I am reading "Elements of the representation theory of associative algebras"'s book of Skowronski, Simson and Assem. I want to compute the global dimension of the example 2.5 c), of chapter 3, page 80.
By Auslander's theorem, I only need to compute the projective dimension of the simple modules.
I want to know how to continue with the resolution. I already know it has infinite global dimension because of the Cartan matrix.