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

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.