Tate Cohomology for Arbitrary Groups via Satellites

We define cohomology groups Ĥn(G;M), n ∈ Z, for an arbitrary group G and G-module M , using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property. The classical Tate cohomology groups of finite groups have been generalized to larger classes of groups by several authors ([BC], [F], [GG], [I]). The definition of Benson and Carlson in [BC] makes sense for an arbitrary group, but no formal properties are discussed there. We propose here a different definition for Tate cohomology groups for an arbitrary group G and G-module M , which takes the form Ĥ(G;M) = lim −→ j≥0 S−jHn+j(G;M) with S−jHn+j(G; ?) denoting the j-th left satellite of the functor H(G; ?). These general Tate groups are shown to agree, if applicable, with the ones obtained by the various generalizations mentioned above. The family Ĥ• = {Ĥ(G; ?);n ∈ Z} forms a connected sequence of functors and is as such characterized by a natural universal property, which identifies it with what we call the completion with respect to projective modules, or short, the P-completion of ordinary cohomology (cf. section 3). In the first section we recall, to fix our terminology and notation, some basic facts on satellites. Section two is devoted to an axiomatic description of the P -completion of a cohomological functor, leading to our definition of the general Tate groups. In section three we compare the definition to various other ones and discuss a few examples. Section four contains a detailed comparison with the definition proposed by Benson and Carlson. We thank Karl Gruenberg for enlightening discussions concerning the concept of projective completion for connected sequences of functors. 1991 Mathematics Subject Classification. Primary 18Gxx, 55Uxx; Secondary 20Jxx.