Tate (Co)homology of invariant group chains

Let [Formula: see text] be a finite group acting on a group [Formula: see text] as a group automorphisms, [Formula: see text] the bar complex, [Formula: see text] the homology of invariant group chains and [Formula: see text] the cohomology invariant, both defined in Knudson’s paper “The homology of invariant group chains”. In this paper, we define the Tate homology of invariants [Formula: see text] and the Tate cohomology of invariants [Formula: see text]. When the coefficient [Formula: see text] is the abelian group of the integers, we proved that these groups are isomorphics, [Formula: see text]. Further, we prove that the homology and cohomology of invariant group chains are duals, [Formula: see text], [Formula: see text].