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Module categories without short cycles are of finite type

Authors: Dieter Happel and Shi Ping Liu
Journal: Proc. Amer. Math. Soc. 120 (1994), 371-375
MSC: Primary 16D90; Secondary 16G10, 16G60
MathSciNet review: 1164144
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Abstract: Let $ A$ be an artin algebra. An indecomposable finitely generated $ A$-module $ X$ is said to be on a short cycle if there exists an indecomposable finitely generated $ A$-module $ Y$ and two nonzero noninvertible maps $ f:X \to Y$ and $ g:Y \to X$. If there are no short cycles we show that there exist only finitely many indecomposable $ A$-modules up to isomorphism.

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  • [1] M. Auslander, Applications of morphisms determined by objects, Representation Theory of Algebras, Proc. of the Philadelphia Conference (R. Gordan, ed.), Lecture Notes in Pure and Appl. Math., vol. 37, Dekker, New York, 1976, pp. 245-294.
  • [2] M. Auslander and I. Reiten, Representation theory of artin algebras. III, Comm. Algebra 5 (1975), 239-294. MR 0379599 (52:504)
  • [3] -, Modules determined by their composition factors, Illinois J. Math. 29 (1985), 280-301. MR 784524 (86i:16032)
  • [4] D. Happel, U. Preiser, and C. M. Ringel, Vinberg's characterization of Dynkin diagrams using subadditive functions with application to $ DTr$-periodic modules, Lecture Notes in Math., vol. 832, Springer, New York, 1980, pp. 280-294. MR 607159 (82g:16027)
  • [5] D. Happel and C. M. Ringel, Directing projective modules, Arch. Math. 60 (1993), 237-249. MR 1201637 (94b:16016)
  • [6] S. Liu, The degrees of irreducible maps and the shapes of Auslander-Reiten quivers, J. London Math. Soc. (2) 45 (1992), 32-54. MR 1157550 (93f:16015)
  • [7] I. Reiten, A. Skowronski, and S. O. Smalø, Short chains and short cycles of modules, Proc. Amer. Math. Soc. 117 (1993), 343-354. MR 1136238 (93d:16013)
  • [8] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer, New York, 1984. MR 774589 (87f:16027)

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