# Sum of all projective dimensions of simple modules

Let $$X_{n,t}$$ be the set of all finite dimensional algebras (we can assume they are given by a connected quiver and admissible relations) that have global dimension equal to $$n$$ and $$t$$ simple modules. For such an algebra $$A$$, define $$\phi(A):= \sum\limits_{k=1}^{t}{pd(S_k)}$$ as the sum of all projective dimension of the simple modules $$S_k$$ of $$A$$.

Question:

What is $$r_{n,t} :=max \{ \phi(A) | A \in X_{n,t} \}$$ and $$s_{n,t} :=min \{ \phi(A) | A \in X_{n,t} \}$$ for $$n \geq 1$$ and $$t \geq 2$$?

For $$n=1$$ we have $$r_{1,t}=t-1$$ and $$s_{1,t}=1$$.

Note that in general there is a simple module $$W$$ that is a submodule of a projective module and thus $$pd(W) \leq t-1$$, which shows that $$r_{n,t} \leq nt-1$$.

For $$n=2$$, we have that the Auslander algebra $$A$$ of $$K[x]/(x^t)$$ has global dimension 2 and $$t$$ simple modules with $$\phi(A)=2t-1$$, which shows that $$r_{2,t}=2t-1$$.

This leads to the question:

Do we have $$r_{n,t}=nt-1$$?