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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On $\Delta $-good module categories without short cycles


Authors: Bangming Deng and Bin Zhu
Journal: Proc. Amer. Math. Soc. 129 (2001), 69-77
MSC (2000): Primary 16G10, 16G60
Published electronically: June 14, 2000
MathSciNet review: 1694857
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Abstract: Let $A$ be a quasi-hereditary algebra, and ${\mathcal{F}}(\Delta )$ the $\Delta $-good module category consisting of $A$-modules which have a filtration by standard modules. An indecomposable module $M$ in ${\mathcal{F}}(\Delta )$ is said to be on a short cycle in ${\mathcal{F}}(\Delta )$ if there exist an indecomposable module $N$ in ${\mathcal{F}}(\Delta )$ and a chain of two nonzero noninvertible maps $M\rightarrow N\rightarrow M$. It is shown that two indecomposable modules in ${\mathcal{F}}(\Delta )$ are isomorphic if they are not on short cycles in ${\mathcal{F}}(\Delta )$ and have the same composition factors. Moreover, if there is no short cycle in ${\mathcal{F}}(\Delta )$, we show that ${\mathcal{F}}(\Delta )$ is finite, that is, there are only finitely many isomorphism classes of indecomposables in ${\mathcal{F}}(\Delta )$. This is an analogue to a result in a complete module category proved by Happel and Liu.


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Additional Information

Bangming Deng
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China
Email: dengbm@bnu.edu.cn

Bin Zhu
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China
Address at time of publication: Department of Mathematics, Tsinghua University, 100084 Beijing, People’s Republic of China
Email: bzhu@math.tsinghua.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05518-0
PII: S 0002-9939(00)05518-0
Received by editor(s): September 21, 1998
Received by editor(s) in revised form: March 31, 1999
Published electronically: June 14, 2000
Dedicated: To our teacher Shaoxue Liu on the occasion of his 70th birthday
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society