Transfer maps and virtual projectivity

Authors:  Jon F. Carlson, Chuang Peng, and Wayne W. Wheeler

Abstract:  When $G$ is a finite group and $H$ is a subgroup of $G$, the
concept of relative $H$-projectivity is of fundamental importance in
modular representation theory.  Unfortunately, in some ways the properties
of relatively $H$-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 as {\it virtual $H$-projectivity\/} that
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 module $W$, and the
main tool is a transfer map $\Tr_W$ that generalizes the usual transfer
map $\Tr_H^G$ with respect to a subgroup $H$.