Convex subquivers and the finitistic dimension

Let $\text{cQ}$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\text{cQ}$ and quotients of the path algebra $K\text{cQ}$. We introduce the homological heart of $\text{cQ}$ which is a particularly nice convex subquiver of $\text{cQ}$. For any algebra of the form $K\text{cQ}\mathrm{/}I$, the algebra associated to $K\text{cQ}\mathrm{/}I$ 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.