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 $i$.



Homological characterisation of standardly stratified algebras using Ext
Such a characterization was already given in the paper where standardly stratified algebras were first defined:



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> Ágoston, István; Dlab, Vlastimil; Lukács, Erzsébet. Stratified algebras. C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 1, 22--28
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[Link](http://www.cs.elte.hu/~agoston/papers/stratified.pdf)


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 $1 \leq i,j \leq n$.


