Profinite direct sums with applications to profinite groups of type ΦR

Abstract We show that the ‘profinite direct sum’ is a good notion of infinite direct sums for profinite modules, having properties similar to those of direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective, and there is a Mackey's formula for profinite modules described using these sums. As an application, we prove that the class of profinite groups of type is closed under subgroups.