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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
Published electronically: April 20, 1999
MathSciNet review: 1467463
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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 $A$.


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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: http://dx.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