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 …