Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the number of terms in the middle
of almost split sequences over tame algebras


Authors: J. A. de la Peña and M. Takane
Journal: Trans. Amer. Math. Soc. 351 (1999), 3857-3868
MSC (1991): Primary 16G60, 16G70
DOI: https://doi.org/10.1090/S0002-9947-99-02137-6
Published electronically: April 20, 1999
MathSciNet review: 1467463
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $A$ be a finite dimensional tame algebra over an algebraically closed field $k$. It has been conjectured that any almost split sequence $0 \to X \to \oplus _{i=1} ^n Y_i \to Z \to 0$ with $Y_i$ indecomposable modules has $n \le 5$ and in case $n=5$, then exactly one of the $Y_i$ is a projective-injective module. In this work we show this conjecture in case all the $Y_i$ are directing modules, that is, there are no cycles of non-zero, non-iso maps $Y_i =M_1 \to M_2 \to \cdots \to M_s=Y_i$ between indecomposable $A$-modules. In case, $Y_1$ and $Y_2$ are isomorphic, we show that $n \le 3$ and give precise information on the structure of $A$.


References [Enhancements On Off] (What's this?)

  • 1. I. Assem and A. Skowronski, Indecomposable modules over multicoil algebras, Math. Scand. 71 (1992), 31-61. MR 94c:16020
  • 2. M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36 (1994). MR 98e:16011
  • 3. R. Bautista and S. Brenner, On the number of terms in the middle of an almost split sequence, In Lecture Notes in Mathematics, vol. 903, Springer, (1981), pp. 3857-3868. MR 83f:16034
  • 4. R. Bautista and S. Smalø, Non-existing cycles, Comm. Algebra 11 (1983), 1755-1767. MR 85d:16010
  • 5. V. Dlab and C. M. Ringel, Eigenvalues of Coxeter transformations and the Gelfand-Kirillov dimension of preprojective algebras, Proc. Amer. Math. Soc. 83 (2) (1981), 228-232. MR 83c:15007
  • 6. P. Gabriel and A. V. Roiter, Representations of finite-dimensional algebras, Algebra VIII, Encyclopaedia of Math. Sc., vol. 73, Springer, 1992. MR 94h:16001b
  • 7. H. J. von Höhne, On weakly positive unit forms, Comment. Math. Helvetici 63 (1988), 312-336. MR 89g:15033
  • 8. O. Kerner, Tilting wild algebras, J. London Math. Soc. Math. Soc. 39 (1989), 29-47. MR 90d:16025
  • 9. S. Liu, Almost split sequences for non-regular modules, Fundamenta Mathematicae 143 (1993), 183-190. MR 94g:16018
  • 10. -, Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. 47 (1993), 405-416. MR 94a:16024
  • 11. J. A. de la Peña, Algebras whose Auslander-Reiten quiver is planar, J. London Math. Soc. (2) 32 (1985), 62-74. MR 87g:16043
  • 12. -, Quadratic forms and the representation type of an algebra, Sonderforschungsbereich Diskrete Strukturen in der Mathematik. Ergänzungsreihe, vol. 003, Bielefeld, 1990.
  • 13. -, Functors preserving tameness, Fundamenta Math. 137 (1991), 177-185. MR 92h:16013
  • 14. J. A. de la Peña and M. Takane, Spectral properties of Coxeter transformations and applications, Arch. Math. 55 (1990), 120-134. MR 91k:16015
  • 15. -, Constructing the directing components of an algebra, Colloquium Math. 74 (1), (1997), 29-46. MR 98d:16026
  • 16. K. Pogorzaly and A. Skowronski, On algebras whose indecomposable modules are multiplicity free, Proc. London Math. Soc. 47 (1983), 463-479. MR 85a:16034
  • 17. C. M. Ringel., The spectral radius of the Coxeter transformation for a generalized Cartan matrix, Math. Ann. 300 (1994), 331-339. MR 95i:17004
  • 18. -, Tame algebras and integral quadratic forms, LNM, vol. 1099, Springer, Berlin, 1984. MR 87f:16027
  • 19. A. Skowronski, Tame algebras with simply connected Galois covering, Compositio Math. (to appear).
  • 20. M. Takane, The Coxeter transformations of representation infinite quivers, In Proceedings Workshop UNAM 1994, CMS Conference Proceedings 19 (1996), 349-371. MR 97d:16021

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 16G60, 16G70

Retrieve articles in all journals with MSC (1991): 16G60, 16G70


Additional Information

J. A. de la Peña
Affiliation: Instituto de Matemáticas, UNAM Ciudad Universitaria 04510 México, D. F. México
Email: jap@penelope.matem.unam.mx

M. Takane
Affiliation: Instituto de Matemáticas, UNAM Ciudad Universitaria 04510 México, D. F. México
Email: takane@gauss.matem.unam.mx

DOI: https://doi.org/10.1090/S0002-9947-99-02137-6
Received by editor(s): August 22, 1996
Received by editor(s) in revised form: April 25, 1997
Published electronically: April 20, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society