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.


1 Answer 1


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 $k$.

Still, it seems unknown whether there exists a counterample among quadratic connected algebras.

  • $\begingroup$ With quiver algebra I meant a finite dimensional algebra. I made this more precise in the question. Your algebra seems to be not finite dimensional. $\endgroup$
    – Mare
    Nov 25 at 1:17

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