I allready asked this on MO, but did not get any answer.


**Given a finite quiver with relations. When is the path algebra modulo relations hereditary?**


If the path algebra is finite dimensional or there are no relations, the answer is well known. **What happens if the path algebra is infinite dimensional and there are relations?**


I know that the question is everything but precise, however I would appreciate any reference dealing with this topic.



When are infinite dimensional path algebras hereditary?
The algebra is $A=kQ/I$ with the assumptions $I \subseteq J^2$ and $I/(JI+IJ) \neq 0$, where $J$ is the ideal in $kQ$ generated by the arrows.


Assume $A$ is hereditary. Then $J/I$ is a projective $A$-module and the exact sequence of $A$-modules $$0 \rightarrow I/IJ \rightarrow J/IJ \rightarrow J/I \rightarrow 0$$ must split. Let $s \colon J/IJ \rightarrow I/IJ$ denote a retraction.


Let $x$ be an element in $I \smallsetminus (JI+IJ)$. Since $I \subseteq J^2$, we can write $$x=p\_1 q\_1 + \cdots + p\_n q\_n$$ with $p\_i,q\_i \in J$ for all $1 \leq i \leq n$. Then $$x+IJ=s(p\_1 q\_1 + \cdots + p\_n q\_n+IJ)=p\_1 s(q\_1) + \cdots + p\_n s(q\_n) +IJ \in JI+IJ,$$ a contradiction.


Therefore $A$ is not hereditary.


