Idempotent reduction for the finitistic dimension conjecture

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.