On certain homological invariants of groups

Abstract

Recently the notions of sfliΓ, the supremum of the flat lengths of injective Γ-modules, and silfΓ, the supremum of the injective lengths of flat Γ-modules have been studied by some authors. These homological invariants are based on spli and silp invariants of Gedrich and Gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups. In this paper we will study these invariants. It turns out that, for any group Γ, the finiteness of silfΓ implies the finiteness of sfliΓ, but the converse is not known. We investigate the situation in which sfli Γ < ∞ implies silf Γ < ∞ . The statement holds for example, for groups Γ with the property that flat Γ-modules have finite projective dimension. Moreover, we show that the Gorenstein flat dimension of the trivial Z Γ -module Z , that will be called Gorenstein homological dimension of Γ, denoted GhdΓ, is completely related to these invariants.