For a quasihereditary algebra $A$, we have a partially ordered set $\Lambda$ parameterizing the simples $L(\lambda)$/projectives indecomposables $P(\lambda)$. It also parameterize a set of special modules called the standard modules $\Delta(\lambda)$ such that $\Delta(\lambda) \to P(\lambda)$ is a surjection such that the kernel has composition factors $L(\mu)$ with $\mu > \lambda$ and $\Delta(\lambda)$ has composition factors with $L(\mu)$, for $\mu\leq \lambda$.


Now let $\Delta = \oplus\_{\lambda\in\Lambda} \Delta(\lambda)$. Apparently, the following Ext-algebra: $\text{Ext}\_A^\ast (\Delta,\Delta)$ is already known to be directed when $A$ is quasihereditary. Does anyone know the reference for this result and proof? Thanks.



Is Ext algebra of standard modules of quasihereditary algebras directed?
Everything you need is written here


<http://www.math.uni-bielefeld.de/~sek/select/K-L.pdf>


(see Lemma 3). I couldn't find the originial source, but it must be one of the references listed.


Let $B=\text{Ext}\_A^\ast (\Delta,\Delta)$ with idempotents $e\_i=\text{id}\_{\Delta(i)}$. Then from Lemma 3 it follows that
$e\_i B e\_j \simeq \text{Ext}\_A^\ast (\Delta(j),\Delta(i))=0$ for $i<j$ and also that
$e\_i B e\_i \cong k$. I suppose this is what you mean by the algebra being directed.


