Excellent extensions and homological conjectures

Abstract

In this paper, we introduce the notion of excellent extension of rings. Let $\Gamma$ be an excellent extension of an artin algebra $\Lambda$, we prove that $\Lambda$ satisfies the Gorenstein symmetry conjecture (resp. finitistic dimension conjecture, Auslander-Gorenstein conjecture, Nakayama conjecture) if and only if so does $\Gamma$. As a special case of excellent extensions, if $G$ is a finite group whose order is invertible in $\Lambda$ acting on $\Lambda$ and $\Lambda$ is $G$-stable, we prove that if the skew group algebras $\Lambda G$ satisfies strong Nakayama conjecture (resp. generalized Nakayama conjecture), then so does $\Lambda$.