# Bounding the projective dimension of modules by the number of points and arrows

Let $$A=KQ/I$$ be a finite dimensional connected quiver algebra with admissible ideal $$I$$ with n points and m arrows and $$M$$ an $$A$$-module.

Question: Is the projective dimension of $$M$$ bounded by $$n+m$$ if it is finite?

Of course it would be surprising to see a proof, but maybe there is a simple counterexample.

A counterexample is the algebra $$k\langle x,y| xyxy^2, xy^2xy^3, \dots ,xy^{k-1} xy^{k}\rangle$$. It is connected ($$n=1$$) with $$m=2$$ generators and have global dimension $$k$$. So, the trivial module $$M=k$$ has projective dimension $$k$$.