Transfer Maps and Virtual Projectivity

WhenGis a finite group andHis a subgroup ofG, the concept of relativeH-projectivity is of fundamental importance in modular representation theory. Unfortunately, in some ways the properties of relativelyH-projective modules are not as closely related to other ideas of representation theory as one might hope. For example, the variety of a module bears only a weak connection to relative projectivity. The purpose of this paper is to introduce and study a related concept known asvirtual H-projectivitythat seems to be much more closely related to the theory of varieties. The study is actually carried out in the more general setting of virtual projectivity with respect to a finitely generated moduleW, and the main tool is a transfer map TrWthat generalizes the usual transfer map TrGHwith respect to a subgroupH.