Algebras and Representation Theory

, Volume 21, Issue 6, pp 1251–1275 | Cite as

Külshammer Ideals of Graded Categories and Hochschild Cohomology

  • Yury Volkov
  • Alexandra ZvonarevaEmail author


We generalize the notion of Külshammer ideals to the setting of a graded category. This allows us to define and study some properties of Külshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of Külshammer ideals are studied in the case where the category is d-Calabi-Yau.


Külshammer ideals Hochschild cohomology Graded center Calabi-Yau categories 

Mathematics Subject Classification (2010)

16E40 16E35 18E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Yury Volkov is supported by the RFBR Grant 17-01-00258 and by the President’s Program ”Support of Young Russian Scientists” (Grant MK-1378.2017.1). Alexandra Zvonareva is supported by the RBFR Grant 16-31-60089.


  1. 1.
    Asashiba, H.: A generalization of Gabriel’s Galois covering functors and derived equivalences. J. Algebra 334(1), 109–149 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benson, D., Iyengar, S., Krause, H.: Local cohomology and support for triangulated categories. Annales Scientifiques de l’É. NS. 41, 4 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bessenrodt, C., Holm, T., Zimmermann, A.: Generalized Reynolds ideals for non-symmetric algebras. J. Algebra 312, 985–994 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bocian, R., Skowronski, A.: Derived equivalence classification of weakly symmetric algebras of domestic type. Colloq. Math. 142, 115–133 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buchweitz, R. -O., Flenner, H.: Global Hochschild (co-)homology of singular spaces. Adv. Math. 217, 205–242 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Happel, D.: Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)Google Scholar
  7. 7.
    Hattori, A.: Rank element of a projective module. Nagoya Math. J. 25, 113–120 (1965)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Herscovich, E., Solotar, A.: Derived invariance of Hochschild-Mitchell (co)homology and one-point extensions. J. Algebra 315(2), 852–873 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Holm, T., Skowronski, A.: Derived equivalence classification of symmetric algebras of domestic type. J. Math. Soc. Jpn 58.4, 1133–1149 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Holm, T., Skowronski, A.: Derived equivalence classification of symmetric algebras of polynomial growth. Glasgow Math. J. 53(2), 277–291 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Holm, T., Zimmermann, A.: Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type. J. Algebra 320, 3425–3437 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Keller, B.: Calabi-Yau triangulated categories. Trends Represent Theory Algebras Related Top. 467–489 (2008)Google Scholar
  13. 13.
    Koenig, S., Liu, Y., Zhou, G.: Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture. Trans. Am. Math. Soc. 364(1), 195–232 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Krause, H., Ye, Y.: On the centre of a triangulated category. Proc. Edinburgh Math. Soc. (Series 2) 54(2), 443–466 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Künzer, M.: On the center of the derived category preprint (2006)Google Scholar
  16. 16.
    Lenzing, H.: Nilpotente Elemente in Ringen von endlicher globaler Dimension. Math. Z. 108, 313–324 (1969)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Linckelmann, M.: On graded centers and block cohomology. Proc. Edinburgh Math. Soc. (2) 52(2), 489–514 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mitchell, B.: Rings with several objects. Adv. Math. 8(1), 1–161 (1972)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rickard, J.: Derived equivalences as derived functors. J. London Math. Soc. (2) 43(1), 37–48 (1991)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Stallings, J.: Centerless groups – an algebraic formulation of Gottlieb’s theorem. Topology 4, 129–134 (1965)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tamarkin, D., Tsygan, B.: The ring of differential operators on forms in noncommutative calculus, Graphs and patterns in mathematics and theoretical physics. Proc. Sympos. Pure Math. 73, 105–131 (2005)CrossRefGoogle Scholar
  22. 22.
    Zimmermann, A.: Fine Hochschild invariants of derived categories for symmetric algebras. J. Algebra 308, 350–367 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zimmermann, A.: Hochschild homology invariants of Külshammer type of derived categories. Commun. Algebra 39(8), 2963–2980 (2011)CrossRefGoogle Scholar
  24. 24.
    Zimmermann, A.: Invariance of generalized Reynolds ideals under derived equivalences. Math. Proc. R. Ir. Acad. 107A(1), 1–9 (2007)CrossRefGoogle Scholar
  25. 25.
    Zimmermann, A: Külshammer ideals of algebras of quaternion type, arXiv: (2016)
  26. 26.
    Zimmermann, A.: On the use of Külshammer type invariants in representation theory. Bull. Iranian Math. Soc. 37(2), 291–341 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySaint PetersburgRussia
  2. 2.Chebyshev LaboratorySt. Petersburg State UniversitySaint PetersburgRussia

Personalised recommendations