Pure Injectives and the Spectrum of the Cohomology Ring of a Finite Group

Let G be a finite group and k be a field of characteristic p. The connection between the modular representations and the cohomology of G has been a subject of major interest over the last several years. For example, recent study of the stable category of kG-modules StMod(kG) led Rickard [27] to introduce certain idempotent modules and functors. This has led to a theory of varieties for infinitely generated modules [3, 4], and the classification of thick subcategories of the stable category stmod(kG) of finitely generated modules, at least in the case of a p-group [5]. The cohomology of the group G is intimately related to the properties of the trivial kG-module k and its shifted copies Ωk, n ∈ Z. It is therefore natural to ask for a description of the kG-modules which arise as a direct summand of a (possibly infinite) direct product of modules of the form Ωk, n ∈ Z. Note that such a module is always pure injective because any module over a finite dimensional algebra is pure injective if and only if it is isomorphic to a direct summand of a direct product of finite dimensional modules. Pure injective modules have been studied for some time in representation theory of finite dimensional algebras, mostly because certain infinitely generated pure injectives (so-called generic modules) control the representation type of an algebra [11]. For example, generic modules have been used to show that the representation type of an algebra is an invariant of the stable module category [20]. More recently, the notion of a phantom map in StMod(kG) was introduced in [16, 6] as an analogue of a classical concept from stable homotopy theory. In this context, it became apparent as well that pure injective modules play an important role, as the modules which receive no phantom maps. In particular, under rather restrictive hypotheses, some of Rickard’s idempotent modules were observed to be pure injective. This paper grew out of an attempt to understand this phenomenon. The purpose of this paper is to investigate a certain functor T from injective modules I over the cohomology ring H∗(G, k) to pure injective modules in the stable category StMod(kG). Most of the arguments work when kG is replaced by any finite dimensional cocommutative Hopf algebra A over k, and so this is the generality in which we work, although at some stages we need to make an assumption about the behavior of negative Tate cohomology which we only know to be true in the finite group context. Namely, either Tate cohomology is periodic, or negative Tate cohomology is nilpotent, in the sense that there exists an integer n > 0 such that every product of at least n elements is zero. We do not know of an example of a finite dimensional cocommutative Hopf algebra for which this fails.