The Weak Finitistic Dimension of a Path Algebra is Finite

TL;DR

This paper proves that the weak finitistic dimension of any path algebra over an arbitrary directed graph is finite, extending known results from acyclic graphs to all directed graphs using flat dimension.

Contribution

It establishes the finiteness of the weak finitistic dimension for path algebras over arbitrary directed graphs, a significant step towards Bass' finitistic dimension conjecture.

Findings

Weak finitistic dimension of path algebras is finite

Uses flat dimension instead of projective dimension

Extends results from acyclic to arbitrary directed graphs

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.