Milnor operations and the generalized Chern character

We have shown that the n–th Morava K –theory K∗(X) for a CW–spectrum X with action of Morava stabilizer group Gn can be recovered from the system of some height–(n + 1) cohomology groups E∗(Z) with Gn+1 –action indexed by finite subspectra Z . In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E∨ ∗ (E)–precomodules to the category of K∗(K)–comodules. Then we show that K∗(X) is naturally isomorphic to the inverse limit of F(E∗(Z)) as a K∗(K)–comodule.