  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.
