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

TL;DR

This paper proves that every K(n)-local spectrum can be represented as the homotopy fixed points of its Morava module, extending previous results from finite spectra to all S-cofibrant spectra and relating spectral sequences.

Contribution

It generalizes the equivalence between K(n)-local spectra and homotopy fixed points from finite spectra to all S-cofibrant spectra and connects the Adams-type spectral sequence with the descent spectral sequence.

Findings

The equivalence holds for all S-cofibrant spectra.

The Adams-type spectral sequence is isomorphic to the descent spectral sequence.

This extends the understanding of the structure of K(n)-local spectra.

Abstract

Let n \geq 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 note, 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 \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, 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 \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).