2003)

  35. Keller, B.: Singular Hochschild cohomology via the singularity category, arXiv:1809.05121 [math.RT]

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

    Article  MathSciNet  Google Scholar 

  37. Keller, B.: Invariance and localization for cyclic homology of DG algebras. J. Pure Appl. Algebra 123(1–3), 223–273 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)

    Article  MathSciNet  Google Scholar 

  40. Loday, J.-L.: Cyclic homology, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by Marí a O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili (1998)

  41. Lowen, W., Van den Bergh, M.: The \(B_\infty \)-structure on the derived endomorphism algebra of the unit in a monoidal category, arXiv:1907.06026 [math.KT]

  42. Lowen, W., Bergh, M.: Hochschild cohomology of abelian categories and ringed spaces. Adv. Math. 198(1), 172–221 (2005)

    Article  MathSciNet  Google Scholar 

  43. Marco Antonio Armenta and Bernhard Keller: Derived invariance of the cap product in Hochschild theory. C. R. Math. Acad. Sci. Paris 355(12), 1205–1207 (2017)

    Article  MathSciNet  Google Scholar 

  44. Mather, J.N., Yau, S.S.T.: Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69(2), 243–251 (1982)

    Article  MathSciNet  Google Scholar 

  45. Michelena, S., Platzeck, M.I.: Hochschild cohomology of triangular matrix algebras. J. Algebra 233(2), 502–525 (2000)

    Article  MathSciNet  Google Scholar 

  46. Mitchell, B.: Rings with several objects. Adv. Math. 8, 1–161 (1972)

    Article  MathSciNet  Google Scholar 

  47. Nystedt, P.: A survey of s-unital and locally unital rings. Rev. Integr. Temas Mat. 37(2), 251–260 (2019)

    Article  MathSciNet  Google Scholar 

  48. Orlov, D.O.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Trans. Mat. Inst. Steklova 246, no. Algebr. Geom. Metody, Svyazi i Prilozh 240–262 (2004)

  49. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39, 436–456 (1989)

    Article  MathSciNet  Google Scholar 

  50. Rickard, Jeremy: Derived equivalences as derived functors. J. Lond. Math. Soc. (2) 43(1), 37–48 (1991)

    Article  MathSciNet  Google Scholar 

  51. Stallings, J.: Centerless groups–an algebraic formulation of Gottlieb’s theorem. Topology 4, 129–134 (1965)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  54. Van den Bergh, M., Völcsey, L.T.: Explicit models for some stable categories of maximal Cohen–Macaulay modules, arXiv:1006.2021 [math.RA]

  55. Wang, Z.: Gerstenhaber algebra and Deligne’s conjecture on Tate–Hochschild cohomology, arXiv:1801.07990

  56. Wang, Z.: Singular Hochschild cohomology and Gerstenhaber algebra structure, arXiv:1508.00190 [math.RT]

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Acknowledgements

I thank the organizers of the ISCRA 2019, and in particular Javad Asadollahi, for the wonderful job they have done and for their exquisite hospitality.

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  1. Université de Paris, UFR de Mathématiques, Institut de Mathématiques de Jussieu–PRG, UMR 7586 du CNRS, Case 7012, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Bernhard Keller

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Correspondence to Bernhard Keller.

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Communicated by Javad Asadollahi.

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Keller, B. Hochschild (Co)homology and Derived Categories. Bull. Iran. Math. Soc. 47 (Suppl 1), 57–83 (2021). https://doi.org/10.1007/s41980-021-00556-0

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