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Let n be any positive integer and p any prime. Also, let X be any spectrumand let K(n) denote the nth Morava K-theory spectrum. Then we construct adescent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal tothe continuous cohomology of G_n, the extended Morava stabilizer group, withcoefficients in a certain discrete G_n-module that is built from varioushomotopy fixed point spectra of the Morava module of X. This spectral sequencecan be contrasted with the K(n)-local E_n-Adams spectral sequence forpi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to acontinuous cohomology group.

A descent spectral sequence for arbitrary K(n)-local spectra with explicit $E_2$-term