Homological Dimensions of Crossed Products

Abstract

Abstract. In this paper we consider several homological dimensions of crossed products A σ α 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 G, we show that A and A σ α G share the same homological dimensions under extra assumptions, extending the main results in [15,16].

Finiteness properties and homological dimensions of Skew Group rings

“…If G is a group acting faithfully as automorphisms of R, then the skew group ring RG is a separable extension of R if and only if there exists a central element in R with trace one. Moreover, if H is a subgroup ofG such that the index [G : H] is invertible in R, by[30, Proposition 3.6], we have that RG is a separable extension over RH. In particular, if the order of G is invertible in R, the skew group ring RG is a separable extension over R.1.1.…”

“…Let G be a cyclic group of order 2 generated by g, which permutes vertices x and y, and arrows β and γ. This action determine a skew group algebra AG which is not a separable extension over A. See[30, Example 4.5].1.4. Induction and restriction functors.…”