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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  be an artin algebra. An indecomposable finitely generated  -module  is said to be on a short cycle if there exists an indecomposable finitely generated  -module  and two nonzero noninvertible maps  and  . If there are no short cycles we show that there exist only finitely many indecomposable  -modules up to isomorphism.
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Dieter
Happel, Udo
Preiser, and Claus
Michael Ringel, Vinberg’s characterization of Dynkin diagrams
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II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979),
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(82g:16027)
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Dieter
Happel and Claus
Michael Ringel, Directing projective modules, Arch. Math.
(Basel) 60 (1993), no. 3, 237–246. MR 1201637
(94b:16016), http://dx.doi.org/10.1007/BF01198807
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Shi
Ping Liu, Degrees of irreducible maps and the shapes of
Auslander-Reiten quivers, J. London Math. Soc. (2) 45
(1992), no. 1, 32–54. MR 1157550
(93f:16015), http://dx.doi.org/10.1112/jlms/s2-45.1.32
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Reiten, A.
Skowroński, and SmaløS.
O., Short chains and short cycles of
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(93d:16013), http://dx.doi.org/10.1090/S0002-9939-1993-1136238-4
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Claus
Michael Ringel, Tame algebras and integral quadratic forms,
Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin,
1984. MR
774589 (87f:16027)
- [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
 -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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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1994-1164144-9
PII:
S 0002-9939(1994)1164144-9
Article copyright:
© Copyright 1994 American Mathematical Society
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