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

The connective $ku$-(co)homology of elementary abelian $2$-groups is determined as a functor of the elementary abelian $2$-group, using the action of the Milnor operations $Q_0, Q_1$ on mod $2$ group cohomology, the Atiyah-Segal theorem for $KU$-cohomology, together with an analysis of the functorial structure of the integral group ring; the functorial structure then reduces calculations to the rank 1 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, giving a conceptual explanation of results of Bruner and Greenlees.

Keywords

connective $K$-theory, elementary abelian group, group cohomology, group homology, local cohomology

2010 Mathematics Subject Classification

19L41, 20J06

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Published 2 June 2014