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 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 ExtUBP∗BP (BP∗,M) and ExtUEn∗En (En∗, En∗⊗BP∗ M) for unstable BP∗BP -comodules that are vn-local and satisfy InM = 0. We show that the latter Ext groups can be interpreted as the continuous cohomology of 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. This in turn has implications for the convergence of the Bousfield-Kan spectral sequence. When p−1 > n and k is odd and sufficiently large we show that the E(n)-based spectral sequence for S has a horizontal vanishing line in E2 and converges to the E(n)-completion of S.