Definitions
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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:


1. *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 $M$,
2. $L\_{\lambda}$ has multiplicity one in $\Delta(\lambda)$, for all $\lambda \in \Lambda$;
3. $P\_{\lambda}$ 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
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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?



Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra
Let $M$ be a module with a $\Delta$-filtration. Let $N$ be a $\Delta$-semisimple submodule of M such that $M/N$ still has a $\Delta$-filtration -- I prove that there is a unique maximal module with this property (and hope that my justification is not immensely stupid).


This will be a consequence of the following Lemma.


**Lemma** Let $M\_1$ and $M\_2$ be submodules of $M$, with $M\_i$ isomorphic to a standard module and such that $M/M\_i$ has a $\Delta$-filtration, $i=1,2$. Then either $M\_1=M\_2$ or $M\_1 \cap M\_2 =0$.


**Proof** Suppose $M\_1 \cap M\_2 \neq 0$ and let $M\_i \cong \Delta(x\_i)$, for some $x\_i \in \Phi$, $i=1,2$. Let $\iota\_i$ be the inclusion map of $M\_1 \cap M\_2$ on $M\_i$, and let $\tau\_i$ be the inclusion of $M\_i$ on $M$,
$$
\iota\_i : M\_1 \cap M\_2 \longrightarrow M\_i,
$$
$$
\tau\_i : M\_i \longrightarrow M.
$$


Consider the (right exact) functor $F:=D\circ \operatorname{Hom}\_A(-,T)$, where $D$ is the standard duality and $T$ the characteristic $A$-module. Denote $\operatorname{End}\_A(T)^{op}$ by $\mathcal{R}(A)$ (the Ringel dual of $A$) -- the functor $F$ induces an equivalence between $\mathcal{F}(\Delta)$ and $\mathcal{F}(\nabla ')$ (where $\nabla '$ is the set of costandard modules over $\mathcal{R}(A)$). 


The functor $F$ maps the monics $\tau\_i$ in $\operatorname{mod}A$ to monics $F(\tau\_i)$ in $\operatorname{mod} \mathcal{R}(A)$ -- this is because the cokernel of $\tau\_i$ is in $\mathcal{F}(\Delta)$. I claim that the functor $F$ maps the monics $\iota\_i$ in $\operatorname{mod}A$ to nonzero morphisms $F(\iota\_i)$ in $\operatorname{mod} \mathcal{R}(A)$. Let $\varepsilon$ be the projective cover of $M\_1 \cap M\_2$. The morphisms $\varepsilon \circ \iota\_i$ are nonzero, with domain and codomain in $\mathcal{F}(\Delta)$, thus $F(\varepsilon) \circ F(\iota\_i)$ is nonzero, and hence $F(\iota\_i) \neq 0$. 


Now the image of $F(\iota\_i)$ is contained in a module isomorphic to $\nabla '(x\_i)$. Because $F(\tau\_i)$ is monic, the image of $F(\tau\_i) \circ F(\iota\_i)$ has simple socle $L\_{x\_i}$. As
$$F(\tau\_1) \circ F(\iota\_1)=F(\tau\_1 \circ \iota\_1)=F(\tau\_2 \circ \iota\_2) =F(\tau\_2) \circ F(\iota\_2),$$
we must have $x\_1=x\_2=:x$.


We want to show that $M\_1$ and $M\_2$ coincide. Suppose (by contradiction) that $M\_1\neq M\_2$. Then the (simple) top $L\_x$ of $M\_1$ is a composition factor of $M/ M\_2$. Indeed, there must be some subquotient of $M/ M\_2$, say $M\_3$, satisfying: $M\_3$ is part of a $\Delta$-filtration of $M/M\_2$; $M\_3$ is isomorphic to $\Delta(y)$ (for some $y \in \Phi$); $M\_3$ contains the top of $M\_1$ as a composition factor. Note that we must have $y \geq x$ in $\Phi$. In fact, by the definition of standard module and by the structure of the module $M$, the module $M\_1$ (which is isomorphic to $\Delta(x)$) would have to be contained in $M\_3\cong \Delta(y)$. On the other hand, we are assuming that $M\_3$ is a subquotient of $M/ M\_2$ and that $M\_1\cap M\_2\neq 0$.


