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

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