Given a quiver algebra $A=KQ/I$ with acyclic $Q$. Then $A$ has finite global dimension, lets say $g$.



> 
> Question: Is there for any $0 \leq i \leq g$ a simple module with projective dimension equal to $i$? Is this true when $Q$ is a linear oriented quiver of Dynkin type $A\_n$ ?
> 
> 
> 


I guess the first question is wrong, but the second one is true for $n \leq 8$.



Projective dimensions of simple modules in acyclic quiver algebras
This is true by induction on the number of vertices.


Let $v$ be a source vertex of $Q$, and $B$ the quiver algebra obtained by removing $v$. Assume that $B$ has simples of all projective dimensions up to $\text{gldim} B$. 


The simples for $B$ have the same projective dimensions as the corresponding simples for $A$, and the remaining simple for $A$ has projective dimension at most $\text{gldim} B+1$.


