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; a Gorenstein version of Serre's theorem is proved, i.e. Gcd R H = Gcd R G for any subgroup H of G with finite index; characterizations for finite groups are given. These generalize the results in literatures, which were obtained over Z or rings of finite global dimension, to more general rings. Moreover, we establish a model structure on the weakly idempotent complete exact category F ib consisting of fibrant RGmodules, and show that the homotopy category Ho(F ib) is triangle equivalent to the stable category Cof (RG) of Benson's cofibrant modules, as well as the stable module category StMod(RG). For any commutative ring R of finite global dimension, if either G is of type Φ R or is in Kropholler's large class such that Gcd R G is finite, then Ho(F ib) is equivalent to the stable category of Gorenstein projective RG-modules, the singular category, and the homotopy category of totally acyclic complexes of projective RG-modules.