Pure injectives and the spectrum of the cohomology ring of a finite group

For a finite group G and a field k of prime characteristic, we study certain pure injective kG-modules in terms of the spectrum of the group cohomology ring H* (G, k). For instance, we construct a map from the projective variety Proj (H* (G, k)) to the Ziegler spectrum of indecomposable pure injective kG-modules. We identify the module corresponding to a generic point for a component of the variety; it is generic in the sense of Crawley-Boevey and closely related to a certain Rickard idempotent module. We include also a complete classification of all kG-modules which arise as a direct summand of a (possibly infinite) product of syzygies of the trivial module k.