$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k with exactly m simple modules up to isomorphism. Let S be a simple left A-module. Suppose the projective dimension $\pd_A(S)$ of $S$ is finite and $\pd_A(S)\geq m$. Then for any nonnegative integer $j\leq \pd_A(S)-m$, we can always find m simple left $A$-modules $S_1, S_2 \cdots S_m$ such that $\Ext_A^{i+j}(S, S_i)\neq 0$ for $1\leq i \leq m$. My question is the following:
In the above situation, can we always choose simple $A$-modules $S_1, S_2 \cdots S_m$ such that there are at least two nonisomorphic ones among them? What if we suppose that A is indecomposable with at least 2 nonisomorphic simple modules and remove the finiteness of $\pd_A(S)$?
