Idempotent reduction for the finitistic dimension conjecture

TL;DR

This paper proves that the finiteness of the finitistic dimension for certain Artin algebras can be reduced to smaller algebras via idempotent elements, providing new insights into the finitistic dimension conjecture.

Contribution

It introduces an idempotent reduction technique for the finitistic dimension conjecture in Artin algebras, extending previous results and offering new criteria for finiteness.

Findings

Finiteness of finitistic dimension of $\Lambda$ implies finiteness of a related subalgebra.

Recovered a known result relating finitistic dimensions of quotient algebras.

Provided conditions under which finitistic dimension finiteness is preserved under algebra reduction.

Abstract

In this note, we prove that if $\Lambda$ is an Artin algebra with a simple module $S$ of finite projective dimension, then the finiteness of the finitistic dimension of $\Lambda$ implies that of $(1-e)\Lambda(1-e)$ where $e$ is the primitive idempotent supporting $S$. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if $\Lambda$ is the quotient of a path algebra by an admissible ideal $I$ whose defining relations do not involve a certain arrow $\alpha$, then the finitistic dimension of $\Lambda$ is finite if and only if the finitistic dimension of $\Lambda/\Lambda\alpha \Lambda$ is finite.