# Convex subquivers and the finitistic dimension

## TL;DR

This paper investigates the homological properties of algebras derived from convex subquivers of a quiver, introduces the homological heart concept, and applies these ideas to the finitistic dimension conjecture, reducing the conjecture's scope.

## Contribution

It introduces the homological heart of a quiver and shows its relevance in relating homological properties of associated algebras, simplifying the finitistic dimension conjecture.

## Key findings

- Homological properties of algebras from convex subquivers are interconnected.
- The homological heart of a quiver is a key convex subquiver with significant properties.
- The finitistic dimension conjecture can be reduced to algebras with path connected quivers.

## Abstract

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