Commutative Algebra in Group Cohomology . 3

We apply constructions from equivariant topology to Benson-Carlson resolutions and hence prove in (2.1) that the group cohomology ring of a nite group enjoys remarkable duality properties based on its global geometry. This recovers and generalizes the result of Benson-Carlson stating that a Cohen-Macaulay cohomology ring is automatically Gorenstein. We give an alternative approach to Tate cohomology of groups and in (4.1) show that the Tate cohomology of a group is close to being the cohomology of the projective space of the group cohomology ring. In this article we investigate the eeect of certain standard constructions from commu-tative algebra when applied to the cohomology ring H (G; k) of a nite group G with coeecients in a eld k. This is a further step in understanding the connection between homological and commutative algebra, which is the algebraic counterpart of completion theorems and their duals from algebraic topology. 1 Our results come by using the methods from 5] in the context of group cohomology, and the present article will esh out certain assertions made there. Fundamental to this application is the work of Benson-Carlson 1, 2] on algebraic analogues of free actions of nite groups on products of spheres. The paper is divided into four sections. In Section 1 we recall some constructions from commutative algebra that we need. In Section 2 we prove that the group homology H (G; M) is essentially the local cohomology of the corresponding cohomology H (G; M) as a module over the graded local ring H (G); this is an unusual duality phenomenon based on the global geometry of the ring H (G). In Section 3 we explain how methods from topology 4, 8] give an alternative approach to the construction of Tate cohomology; this is simply homotopy theory of chain complexes over kG and may be of independent interest. Section 4 gives an analogue for Tate cohomology of the results of Section 2: the Tate cohomology ^ H (G; M) is essentially the Cech cohomology of H (G; M) as a module over H (G). Geometrically speaking, this says that the Tate cohomology of the group is the cohomology (with all twists) of the projective space of the group cohomology ring H (G). In the next section we shall make certain constructions that are analogous to classical constructions in commutative algebra. To establish notation and to ensure the reader is 1 1991 AMS subject …