## Definitions

Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a partially ordered set $(\Lambda, \leq)$.

Denote the simple $A$-modules (up to isomorphism) by $L_{\lambda}$, $\lambda \in \Lambda$, and let $P_{\lambda}$ be the projective cover of $L_{\lambda}$.

For every $\lambda \in \Lambda$, denote by $\Delta(\lambda)$ the largest factor module of $P_{\lambda}$ all of whose composition factors are of the form $L_{\mu}$, $\mu \leq \lambda$. Call this module the *standard module* with weight $\lambda$.

We say that $A$ is *quasihereditary* with respect to the poset $(\Lambda, \leq)$ if:

*The poset $(\Lambda, \leq)$ is adapted to $B$*: for every finitely generated module $M$ with simple socle $L_{\lambda}$ and with simple top $L_{\mu}$, where $\mu$ and $\lambda$ uncomparable in $(\Lambda, \leq)$, there is $\nu \in \Lambda$, $\nu > \mu$ or $\nu > \lambda$, such that $L_{\nu}$ is a composition factor of $M$,$L_{\lambda}$ has multiplicity one in $\Delta(\lambda)$, for all $\lambda \in \Lambda$;

$P_{\lambda}$ has a

*$\Delta$-filtration*(i.e. a filtration whose factors are standard modules), for all $\lambda \in \Lambda$.

Finally, call a module *$\Delta$-semisimple* if it is a direct sum of standard modules.

## Questions

Let $A$ be quasihereditary and let $M$ be some finitely genereted $A$-module with a $\Delta$-filtration. I can convince myself that the $\Delta$-filtrations of $M$ are essentially unique (à la Jordan-Holder), up to possible permutation of the factors (I could not find this in literature though -- does anyone know a reference?).

Let $N$ be a $\Delta$-semisimple submodule of $M$ such that $M/N$ still has a $\Delta$-filtration -- I have the feeling that $M$ has a unique maximal submodule $N$ with this property. I have not been able to prove this nor found a reference. Does anyone know if this is true? And would this be "real" uniqueness or just uniqueness up to isomorphism?