Finiteness properties and homological dimensions of Skew Group rings

TL;DR

This paper investigates the homological properties and dimensions of skew group rings formed by a finite group acting on a ring, establishing conditions under which these properties are preserved between different subgroup-related skew group rings.

Contribution

It compares homological dimensions of skew group rings and shows that certain properties are shared under separability conditions.

Findings

Homological dimensions over $RG$ and $RH$ are compared.

Shared properties like $n$-Gorenstein and $n$-coherent are established under separability.

Conditions for properties to be preserved between skew group rings are identified.

Abstract

Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over $RH$, we show that the skew group rings $RG$ and $RH$ share some properties such as being $n$-Gorenstein, $n$-perfect, $n$-coherent, $(n,d)$, Ding-Chen or IF-rings.