GORENSTEIN HOMOLOGICAL DIMENSION AND SOME INVARIANTS OF GROUPS

For any group G, the Gorenstein homological dimension GhdR G is defined to be the Gorenstein flat dimension of the coefficient ring R, which is considered as an RG-module with trivial group action. We prove that GhdR G<∞ if and only if the Gorenstein flat dimension of any RG-module is finite, if and only if there exists an R-pure RG-monic R→A with A being R-flat and fdRG A<∞, where R is a commutative ring with finite Gorenstein weak global dimension. As applications, properties of Ghd on subgroup, quotient group, extension of groups as well as Weyl group are investigated. Moreover, we compare the relations between some invariants such as sfli RG, silf RG, spli RG, silp RG, and Gorenstein projective, Gorenstein flat and PGF dimensions of RG-modules; a sufficient condition for the Gorenstein projective-flat problem over group rings is given.