Composition factors for indecomposable modules
HTML articles powered by AMS MathViewer
- by Dieter Happel PDF
- Proc. Amer. Math. Soc. 86 (1982), 29-31 Request permission
Abstract:
Let $k$ be a field and $A$ be a finite-dimensional algebra over $k$ having only a finite number of isomorphism classes of indecomposable $A$-modules. Let $M$, $N$ be two indecomposable $A$-modules. Then a homomorphism $f:M \to N$ is said to be irreducible if for every factorization $f = gh$, $g$ is split mono or $h$ is split epi [2]. The aim of this note is to give an elementary proof of the fact that the indecomposable $A$-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.References
- Maurice Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Lecture Notes in Pure Appl. Math., Vol. 37, Dekker, New York, 1978, pp. 1–244. MR 0480688
- 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 439881, DOI 10.1080/00927877708822180 R. Bautista and F. Larion, Auslander-Reiten quivers for certain algebras of finite representation type (to appear).
- Klaus Bongartz and Claus Michael Ringel, Representation-finite tree algebras, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 39–54. MR 654702
- Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 86 (1982), 29-31
- MSC: Primary 16A64; Secondary 16A46
- DOI: https://doi.org/10.1090/S0002-9939-1982-0663860-4
- MathSciNet review: 663860