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.

Recently the notions of sfli , the supremum of the flat lengths of injective -modules, and silf , the supremum of the injective lengths of flat -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 , the finiteness of silf implies the finiteness of sfli , but the converse is not known. We investigate the situation in which sfli < ∞ implies silf < ∞. The statement holds for example, for groups with the property that flat -modules have finite projective dimension. Moreover, we show that the Gorenstein flat dimension of the trivial Z -module Z, that will be called Gorenstein homological dimension of , denoted Ghd , is completely related to these invariants.

Communicated by C. Kassel MSC: 13D05 13D45 13H10 13D03 55N35a b s t r a c t Using Auslander's G-dimension, we assign a numerical invariant to any group Γ . It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to Γ . We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups.

Abstract. Let (R, m, k) be a commutative noetherian local ring of Krull dimension d. We prove that the cohomology annihilator ca(R) of R is m-primary if and only if for some n ≥ 0 the n-th syzygies in mod R are constructed from syzygies of k by taking direct sums/summands and a fixed number of extensions. These conditions yield that R is an isolated singularity such that the bounded derived category D b (R) and the singularity category D sg (R) have finite dimension, and the converse holds when R is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and d extensions. This result is exploited to investigate several ascent and descent problems between R and its completion R.

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