Phantom Maps in the Stable Module Category

We study phantom maps in the stable module category (kG), wherekis a field andGis a finite group. In this article we almost exclusively deal with maps out of countably generated modules. We show that the space kG(M, N) of stabilized phantom mapsM → Nhas an expression as a 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.