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

Robert Thompson

Algebraic & Geometric Topology 20 (2020) 2145–2176

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 En. For nice spaces the E2–term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between Ext𝒰Γ B(B,M) and Ext𝒰E nEnIn(EnIn,EnBPM), where (B,ΓB) denotes the Hopf algebroid (vn1BPIn,vn1BPBPIn). 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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unstable Adams spectral sequence, Morava changes of rings theorem
Mathematical Subject Classification 2010
Primary: 55N20, 55Q51, 55T15
Received: 20 September 2014
Revised: 24 August 2019
Accepted: 8 September 2019
Published: 4 November 2020
Robert Thompson
Department of Mathematics and Statistics
Hunter College and the Graduate Center, CUNY
New York, NY
United States