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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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1 Answer 1

up vote 3 down vote accepted

The global dimension of a noetherian ring with finite global dimension is equal to the supremum of the projective dimensions of its simple modules. This is proved in most textbooks dealing with the subject. For example, this is proved in McConnell and Robson's Noncommutative Noetherian rings (Corollary 7.1.14)

If the ring is semiprimary (that is, if its Jacobson ideal is nilpotent and the corresponding quotient semisimple) then you can drop the hypothesis that the globaldimension be finite. This covers your case. You can find this theorem of Auslander in Lam's Lectures on modules and rings.

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