The global dimension of a $k$-algebra depends only on the simple modules

Question

I read about a theorem a while ago that said something along the lines that the global dimension of a finite dimensional, associative unitary $k$-algebra is the maximum of the projective dimension of its simple modules. I can't find a reference for this. Anyone can point me to it? Thanks!

Answer 1

In [McConnell, Robson: Noncommutative noetherian rings], 7.1.8, you can find a proof that

the global dimension of a ring is equal to the supremum of the projective dimensions of its cyclic modules.

If the ring is a finite dimensional algebra, then a cyclic module is of course also finite dimensional. Such a module has a finite composition series, and a little homological algebra involving long exact sequences for Ext and induction on the length of the composition series shows that

if all simple finite dimensional modules have finite projective dimension, then all cyclic modules have finite projective dimension.

Together with the previous observation, we see that

if all simple finite dimensional modules have finite projective dimension, then the global dimension of the algebra is finite.

Now Corollary 7.1.14 in that book tells us that if $A$ has finite global dimension, then that global dimension is equal to the supremum of the projective dimensions of the simples. On the other hand, if the global dimension is infinite, the last displayed result tells us that at least one simple has infinite projective dimension. In both cases, therefore, we have that

the global dimesion is the supremum of the projective dimensions of the simples.