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)

 
 

 

Short chains and short cycles of modules


Authors: I. Reiten, A. Skowroński and S. O. Smalø
Journal: Proc. Amer. Math. Soc. 117 (1993), 343-354
MSC: Primary 16G10; Secondary 16G70
DOI: https://doi.org/10.1090/S0002-9939-1993-1136238-4
MathSciNet review: 1136238
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module $ M$ is not the middle of a short chain if and only if there is no short cycle $ M \to N \to M$ of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable $ \Lambda $-module that is not the middle of a short chain, then $ \Lambda $ is a tilted algebra.


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

  • [1] M. Auslander and I. Reiten, Modules determined by their composition factors, Illinois J. Math. 29 (1985), 280-301. MR 784524 (86i:16032)
  • [2] Ø. Bakke, Some characterizations of tilted algebras, Math. Scand. 63 (1988), 43-50. MR 994969 (90j:16044)
  • [3] Ø. Bakke and S. O. Smalø, Modules with the same socles and tops as directing module are isomorphic, Comm. Algebra 15 (1987), 1-9. MR 876969 (88c:16024)
  • [4] R. Bautista and S. O. Smalø, Nonexistent cycles, Comm. Algebra 11 (1983), 1755-1767. MR 703234 (85d:16010)
  • [5] K. Bongartz, On a result of Bautista and Smaløon cycles, Comm. Algebra 11 (1983), 2123-2124. MR 710213 (85b:16012)
  • [6] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer-Verlag, Berlin and New York, 1984. MR 774589 (87f:16027)

Similar Articles

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

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


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1136238-4
Keywords: Short chains, short cycles, tilted algebras
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society