The action of the Steenrod algebra on Tate cohomology

Abstract

Let G be a finite group and k a field of characteristic p dividing | G|. In this paper, we examine the action of the Steenrod operations on the Tate cohomology Ĥ∗( G,k). In particular, we prove that all Steenrod operations from negative to positive degree Tate cohomology vanish if and only if all products in negative degree Tate cohomology vanish.