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

de la Peña, J.; Takane, M.

doi:10.1090/S0002-9947-99-02137-6

On the number of terms in the middle of almost split sequences over tame algebras
HTML articles powered by AMS MathViewer

by J. A. de la Peña and M. Takane PDF

Trans. Amer. Math. Soc. 351 (1999), 3857-3868 Request permission

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

Ibrahim Assem and Andrzej Skowroński, Indecomposable modules over multicoil algebras, Math. Scand. 71 (1992), no. 1, 31–61. MR 1216102, DOI 10.7146/math.scand.a-12409
Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
Raymundo Bautista and Sheila Brenner, On the number of terms in the middle of an almost split sequence, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 1–8. MR 654699
R. Bautista and S. O. Smalø, Nonexistent cycles, Comm. Algebra 11 (1983), no. 16, 1755–1767. MR 703234, DOI 10.1080/00927878308822931
Vlastimil Dlab and Claus Michael Ringel, Eigenvalues of Coxeter transformations and the Gel′fand-Kirillov dimension of the preprojective algebras, Proc. Amer. Math. Soc. 83 (1981), no. 2, 228–232. MR 624903, DOI 10.1090/S0002-9939-1981-0624903-6
I. R. Shafarevich (ed.), Algebra. VIII, Encyclopaedia of Mathematical Sciences, vol. 73, Springer-Verlag, Berlin, 1992. Representations of finite-dimensional algebras; A translation of Algebra, VIII (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow; Translation edited by A. I. Kostrikin and I. R. Shafarevich. MR 1239446
Hans-Joachim von Höhne, On weakly positive unit forms, Comment. Math. Helv. 63 (1988), no. 2, 312–336. MR 948786, DOI 10.1007/BF02566771
Otto Kerner, Tilting wild algebras, J. London Math. Soc. (2) 39 (1989), no. 1, 29–47. MR 989917, DOI 10.1112/jlms/s2-39.1.29
Shi Ping Liu, Almost split sequences for nonregular modules, Fund. Math. 143 (1993), no. 2, 183–190. MR 1240634, DOI 10.4064/fm-143-2-183-190
Shi Ping Liu, Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. (2) 47 (1993), no. 3, 405–416. MR 1214905, DOI 10.1112/jlms/s2-47.3.405
J. A. de la Peña, Representation-finite algebras whose Auslander-Reiten quiver is planar, J. London Math. Soc. (2) 32 (1985), no. 1, 62–74. MR 813386, DOI 10.1112/jlms/s2-32.1.62
—, Quadratic forms and the representation type of an algebra, Sonderforschungsbereich Diskrete Strukturen in der Mathematik. Ergänzungsreihe, vol. 003, Bielefeld, 1990.
J. A. de la Peña, Functors preserving tameness, Fund. Math. 137 (1991), no. 3, 177–185. MR 1110031, DOI 10.4064/fm-137-3-177-185
J. A. de la Peña and M. Takane, Spectral properties of Coxeter transformations and applications, Arch. Math. (Basel) 55 (1990), no. 2, 120–134. MR 1064377, DOI 10.1007/BF01189130
J. A. de la Peña and M. Takane, Constructing the directing components of an algebra, Colloq. Math. 74 (1997), no. 1, 29–46. MR 1455454, DOI 10.4064/cm-74-1-29-46
Zygmunt Pogorzały and Andrzej Skowroński, On algebras whose indecomposable modules are multiplicity-free, Proc. London Math. Soc. (3) 47 (1983), no. 3, 463–479. MR 716798, DOI 10.1112/plms/s3-47.3.463
Claus Michael Ringel, The spectral radius of the Coxeter transformations for a generalized Cartan matrix, Math. Ann. 300 (1994), no. 2, 331–339. MR 1299066, DOI 10.1007/BF01450490
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
A. Skowroński, Tame algebras with simply connected Galois covering, Compositio Math. (to appear).
Martha Takane, The Coxeter transformations of representation infinite quivers, Representation theory of algebras and related topics (Mexico City, 1994) CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 349–371. MR 1388569