Gorenstein Injective and Injective Complete Cohomological Dimensions of Groups

Abstract

Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group Γ, called the injective complete cohomological dimension of Γ, denoted by iccd Γ. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group Γ, which is denoted by Gid Γ. We show that Gid Γ is related to iccd Γ, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd Γ is a refinement of Gid Γ. In addition, we show that silp Γ = spli Γ < ∞ if and only if the Shapiro lemma holds for injective complete cohomology.