Equivariance of generalized Chern characters

TL;DR

This paper generalizes the Chern character within chromatic homotopy theory, constructing equivariant transformations between height (n+1) and height n cohomology theories, revealing new structural insights.

Contribution

It introduces a G_{n+1}-equivariant natural transformation from height (n+1) to height n cohomology theories, and shows how to recover K^*(X) from E^*(X).

Findings

Constructed a multiplicative G_{n+1}-equivariant transformation  from height (n+1) to height n theories.

Demonstrated that K^*(X) can be recovered from E^*(X) as G_n-modules.

Lifted the transformation  to characteristic zero cohomology theories.

Abstract

In this note some generalization of the Chern character is discussed from the chromatic point of view. We construct a multiplicative G_{n+1}-equivariant natural transformation \Theta from some height (n+1) cohomology theory E^*(-) to the height n cohomology theory K^*(-)\hat{\otimes}_F L, where K^*(-) is essentially the n-th Morava K-theory. As a corollary, it is shown that the G_n-module K^*(X) can be recovered from the G_{n+1}-module E^*(X). We also construct a lift of \Theta to a natural transformation between characteristic zero cohomology theories.