3
$\begingroup$

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.

$\endgroup$

1 Answer 1

0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.