Every K(n)-local spectrum is the homotopy fixed points of its Morava module

Let n ≥ 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this paper show that if X is a finite spectrum, then the localization L_K(n)(X) is equivalent to the homotopy fixed point spectrum (L_K(n)(E_n ∧ X))^hG_n, which is formed with respect to the continuous action of G_n on L_K(n)(E_n ∧ X). In this paper, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π_*(L_K(n)(X)) is isomorphic to the descent spectral sequence that abuts to π_*((L_K(n)(E_n ∧ X))^hG_n).