On $\Delta$–good module categories without short cycles
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- by Bangming Deng and Bin Zhu PDF
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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.References
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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
- MR Author ID: 262817
- Email: bzhu@math.tsinghua.edu.cn
- Received by editor(s): September 21, 1998
- Received by editor(s) in revised form: March 31, 1999
- Published electronically: June 14, 2000
- Communicated by: Ken Goodearl
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 69-77
- MSC (2000): Primary 16G10, 16G60
- DOI: https://doi.org/10.1090/S0002-9939-00-05518-0
- MathSciNet review: 1694857
Dedicated: To our teacher Shaoxue Liu on the occasion of his 70th birthday