This paper deals with several applications of functorial methods to the study of group-graded rings. The general framework for this study is provided by the categories of modules graded by \(G\)-sets (where \(G\) is a not necessarily finite group) and the main tools used are pairs of adjoint functors between categories of this type, whose relevant role had been noticed by C. Menini [Functors between categories of graded modules. Applications (preprint)].
In the first part of the paper, the author obtains some general results about these functors, which are subsequently applied to the description of equivalences and Morita dualities between categories of modules graded by \(G\)-sets. He then proceeds to obtain a general version of the Cohen- Montgomery duality theorems [M. Cohen and S. Montgomery, Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001)] using as main ingredient a representation of the category of modules graded by \(G\)-sets as a category of modules generated by an ideal, and containing previous extensions of these theorems to the case of a possibly infinite grading group.
In the last sections, the author studies separable adjoint functors between categories of modules graded by \(G\)-sets and, as an application, he gives a sufficient condition (in terms of separability) for the graded and ungraded weak global dimensions of a group-graded ring to coincide.