Operations and co-operations in Morava \(E\)-theory
type: Item
instance of: scholarly article
title: Operations and co-operations in Morava \(E\)-theory
zbMATH Open document ID: 1063.55003
published in: Homology, Homotopy and Applications
publication date: 2005-04-05T00:00:00Z
full work available at URL: https://eudml.org/doc/51320
full work available at URL: volume6-1.htm
review text: Let \(p\) be a prime and \(n\) a non-negative integer. Let \(E=E(p,n)\) be the complement of the Johnson--Wilson spectrum, see [\textit{M. Hovey} and \textit{N. Strickland}, Morava \(K\)-theories and localisation, Mem. Am. Math. Soc. 139, No. 666 (1999; Zbl 0929.55010)] for the precise definition. The author studies the ring of cohomology operations \(E^*(E)\) and proves that it is isomorphic to the completed twisted group ring \(\pi_*(E)\Gamma\) where \(\Gamma\) is the group of automorphisms (i.e. homotopy self-equivalences) of the spectrum \(E\). Furthermore, let \(C(\Gamma, \pi_*(E))\) be the Hopf algebroid of continuous maps from profinite group \(\Gamma\) to \(\pi_*(E)\) where \(\pi_*(E)\) is equipped with the {\(\mathfrak m\)}-adic topology and {\(\mathfrak m\)} is the unique homogeneous maximal ideal. The author establishes an isomorphism of graded formal Hopf algebroids \(\pi_*L_{K(n)}(E \wedge E) \simeq C(\Gamma, \pi_*(E))\).
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55S25
Mathematics Subject Classification ID: 57T05
zbMATH DE Number: 2153111
zbMATH Keywords: Morava \(E\)-theory
zbMATH Keywords: Morava stabilizer group
zbMATH Keywords: cohomology operations
zbMATH Keywords: localization
zbMATH Keywords: twisted completed group ring
zbMATH Keywords: Hopf algebroid
author: Mark Hovey
reviewed by: Yuli Rudyak
MaRDI profile type: Publication
OpenAlex ID: W2112425511
DOI: 10.4310/HHA.2004.V6.N1.A13
description: scientific article; zbMATH DE number 2153111