Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Regular Auslander-Reiten components containing directing modules


Author: Andrzej Skowroński
Journal: Proc. Amer. Math. Soc. 120 (1994), 19-26
MSC: Primary 16G70; Secondary 16D80, 16G10
MathSciNet review: 1156473
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe completely all regular Auslander-Reiten components of artin algebras containing directing modules. We prove also that the Auslander-Reiten quiver of an artin algebra admits at most finitely many DTr-orbits containing directing modules.


References [Enhancements On Off] (What's this?)

  • [1] M. Auslander and I. Reiten, Representation theory of artin algebras. III; IV, Comm. Algebra 3 (1975), 239-294; 5 (1977), 443-518. MR 0439881 (55:12762)
  • [2] R. Bautista and S. O. Smalø, Nonexistent cycles, Comm. Algebra 11 (1983), 1755-1767. MR 703234 (85d:16010)
  • [3] K. Bongartz, Tilted algebras, Representations of Algebras, Lecture Notes in Math., vol. 903, Springer, Berlin, 1981, pp. 26-38. MR 654701 (83g:16053)
  • [4] D. Happel, U. Preiser, and C. M. Ringel, Vinberg's characterization of Dynkin diagrams using subadditive functions with applications to DTr-periodic modules, Representation Theory. II, Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 280-294. MR 607159 (82g:16027)
  • [5] S. Liu, The degrees of irreducible maps and shapes of Auslander-Reiten quivers, J. London Math. Soc. (2) 45 (1992), 32-54. MR 1157550 (93f:16015)
  • [6] -, Semi-stable components of an Auslander-Reiten quiver, Preprint No. 12/91, University of Trondheim.
  • [7] I. Reiten, S. O. Smalø, and A. Skowroński, Short chains and short cycles of indecomposable modules, Proc. Amer. Math. Soc. 117 (1993), 343-354. MR 1136238 (93d:16013)
  • [8] -, Short chains and regular components, Proc. Amer. Math. Soc. 117 (1993), 601-612. MR 1124149 (93d:16018)
  • [9] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer, Berlin, 1984. MR 774589 (87f:16027)
  • [10] -, Representation theory of finite-dimensional algebras, Representations of Algebras (Proceedings, Durham 1985) Cambridge Univ. Press, London and New York, 1986, pp. 7-80. MR 897319 (89c:16030)
  • [11] -, The regular components of the Auslander-Reiten quiver of a tilted algebra, Chinese Ann. Math. Ser. B 1 (1988), 1-18. MR 943675 (89e:16036)
  • [12] A. Skowroński, Cycles of indecomposable modules, Representation Theory of Finite Groups and Finite Dimensional Algebras (Proc. Res. Inst. Math. Sci., Kyoto 1991) J. Math. Soc. Japan (to appear).
  • [13] -, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan (to appear). MR 1276836 (95d:16022)
  • [14] A. Skowroński and S. O Smalø, Directing modules, J. Algebra 147 (1992), 137-146. MR 1154679 (93a:16011)
  • [15] Y. Zhang, The structure of regular components, Canad. J. Math. 43 (1991), 652-672. MR 1118014 (92f:16017)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16G70, 16D80, 16G10

Retrieve articles in all journals with MSC: 16G70, 16D80, 16G10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1156473-X
Article copyright: © Copyright 1994 American Mathematical Society