An Unstable Change of Rings for Morava E-Theory

TL;DR

This paper establishes an unstable change of rings isomorphism for Morava E-theory, linking Ext groups in unstable comodules to continuous modules over endomorphism monoids, and proves a vanishing theorem under certain conditions.

Contribution

It introduces a novel unstable Morava change of rings isomorphism and relates Ext groups to continuous modules over endomorphism monoids, advancing understanding of Morava E-theory.

Findings

Established an unstable Morava change of rings isomorphism.

Interpreted Ext groups as modules over endomorphism monoids.

Proved an unstable Morava vanishing theorem when p-1 does not divide n.

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 $\text{Ext}_{\mathcal{U}_{\Gamma_B}}(B,M)$ and $\text{Ext}_{\mathcal{U}_{E_{n*}E_n}/I_{n}}(E_{n*}/I_{n},E_{n*}\otimes_{BP_*} M)$ where $(B,\Gamma_B)$ 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 $\text{Ext}$ 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 $p-1 \nmid n$.