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

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}).


Introduction
In this note, we extend a result about the K(n)-localization of finite spectra, which is due to a combination of work by Devinatz and Hopkins (in [4]) and Behrens and the first author of this note (in [1], [2]) (with most of the hard work being done by Devinatz and Hopkins), to all K(n)-local spectra.
In more detail, let n ≥ 1 and let p be a prime. Above and elsewhere, K(n) denotes the nth Morava K-theory spectrum, G n = S n ⋊ Gal(F p n /F p ) is the nth extended Morava stabilizer group, and E n is the nth Lubin-Tate spectrum, with π * (E n ) = W (F p n ) u 1 , ..., u n−1 [u ±1 ], where W (F p n ) is the ring of Witt vectors with coefficients in the field F p n , the degree of u is −2, and the complete power series ring is in degree zero. Given a spectrum X, we define the Bousfield localization of E n ∧ X with respect to K(n), to be the point-set level Morava module of X. When π * (L K(n) (E n ∧ X)) satisfies certain hypotheses, it is common for these stable homotopy groups to be referred to as the Morava module of X. However, in this note, since we never use the term "Morava module" in this algebraic sense, we will henceforth always refer to the point-set level Morava module of X as just its Morava module.
By [5], G n acts on E n through maps of commutative S-algebras and, by regarding X as having trivial G n -action and then giving its Morava module the diagonal G naction, L K(n) (E n ∧ X) is a G n -spectrum.
Since the time of [10] and the circulation of the results of [11], it has been believed by many experts in chromatic homotopy theory that it ought to be possible to realize the K(n)-localization of any spectrum X as the G n -homotopy fixed points of some E n -module spectrum that is built out of E n and X. However, to date, it has not been clear how to do this.
Tremendous progress towards such a result was made by [4], which showed that where E dhGn n is a commutative S-algebra that behaves like a G n -homotopy fixed point spectrum (e.g., the associated K(n)-local E n -Adams spectral sequence looks like a descent spectral sequence, with E 2 -term equal to continuous cohomology).
Notice that equivalence (1.1) implies that whenever X is a finite spectrum, Additionally, by [2, Theorem 1.3], the Morava module L K(n) (E n ∧ X) ≃ E n ∧ X (since X is finite) is a continuous G n -spectrum, so that its G n -homotopy fixed point spectrum (L K(n) (E n ∧ X)) hGn can be formed. Also, by [ Taken together, the preceding conclusions imply that E hGn n ∧ X ≃ L K(n) (X), and hence, since X is a finite spectrum, by [2,Theorem 9.9]. Therefore, the K(n)-localization of any finite spectrum can be realized as the G n -homotopy fixed points of its Morava module. Now we are ready to explain how our last conclusion is generalized in this note. First, we remark that, from this point onward, we always work in the stable model category of symmetric spectra of simplicial sets or in its homotopy category. Thus, a "spectrum" is a symmetric spectrum of simplicial sets and, when we work with, for example, commutative algebras, these objects are always to be understood as referring to commutative algebras in the setting of symmetric spectra.
Recall that [2, Theorem 1.3] shows that for any spectrum X, the Morava module L K(n) (E n ∧ X) is a continuous G n -spectrum, where, as before, X is regarded as having the trivial G n -action and the Morava module has the diagonal G n -action. Then, in this note, we generalize equivalence (1.2) in the following way.
We quickly make a technical (but useful) comment about Theorem 1.3. By using cofibrant replacement in the S model structure on the category of symmetric spectra, given any spectrum Z, there is a weak equivalence Z c → Z in the usual stable model category of symmetric spectra, with Z c S-cofibrant (see [7,Section 5.3]). Thus, there is no loss of generality in Theorem 1.3 in requiring that X be S-cofibrant, so that the theorem can be thought of as being valid for an arbitrary spectrum X. Theorem 1.3 shows that the K(n)-localization of any (S-cofibrant) spectrum is the G n -homotopy fixed points of its Morava module, answering the relatively old question of how to show that every K(n)-local spectrum can be obtained from a homotopy fixed point construction involving E n and G n . The proof of Theorem 1.3 is given in Section 2.
We give Theorem 1.3 the desired (but generally unwieldy) "computational legs" with the following result.
Theorem 1.4. If X is any S-cofibrant spectrum, then the strongly convergent K(n)-local E n -Adams 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)) hGn ), from the E 2terms onward.
If X is a finite spectrum, then Theorem 1.4 reduces to [1,Theorem 8.2.5]. We refer the reader to [4, Appendix A] for an exposition of the aforementioned Adamstype spectral sequence. The descent spectral sequence that Theorem 1.4 refers to is defined in [1,Section 4.6]. The proof of Theorem 1.4 is given in Section 3.
Acknowledgements. The proof of Theorem 1.3 that appears in this note is a simplified version of an argument that relied more heavily on some results from [1]. Thus, the first author thanks Mark Behrens for various things that he learned from him during their collaboration on [1].

The proof of Theorem 1.3
We begin this section by establishing some notation. We let be a cofibrant replacement of L K(n) (S 0 ) in the model category of commutative symmetric ring spectra (see [8, the discussion just before Theorem 19.6]); the map c is a weak equivalence in the stable model category of symmetric spectra. (We need the cofibrant commutative symmetric ring spectrum S because later we will regard it as the ground ring of a profinite Galois extension.) Also, for the remainder of this section, we write K in place of K(n), so that our notation does not become too cumbersome. We will sometimes use the terminology of [9, Section 1], adapted to the K-local category, as in [4, Appendix A].
As in [1, Section 5.2], let Alg be the model category of discrete commutative G n -S-algebras: the objects of Alg are discrete G n -spectra that are also commutative Salgebras, and the morphisms are G n -equivariant maps of commutative S-algebras. Let (−) F : Alg → Alg be a fibrant replacement functor for the model category Alg , and let N ⊳ o G n denote an open normal subgroup of G n . Also, recall from [4] that each E dhN n -the commutative S-algebra that is written as E hN n in [4] and behaves like the N -homotopy fixed point spectrum of E n -is a G n /N -spectrum that is K-local. Then, as in [2] and [1], let by construction, F n is a discrete G n -spectrum and a commutative symmetric ring spectrum that is E(n)-local. Here, E(n) is the usual Johnson-Wilson spectrum, with Let X be any S-cofibrant spectrum. By [2,Theorem 9.7], where (F n ∧ X) hGn is the G n -homotopy fixed points of the discrete G n -spectrum F n ∧ X. Thus, to prove Theorem 1.3, we only have to show that Let L K (E n ) ∧(•+1) be the usual cosimplicial spectrum that is built from the unit map S 0 → E n and the multiplication E n ∧ E n → E n . Here, for each k ≥ 0, and there is the associated augmented resolution which is the canonical K-local E n -resolution of S. Given S-modules M and N , we let M ∧ S N denote their smash product in the category of S-modules. Then resolution (2.1) can be identified with the resolution since, for each k ≥ 0, Since E n ≃ L K (F n ) (this equivalence is due to [4], but the reader might find the proof of it in [2, Theorem 6.3] useful), resolution (2.2) can be identified with the resolution For the next step in our proof, we make a few recollections. We define the cosimplicial spectrum Map c (G • n , F n ) as in [1, Section 3.2]: if K is a profinite group, then Map c (K, F n ) = colim U⊳oK Map(K/U, F n ) ∼ = colim U⊳oK K/U F n ; Map c (G n , −) is a coaugmented comonad on the category of spectra; and, via the comonadic cobar construction, Map c (G • n , F n ) is the associated cosimplicial spectrum, which, in each codegree k, satisfies the isomorphism We recall that by [12,Theorem 5.4.4] and [1, pg. 5034], the work of [4] and [3] implies that F n is a K-local profinite G n -Galois extension of S of finite vcd (in the sense of [1]). Thus, as in the proof of [1, Proposition 6.3.1], there is an equivalence n , F n )) of cosimplicial spectra. (Here, if a map X • → Y • of cosimplicial spectra is a weak equivalence in each codegree, then we regard it as a weak equivalence of cosimplicial spectra.) Therefore, resolution (2.3) is equivalent to the resolution n , F n )) → · · · , and hence, resolution (2.4) can be regarded as the canonical K-local E n -resolution of S.
Below, to save space, we sometimes use the notation Y ∧ Z to denote L K (Y ∧Z), where Y and Z are arbitrary spectra, and, for the same reason, we occasionally write (−) K in place of L K (−). By smashing resolution (2.4) with X and then localizing with respect to K, it follows from [4,Remark A.9] that * X K (F n ) K ∧ X Map c (G n , F n ) K ∧ X Map c (G 2 n , F n ) K ∧ X · · · is a K-local E n -resolution of L K (X). Thus, the equivalent resolution , so that, since (2.5) is the resolution associated to the canonical map Φ is a weak equivalence, by [4,Corollary A.8].
Since F n is E(n)-local and L E(n) (−) is a smashing localization, F n ∧ X is also E(n)-local. Then the proof of Theorem 1.3 is finished by noting that where the first equivalence follows immediately from [1, Theorem 3. F n ∧ X ∼ = Map c (G k n , F n ) ∧ X, where the third equivalence (just above, between the two colimits) uses the fact that, since X is S-cofibrant, the functor (−) ∧ X preserves weak equivalences, by [7, Corollary 5.3.10].
3. The proof of Theorem 1.4 Let X be any S-cofibrant spectrum. In Section 2, we showed that resolution (2.5) is a K(n)-local E n -resolution of L K(n) (X). Thus, by [4, discussion preceding Proposition A.5], there is a map φ from the strongly convergent K(n)-local E n -Adams spectral sequence for π * (L K(n) (X)), which we denote by A E * , * r (X), to the homotopy spectral sequence for π * holim ∆ L K(n) (Map c (G • n , F n ) ∧ X) , which we denote by I E * , * r (X). Furthermore, by [4,Proposition A.5], the map φ of spectral sequences is an isomorphism, from the E 2 -terms onward.