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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 $n \leq 8$.

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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 $\text{gldim} B+1$.

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  • $\begingroup$ This gives also a very nice and quick proof that the global dimension is bounded by $n-1$ when $n$ is the number of simples. See for example math.uni-bielefeld.de/~sek/select/Acyclic.pdf . $\endgroup$
    – Mare
    Sep 7, 2019 at 12:07

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