Local fields and extraordinary K-theory

TL;DR

This paper constructs integral lifts of extraordinary cohomology theories associated with local fields and uses generalized character theory to identify their values on classifying spaces as rings of functions on schemes related to conjugacy classes of homomorphisms.

Contribution

It introduces integral lifts of K(n) theories for local fields and applies generalized character theory to describe their values on classifying spaces.

Findings

K(L) theories are integral lifts of K(n) for local fields L.

K(L)(BG) tensor Q is identified with functions on a scheme of conjugacy classes.

Special case recovers classical theorem of Artin and Atiyah.

Abstract

We describe integral lifts K(L), indexed by local fields L of degree n = [L:\Q_p], of the extraordinary cohomology theories K(n), and apply the generalized character theory of Hopkins, Kuhn and Ravenel to identify K(L)(BG) \otimes \Q$, for a finite group G, as a ring of functions on a certain scheme \frak C_LG \'etale over L, whose points are conjugacy classes of homomorphisms from the valuation ring of L to G. When L is \Q_p this specializes to a classical theorem of Artin and Atiyah.