Idempotent reduction for the finitistic dimension conjecture

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.