# Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$

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.

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