Homological dimensions of crossed products

Abstract

In this paper we consider several homological dimensions of crossed products $A _α ^σ G$, 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 $A ^σ _α G$ are classified: global dimension of $A ^σ _α G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^σ _α G$ 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 $A _α ^σ G$ share the same homological dimensions under extra assumptions, extending the main results of the author in some previous papers.