Some properties of Lubin-Tate cohomology for classifying spaces of finite groups

TL;DR

This paper investigates properties of Lubin-Tate and Morava K-theory cochain extensions for classifying spaces of finite groups, revealing conditions under which these extensions are faithful or ramified, impacting spectral sequence convergence.

Contribution

It demonstrates that for Lubin-Tate and Morava K-theory, cochain extensions are always faithful in the K(n)-local category, but can ramify and fail to be Galois for cyclic p-groups.

Findings

Extensions are always faithful in the K(n)-local category.

Cochain extensions ramify for cyclic p-groups, preventing Galois property.

Spectral sequences may not converge as expected in certain cases.

Abstract

We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory $K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a $G$-Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it ramifies. As a consequence, it follows that the $E_n$-theory Eilenberg-Moore spectral sequence for $G$ and $BG$ does not always converge to its expected target.