An unstable change of rings for Morava E–theory

Thompson, Robert

doi:10.2140/agt.2020.20.2145

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An unstable change of rings for Morava $E$–theory

Robert Thompson

Algebraic & Geometric Topology 20 (2020) 2145–2176

DOI: 10.2140/agt.2020.20.2145

Abstract

The Bousfield–Kan (or unstable Adams) spectral sequence can be constructed for various homology theories, such as Brown–Peterson homology theory $BP$ , Johnson–Wilson theory $E (n)$ or Morava $E$ –theory $E_{n}$ . For nice spaces the $E_{2}$ –term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between ${Ext}_{𝒰_{Γ_{B}}} (B, M)$ and ${Ext}_{𝒰_{E_{n *} E_{n}} ∕ I_{n}} (E_{n *} ∕ I_{n}, E_{n *} \otimes_{{BP}_{*}} M)$ , where $(B, Γ_{B})$ denotes the Hopf algebroid $(v_{n}^{- 1} {BP}_{*} ∕ I_{n}, v_{n}^{- 1} {BP}_{*} BP ∕ I_{n})$ . We show that the latter groups can be interpreted as $Ext$ in the category of continuous modules over the profinite monoid of endomorphisms of the Honda formal group law. By comparing this with the cohomology of the Morava stabilizer group we obtain an unstable Morava vanishing theorem when $p - 1 ∤ n$ .

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Keywords

unstable Adams spectral sequence, Morava changes of rings theorem

Mathematical Subject Classification 2010

Primary: 55N20, 55Q51, 55T15

References

Publication

Received: 20 September 2014

Revised: 24 August 2019

Accepted: 8 September 2019

Published: 4 November 2020

Authors

Robert Thompson
Department of Mathematics and Statistics Hunter College and the Graduate Center, CUNY New York, NY United States

Volume 20, issue 5 (2020)