# Excellent extensions and homological conjectures

## TL;DR

This paper introduces the concept of excellent extensions of rings and demonstrates their preservation of several important homological conjectures in algebra, including Gorenstein symmetry and Nakayama conjectures.

## Contribution

It establishes that excellent extensions preserve key homological conjectures for artin algebras, extending the understanding of their structural properties.

## Key findings

- Excellent extensions preserve the Gorenstein symmetry conjecture.
- Excellent extensions preserve the finitistic dimension conjecture.
- Skew group algebras inherit strong Nakayama conjecture from the base algebra.

## 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 $\Lambda$.