Convex subquivers and the finitistic dimension

Abstract

Let $\mathcal{Q}$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\mathcal{Q}$ and quotients of the path algebra $K\mathcal{Q}$. We introduce the homological heart of $\mathcal{Q}$ which is a particularly nice convex subquiver of $\mathcal{Q}$. For any algebra of the form $K\mathcal{Q}/I$, the algebra associated to $K\mathcal{Q}/I$ and the homological heart have similar homological properties. We give an application showing that the finitistic dimension conjecture need only be proved for algebras with path connected quivers.