Let $A$ be finite dimensional $\mathbb{C}$ algebra. Suppose $A$ is representation finite.(i.e. $A$'s has a finite list of isomorphism classes of indecomposables.) Set $mod-A$ to be category of finitely generated module over $A$.
$\max\{pd(M)|pd(M)<\infty, M\in mod-A\}=\max\{pd(M)|M$ indecomposable $,pd(M)<\infty\}$
My reasoning is the following. Take $M\in mod-A$. Since $A$ is representation finite, we can write $M\cong\oplus_{i\leq s}N_i^{m_i}$ where $N_i$ are indecomposable. We can replace $M$ by $RHS$ of isomorphism instead and perform resolution against $RHS$. $M$'s projective dimension is determined by length of its minimal projective resolution which exists by $A$ Artinian $\mathbb{C}$ algebra. Minimal projective resolution of $M$ is determined by direct sum of those of minimal projective resolutions of $N_i$. Thus $pd(M)=max\{pd(N_i^{m_i})\}$. However, $pd(N_i^{m_i})=pd(N_i)$ by the same reasoning.
$\{pd(M)|pd(M)<\infty, M\in mod-A\}$
$=\max\{\max(pd(N_i^{m_i})|pd(N_i)<\infty,m_i\geq 0,N_i\in mod-A\}$
$=\max\{pd(N_i)|pd(N_i)<\infty,N_i$ indecomposable$\}$
Is this correct?