We conjecture that a group G admits a finite‐dimensional classifying space for proper actions if and only if the Gorenstein projective dimension of G is finite. We verify the one‐dimensional case of this conjecture. Some evidence are given for the hypothesis that the Gorenstein projective ℤG‐modules are precisely Benson's class of cofibrant modules.
A group G is said to have periodic cohomology with period q after k steps, if the functors H i (G, −) and H i+q (G, −) are naturally equivalent for all i > k. Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups G that admit a finite-dimensional, free G-CW-complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in H q (G, Z). It is expected that the periodicity isomorphisms will always be given by the cup product with an element in H q (G, Z); this paper shows that this is the case if and only if the group G admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in H q (G, Z) is equivalent to silp G being finite, where silp G is the supremum of the injective lengths of the projective ZG-modules.
We define a group G to be of type Φ if it has the property that for every ZG-module G, proj. dim ZG G < ∞ iff proj. dim Z H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for EG, the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spliZG is the supremum of the projective lengths of the injective ZG-modules. Here we prove certain parts of these conjectures.
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.
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