Tate cohomology of connected k-theory for elementary abelian groups revisited

Abstract: Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for $G=(\mathbb{Z}/2)^n$ was completely calculated by Bruner and Greenlees. In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to $p>2$ prime. We also identify the resulting spectra, which are products of Eilenberg-Mac Lane spectra, and finitely many finite Postnikov towers. For $p=2$, we also reconcile our answer completely with the result of Bruner and Greenlees, which is in a different form, and hence the comparison involves some non-trivial combinatorics.