Pure injectives and the spectrum of the cohomology ring of a finite group (Q2769330)

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 \(\text{Proj}(H^*(G;k))\) 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 \(k\).