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

Abstract

We consider brave new cochain extensions $F(B{G}_{+},R)\to F(E{G}_{+},R)$, 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 $F({B{C}_{p^r}}_{+},E_n)\to F({E{C}_{p^r}}_{+},E_n)$ 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.

Minor changes, section on Frobenius algebra structure removed. Final version: to appear in Central European Journal of Mathematics under title `Galois theory and Lubin-Tate cochains on classifying spaces'

References in corpus (2)

• Galois extensions of structured ring spectra
• Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories