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

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.