On the cohomology of some Hopf algebroids and Hattori-Stong theorems

Abstract

We apply group cohomological methods to calculate the cohomology of $K(n)_*BP$ as a $K(n)_*K(n)$-comodule, recovering recent results of Hovey and Sadofsky. As applications we determine the Chromatic Spectral Sequence for $BP$ based on Johnson and Wilson's $E(n)$, showing the relationship to some generalizations of the classical Hattori-Stong Theorem and determine the change of Hopf algebroid spectral sequence associated with the natural map $BP\rightarrow E(n)$, extending calculations of Clarke for the Todd orientation $MU\rightarrow KU$.