Given a finite dimensional algebra $A$ and a generator-cogenerator $M$ and let $B:=End_A(M)$. $B$ has finite global dimension iff every $A$-module has finite $add(M)$-resolution dimension, which is usually hard to check.

But $B$ also has finite global dimension iff every simple module $S_i$ has finite projective dimension iff $\Omega^2(S_i)$ has finite projective dimension.

Now $\Omega^2(S)$ is in the image of the functor $F:=Hom_A(M,-)$.

Can one somehow find $A$-modules $N_i$ such that $F(N_i)=\Omega^2(S_i)$? Here I mean a more or less explicit description in terms of $M$. Having those $N_i$, $B$ has finite global dimesnion iff all $N_i$ have finite $add(M)$-resolution dimension.