The Weak Finitistic Dimension of a Path Algebra is Finite

Abstract

We prove a version of Bass' finitistic dimension conjecture for path algebras over arbitrary directed graphs. It is known that the path algebra of a finite directed graph is hereditary, hence it has finite finitistic dimension, when the graph is acyclic. The case for arbitrary directed graphs is still open. We use flat dimension instead of projective dimension (hence the designation "weak") and show that the weak finitistic dimension of an arbitrary path algebra is finite.

This paper has been withdrawn by the authors. A stronger result is already known. All path algebras are hereditary regardless of existence of oriented cycles. See Crawley-Boevey "Lectures on Representations of Quivers" (Page 8, Consequences (2))