An Unstable Change of Rings for Morava E-Theory

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 $p-1 \nmid n$.