On the hierarchy of cohomological dimensions of groups
In this paper, several numerical invariants associated to a group \(\Gamma\) are compared. The invariants in question are: The classical cohomological dimension (\(\text{cd}\,\Gamma\)), the virtual cohomological dimension (\(\text{vcd}\,\Gamma\)), the Gorenstein cohomological dimension (\(\text{Gcd}\,\Gamma\)), the generalized cohomological dimension (\(\underline{\text{cd}}\,\Gamma\)), and the projective complete cohomological dimension (\(\text{pccd}\,\Gamma\)). Particular attention is given to the Gorenstein cohomological dimension \(\text{Gcd}\,\Gamma\) of the group \(\Gamma\). For a module \(M\) over an arbitrary ring \(R\), the Gorenstein projective dimension \(\text{Gpd}_RM\) was introduced by \textit{M. Auslander} and \textit{M. Bridger} [''Stable module theory'', Mem. Am. Math. Soc. 94, 146 p. (1969; Zbl 0204.36402)] and by \textit{E. E. Enochs} and \textit{O. M. G. Jenda} [Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)]. This dimension has been studied by several authors. With the authors' definition, \(\text{Gcd}\,\Gamma = \text{Gpd}_{\mathbb{Z}\Gamma}\mathbb{Z}\), where \(\mathbb{Z}\) is the trivial module over the group algebra \(\mathbb{Z}\Gamma\).
scientific article; zbMATH DE number 5561731
scientific article