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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.

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  • $\begingroup$ The surjection is from P(λ) to Δ(λ) rather than the other way around. $\endgroup$ Jul 9, 2016 at 1:14

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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.

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  • $\begingroup$ Thank you! That's exactly what I am looking for. I found the statement for $\mathrm{Ext}^1$ in Dlab-Ringel's "The module-theoretic approach to quasi-hereditary algebras" and thought the proof for all higher Ext should be similar, and this document is what I need. $\endgroup$
    – Aaron Chan
    Oct 28, 2011 at 10:21

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