Let A be a finite dimensional algebra and $S_1,S_2,...,S_n$ the simple $A$-modules and $P_1,..,P_n$ the indecomposable projective $A$-modules. For $i=1,...,n$, define the standard module $\Delta_i$ as the largest factor module of $P_i$ with no composition factors $S_j$ for a $j>i$. For $i=1,...,n$, define the proper standard module $\hat{\Delta_i}$ as the largest factor of $\Delta_i$ where $S_i$ occurs only once as a composition factor. Define the the costandard modules $\nabla_i$ and the proper costandard modules $\hat{\nabla_i}$ dually. Denote by $F(\Delta)$ the subcategory of all finitely generated $A$-modules filtered by standard modules. $A$ is called standardly stratified in case the regular module is contained in $F(\Delta)$. My question is the following: Is there a homological characterisation of standardly stratified algebras involving fintely many Ext-groups of combinations of the standand, proper standard, costandard or proper costandard modules? So it could look something like this: $A$ is standardly stratified iff $Ext^{2}(\Delta_i, \hat{\nabla_j})=0$ for any $i,j$ + maybe some other conditions. Here is the motivation: A standardly stratified algebra is quasi-hereditary iff it has finite global dimension. There is the following homological characterisation of quasi-hereditary algebras: $A$ is quasi-hereditary iff $Ext^{2}(\Delta_i,\nabla_j)=0$ for any $i,j$ and $End_(\Delta_i)$ is a division ring for every $i$.
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Such a characterization was already given in the paper where standardly stratified algebras were first defined:
Ágoston, István; Dlab, Vlastimil; Lukács, Erzsébet. Stratified algebras. C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 1, 22--28
According to Theorem 3.1, the algebra $A$ is standardly stratified if and only if $$\operatorname{Ext}^{2}_A(\hat \Delta_i, \nabla_j)=0$$ for all $1 \leq i,j \leq n$.
answered Dec 22, 2015 at 1:44
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1$\begingroup$ @Mare Something seems a bit strange, though, so you should check the details of the proof. Later papers claim that for standardly stratified algebras, $\operatorname{Ext}^{2}_A(\Delta_i,\hat \nabla_j)=0$, but the standardly stratified property is not supposed to be left-right symmetric. $\endgroup$ Commented Dec 22, 2015 at 9:11
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1$\begingroup$ Actually the explanation could be that some papers consider left modules, while other papers consider right modules. $\endgroup$ Commented Dec 22, 2015 at 9:16
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1$\begingroup$ For instance Theorem 2 in www2.math.uu.se/research/pub/Frisk3.pdf $\endgroup$ Commented Dec 22, 2015 at 9:18
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$\begingroup$ thanks again. luckily im mainly interested in properly stratified algebras, which should be left-right symmetric. $\endgroup$– MareCommented Dec 22, 2015 at 10:22