I am looking for an example of finite acyclic quiver path algebra with a simple module of infinite projective dimension.
I have considered the following. Consider quiver $1\to 2\to 3\to 4$. Form path algebra $kQ$ over algebraic closed field $k$. After a few line computation, it seems that all simple modules have finite projective dimension. So I have to conclude that all finitely generated modules must have finite projective dimension.(As any f.g module $M\supset MJac(KQ)\supset MJac(KQ)^2\supset\dots 0$ gives a filtration of simples and those filtration pastes the resolution where $Jac(KQ)$ denotes Jacobson radical, any f.g. $M$ has finite projective resolution.) And the computed resolution length for simples is bounded by $4$.
$\textbf{Q1:}$ Is it true that simple modules' projective dimension bounded by $n$ for type $A_n$? Furthermore, if that is the case, I would deduce every f.g. module $M$ say $dim_k(M)=l$, has projective dimension at most $ln$.
$\textbf{Q2:}$ How can I find a simple module of a finite acyclic quiver path algebra with infinite projective dimension?