Idempotent reduction for the finitistic dimension conjecture

Abstract

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