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Auslander–Reiten conjecture for symmetric algebras of polynomial growth

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Abstract

This paper studies self-injective algebras of polynomial growth. We prove that two indecomposable weakly symmetric algebras of domestic type are derived equivalent if and only if they are stably equivalent. Furthermore we prove that for indecomposable weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander–Reiten conjecture for stable equivalences between weakly symmetric algebras of domestic type and for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth.

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References

  • Al-Kadi, D.: Distinguishing derived equivalence classes using the second hochschild cohomology group. Preprint (2010). arxiv 0912.2971v4

  • Asashiba H.: The derived equivalence classification of representation-finite self-injective algebras. J. Algebra 214, 182–221 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Auslander M., Reiten I., Smalø S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics. vol. 36. Cambridge University Press, London (1995)

    Book  Google Scholar 

  • Białkowski J., Holm T., Skowroński A.: Derived equivalences for tame weakly symmetric algebras having only periodic modules. J. Algebra 269(2), 652–668 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Białkowski J., Holm T., Skowronski A.: On nonstandard tame self-injective algebras having only periodic modules. Colloq. Math. 97(1), 33–47 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Bocian R., Holm T., Skowroński A.: Derived equivalence classification of weakly symmetric algebras of Euclidean type. J. Pure Appl. Algebra 191(1–2), 43–74 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Bocian R., Holm T., Skowroński A.: Derived equivalence classification of nonstandard self-injective algebras of domestic type. Commun. Algebra 35(2), 515–526 (2007)

    Article  MATH  Google Scholar 

  • Drozd, Y.A.: Tame and wild matrix problems. Representation theory. In: (Proceedings of the Second International Conference, Carleton University, Ottawa, ON, 1979) Lecture Notes in Mathematics, vol. 832, pp. 242–258. Springer, Berlin (1980)

  • Erdmann K.: Blocks of Tame Representation Type and Eelated Algebras. LNM, vol. 1428. Springer, Berlin (1980)

    Google Scholar 

  • Erdmann, K., Skowronski, A.: Classification of tame symmetric algebras with periodic modules. Preprint (2011)

  • Erdmann, K., Skowroński, A.: Periodic algebras. In: Trends in Representation Theory of Algebras and Related Topics, pp. 201–251. European Mathematical Society, Series of Congress Reports (2008)

  • Holm T., Skowroński A.: Derived equivalence classification of symmetric algebras of domestic type. J. Math. Soc. Jpn. 58(4), 1133–1149 (2006)

    Article  MATH  Google Scholar 

  • Holm T., Skowroński A.: Derived equivalence classification of symmetric algebras of polynomial growth. Glasg. Math. J. 53, 277–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Keller B., Vossieck D.: Sous les catégories dérivées. C. R. Acad. Sci. Paris Série I Math. 305(6), 225–228 (1987)

    MathSciNet  MATH  Google Scholar 

  • König, 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. To appear, preprint (2009)

  • Krause H., Zwara G.: Stable equivalence and generic modules. Bull. Lond. Math. Soc. 32, 615–618 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Liu, Y., Zhou, G., Zimmermann, A.: Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture. Math. Z. (2011, in press)

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

    Article  MATH  Google Scholar 

  • Reiten I.: Stable equivalence of self-injective algebras. J. Algebra 40(1), 63–74 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Skowroński A.: Selfinjective algebras: finite and tame type. Contemp. Math. Am. Math. Soc. 406, 169–238 (2006)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou, G., Zimmermann, A.: Classifying tame blocks and related algebras up to stable equivalences of Morita type. J. Pure Appl. Algebra (2011) doi:10.1016/j.jpaa.2011.04.017

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Authors and Affiliations

  1. Ecole Polytechnique Féderale de Lausanne, SB IGAT CTG, Batîment MA B3 424, 1015, Lausanne, Switzerland

    Guodong Zhou

  2. 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. Guodong Zhou
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  2. Alexander Zimmermann
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Correspondence to Alexander Zimmermann.

Additional information

G. Zhou was supported by DFG SPP 1388. Part of this work was done during the visit of G. Zhou to the Univerisité de Picardie Jules Verne in March 2010. G. Zhou would like to thank A. Zimmermann for his kind hospitality.

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Zhou, G., Zimmermann, A. Auslander–Reiten conjecture for symmetric algebras of polynomial growth. Beitr Algebra Geom 53, 349–364 (2012). https://doi.org/10.1007/s13366-011-0036-8

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  • DOI: https://doi.org/10.1007/s13366-011-0036-8

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