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‐.nC 1/ cohomology groups E .Z/ with GnC1 ‐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. 55N22; 55N20, 55S05 Dedicated to Professor Nishida on the occasion of his 60th birthday