Homological finiteness conditions for modules over strongly group-graded rings

Throughout this paper, k denotes a commutative ring. We will develop a theory of homological finiteness conditions for modules over certain graded k -algebras which generalizes known theory for group algebras. The simplest of our results, Theorem A below, generalizes certain results of Aljadeff and Yi on crossed products of polycyclic-by-finite groups (cf. [1, 11]), but also applies to many other crossed products in cases where little was previously known. Before stating the results, we recall definitions of graded and strongly graded rings. Let G be a monoid. Naively, a G -graded k -algebra is a k -algebra R which admits a k -module decomposition, in such a way that R g R h ⊆ for all g , h ∈ G . If R is a G -graded k -algebra and X is any subset of G , then we write R x for the k -submodule of R supported on X ; that is Note that if H is a submonoid of G then R H is a subalgebra of R .