On connective K-theory of elementary abelian 2-groups and local duality

TL;DR

This paper computes the connective ku-(co)homology of elementary abelian 2-groups using functorial methods, local duality, and the Atiyah-Segal theorem, providing new insights into the structure and calculations of these cohomology theories.

Contribution

It introduces a functorial approach to determine ku-(co)homology of elementary abelian 2-groups and applies local duality to analyze the associated spectral sequence, extending previous results.

Findings

Explicit calculation of ku-(co)homology for elementary abelian 2-groups.

Application of local duality to the local cohomology spectral sequence.

Extension of methods to the odd primary case.

Abstract

The connective ku-(co)homology of elementary abelian 2-groups is determined as a functor of the elementary abelian 2-group. The argument requires only the calculation of the rank one case and the Atiyah-Segal theorem for KU-cohomology together with an analysis of the functorial structure of the integral group ring. The methods can also be applied to the odd primary case. These results are used to analyse the local cohomology spectral sequence calculating ku-homology, via a functorial version of local duality for Koszul complexes. This gives a conceptual explanation of results of Bruner and Greenlees.