Every \(K(n)\)-local spectrum is the homotopy fixed points of its Morava module (Q2880671)

This paper is concerned with the localization of a spectrum \(X\) with respect to the Morava \(K\)-theory spectrum \(K(n)\). In particular, it was conjectured that such a localization should be realized by building a module spectrum from \(X\) and the Lubin-Tate spectrum \(E_n\), then taking the fixed points with respect to the action of the Morava stabilizer group \(G_n\). Building on work of Devinatz and Hopkins, the first-named author together with Behrens was able to prove this result in the case that \(X\) is a finite spectrum. In the current paper, the authors settle the conjecture for any cofibrant spectrum (in the sense of symmetric spectra), from which it follows that it holds for any spectrum. This result is proved by establishing that the Adams spectral sequences converging to the homotopy groups of each of these spectra are isomorphic to one another.