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

## TL;DR

This paper revisits the Tate cohomology of connective K-theory for elementary abelian groups, providing a more elementary calculation method, extending results to odd primes, and identifying the spectra involved.

## Contribution

It offers a new, simpler approach to calculating Tate cohomology for these groups and extends the results to primes greater than two, clarifying the spectral structure.

## Key findings

- Elementary method for Tate cohomology calculation
- Extension to primes p > 2
- Identification of spectra as products of Eilenberg-Mac Lane spectra

## 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.