Infinitely generated modules

Abstract

In this section we wish to discuss some of the problems involved with extending some of the results to the category _kG Mod of all left kG-modules. Let _kG stMod denote the stable category of all left kG-modules (even infinitely generated ones) modulo projectives. The main thing we want in this section is to show that _kG stMod is a triangulated category. We should recall that the triangulated structure of _kG stMod depended very much on the fact that injective modules are projective and vice versa. This result can be proved for large classes of rings using various sorts of sophisticated machinery. For group algebras, however, we can stick to reasonably elementary methods.