Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that means that all proj. indecomposable $A$-modules and all inj. indecomposable $A$-modules occur as direct summands of $M$.
For any indecomposable direct summand $N$ of $M$, denote the corresponding simple End$_A(M)$-module by $E_N$.
My question is:
Why is it enough to construct a proj. resolution with length $\leq 3$ for every simple module $E_N$ in order to prove that the global dimension of End$_A(M)$ is $\leq 3$?
Is there a general theorem which states that fact?
I would be very grateful for any hints and references concerning literature, respectively.
Thank you very much.