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The Gorenstein cohomological dimension of a group G generalizes the ordinary cohomological dimension of G, in the sense that the two invariants coincide when the latter one is finite. In this paper, we show that the Gorenstein cohomological dimension GcdkG of G over a commutative ring k shares many properties with the cohomological dimension of G over k. For example, if k has finite global dimension, the finiteness of GcdkG implies that GpdkGM is finite for any kG‐module M. Other properties concern the dependence of the Gorenstein cohomological dimension upon the coefficient ring, its behavior with respect to subgroups, the subadditivity with respect to extensions and the formula for the dimension of an ascending union. We also prove that if k has finite global dimension then the finiteness of GcdkG gives a criterion for the conditions cdkG<∞ and hdkG<∞ to be equivalent. Finally, we show that if k is a countable ring and G is a countable group of finite Gorenstein cohomological dimension over k, then GcdkG=supfalse{n:Hn(G,kG)≠0false}. The latter two results are special cases for group algebras of certain assertions that are valid over more general rings.

In this paper, we consider for any free presentation G = F /R of a group G the coinvariance H 0 (G, R ⊗n ab ) of the nth tensor power of the relation module R ab and show that the homology group H 2n (G, Z) may be identified with the limit of the groups H 0 (G, R ⊗n ab ), where the limit is taken over the category of these presentations of G. We also consider the free Lie ring generated by the relation module R ab , in order to relate the limit of the groups γ n R/[γ n R, F ] to the n-torsion subgroup of H 2n (G, Z).

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