Composition factors for indecomposable modules
Author:
Dieter Happel
Journal:
Proc. Amer. Math. Soc. 86 (1982), 2931
MSC:
Primary 16A64; Secondary 16A46
MathSciNet review:
663860
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Abstract: Let be a field and be a finitedimensional algebra over having only a finite number of isomorphism classes of indecomposable modules. Let , be two indecomposable modules. Then a homomorphism is said to be irreducible if for every factorization , is split mono or is split epi [2]. The aim of this note is to give an elementary proof of the fact that the indecomposable modules are completely determined, up to isomorphism, by their composition factors if there is no chain of irreducible maps from an indecomposable module to itself. This theorem was first proved in [5] involving the theory of tilted algebras.
 [1]
Maurice
Auslander, Functors and morphisms determined by objects,
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Pa., 1976), Dekker, New York, 1978, pp. 1–244. Lecture Notes in
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(58 #844)
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Maurice
Auslander and Idun
Reiten, Representation theory of Artin algebras. IV. Invariants
given by almost split sequences, Comm. Algebra 5
(1977), no. 5, 443–518. MR 0439881
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R. Bautista and F. Larion, AuslanderReiten quivers for certain algebras of finite representation type (to appear).
 [4]
Klaus
Bongartz and Claus
Michael Ringel, Representationfinite tree algebras,
Representations of algebras (Puebla, 1980) Lecture Notes in Math.,
vol. 903, Springer, Berlin, 1981, pp. 39–54. MR 654702
(83g:16054)
 [5]
Dieter
Happel and Claus
Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063
(84d:16027), http://dx.doi.org/10.1090/S00029947198206750632
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 M. Auslander, Functors and morphisms determined by objects, Proc. Conf. on Representation Theory (Philadelphia 1976), Dekker, New York, 1978, pp. 1244. MR 0480688 (58:844)
 [2]
 M. Auslander and I. Reiten, Representation theory of Artin algebras. III, VI, Comm. Algebra 3 (1975), 239294; 5 (1977), 443518. MR 0439881 (55:12762)
 [3]
 R. Bautista and F. Larion, AuslanderReiten quivers for certain algebras of finite representation type (to appear).
 [4]
 K. Bongartz and C. M. Ringel, Representationfinite tree algebras (to appear). MR 654702 (83g:16054)
 [5]
 D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. (to appear). MR 675063 (84d:16027)
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DOI:
http://dx.doi.org/10.1090/S00029939198206638604
PII:
S 00029939(1982)06638604
Article copyright:
© Copyright 1982 American Mathematical Society
