Homological dimensions of crossed products

TL;DR

This paper investigates the homological dimensions of crossed product rings formed from a Noetherian ring and a finite group, providing classifications, criteria, and extensions of previous results in algebra.

Contribution

It classifies the global and finitistic dimensions of crossed products and extends prior results to semiprimary algebras with specific idempotent structures.

Findings

Global dimension of crossed product is either infinite or equals that of the base ring.

Finitistic dimension of crossed product coincides with that of the base ring.

Criteria established for finite global dimension of skew group rings.

Abstract

In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions, extending the main results of the author in some previous papers.