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 (The connective K-theory of finite groups, 2003). 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 [2], which is in a different form, and hence the comparison involves some non-trivial combinatorics.