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The title says mostly everything.

Suppose we have a quiver, maybe with relations and cycles. Is it known when the path algebra modulo relations is hereditary? Especially in the case that the path algebra is infinite dimensional.

I would appreciate any reference.

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If an associative algebra $A$ is finite dimensional, basic, connected and defined over an algebraically closed field $K$, then $A$ is hereditary if and only if it is isomorphic to the path algebra of a finite, connected and acyclic quiver (see Thm. VII.1.7 of Assem-Simson-Skowroński). However, this still leaves the more interesting cases of your question unanswered... – Matthew Pressland Aug 29 '13 at 15:27
up vote 2 down vote accepted

I assume that you want to unital algebras, so the quivers have only finitely many points. Now let kQ be the path algebra and I an ideal contained in J^2. Then kQ/I being hereditary implies I=0, see Lemma 4.2.1 in the representation theory book of Benson. Now kQ is always hereditary, that is mentioned in theorem 4.1.4 in that book and there it quotes a paper by Bergman: Modules over coproducts of rings. So the general answer might be the same as in the finite dimensional case: kQ/I is hereditary (for an admissible ideal I) for a quiver with finitely many points iff I=0

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