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2004 | 2 | 1 | 67-75
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The representation dimension of domestic weakly symmetric algebras

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EN
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EN
Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Strony
67-75
Opis fizyczny
Daty
wydano
2004-03-01
online
2004-03-01
Twórcy
Bibliografia
  • [1] M. Auslander: Representation dimension of artin algebras, Queen Mary College, Mathematics Notes, University of London, 1971, 1–179. Also in: Selected works of Maurice Auslander (eds. I. Reiten, S. Smalø and Ø. Solberg) Part I, Amer. Math. Soc., 1999, pp. 505–574.
  • [2] R. Bocian, T. Holm and A. Skowroński: “Derived equivalence classification of weakly symmetric algebras of Euclidean type”, Preprint, (2003).
  • [3] R. Bocian and A. Skowroński: “Symmetric special biserial algebras of Euclidean type”, Colloq. Math., Vol. 96, (2003), pp. 121–148. http://dx.doi.org/10.4064/cm96-1-11
  • [4] R. Bocian and A. Skowroński: “Weakly symmetric algebras of Euclidean type”, Preprint, (2003).
  • [5] R. Bocian and A. Skowroński: “Socle deformations of selfinjective algebras of Euclidean type”, Preprint, (2003).
  • [6] K. Erdmann, T. Holm, O. Iyama and J. Schröer: “Radical embeddings and representation dimension”, Advances Math., to appear. (arXiv:math.RT/0210362).
  • [7] D. Happel: Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, 1988.
  • [8] D. Happel and D. Vossieck: “Minimal algebras of infinite representation type with preprojective component”, Manuscr. Math., Vol. 42, (1983), pp. 221–243. http://dx.doi.org/10.1007/BF01169585
  • [9] T. Holm: “The representation dimension of Schur algebras: the tame case”, Preprint, (2003).
  • [10] T. Holm: “Representation dimension of some tame blocks of finite groups”, Algebra Colloquium, Vol. 10:3, (2003), pp. 275–284.
  • [11] K. Igusa and G. Todorov: “On the finitistic global dimension conjecture for artin algebras”, Preprint, (2002).
  • [12] O. Iyama: “Finiteness of representation dimension”, Proc. Amer. Math. Soc., Vol. 131, (2003), pp. 1011–1014. http://dx.doi.org/10.1090/S0002-9939-02-06616-9
  • [13] H. Krause: “Stable equivalence preserves representation type”, Comment. Math. Helv., Vol. 72, (1997), pp. 266–284. http://dx.doi.org/10.1007/s000140050016
  • [14] H. Krause and G. Zwara: “Stable equivalence and generic modules”, Bull. London Math. Soc., Vol. 32, (2000), pp. 615–618. http://dx.doi.org/10.1112/S0024609300007293
  • [15] H. Lenzing and A. Skowroński: “On selfinjective algebras of Euclidean type”, Colloq. Math., Vol. 79, (1999), pp. 71–76.
  • [16] J. Rickard: “Derived categories and stable equivalence”, J. Pure Appl. Algebra, Vol. 61, (1989), pp. 303–317. http://dx.doi.org/10.1016/0022-4049(89)90081-9
  • [17] C. M. Ringel: Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer Verlag, Berlin, 1984.
  • [18] A. Skowroński: “Selfinjective algebras of polynomial growth”, Math. Ann., Vol. 285, (1989), pp. 177–199. http://dx.doi.org/10.1007/BF01443513
  • [19] A. Skowroński and J. Waschbüsch: “Representation-finite biserial algebras”, J. Reine Angew. Math., Vol. 345, (1983), pp. 172–181.
  • [20] C. Xi: “Representation dimension and quasi-hereditary algebras”, Advances Math., Vol. 168, (2002), pp. 193–212. http://dx.doi.org/10.1006/aima.2001.2046
  • [21] B. Zimmermann Huisgen, The finitistic dimension conjectures-a tale of 3.5 decades. In: Facchini, Alberto (ed.) et al., Abelian groups and modules. Proceedings of the Padova conference, Padova, Italy, June 23–July 1, 1994. Kluwer (1995).
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475951
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