Phantom maps in the stable module category

We study phantom maps in the stable module category StMod(kG), where k is a field and G is a finite group. In this article we almost exclusively deal with maps out of countably generated modules. We show that the space Ph stabilized phantom maps M --> N has an expression as a (lim)under left arrow(1) space which allows some control on its vanishing. Then we present a situation where all maps are phantom and need not be trivial. We provide explicit details for such a particular situation. Finally we construct a universal phantom map. We use it to show that the composite of two phantom maps is trivial and to characterize the modules with no nontrivial outbound phantom maps. (C) 1998 Academic Press.