Gorenstein cohomological dimension and stable categories for groups

Abstract

First we study the Gorenstein cohomological dimension Gcd R G of groups G over coefficient rings R, under changes of groups and rings; a characterization for finiteness of Gcd R G is given. Some results in literature obtained over the coefficient ring Z or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category F ib consisting of fibrant RG-modules, and show that the homotopy category Ho ( F ib ) is triangle equivalent to both the stable category C of ¯ ( RG ) of Benson’s cofibrant modules, and the stable module category StMod ( RG ) . The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that Gcd R G < ∞ , Ho ( F ib ) is equivalent to the stable category of Gorenstein projective RG-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective RG-modules.