Milnor operations and the generalized Chern character

TL;DR

This paper demonstrates how Morava K-theory of a CW-spectrum with a G_n-action can be reconstructed from a system of higher height cohomology groups with G_{n+1}-action, extending previous results.

Contribution

It introduces a symmetric monoidal functor connecting E^{vee}_*(E)-precomodules to K_{*}(K)-comodules and shows the Morava K-theory can be obtained as an inverse limit of this functor applied to certain cohomology groups.

Findings

Morava K-theory can be recovered from higher height cohomology groups.

A new symmetric monoidal functor relates different comodule categories.

K-theory is expressed as an inverse limit of functor images.

Abstract

We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite subspectra Z. In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E^{vee}_*(E)-precomodules to the category of K_{*}(K)-comodules. Then we show that K^*(X) is naturally isomorphic to the inverse limit of F(E^*(Z)) as a K_{*}(K)-comodule.