On indecomposable modules over directed algebras
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- by Peter Dräxler PDF
- Proc. Amer. Math. Soc. 112 (1991), 321-327 Request permission
Abstract:
Generalizing a result of Bongartz we show that any nonsimple indecomposable module over a finite-dimensional $k$-algebra $A$ is an extension of an indecomposable and a simple module provided $k$ is a field with more than two elements and $A$ is representation directed. Our proof is based on fibre sums over simple modules and some known classification results on socle projective modules over peak algebras. In case the global dimension of $A$ is at most 2 our methods also yield a description of the dimension vectors of the indecomposable $A$-modules by the roots of the associated quadratic form.References
- Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. III. Almost split sequences, Comm. Algebra 3 (1975), 239–294. MR 379599, DOI 10.1080/00927877508822046
- Klaus Bongartz, Algebras and quadratic forms, J. London Math. Soc. (2) 28 (1983), no. 3, 461–469. MR 724715, DOI 10.1112/jlms/s2-28.3.461
- Klaus Bongartz, Indecomposables over representation-finite algebras are extensions of an indecomposable and a simple, Math. Z. 187 (1984), no. 1, 75–80. MR 753421, DOI 10.1007/BF01163167
- K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
- Dieter Bünermann, Hereditary torsion theories and Auslander-Reiten sequences, Arch. Math. (Basel) 41 (1983), no. 4, 304–308. MR 731602, DOI 10.1007/BF01371402
- Vlastimil Dlab and Claus Michael Ringel, On algebras of finite representation type, J. Algebra 33 (1975), 306–394. MR 357506, DOI 10.1016/0021-8693(75)90125-8
- Vlastimil Dlab and Claus Michael Ringel, A module theoretical interpretation of properties of the root system, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978) Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, 1979, pp. 435–451. MR 563306
- Peter Dräxler, Eine Bemerkung zur Eulercharakteristik von Algebren, Comm. Algebra 14 (1986), no. 4, 707–716 (German). MR 830815, DOI 10.1080/00927878608823330
- Peter Dräxler, ${\mathfrak {U}}$-Fasersummen in darstellungsendlichen Algebren, J. Algebra 113 (1988), no. 2, 430–437 (German). MR 929771, DOI 10.1016/0021-8693(88)90170-6
- Peter Dräxler, ${\mathfrak {U}}$-Fasersummen in artinschen Ringen, J. Algebra 123 (1989), no. 2, 496–499 (German). MR 1000499, DOI 10.1016/0021-8693(89)90058-6
- Peter Dräxler, Fasersummen über dünnen $s$-Startmoduln, Arch. Math. (Basel) 54 (1990), no. 3, 252–257 (German). MR 1037614, DOI 10.1007/BF01188520
- Peter Gabriel, Auslander-Reiten sequences and representation-finite algebras, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin, 1980, pp. 1–71. MR 607140
- Dieter Happel, Composition factors for indecomposable modules, Proc. Amer. Math. Soc. 86 (1982), no. 1, 29–31. MR 663860, DOI 10.1090/S0002-9939-1982-0663860-4 M. M. Kleiner, On the exact representations of partially ordered sets of finite type, J. Soviet Math. 23 (1975), 607-615.
- Bogumiła Klemp and Daniel Simson, Schurian sp-representation-finite right peak PI-rings and their indecomposable socle projective modules, J. Algebra 134 (1990), no. 2, 390–468. MR 1074337, DOI 10.1016/0021-8693(90)90061-R
- Wolfgang Müller, Verallgemeinerte Fasersummen von unzerlegbaren Moduln, Math. Ann. 255 (1981), no. 4, 549–564 (German). MR 618184, DOI 10.1007/BF01451933
- Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
- Claus Michael Ringel, Hall polynomials for the representation-finite hereditary algebras, Adv. Math. 84 (1990), no. 2, 137–178. MR 1080975, DOI 10.1016/0001-8708(90)90043-M
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 112 (1991), 321-327
- MSC: Primary 16G60; Secondary 16D60
- DOI: https://doi.org/10.1090/S0002-9939-1991-1062830-X
- MathSciNet review: 1062830