Skip to main content
SpringerLink
Log in

Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH 0(A) and HH 0(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auslander M., Reiten I., Smalø S.O.: Representation Theory of Artin Algebras. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  2. Bessenrodt Ch., Holm Th., Zimmermann A.: Generalized Reynolds ideals for non-symmetric algebras. J. Algebra 312, 985–994 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouc, S.: Bimodules, trace généralisée, et transferts en homologie de Hochschild. http://www.lamfa.u-picardie.fr/bouc/ (1997, preprint)

  4. Broué M.: Equivalences of blocks of group algebras. In: Dlab, V., Scott, L.L. (eds) Finite Dimensional Algebras and Related Topics, pp. 1–26. Kluwer, Dordrecht (1994)

    Google Scholar 

  5. Broué M.: Higman criterion revisited. Mich. J. Math. 58, 125–179 (2009)

    Article  MATH  Google Scholar 

  6. Curtis C.W., Reiner I.: Methods of Representation Theory I. Wiley, New York (1981)

    MATH  Google Scholar 

  7. Erdmann, K.: Blocks of tame representation type and related algebras. Springer Lecture Notes in Mathematics, vol. 1428 (1980)

  8. Eu C.H., Schedler T.: Calabi-Yau Frobenius algebras. J. Algebra 321, 774–815 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Héthelyi L., Horváth E., Külshammer B., Murray J.: Central ideals and cartan invariants of symmetric algebras. J. Algebra 293, 243–260 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Higman D.G.: On orders in separable algebras. Can. J. Math. 7, 509–515 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  11. Holm Th., Zimmermann A.: Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type. J. Algebra 320, 3425–3437 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Keller B., Vossieck D.: Sous les catégories dérivées. C. R. Acad. Sci. Paris 305, 225–228 (1987)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. König S., Liu Y.M.: Gluing of idempotents, radical embeddings and two classes of stable equivalences. J. Algebra 319, 5144–5164 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. König, S., Liu. Y.M., Zhou. G.D.: Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture. Trans. Am. Math. Soc (to appear)

  16. König S., Zimmermann A.: Derived Equivalences for Group Rings. Lecture Notes in Mathematics, vol. 1685. Springer, Berlin (1998)

    Google Scholar 

  17. Külshammer B.: Group-theoretical descriptions of ring theoretical invariants of group algebras. Prog. Math. 95, 425–441 (1991)

    Google Scholar 

  18. Lenzing H.: Nilpotente Elemente in Ringen von endlicher globaler Dimension. Math. Zeit. 108, 313–324 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  19. Linckelmann M.: Stable equivalences of Morita type for selfinjective algebras and p-groups. Math. Zeit. 223, 87–100 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu Y.M.: On stable equivalences induced by exact functors. Proc. Am. Math. Soc. 134, 1605–1613 (2006)

    Article  MATH  Google Scholar 

  21. Liu Y.M.: Summands of stable equivalences of Morita type. Commun. Algebra 36(10), 3778–3782 (2008)

    Article  MATH  Google Scholar 

  22. Liu Y.M., Xi C.C.: Constructions of stable equivalences of Morita type for finite dimensional algebras II. Math. Zeit. 251, 21–39 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu Y.M., Xi C.C.: Constructions of stable equivalences of Morita type for finite dimensional algebras I. Trans. Am. Math. Soc. 358, 2537–2560 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu Y.M., Xi C.C.: Constructions of stable equivalences of Morita type for finite dimensional algebras III. J. Lond. Math. Soc. 76(3), 567–585 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martinez-Villa R.: Algebras stably equivalent to l-hereditary algeras. In: Dlab, V., Gabriel, P. (eds) Representation Theory II. Lecture Notes in Mathematics, vol. 832, pp. 396–431. Springer, Berlin (1980)

    Google Scholar 

  26. Martinez-Villa R.: The stable equivalence for algebras of finite representation type. Commun. Algebra 13(5), 991–1018 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Martinez-Villa R.: Properties that are left invariant under stable equivalence. Commun. Algebra 18(12), 4141–4169 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Martinez-Villa R.: The stable group of a selfinjective Nakayama algebra. Commun. Algebra 19(2), 509–517 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ohtake K., Fukushima H.: A remark on Higman’s result about separable algebras. Tsukuba J. Math. 10, 35–46 (1986)

    MathSciNet  Google Scholar 

  30. Pogorzały Z.: Algebras stably equivalent to self-injective special biserial algebras. Commun. Algebra 22(4), 1127–1160 (1994)

    Article  MATH  Google Scholar 

  31. Pogorzały Z.: Invariance of Hochschild cohomology algebras under stable equivalences of Morita type. J. Math. Soc. Jpn. 53(4), 913–918 (2001)

    Article  MATH  Google Scholar 

  32. Rickard J.: Derived categories and stable equivalence. J. Pure Appl. Algebra 61(3), 303–317 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Riedtmann Ch.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55(2), 199–224 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. Roggenkamp K.W., Huber-Dyson V.: Lattices over Orders. Lecture Notes in Mathematics, vol. 115. Springer, Berlin (1970)

    Google Scholar 

  36. Rouquier, R.: Derived equivalences and finite dimensional algebras. Proceedings of the International Congress of Mathematicians (Madrid, 2006), vol. II, pp. 191–221. EMS Publishing House, Zurich (2006)

  37. Tang A.P.: A note on the stable equivalence conjecture. Commun. Algebra 24(9), 2793–2809 (1996)

    Article  MATH  Google Scholar 

  38. Tachikawa H., Wakamatsu T.: Cartan matrices and Grothendieck groups of stable categories. J. Algebra 144(2), 390–398 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xi C.C.: Stable equivalences of adjoint type. Forum Math. 20, 81–97 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhou, G.D., Zimmermann, A.: Classifying tame blocks and related algebras up to stable equivalences of Morita type (2010, preprint)

  41. Zhou, G.D., Zimmermann, A.: Auslander-Reiten conjecture for symmetric algebras of polynomial growth (2010, preprint)

  42. Zimmermann A.: Invariance of generalized Reynolds ideals under derived equivalences. Math. Proc. R. Ir. Acad. 107(1), 1–9 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Yuming Liu

  2. Ecole Polytechnique Fédérale de Lausanne, SB/IGAT/CTG, Batîment MA B3 424, 1015, Lausanne, Switzerland

    Guodong Zhou

  3. Département de Mathématiques et LAMFA (UMR 6140 du CNRS), Université de Picardie, 33 rue St Leu, 80039, Amiens Cedex 1, France

    Alexander Zimmermann

Authors
  1. Yuming Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guodong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Zimmermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guodong Zhou.

Additional information

The first author was supported by Marie Curie Fellowship IIF. The second and the third authors were supported by a German-French grant “PROCOPE” of the DAAD, respectively a French-German grant “partenariat Hubert Curien PROCOPE dossier 14765WB”. The second author benefits also from financial support via postdoctoral fellowship from the network “Representation theory of algebras and algebraic Lie theory” and the DAAD.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y., Zhou, G. & Zimmermann, A. Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture. Math. Z. 270, 759–781 (2012). https://doi.org/10.1007/s00209-010-0825-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-010-0825-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation