A descent spectral sequence for arbitrary K(n)-local spectra with explicit e ₂-term

Let n be any positive integer and p be any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment π* (LK(n) (X)) and E ₂-term equal to the continuous cohomology of Gn , the extended Morava stabilizer group, with coefficients in a certain discrete Gn -module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local En -Adams spectral sequence for π* (LK(n) (X)), whose E ₂-term is not known to always be equal to a continuous cohomology group.