HOMOTOPY FIXED POINTS FOR L_{K(n)}(E_{n} ∧ X) USING THE CONTINUOUS ACTION^{1}

DANIEL G. DAVIS^{2}

Abstract. Let K(n) be the nth Morava K-theory spectrum. Let E_{n} be the Lubin-Tate spectrum, which plays a central role in understanding L_{K(n)}(S^{0}), the K(n)-local sphere. For any spectrum X, define E^{∨}(X) to be the spectrum L_{K(n)}(E_{n} ∧ X). Let G be a closed subgroup of the profinite group G_{n}, the group of ring spectrum automorphisms of E_{n} in the stable homotopy category. We show that E^{∨}(X) is a continuous G-spectrum, with homotopy fixed point spectrum (E^{∨}(X))^{hG}. Also, we construct a descent spectral sequence with abutment π_{∗}((E^{∨}(X))^{hG}).

1. Introduction

Let p be a fixed prime. For each n ≥ 0, let K(n) be the nth Morava K-theory spectrum, where K(0) is the Eilenberg-Mac Lane spectrum Hℚ, and, for n ≥ 1, K(n)_{∗} = 𝔽_{p}[v_{n}, v_{n}^{−1}], where the degree of v_{n} is 2(p^{n}−1). Let X be a finite spectrum. There are maps L_{n}X → L_{n−1}X, where L_{n} denotes Bousfield localization with respect to K(0) ∨ K(1) ∨ ⋯ ∨ K(n). Then the chromatic convergence theorem [33, Theorem 7.5.7] says that X_{(p)} ≃ holim_{n≥0} L_{n}X, where X_{(p)} is the p-localization of X. Thus, to understand X_{(p)}, it is important that one understands each localization L_{n}X.

Henceforth, let n ≥ 1, and let L̂ denote Bousfield localization with respect to K(n). Then there is a homotopy pullback square [15]

\xymatrix{L_{n}X \ar[r] \ar[d] & L̂(X) \ar[d] \\ L_{n−1}(X_{(p)}) \ar[r] & L_{n−1}L̂(X),}

which shows that, to understand the localizations L_{n}X, it is very helpful to understand each L̂(X).

In attempting to understand L̂(X), one of the main tools is a certain spectral sequence, which we now recall. Let E_{n} be the Lubin-Tate spectrum with E_{n∗} =

Date: July 26, 2005. 1This version of the paper is identical to the version accepted for publication in the Journal of

Pure and Applied Algebra (except for minor changes required by a change in document format). The main differences between this version and the version dated 1/12/05 are: the present version is a little shorter; pg. 5 begins with two new paragraphs; Cor. 3.7 was added; H_{cont}^{s}(G; −) is used in places where H_{cts}^{s}(G; −) was used earlier (this is only a change in presentation and not a correction - see the last sentence of Def. 2.15); and there are some changes in notation.

2The author was supported by an NSF grant. Some revision of this paper was done while the author was visiting the Notre Dame math department.

W [[u_{1}, ..., u_{n−1}]][u^{±1}], where the degree of u is −2, and the complete power series ring over the Witt vectors W = W (𝔽_{p^{n}}) is in degree zero. Let G_{n} be the profinite group of ring spectrum automorphisms of E_{n} in the stable homotopy category [17, Thm. 1.4]. (There is an isomorphism G_{n} ≅ S_{n} ⋊ Gal, where S_{n} is the nth Morava stabilizer group - the automorphism group of the Honda formal group law Γ_{n} of height n over 𝔽_{p^{n}}, and Gal is the Galois group Gal(𝔽_{p^{n}}/𝔽_{p}) (see [38, Prop. 4]).) By [32] and [14, Prop. 7.4], Morava’s change of rings theorem yields a spectral sequence

H_{c}^{∗}(G_{n}; π_{∗}(E_{n} ∧ X)) ⇒ π_{∗}L̂(X), (1.1)

where the E_{2}-term is the continuous cohomology of G_{n}, with coefficients in the continuous G_{n}-module π_{∗}(E_{n} ∧ X) (see Definition 2.17). Thus, we see that it is critical to study the relationship between E_{n} and G_{n}.

The above action of G_{n} on π_{∗}(E_{n}∧X) is induced by a point-set level action of G_{n} on E_{n} (work of Goerss and Hopkins ([12], [9]), and Hopkins and Miller [34]). Let G be a closed subgroup of G_{n}. Using the G_{n}-action on E_{n}, Devinatz and Hopkins [5] construct spectra E_{n}^{dhG} with spectral sequences

H_{c}^{s}(G; π_{t}(E_{n} ∧ X)) ⇒ π_{t−s}(E_{n}^{dhG} ∧ X). (1.2)

Also, they show that E_{n}^{dhG_{n}} ≃ L̂(S^{0}), so that E_{n}^{dhG_{n}} ∧ X ≃ L̂(X), and (1.1) is a special case of (1.2).

We compare the spectrum E_{n}^{dhG} and spectral sequence (1.2) with constructions for homotopy fixed point spectra. When K is a discrete group and Y is a Kspectrum of topological spaces, there is a homotopy fixed point spectrum Y^{hK} = Map_{K}(EK_{+}, Y), where EK is a free contractible K-space. Also, there is a descent spectral sequence

E_{2}^{s,t} = H^{s}(K; π_{t}(Y)) ⇒ π_{t−s}(Y^{hK}),

where the E_{2}-term is group cohomology [29, §1.1].

Now let K be a profinite group. If S is a K-set, then S is a discrete K-set if the action map K × S → S is continuous, where S is given the discrete topology. Then, a discrete K-spectrum Y is a K-spectrum of simplicial sets, such that each simplicial set Y_{k} is a simplicial discrete K-set (that is, for each l ≥ 0, Y_{k,l} is a discrete K-set, and all the face and degeneracy maps are K-equivariant). Then, due to work of Jardine (e.g. [21], [22], [23], [24]) and Thomason [41], as explained in Sections 5 and 7, there is a homotopy fixed point spectrum Y^{hK} defined with respect to the continuous action of K, and, in nice situations, a descent spectral sequence

H_{c}^{s}(K; π_{t}(Y)) ⇒ π_{t−s}(Y^{hK}),

where the E_{2}-term is the continuous cohomology of K with coefficients in the discrete K-module π_{t}(Y).

Notice that we use the notation E_{n}^{dhG} for the construction of Devinatz and Hopkins (which they denote as E_{n}^{hG} in [5]), and (−)^{hK} for homotopy fixed points with respect to a continuous action, although henceforth, when K is finite and Y is a K-spectrum of topological spaces, we write Y^{h′K} for holim_{K} Y, which is an equivalent definition of the homotopy fixed point spectrum Map_{K}(EK_{+}, Y).

HOMOTOPY FIXED POINTS FOR L_{K(n)}(E_{n} ∧ X) 3

After comparing the spectral sequence for E_{n}^{dhG} ∧ X with the descent spectral sequence for Y^{hK}, E_{n} ∧ X appears to be a continuous G_{n}-spectrum with “descent” spectral sequences for “homotopy fixed point spectra” E_{n}^{dhG} ∧ X. Indeed, we apply [5] to show that E_{n} ∧ X is a continuous G_{n}-spectrum; that is, E_{n} ∧ X is the homotopy limit of a tower of fibrant discrete G_{n}-spectra. Using this continuous action, we define the homotopy fixed point spectrum (E_{n} ∧ X)^{hG} and construct its descent spectral sequence.

In more detail, G_{n} acts on the K(n)-local spectrum E_{n} through maps of commutative S-algebras. The spectrum E_{n}^{dhG}, a K(n)-local commutative S-algebra, is referred to as a “homotopy fixed point spectrum” because it has the following desired properties: (a) spectral sequence (1.2), which has the form of a descent spectral sequence, exists; (b) when G is finite, there is a weak equivalence E_{n}^{dhG} → E_{n}^{h′G}, and the descent spectral sequence for E_{n}^{h′G} is isomorphic to spectral sequence (1.2) (when X = S^{0}) [5, Thm. 3]; and (c) E_{n}^{dhG} is an N(G)/G-spectrum, where N(G) is the normalizer of G in G_{n} [5, pg. 5]. These properties suggest that G_{n} acts on E_{n} in a continuous sense.

However, in [5], the G_{n}-action on E_{n} is not proven to be continuous, and E_{n}^{dhG} is not defined with respect to a continuous G-action. Also, when G is profinite, homotopy fixed points should always be the total right derived functor of fixed points, in some sense, and [5] does not show that the “homotopy fixed point spectrum” E_{n}^{dhG} can be obtained through such a total right derived functor.

After introducing some notation, we state the main results of this paper. Let BP be the Brown-Peterson spectrum with BP_{∗} = ℤ_{(p)}[v_{1}, v_{2}, ...], where the degree of v_{i} is 2(p^{i} − 1). The ideal (p^{i_{0}}, v_{1}^{i_{1}}, ..., v_{n−1}^{i_{n−1}}) ⊂ BP_{∗} is denoted by I; M_{I} is the corresponding generalized Moore spectrum M(p^{i_{0}}, v_{1}^{i_{1}}, ..., v_{n−1}^{i_{n−1}}), a spectrum with trivial G_{n}-action. Given I, M_{I} need not exist; however, enough exist for our constructions. Each M_{I} is a finite type n spectrum with BP_{∗}(M_{I}) ≅ BP_{∗}/I. The set {i_{0}, ..., i_{n−1}} of superscripts varies so that there is a family of ideals {I}. ([3, §4], [19, §4], and [27, Prop. 3.7] provide details for our statements about the spectra M_{I}.) The map r: BP_{∗} → E_{n∗} - defined by r(v_{i}) = u_{i}u^{1−p^{i}}, where u_{n} = 1 and u_{i} = 0, when i > n - makes E_{n∗} a BP_{∗}-module. By the Landweber exact functor theorem, π_{∗}(E_{n} ∧ M_{I}) ≅ E_{n∗}/I.

The collection {I} contains a descending chain of ideals {I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ ⋯ }, such that there exists a corresponding tower of generalized Moore spectra

{M_{I_{0}} ← M_{I_{1}} ← M_{I_{2}} ← ⋯ }.

In this paper, the functors lim_{I} and holim_{I} are always taken over the tower of ideals {I_{i}}, so that lim_{I} and holim_{I} are really lim_{I_{i}} and holim_{I_{i}}, respectively. Also, in this paper, if {X_{α}}_{α} is a diagram of spectra (even if each X_{α} has additional structure), then holim_{α} X_{α} always denotes the version of the homotopy limit of spectra that is constructed levelwise in S, the category of simplicial sets, as defined in [2] and [41, 5.6].

As in [5, (1.4)], let G_{n} = U_{0} ⪈ U_{1} ⪈ ⋯ ⪈ U_{i} ⪈ ⋯ be a descending chain of open normal subgroups, such that ⋂_{i} U_{i} = {e} and the canonical map G_{n} → lim_{i} G_{n}/U_{i} is a homeomorphism. We define

F_{n} = colim_{i} E_{n}^{dhU_{i}}.

Then the key to getting our work started is knowing that

E_{n} ∧ M_{I} ≃ F_{n} ∧ M_{I},

and thus, E_{n} ∧ M_{I} has the homotopy type of the discrete G_{n}-spectrum F_{n} ∧ M_{I}. This result (Corollary 6.5) is not difficult, thanks to [5].

Given a tower {Z_{I}} of discrete G_{n}-spectra, there is a tower {(Z_{I})_{f}}, with G_{n}- equivariant maps Z_{I} → (Z_{I})_{f} that are weak equivalences, and (Z_{I})_{f} is a fibrant discrete G_{n}-spectrum (see Definition 4.1). For the remainder of this section, X is any spectrum with trivial G_{n}-action, and, throughout this paper,

E^{∨}(X) = L̂(E_{n} ∧ X).

We use ≅ to signify an isomorphism in the stable homotopy category.

Theorem 1.3. As the homotopy limit of a tower of fibrant discrete G_{n}-spectra, E_{n} ≅ holim_{I}(F_{n} ∧ M_{I})_{f} is a continuous G_{n}-spectrum. Also, for any spectrum X, E^{∨}(X) ≅ holim_{I}(F_{n} ∧ M_{I} ∧ X)_{f} is a continuous G_{n}-spectrum.

We define homotopy fixed points for towers of discrete G-spectra; we show that these homotopy fixed points are the total right derived functor of fixed points in the appropriate sense; and we construct the associated descent spectral sequence. This enables us to define the homotopy fixed point spectrum (E^{∨}(X))^{hG}, using the continuous G_{n}-action, and construct its descent spectral sequence. More specifically, we have the following results.

Definition 1.4. Given a profinite group G, let O_{G} be the orbit category of G. The objects of O_{G} are the continuous left G-spaces G/K, for all K closed in G, and the morphisms are the continuous G-equivariant maps.

Let Spt be the model category (spectra)^{stable} of Bousfield-Friedlander spectra.

Theorem 1.5. There is a functor P: (O_{G_{n}})^{op} → Spt, defined by P (G_{n}/G) = E_{n}^{hG}, where G is any closed subgroup of G_{n}.

We also show that the G-homotopy fixed points of E^{∨}(X) can be obtained by taking the K(n)-localization of the G-homotopy fixed points of the discrete Gspectrum (F_{n} ∧ X). This result shows that the spectrum F_{n} is an interesting spectrum that is worth further study.

Theorem 1.6. For any closed subgroup G and any spectrum X, there is an isomorphism (E^{∨}(X))^{hG} ≅ L̂((F_{n} ∧ X)^{hG}). In particular, E_{n}^{hG} ≅ L̂(F_{n}^{hG}).

Theorem 1.7. Let G be a closed subgroup of G_{n} and let X be any spectrum. Then there is a conditionally convergent descent spectral sequence

E_{2}^{s,t} ⇒ π_{t−s}((E^{∨}(X))^{hG}). (1.8)

If the tower of abelian groups {π_{t}(E_{n} ∧ M_{I} ∧ X)}_{I} satisfies the Mittag-Leffler condition, for each t ∈ ℤ, then E_{2}^{s,t} ≅ H_{cont}^{s}(G; {π_{t}(E_{n} ∧ M_{I} ∧ X)}) (see Definition 2.15). If X is a finite spectrum, then (1.8) has the form

H_{c}^{s}(G; π_{t}(E_{n} ∧ X)) ⇒ π_{t−s}((E_{n} ∧ X)^{hG}), (1.9)

where the E_{2}-term is the continuous cohomology of (1.2).

Also, Theorem 9.9 shows that, when X is finite, (E_{n} ∧ X)^{hG} ≅ E_{n}^{hG} ∧ X, so that descent spectral sequence (1.9) has the same form as spectral sequence (1.2). It is natural to wonder if these two spectral sequences are isomorphic to each other. Also, the spectra E_{n}^{dhG} and E_{n}^{hG} should be the same. We plan to say more about the relationship between E_{n}^{dhG} and E_{n}^{hG} and their associated spectral sequences in future work.

While reading this Introduction (and taking a quick look at §9), the reader might notice that, due to the definition of F_{n}, we use the “homotopy fixed point spectra” E_{n}^{dhU_{i}} to construct the homotopy fixed point spectra E_{n}^{hG}. We discuss the degree to which this method is circular. To obtain the results of this paper, we require a tower {E_{n}/I}_{I} of discrete G_{n}-spectra such that holim_{I}(E_{n}/I)_{f} is the G_{n}-spectrum E_{n}, and, for each I, E_{n}/I and E_{n} ∧ M_{I} have the same stable homotopy type. Any tower with the stated properties will work (and, given such a tower, one defines F_{n} = colim_{i} holim_{I}(E_{n}/I)^{hU_{i}}). We obtained such a tower by using the spectra E_{n}^{dhU_{i}} to form the tower {F_{n} ∧ M_{I}}_{I}.

We believe that one could probably use obstruction theory to construct the tower {E_{n}/I}_{I}. This would yield the above results independently of [5], so that, presumably, [5] is not required to build (E^{∨}(X))^{hG}. However, to date, no one has obtained the requisite tower using obstruction theory, and we suspect that such work would be quite difficult.

We outline the contents of this paper. In §2, we establish some notation and terminology, and we provide some background material. In §3, we study the model category of discrete G-spectra. In §4, we study towers of discrete G-spectra and give a definition of continuous G-spectrum. Homotopy fixed points for discrete G-spectra are defined in §5, and §6 shows that E_{n} is a continuous G_{n}-spectrum, proving the first half of Theorem 1.3. In §7, two useful models of the G-homotopy fixed point spectrum are constructed, when G has finite virtual cohomological dimension. In §8, we define homotopy fixed points for towers of discrete G-spectra, build a descent spectral sequence in this setting, and show that these homotopy fixed points are a total right derived functor, in the appropriate sense. In §9, we complete the proof of Theorem 1.3, study (E^{∨}(X))^{hG}, and prove Theorems 1.5 and 1.6. In §10, we consider the descent spectral sequence for (E^{∨}(X))^{hG} and prove Theorem 1.7.

Acknowledgements. This paper is a development of part of my thesis. I am very grateful to my thesis advisor, Paul Goerss, for many helpful conversations and useful suggestions regarding this paper. Also, I thank Ethan Devinatz for very helpful answers to my questions about his work [5] with Mike Hopkins. I am grateful to Halvard Fausk, Christian Haesemeyer, Rick Jardine, and Charles Rezk for useful conversations. I extend my appreciation to Charles Weibel and the referees for many helpful comments.

2. Notation, Terminology, and Preliminaries

We begin by establishing some notation and terminology that will be used throughout the paper. Ab is the category of abelian groups. Outside of Ab, all groups are assumed to be profinite, unless stated otherwise. For a group G, we write G ≅ lim_{N} G/N, the inverse limit over the open normal subgroups. The notation H <_{c} G means that H is a closed subgroup of G. We use G to denote arbitrary profinite groups and, specifically, closed subgroups of G_{n}.

Let C be a category. A tower {C_{i}} of objects in C is a diagram in C of the form ⋯ → C_{i} → C_{i−1} → ⋯ → C_{1} → C_{0}. We always use Bousfield-Friedlander spectra [1], except when another category of spectra is specified. If C is a model category, then Ho(C) is its homotopy category. The phrase “stable category” always refers to Ho(Spt).

In S, the category of simplicial sets, S^{n} = Δ^{n}/∂Δ^{n} is the n-sphere. Given a spectrum X, X^{(0)} = S^{0}, and for j ≥ 1, X^{(j)} = X ∧ X ∧ ⋯ ∧ X, with j factors.

Definition 2.1. [16, Def. 1.3.1] Let C and D be model categories. The functor F: C → D is a left Quillen functor if F is a left adjoint that preserves cofibrations and trivial cofibrations. The functor P: D → C is a right Quillen functor if P is a right adjoint that preserves fibrations and trivial fibrations. Also, if F and P are an adjoint pair and left and right Quillen functors, respectively, then (F, P) is a Quillen pair for the model categories (C, D).

Recall [16, Lemma 1.3.10] that a Quillen pair (F, P) yields total left and right derived functors LF and RP, respectively, which give an adjunction between the homotopy categories Ho(C) and Ho(D).

We use Map_{c}(G, A) = Γ_{G}(A) to denote the set of continuous maps from G to the topological space A, where A is often a set, equipped with the discrete topology, or a discrete abelian group. Instead of Γ_{G}(A), sometimes we write just Γ(A), when the G is understood from context. Let (Γ_{G})^{k}(A) denote (Γ_{G} Γ_{G} ⋯ Γ_{G})(A), the application of Γ_{G} to A, iteratively, k + 1 times, where k ≥ 0. Let G^{k} be the k-fold product of G and G^{0} = ∗. Then, if A is a discrete set (discrete abelian group), there is a G-equivariant isomorphism (Γ_{G})^{k}(A) ≅ Map_{c}(G^{k+1}, A) of discrete G-sets (modules), where Map_{c}(G^{k+1}, M) has the G-action defined by (g′·f)(g_{1}, ..., g_{k+1}) = f(g_{1}g′, g_{2}, g_{3}, ..., g_{k+1}). Also, we often write Γ_{G}^{k}(A), or Γ^{k}A, for Map_{c}(G^{k}, A).

Let A be a discrete abelian group. Then Map_{c}^{ℓ}(G_{n}^{k}, A) is the discrete G_{n}- module of continuous maps G_{n}^{k} → A, with action defined by (g′ · f)(g_{1}, ..., g_{k}) = f((g′)^{−1}g_{1}, g_{2}, g_{3}, ..., g_{k}). Note that there is a G_{n}-equivariant isomorphism of discrete G_{n}-modules

p: Map_{c}^{ℓ}(G_{n}^{k}, A) → Map_{c}(G_{n}^{k}, A),

which is defined by p(f)(g_{1}, g_{2}, ..., g_{k}) = f(g_{1}^{−1}, g_{2}, ..., g_{k}). Map_{c}^{ℓ}(G_{n}^{k}, A) is also defined when A is an inverse limit of discrete abelian groups.

By a topological G-module, we mean an abelian Hausdorff topological group that is a G-module, with a continuous G-action. Note that if M = lim_{i} M_{i} is the limit of a tower {M_{i}} of discrete G-modules, then M is a topological G-module.

For the remainder of this section, we recall some frequently used facts and discuss background material, to help get our work started.

As explained in [5], Goerss and Hopkins ([12], [9]), building on work by Hopkins and Miller [34], proved that the action of G_{n} on E_{n} is by maps of commutative Salgebras. Previously, Hopkins and Miller had shown that G_{n} acts on E_{n} by maps of A_{∞}-ring spectra. However, the continuous action presented here is not structured. As already mentioned, the starting point for the continuous action is the spectrum F_{n}∧M_{I}, which is not known to be an A_{∞}-ring object in the category of discrete G_{n}- spectra. Thus, we work in the unstructured category Spt of Bousfield-Friedlander spectra of simplicial sets, and the continuous action is simply by maps of spectra.

As mentioned above, [5] is written using E_{∞}, the category of commutative Salgebras, and M_{S}, the category of S-modules (see [7]). However, [18, §4.2], [28, §14, §19], and [36, pp. 529-530] show that M_{S} and Spt are Quillen equivalent model categories [16, §1.3.3]. Thus, we can import the results of Devinatz and Hopkins from M_{S} into Spt. For example, [5, Thm. 1] implies the following result, where R_{G_{n}}^{+} is the category whose objects are finite discrete left G_{n}-sets and G_{n} itself (a continuous profinite left G_{n}-space), and whose morphisms are continuous G_{n}-equivariant maps.

Theorem 2.2 (Devinatz, Hopkins). There is a presheaf of K(n)-local spectra F: (R_{G_{n}}^{+})^{op} → Spt, such that (a) F (G_{n}) = E_{n}; (b) for U an open subgroup of G_{n}, E_{n}^{dhU} := F (G_{n}/U); and (c) F (∗) = E_{n}^{dhG_{n}} ≃ L̂S^{0}.

Now we define a spectrum that is essential to our constructions.

Definition 2.3. Let F_{n} = colim_{i} E_{n}^{dhU_{i}}, where the direct limit is in Spt. Because Hom_{G_{n}}(G_{n}/U_{i}, G_{n}/U_{i}) ≅ G_{n}/U_{i}, F makes E_{n}^{dhU_{i}} a G_{n}/U_{i}-spectrum. Thus, F_{n} is a G_{n}-spectrum, and the canonical map η: F_{n} → E_{n} is G_{n}-equivariant.

The following useful fact is stated in [5, pg. 9] (see also [38, Lemma 14]).

Theorem 2.4. For j ≥ 0, let X be a finite spectrum and regard L̂(E_{n}^{(j+1)} ∧ X) as a G_{n}-spectrum, where G_{n} acts only on the leftmost factor of the smash product. Then there is a G_{n}-equivariant isomorphism

π_{∗}(L̂(E_{n}^{(j+1)} ∧ X)) ≅ Map_{c}^{ℓ}(G_{n}^{j}, π_{∗}(E_{n} ∧ X)).

We review some frequently used facts about the functor L_{n} and homotopy limits of spectra. First, L_{n}, defined earlier, can be equivalently defined as Bousfield localization with respect to the spectrum E(n), where E(n)_{∗} =ℤ_{(p)}[v_{1}, ..., v_{n−1}][v_{n}, v_{n}^{−1}]. Also, L_{n} is smashing, e.g. L_{n}X ≃ X ∧ L_{n}S^{0}, for any spectrum X, and E(n)- localization commutes with homotopy direct limits [33, Thms. 7.5.6, 8.2.2]. Note that this implies that F_{n} is E(n)-local.

Definition 2.5. If ⋯ → X_{i} → X_{i−1} → ⋯ → X_{1} → X_{0} is a tower of spectra such that each X_{i} is fibrant in Spt, then {X_{i}} is a tower of fibrant spectra.

If {X_{i}} is a tower of fibrant spectra, then there is a short exact sequence

0 → lim^{1}_{i} π_{m+1}(X_{i}) → π_{m}(holim_{i} X_{i}) → lim_{i} π_{m}(X_{i}) → 0.

Also, if each map in the tower is a fibration, the map lim_{i} X_{i} → holim_{i} X_{i} is a weak equivalence. If J is a small category and the functor P: J → Spt is a diagram of spectra, such that P_{j} is fibrant for each j ∈ J, then holim_{j} P_{j} is a fibrant spectrum.

Definition 2.6. There is a functor (−)_{f} : Spt → Spt, such that, given Y in Spt, Y_{f} is a fibrant spectrum, and there is a natural transformation id_{Spt} → (−)_{f}, such that, for any Y, the map Y → Y_{f} is a trivial cofibration. For example, if Y is a G-spectrum, then Y_{f} is also a G-spectrum, and the map Y → Y_{f} is G-equivariant.

The following statement says that smashing with a finite spectrum commutes with homotopy limits.

Lemma 2.7 ([42, pg. 96]). Let J be a small category, {Z_{j}} a J-shaped diagram of fibrant spectra, and let Y be a finite spectrum. Then the composition

(holim_{j} Z_{j}) ∧ Y → holim_{j}(Z_{j} ∧ Y) → holim_{j}(Z_{j} ∧ Y)_{f}

is a weak equivalence.

We recall the result that is used to build towers of discrete G-spectra.

Theorem 2.8 ([15, §2], [4, Remark 3.6]). If X is an E(n)-local spectrum, then, in the stable category, there is an isomorphism

L̂X ≅ holim_{I}(X ∧ M_{I})_{f}.

Lemma 2.9 ([19, Lemma 7.2]). If X is any spectrum, and Y is a finite spectrum of type n, then L̂(X ∧ Y) ≃ L̂(X) ∧ Y ≃ L_{n}(X) ∧ Y.

We recall some useful facts about compact p-adic analytic groups. Since S_{n} is compact p-adic analytic, and G_{n} is an extension of S_{n} by Gal, G_{n} is a compact p-adic analytic group [37, Cor. of Thm. 2]. Any closed subgroup of a compact p-adic analytic group is also compact p-adic analytic [6, Thm. 9.6]. Also, since the subgroup in S_{n} of strict automorphisms is finitely generated and pro-p, [35, pp. 76, 124] implies that all subgroups in G_{n} of finite index are open.

Let the profinite group G be a compact p-adic analytic group. Then G contains an open subgroup H, such that H is a pro-p group with finite cohomological pdimension; that is, cd_{p}(H) = m, for some non-negative integer m (see [25, 2.4.9] or the exposition in [39]). Since H is pro-p, cd_{q}(H) = 0, whenever q is a prime different from p [45, Prop. 11.1.4]. Also, if M is a discrete H-module, then, for s ≥ 1, H_{c}^{s}(H; M) is a torsion abelian group [35, Cor. 6.7.4]. These facts imply that, for any discrete H-module M, H_{c}^{s}(H; M) = 0, whenever s > m + 1. We express this conclusion by saying that G has finite virtual cohomological dimension and we write vcd(G) ≤ m. Also, if K is a closed subgroup of G, H ∩ K is an open pro-p subgroup of K with cd_{p}(H ∩ K) ≤ m, so that vcd(K) ≤ m, and thus, m is a uniform bound independent of K.

Now we state various results related to towers of abelian groups and continuous cohomology. The lemma below follows from the fact that an exact additive functor preserves images.

Lemma 2.10. Let F: Ab → Ab be an exact additive functor. If {A_{i}}_{i≥0} is a tower of abelian groups that satisfies the Mittag-Leffler condition, then so does the tower {F (A_{i})}.

Remark 2.11. Let G be a profinite group. The functor Map_{c}(G, −): Ab → Ab, which sends A to Map_{c}(G, A), is defined by giving A the discrete topology. The isomorphism Map_{c}(G, A) ≅ colim_{N} ∏ _{G/N} A shows that Map_{c}(G, −) is an exact additive functor. Later, we will use Lemma 2.10 with this functor.

The next lemma is a consequence of the fact that limits in Ab and in topological spaces are created in Sets.

Lemma 2.12. Let M = lim_{α} M_{α} be an inverse limit of discrete abelian groups, so that M is an abelian topological group. Let H be any profinite group. Then Map_{c}(H, M) → lim_{α} Map_{c}(H, M_{α}) is an isomorphism of abelian groups.

The lemma below follows from the fact that π_{t}(E_{n} ∧ M_{I}) ≅ π_{t}(E_{n})/I is finite.

Lemma 2.13. If X is a finite spectrum and t is any integer, then the abelian group π_{t}(E_{n} ∧ M_{I} ∧ X) is finite.

Corollary 2.14 ([14, pg. 116]). If X is a finite spectrum, then π_{t}(E_{n} ∧ X) ≅ lim_{I} π_{t}(E_{n} ∧ M_{I} ∧ X).

We recall the definition of a second version of continuous cohomology [20].

Definition 2.15. Let C_{G} be the category of discrete G-modules and tow(C_{G}) the category of towers in C_{G}. Then H_{cont}^{s}(G; {M_{i}}), the continuous cohomology of G with coefficients in the tower {M_{i}}, is the sth right derived functor of the left exact functor lim_{i}(−)^{G}: tow(C_{G}) → Ab, which sends {M_{i}} to lim_{i} M_{i}^{G}. By [20, Theorem 2.2], if the tower of abelian groups {M_{i}} satisfies the Mittag-Leffler condition, then H_{cont}^{s}(G; {M_{i}}) ≅ H_{cts}^{s}(G; lim_{i} M_{i}), for s ≥ 0, where H_{cts}^{s}(G; M) is the cohomology of continuous cochains with coefficients in the topological G-module M (see [40, §2]).

Theorem 2.16 ([20, (2.1)]). Let {M_{i}}_{i≥0} be a tower of discrete G-modules satisfying the Mittag-Leffler condition. Then, for each s ≥ 0, there is a short exact sequence

0 → lim^{1}_{i} H_{c}^{s−1}(G; M_{i}) → H_{cont}^{s}(G; {M_{i}}) → lim_{i} H_{c}^{s}(G; M_{i}) → 0,

where H_{c}^{−1}(G; −) = 0.

Definition 2.17. Let G be a closed subgroup of G_{n}, let X be a finite spectrum, and let I_{n} = (p, u_{1}, ..., u_{n−1}) ⊂ E_{n∗}. Then, by [5, Rk. 1.3],

π_{t}(E_{n} ∧ X) ≅ lim_{k} π_{t}(E_{n} ∧ X)/I_{n}^{k}π_{t}(E_{n} ∧ X)

is a profinite continuous ℤ_{p}[[G]]-module (since it is the inverse limit of finite discrete G-modules), and the definition of H_{c}^{s}(G; π_{t}(E_{n} ∧ X)) is given by

H_{c}^{s}(G; π_{t}(E_{n} ∧ X)) = lim_{k} H_{c}^{s}(G; π_{t}(E_{n} ∧ X)/I_{n}^{k}π_{t}(E_{n} ∧ X)).

By [5, Rk. 1.3], for s ≥ 0, there are isomorphisms

H_{c}^{s}(G; π_{t}(E_{n}∧X)) ≅ H_{cts}^{s}(G; π_{t}(E_{n}∧X)) ≅ H_{cont}^{s}(G; {π_{t}(E_{n}∧X)/I_{n}^{k}π_{t}(E_{n} ∧ X)}).

3. The model category of discrete G-spectra

In this section, we begin explaining the theory of homotopy fixed points for discrete G-spectra. We note that much of this theory (in this section and in Sections 5, 7, and 8, through Theorem 8.5) is already known, in some form, especially in the work of Jardine mentioned above, in the excellent article [29], by Mitchell (see also the opening remark of [31, §5]), and in Goerss’s paper [11]. However, since the above theory has not been explained in detail before, using the language of homotopy fixed points for discrete G-spectra, we give a presentation of it.

A pointed simplicial discrete G-set is a pointed simplicial set that is a simplicial discrete G-set, such that the G-action fixes the basepoint.

Definition 3.1. A discrete G-spectrum X is a spectrum of pointed simplicial sets X_{k}, for k ≥ 0, such that each simplicial set X_{k} is a pointed simplicial discrete G-set, and each bonding map S^{1} ∧ X_{k} → X_{k+1} is G-equivariant (S^{1} has trivial Gaction). Let Spt_{G} denote the category of discrete G-spectra, where the morphisms are G-equivariant maps of spectra.

As with discrete G-sets, if X ∈ Spt_{G}, there is a G-equivariant isomorphism X ≅ colim_{N} X^{N}. Also, a discrete G-spectrum X is a continuous G-spectrum since, for all k, l ≥ 0, the set X_{k,l} is a continuous G-space with the discrete topology, and all the face and degeneracy maps are (trivially) continuous.

Definition 3.2. As in [23, §6.2], let G − Sets_{df} be the canonical site of finite discrete G-sets. The pretopology of G−Sets_{df} is given by covering families of the form {f_{α}: S_{α} → S}, a finite set of G-equivariant functions in G−Sets_{df} for a fixed S ∈ G−Sets_{df}, such that ∐ _{α} S_{α} → S is a surjection.

Let Shv be the Grothendieck topos consisting of sheaves of sets on the site G−Sets_{df}. The topos Shv has a unique point u: Sets → Shv. The left and right adjoints, respectively, of the topos morphism u are

u^{∗}: Shv → Sets, F ↦ colim_{N} F(G/N), and

u_{∗}: Sets → Shv, X ↦ Hom_{G}(−, Map_{c}(G, X))

[23, Rk. 6.25]. The G-action on the discrete G-set Map_{c}(G, X) is defined by (g · f)(g′) = f(g′g), for g, g′ in G, and f a continuous map G → X, where X is given the discrete topology.

Recall that the topos Shv is equivalent to T_{G}, the category of discrete Gsets (see [23, Prop. 6.20] or [26, III-9, Thm. 1], for example). The functor Map_{c}(G, −): Sets → T_{G} prolongs to the functor Map_{c}(G, −): Spt → Spt_{G}. Thus, if X is a spectrum, then Map_{c}(G, X) ≅ colim_{N} ∏ _{G/N} X is the discrete G-spectrum with (Map_{c}(G, X))_{k} = Map_{c}(G, X_{k}), where Map_{c}(G, X_{k}) is a pointed simplicial set, with l-simplices Map_{c}(G, X_{k,l}) and basepoint G → ∗, where X_{k,l} is regarded as a discrete set. The G-action on Map_{c}(G, X) is defined by the G-action on the sets Map_{c}(G, X_{k,l}). It is not hard to see that Map_{c}(G, −) is right adjoint to the forgetful functor U: Spt_{G} → Spt.

Note that if X ∈ Spt_{G}, then Hom_{G}(−, X): (G−Sets_{df})^{op} → Spt is a presheaf, such that, for S ∈ G − Sets_{df}, Hom_{G}(S, X) ∈ Spt satisfies Hom_{G}(S, X)_{k,l} = Hom_{G}(S, X_{k,l}), a pointed set with basepoint S → ∗.

Let ShvSpt be the category of sheaves of spectra on the site G−Sets_{df}. A sheaf of spectra F is a presheaf F: (G−Sets_{df})^{op} → Spt, such that, for any S ∈ G−Sets_{df} and any covering family {f_{α}: S_{α} → S}, the usual diagram (of spectra) is an equalizer. Equivalently, a sheaf of spectra F consists of pointed simplicial sheaves F^{n}, together with pointed maps of simplicial presheaves σ: S^{1} ∧ F^{n} → F^{n+1}, for n ≥ 0, where S^{1} is the constant simplicial presheaf. A morphism between sheaves of spectra is a natural transformation between the underlying presheaves.

We equip the category PreSpt of presheaves of spectra on the site G−Sets_{df} with the stable model category structure (see [22], [23, §2.3]). Recall that, in this model category structure, a map h: F → G of presheaves of spectra is a weak equivalence if and only if the associated map of stalks u^{∗}(F) → u^{∗}(G) is a weak equivalence of spectra, where u^{∗}(F) = colim_{N} F(G/N), by prolongation.

Definition 3.3. In the stable model category structure, fibrant presheaves are often referred to as globally fibrant, and if F → G is a weak equivalence of presheaves, with G globally fibrant, then G is a globally fibrant model for F.

We recall the following fact, which is especially useful when S = ∗.

Lemma 3.4. Let S ∈ G−Sets_{df}. The S-sections functor PreSpt → Spt, defined by F ↦ F(S), preserves fibrations, trivial fibrations, and weak equivalences between fibrant objects.

Proof. The S-sections functor has a left adjoint, obtained by left Kan extension, that preserves cofibrations and weak equivalences. See [23, pg. 60] and [29, Cor. 3.16] for the details. □

Let L^{2} denote the sheafification functor for presheaves of sets, simplicial presheaves, and presheaves of spectra, so that L^{2}F ≅ Hom_{G}(−, u^{∗}(F)), by [23, Cor. 6.22]. Then i: ShvSpt → PreSpt, the inclusion functor, is right adjoint to L^{2}. By [10, Rk. 3.11], ShvSpt has the following model category structure. A map h: F → G of sheaves of spectra is a weak equivalence (fibration) if and only if i(f) is a weak equivalence (fibration) of presheaves. Also, h is a cofibration of sheaves of spectra if the following holds:

(1) the map h^{0} : F^{0} → G^{0} is a cofibration of simplicial presheaves; and

(2) for each n ≥ 0, the canonical map L^{2}((S^{1} ∧ G^{n}) ∪_{S^{1}∧F^{n}} F^{n+1}) → G^{n+1} is a cofibration of simplicial presheaves.

Since i preserves weak equivalences and fibrations, (L^{2}, i) is a Quillen pair for (PreSpt, ShvSpt). Thus, for F ∈ PreSpt, F → L^{2}F is a weak equivalence, and Ho(PreSpt) ≅ Ho(ShvSpt) is a Quillen equivalence.

Since Shv ≅ T_{G}, prolongation gives an equivalence of categories ShvSpt ≅ Spt_{G}, via the functors u^{∗}: ShvSpt → Spt_{G}, and R: Spt_{G} → ShvSpt, where R(X) = Hom_{G}(−, X).

Definition 3.5. For the remainder of this paper, if X is a discrete G-set, a simplicial discrete G-set, or a discrete G-spectrum, we let u^{∗}X = colim_{N} X^{N}.

Exploiting the above equivalence, we make Spt_{G} a model category in the following way. Define a map f of discrete G-spectra to be a weak equivalence (fibration) if and only if Hom_{G}(−, f) is a weak equivalence (fibration) of sheaves of spectra. Also, define f to be a cofibration if and only if f has the left lifting property with respect to all trivial fibrations. Thus, f is a cofibration if and only if Hom_{G}(−, f) is a cofibration in ShvSpt. Using this, it is immediate that Spt_{G} is a model category, and there is a Quillen equivalence Ho(ShvSpt) ≅ Ho(Spt_{G}).

In the theorem below, we define the model category structure of Spt_{G} without reference to sheaves of spectra. This extends the model category structure on the category S_{G} (the category of simplicial objects in T_{G}), that is given in [11, Thm. 1.12], to Spt_{G}.

Theorem 3.6. Let f: X → Y be a map in Spt_{G}. Then f is a weak equivalence (cofibration) in Spt_{G} if and only if f is a weak equivalence (cofibration) in Spt.

Proof. For weak equivalences, the statement is clearly true. Assume that f is a cofibration in Spt_{G}. Since Hom_{G}(−, X_{0}) → Hom_{G}(−, Y_{0}) is a cofibration of simplicial presheaves, evaluation at G/N implies that X_{0}^{N} → Y_{0}^{N} is a cofibration in S. Thus, X_{0} ≅ u^{∗}X_{0} → u^{∗}Y_{0} ≅ Y_{0} is a cofibration in S.

Since colimits commute with pushouts,

Hom_{G}(−, (S^{1} ∧ Y_{n}) ∪_{S^{1}∧X_{n}} X_{n+1}) → Hom_{G}(−, Y_{n+1})

is a cofibration of simplicial presheaves, and hence, the map of simplicial sets u^{∗}((S^{1} ∧ Y_{n}) ∪_{S^{1}∧X_{n}} X_{n+1}) → Y_{n+1} is a cofibration.

Let W be a simplicial pointed discrete G-set. Then S^{1} ∧W ≅ colim_{N}(S^{1} ∧W^{N}), so that S^{1}∧W is also a simplicial pointed discrete G-set. Since the forgetful functor U: T_{G} → Sets is a left adjoint, pushouts in T_{G} are formed in Sets, and thus, there is an isomorphism

u^{∗}((S^{1} ∧ Y_{n}) ∪_{S^{1}∧X_{n}} X_{n+1}) ≅ (S^{1} ∧ Y_{n}) ∪_{S^{1}∧X_{n}} X_{n+1}

of simplicial discrete G-sets. Hence, (S^{1} ∧Y_{n})∪_{S^{1}∧X_{n}} X_{n+1} → Y_{n+1} is a cofibration in S, and f is a cofibration in Spt.

The converse follows from the fact that if j is an injection of simplicial discrete G-sets, then Hom_{G}(−, j) is a cofibration of simplicial presheaves. □

The preceding theorem implies the following two corollaries.

Corollary 3.7. If f: X → Y is a weak equivalence (cofibration) in Spt_{G}, then, for any K <_{c} G, f is a weak equivalence (cofibration) in Spt_{K}.

Corollary 3.8. The functors (U, Map_{c}(G,−)) are a Quillen pair for the categories (Spt_{G}, Spt).

Let t: Spt → Spt_{G} give a spectrum trivial G-action, so that t(X) = X. The right adjoint of t is the fixed points functor (−)^{G}. Clearly, t preserves all weak equivalences and cofibrations, giving the next result.

Corollary 3.9. The functors (t, (−)^{G}) are a Quillen pair for (Spt, Spt_{G}).

We conclude this section with a few more useful facts about discrete G-spectra.

Lemma 3.10. If f: X → Y is a fibration in Spt_{G}, then it is a fibration in Spt. In particular, if X is fibrant in Spt_{G}, then X is fibrant in Spt.

Proof. Since Hom_{G}(−, f) is a fibration of presheaves of spectra, Hom_{G}(G/N, f) is a fibration of spectra for each open normal subgroup N. Thus, colim_{N} Hom_{G}(G/N, f) is a fibration of spectra. Then the lemma follows from factoring f as X ≅ u^{∗}X → u^{∗}Y ≅ Y. □

The next lemma and its corollary, whose elementary proofs are omitted, show that the homotopy groups of a discrete G-spectrum are discrete G-modules.

Lemma 3.11. If X is a pointed Kan complex and a simplicial discrete G-set, then π_{n}(X) is a discrete G-module, for all n ≥ 2.

Corollary 3.12. If X is a discrete G-spectrum, then π_{n}(X) is a discrete G-module for any integer n.

The following observation says that certain elementary constructions with discrete G-spectra yield discrete G-spectra.

Lemma 3.13. Given a profinite group G ≅ lim_{N} G/N, let {X_{N}}_{N} be a directed system of spectra, such that each X_{N} is a G/N-spectrum and the maps are Gequivariant. Then colim_{N} X_{N} is a discrete G-spectrum. If Y is a spectrum with trivial G-action, then (colim_{N} X_{N}) ∧ Y ≅ colim_{N}(X_{N} ∧ Y) is a G-equivariant isomorphism of discrete G-spectra. Thus, if X is a discrete G-spectrum, then X ∧Y is a discrete G-spectrum.

The corollary below is very useful later on.

Corollary 3.14. The spectra F_{n}, F_{n} ∧ M_{I}, and F_{n} ∧ M_{I} ∧ X, for any spectrum X, are discrete G_{n}-spectra.

4. Towers of discrete G-spectra and continuous G-spectra

Let tow(Spt_{G}) be the category where a typical object {X_{i}} is a tower

⋯ → X_{i} → X_{i−1} → ⋯ → X_{1} → X_{0}

in Spt_{G}. The morphisms are natural transformations {X_{i}} → {Y_{i}}, such that each X_{i} → Y_{i} is G-equivariant. Since Spt_{G} is a simplicial model category, [13, VI, Prop. 1.3] shows that tow(Spt_{G}) is also a simplicial model category, where {f_{i}} is a weak equivalence (cofibration) if and only if each f_{i} is a weak equivalence (cofibration) in Spt_{G}. By [13, VI, Rk. 1.5], if {X_{i}} is fibrant in tow(Spt_{G}), then each map X_{i} → X_{i−1} in the tower is a fibration and each X_{i} is fibrant, all in Spt_{G}.

Definition 4.1. Let {X_{i}} be in tow(Spt_{G}). Then {X′_{i}} denotes the target of a trivial cofibration {X_{i}} → {X′_{i}}, with {X′_{i}} fibrant, in tow(Spt_{G}).

Remark 4.2. Let {X_{α}} be a diagram in Spt_{G}. Since there is an isomorphism lim_{α}^{Spt_{G}} X_{α} ≅ u^{∗}(lim_{α}^{Spt} X_{α}), limits in Spt_{G} are not formed in Spt. For the rest of this paper, lim is always formed in Spt and not in Spt_{G}.

The functor lim_{i}(−)^{G}: tow(Spt_{G}) → Spt, given by {X_{i}} ↦ lim_{i} X_{i}^{G}, is right adjoint to the functor \underline{t}: Spt → tow(Spt_{G}) that sends a spectrum X to the constant diagram \{X\}, where X has trivial G-action. Since \underline{t} preserves all weak equivalences and cofibrations, we have the following fact.

Lemma 4.3. The functors (\underline{t}, lim_{i}(−)^{G}) are a Quillen pair for the categories (Spt, tow(Spt_{G})).

This lemma implies the existence of the total right derived functor

R(lim_{i}(−)^{G}): Ho(tow(Spt_{G})) → Ho(Spt), {X_{i}} ↦ lim_{i}(X′_{i})^{G}.

Lemma 4.4. If {X_{i}} in tow(Spt_{G}) is a tower of fibrant spectra, then there are weak equivalences holim_{i} X_{i} ⟶^{p} holim_{i} X′_{i} ⟵^{q} lim_{i} X′_{i}.

Proof. Since each X_{i} → X′_{i} is a weak equivalence between fibrant spectra, p is a weak equivalence. Since X′_{i} → X′_{i−1} is a fibration in Spt, for i ≥ 1, q is a weak equivalence. □

We use this lemma to define the continuous G-spectra that we will study.

Definition 4.5. If {X_{i}} ∈ tow(Spt_{G}), then the inverse limit lim_{i} X_{i} is a continuous G-spectrum. Also, if {X_{i}} ∈ tow(Spt_{G}) is a tower of fibrant spectra, we call holim_{i} X_{i} a continuous G-spectrum, due to the zigzag of Lemma 4.4 relating holim_{i} X_{i} to the continuous G-spectrum lim_{i} X′_{i}. Note that if X ∈ Spt_{G}, then, using the constant tower on X, X ≅ lim_{i} X is a continuous G-spectrum.

Remark 4.6. Sometimes we will use the term “continuous G-spectrum” more loosely. Let X be a continuous G-spectrum, as in Definition 4.5. If Y is a Gspectrum that is isomorphic to X, in the stable category, with compatible G-actions, then we call Y a continuous G-spectrum.

We make a few more comments about Definition 4.5. Though the definition is not as general as it could be, it is sufficient for our applications. The inverse limit is central to the definition since the inverse limit of a tower of discrete G-sets is a topological G-space.

Given any tower X_{∗} = {X_{i}} in Spt_{G}, holim_{i} X_{i} = Tot(∏^{∗}X_{∗}) ≅ lim_{n} T(n), where T(n) = Tot_{n}(∏^{∗} X_{∗}). (See §5 for the definition of ∏^{∗} X_{∗}, and [2] is a reference for any undefined notation in this paragraph.) Then it is natural to ask if T (n) is in Spt_{G}, so that holim_{i} X_{i} is canonically a continuous G-spectrum. For this to be true, it must be that, for any m ≥ 0, the simplicial set

T(n)_{m} = Map_{cS}(sk_{n}Δ[−], ∏^{∗}(X_{∗})_{m}) ∈ S_{G}.

If, for all s ≥ 0, ∏^{s}(X_{∗})_{m} ∈ S_{G}, then T(n)_{m} ∈ S_{G}, by [11, pg. 212]. However, the infinite product ∏^{s}(X_{∗})_{m} need not be in S_{G}, and thus, in general, T(n) ∉ Spt_{G}. Therefore, holim_{i} X_{i} is not always identifiable with a continuous G-spectrum, in the above way.

5. Homotopy fixed points for discrete G-spectra

In this section, we define the homotopy fixed point spectrum for X ∈ Spt_{G}. We begin by recalling the homotopy spectral sequence, since we use it often.

If J is a small category, P: J → Spt a diagram of fibrant spectra, and Z any spectrum, then there is a conditionally convergent spectral sequence

(5.1) E_{2}^{s,t} = lim^{s}_{J}[Z, P]_{t} ⇒ [Z, holim_{J} P]_{t−s},

where lim^{s} is the sth right derived functor of lim_{J}: Ab^{J} → Ab (see [2], [41, Prop. 5.13, Lem. 5.31]).

Associated to P is the cosimplicial spectrum ∏^{∗} P, with ∏^{n}P = ∏_{(BΔ)_{n}} P (j_{n}), where the n-simplices of the nerve BΔ consist of all strings [j_{0}] → ⋯ → [j_{n}] of n morphisms in Δ. For any k ≥ 0, (∏^{∗}P)_{k} is a fibrant cosimplicial pointed simplicial set, and holim_{J} P = Tot(∏^{∗}P).

If P is a cosimplicial diagram of fibrant spectra (a cosimplicial fibrant spectrum), then E_{2}^{s,t} = π^{s}[Z, P]_{t}, the sth cohomotopy group of the cosimplicial abelian group [Z, P]_{t}.

Definition 5.2. Let X ∈ Spt_{G}. In Spt_{G}, factor X → ∗ as X → X_{f,G} → ∗, the composition of a trivial cofibration in Spt_{G}, followed by a fibration in Spt_{G}. Then we define X^{hG} = (X_{f,G})^{G}, and X^{hG} is the homotopy fixed point spectrum of X with respect to the continuous action of G. We write X_{f} instead of X_{f,G} when doing so causes no confusion.

Note that X^{hG} = Hom_{G}(∗, X_{f,G}), the global sections of the presheaf of spectra Hom_{G}(−, X_{f,G}). This definition has been developed in other categories: see [11] for simplicial discrete G-sets, [21] for simplicial presheaves, and [23, Ch. 6] for presheaves of spectra.

As expected, the definition of homotopy fixed points for a profinite group generalizes the definition for a finite group. Let G be a finite group, X a G-spectrum, and let X → X_{f} be a weak equivalence that is G-equivariant, with X_{f} a fibrant spectrum. Then, since G is finite, the homotopy fixed point spectrum X^{h′G} = Map_{G}(EG_{+}, X_{f}) can be defined to be holim_{G} X_{f} (as in the Introduction). Note that there is a descent spectral sequence

E_{2}^{s,t} = lim^{s}_{G} π_{t}(X) ≅ H^{s}(G; π_{t}(X)) ⇒ π_{t−s}(X^{h′G}).

Since G is profinite, X is a discrete G-spectrum, and X_{f,G} is a fibrant spectrum, so that X^{h′G} = holim_{G} X_{f}. Then, by [23, Prop. 6.39], the canonical map X^{hG} = lim_{G} X_{f} → holim_{G} X_{f} = X^{h′G} is a weak equivalence, as desired.

The next lemma follows from Corollary 3.9.

Lemma 5.3. The homotopy fixed points functor (−)^{hG}: Ho(Spt_{G}) → Ho(Spt) is the total right derived functor of the fixed points functor (−)^{G}: Spt_{G} → Spt. In particular, if X → Y is a weak equivalence of discrete G-spectra, then X^{hG} → Y^{hG} is a weak equivalence.

6. E_{n} is a continuous G_{n}-spectrum

We show that the Lubin-Tate spectrum E_{n} is a continuous G_{n}-spectrum by successively eliminating simpler ways of constructing a continuous action, and by applying the theory of the previous section.

First of all, since the profinite ring π_{0}(E_{n}) is not a discrete G_{n}-module, Corollary 3.12 implies the following observation.

Lemma 6.1. E_{n} is not a discrete G_{n}-spectrum.

However, note that, for k ∈ ℤ, π_{2k}(E_{n} ∧ M_{I}) is a finite discrete G_{n}-module so that the action factors through a finite quotient G_{n}/U_{I}, where U_{I} is some open normal subgroup (see [35, Lem. 1.1.16]). Thus, π_{2k}(E_{n} ∧ M_{I}) is a G_{n}/U_{I}-module, and one is led to ask if E_{n} ∧ M_{I} is a G_{n}/U_{I}-spectrum. If so, then E_{n} ∧ M_{I} is a discrete G_{n}-spectrum, and E_{n} is easily seen to be a continuous G_{n}-spectrum.

However, E_{n} ∧ M_{I} is not a G_{n}/U-spectrum for all open normal subgroups U of G_{n}, as the lemma below shows. As far as the author knows, this lemma is due to Hopkins; the author learned the proof from Hal Sadofsky.

Lemma 6.2. There is no open normal subgroup U of G_{n} such that the G_{n}-action on E_{n} ∧ M_{I} factors through G_{n}/U.

Proof. Suppose the G_{n}-spectrum E_{n}∧M_{I} is a G_{n}/U-spectrum. Then the G_{n}-action on the middle factor of E_{n}∧E_{n}∧M_{I} factors through G_{n}/U, so that π_{∗}(E_{n}∧E_{n}∧M_{I}) is a G_{n}/U-module. Note that π_{∗}(E_{n} ∧ E_{n} ∧ M_{I}) ≅ Map_{c}^{ℓ}(G_{n}, E_{n∗}/I).

Since the G_{n}-module Map_{c}^{ℓ}(G_{n}, E_{n∗}/I) is a G_{n}/U-module, there is an isomorphism of sets Map_{c}^{ℓ}(G_{n}, E_{n,0}/I) = Map_{c}^{ℓ}(G_{n}, E_{n,0}/I)^{U} ≅ Map_{c}(G_{n}/U, E_{n,0}/I). But the first set is infinite and the last is finite, a contradiction. □

Since π_{∗}(E_{n} ∧ M_{I}) is a discrete G_{n}-module, one can still hope for a spectrum E_{n}/I ≃ E_{n} ∧ M_{I}, such that E_{n}/I is a discrete G_{n}-spectrum.

To produce E_{n}/I, we make the following observation. By [23, Remark 6.26], since U_{i} is an open normal subgroup of G_{n}, the presheaf Hom_{U_{i}}(−, (E_{n}/I)_{f,G_{n}}) is fibrant in the model category of presheaves of spectra on the site U_{i}−Sets_{df}. Thus, for each i, the map E_{n}/I → (E_{n}/I)_{f,G_{n}} is a trivial cofibration, with fibrant target, all in Spt_{U_{i}}, so that ((E_{n}/I)_{f,G_{n}})^{U_{i}} = (E_{n}/I)^{hU_{i}}.

Combining this fact with the idea, discussed in §1, that E_{n} ∧ M_{I} has homotopy fixed point spectra (E_{n} ∧ M_{I})^{hU_{i}} ≃ E_{n}^{dhU_{i}} ∧ M_{I} ≃ (E_{n}/I)^{hU_{i}}, we have:

E_{n}/I ≃ (E_{n}/I)_{f,G_{n}} ≅ colim_{i}((E_{n}/I)_{f,G_{n}})^{U_{i}} = colim_{i}(E_{n}/I)^{hU_{i}}

≃ colim_{i}(E_{n}^{dhU_{i}} ∧ M_{I}) ≅ F_{n} ∧ M_{I}.

This argument suggests that E_{n} ∧ M_{I} has the homotopy type of the discrete G_{n}- spectrum F_{n} ∧ M_{I}. To show that this is indeed the case, we consider the spectrum F_{n} in more detail. The key result is the following theorem, due to Devinatz and Hopkins.

Theorem 6.3 ([5]). There is a weak equivalence E_{n} ≃ L̂(F_{n}).

Proof. By [5, Thm. 3], E_{n}^{h′{e}} ≃ E_{n}^{dh{e}}. (We remark that this weak equivalence is far from obvious.) By [5, Definition 1.5], E_{n}^{dh{e}} = L̂(hocolim_{i} E_{n}^{dhU_{i}}), where the homotopy colimit is in the category E_{∞}. Then, by [5, Remark 1.6, Lemma 6.2], hocolim_{i} E_{n}^{dhU_{i}} ≃ colim_{i} E_{n}^{dhU_{i}}, where the colimit is in M_{S}. Thus, as spectra in Spt, E_{n}^{dh{e}} ≃ L̂(F_{n}), so that E_{n} ≃ E_{n}^{h′{e}} ≃ L̂(F_{n}). □

Corollary 6.4. In the stable category, there are isomorphisms

E_{n} ≅ holim_{I}(F_{n} ∧ M_{I})_{f} ≅ holim_{I}(E_{n} ∧ M_{I})_{f}.

The following result shows that E_{n} ∧ M_{I} ≃ F_{n} ∧ M_{I}. This weak equivalence and vcd(G_{n}) < ∞ are the main facts that make it possible to construct the homotopy fixed point spectra of E_{n}.

Corollary 6.5. If Y is a finite spectrum of type n, then the G_{n}-equivariant map F_{n} ∧ Y → E_{n} ∧ Y is a weak equivalence. In particular, E_{n} ∧ M_{I} ≃ F_{n} ∧ M_{I}.

Proof. We have E_{n} ∧ Y ≃ L̂(F_{n}) ∧ Y ≃ L_{n}(F_{n}) ∧ Y ≃ F_{n} ∧ Y. □

Now we show that E_{n} is a continuous G_{n}-spectrum.

Theorem 6.6. There is an isomorphism E_{n} ≅ holim_{I}(F_{n} ∧ M_{I})_{f,G_{n}}. Thus, E_{n} is a continuous G_{n}-spectrum.

Proof. By Corollary 6.4, E_{n} ≅ holim_{I}(F_{n}∧M_{I})_{f}. By functorial fibrant replacement, the map of towers {F_{n} ∧ M_{I}} → {(F_{n} ∧ M_{I})_{f,G_{n}}} induces a map of towers

{(F_{n} ∧ M_{I})_{f}} → {((F_{n} ∧ M_{I})_{f,G_{n}})_{f}}

and, hence, weak equivalences

holim_{I}(F_{n} ∧ M_{I})_{f,G_{n}} → holim_{I}((F_{n} ∧ M_{I})_{f,G_{n}})_{f} ← holim_{I}(F_{n} ∧ M_{I})_{f}.

Thus, holim_{I}(F_{n} ∧ M_{I})_{f,G_{n}} is isomorphic to holim_{I}(F_{n} ∧ M_{I})_{f} and E_{n}. Since {(F_{n} ∧ M_{I})_{f,G_{n}}} is a tower of fibrant spectra, holim_{I}(F_{n} ∧ M_{I})_{f,G_{n}} is a continuous G_{n}-spectrum. Then, by Remark 4.6, E_{n} is a continuous G_{n}-spectrum. □

We conclude this section with some observations about F_{n}.

Lemma 6.7. The map η: F_{n} → E_{n} is not a weak equivalence and F_{n} is not K(n)- local.

Proof. If η is a weak equivalence, then π_{0}(η) is a G_{n}-equivariant isomorphism from a discrete G_{n}-module (with all orbits finite) to a non-finite profinite G_{n}-module, which is impossible. If F_{n} is K(n)-local, then F_{n} ≃ L̂(F_{n}) ≃ E_{n}, and η is a weak equivalence, a contradiction. □

Lemma 6.8. The maps L̂(F_{n} ∧ F_{n}) → L̂(E_{n} ∧ F_{n}) → L̂(E_{n} ∧ E_{n}) are weak equivalences.

Proof. Since F_{n} ∧ M_{I} ≃ E_{n} ∧ M_{I}, F_{n} ∧ F_{n} ∧ M_{I} ≃ E_{n} ∧ E_{n} ∧ M_{I}. Since F_{n} ∧ F_{n}, E_{n} ∧ F_{n} and E_{n} ∧ E_{n} are E(n)-local, the result follows from Theorem 2.8. □

7. Homotopy fixed points when vcd(G) < ∞

In this section, G always has finite virtual cohomological dimension. Thus, there exists a uniform bound m, such that for all K <_{c} G, vcd(K) ≤ m. For X ∈ Spt_{G}, we use this fact to give a model for X^{hG} that eases the construction of its descent spectral sequence. Also, this fact yields a second model for X^{hK} that is functorial in K.

Definition 7.1. Consider the functor

Γ_{G} = Map_{c}(G, −) ∘ U: Spt_{G} → Spt_{G}, X ↦ Γ_{G}(X) = Map_{c}(G, X),

where Γ_{G}(X) has the G-action defined in §3. We write Γ instead of Γ_{G}, when G is understood from context. There is a G-equivariant monomorphism i: X → ΓX defined, on the level of sets, by i(x)(g) = g · x. As in [44, 8.6.2], since U and Map_{c}(G, −) are adjoints, Γ forms a triple and there is a cosimplicial discrete Gspectrum Γ^{•}X, with (Γ^{•}X)^{k} ≅ Map_{c}(G^{k+1}, X).

We recall the construction of Thomason’s hypercohomology spectrum for the topos Shv of sheaves of sets on the site G − Sets_{df} (see Definition 3.2). ([41, 1.31-1.33] and [29, §1.3, §3.2] give more details about the hypercohomology spectrum.) Consider the functor T = u_{∗}u^{∗}: ShvSpt → ShvSpt, which sends F to Hom_{G}(−, Map_{c}(G, colim_{N} F(G/N))), obtained by composing the adjoints in the point of the topos. For X ∈ Spt_{G}, T (Hom_{G}(−, X)) ≅ Hom_{G}(−, Map_{c}(G, X)). By iterating this isomorphism, the cosimplicial sheaf of spectra T^{•}Hom_{G}(−, X) gives rise to the cosimplicial sheaf Hom_{G}(−, Γ^{•}X).

Definition 7.2. Given X ∈ Spt_{G}, the presheaf of hypercohomology spectra of G− Sets_{df} with coefficients in X is the presheaf of spectra

ℍ^{•}(−; X) = holim_{Δ} Hom_{G}(−, Γ^{•}X): (G−Sets_{df})^{op} → Spt,

and ℍ^{•}(S; X) = holim_{Δ} Hom_{G}(S, Γ^{•}X) is the hypercohomology spectrum of S with coefficients in X.

The map X_{f} → Γ^{•}X_{f}, induced by i, out of the constant cosimplicial diagram, and Hom_{G}(−, X_{f}) → lim_{Δ} Hom_{G}(−, X_{f}) → holim_{Δ} Hom_{G}(−, X_{f}) induce a canonical map Hom_{G}(−, X_{f}) → ℍ^{•}(−, X_{f}).

Now we show that ℍ^{•}(∗, X_{f}) is a model for X^{hG}. Below, a cosimplicial globally fibrant presheaf is a cosimplicial presheaf of spectra that is globally fibrant at each level.

Lemma 7.3. If F^{•} is a cosimplicial globally fibrant presheaf, then holim_{Δ} F^{•} is a globally fibrant presheaf.

Proof. By [23, Rk. 2.35], this is equivalent to showing that, for each n ≥ 0, (a) holim_{Δ}(F^{•})^{n} is a globally fibrant simplicial presheaf; and (b) the adjoint of the bonding map, the composition

γ: holim_{Δ}(F^{•})^{n} → Ω(holim_{Δ}(F^{•})^{n+1}) ≅ holim_{Δ} Ω(F^{•})^{n+1}

is a local weak equivalence of simplicial presheaves. Part (a), the difficult part of this lemma, is proven in [21, Prop. 3.3].

The map γ is a local weak equivalence if the map of stalks u^{∗}γ is a weak equivalence in S, which is true if each γ(G/N) is a weak equivalence. By [21, pg. 74], if P is a globally fibrant simplicial presheaf, then ΩP is too, so that σ: (F^{•})^{n} → Ω(F^{•})^{n+1} is a cosimplicial diagram of local weak equivalences between globally fibrant simplicial presheaves. Thus, σ(G/N) is a cosimplicial diagram of weak equivalences between Kan complexes, so that γ(G/N) = holim_{Δ} σ(G/N) is indeed a weak equivalence. □

The following result is not original: it is basically a special case of [21, Prop. 3.3], versions of which appear in [11, §5], [29, Prop. 3.20], and [30, Prop. 6.1]. Since the result is central to our work, for the benefit of the reader, we give the details of the proof. For use now and later, we recall that, for any group G and any closed subgroup K, H_{c}^{s}(K; Map_{c}(G, A)) = 0, when s > 0 and A is any discrete abelian group [45, Lemma 9.4.5]. Also, recall that X_{f} = X_{f,G} (Definition 5.2).

Theorem 7.4. Let G be a profinite group with vcd(G) ≤ m, and let X be a discrete G-spectrum. Then there are weak equivalences

Hom_{G}(−, X) ⟶^{≃} Hom_{G}(−, X_{f}) ⟶^{≃} holim_{Δ} Hom_{G}(−, Γ^{•}X_{f}),

and holim_{Δ} Hom_{G}(−, Γ^{•}X_{f}) is a globally fibrant model for Hom_{G}(−, X). Thus, evaluation at ∗ ∈ G−Sets_{df} gives a weak equivalence X^{hG} → holim_{Δ}(Γ^{•}X_{f})^{G}.

Remark 7.5. The weak equivalence X ≅ colim_{N} X^{N} → colim_{N}(X^{N})_{f}, whose target is a discrete G-spectrum that is fibrant in Spt, induces a weak equivalence X_{f,G} → (colim_{N}(X^{N})_{f})_{f,G}. Thus, there are weak equivalences

ℍ^{•}(∗, X_{f}) → ℍ^{•}(∗, (colim_{N}(X^{N})_{f})_{f,G}) ← ℍ^{•}(∗, colim_{N}(X^{N})_{f}),

so that ℍ^{•}(∗, colim_{N}(X^{N})_{f}) is a model for X^{hG} that does not require the model category Spt_{G} for its construction.

Proof of Theorem 7.4. Since X_{f} is fibrant in Spt, ΓX_{f} is fibrant in Spt_{G} by Corollary 3.8. By iteration, Hom_{G}(−, Γ^{•}X_{f}) is a cosimplicial globally fibrant presheaf, so that holim_{Δ} Hom_{G}(−, Γ^{•}X_{f}) is globally fibrant, by Lemma 7.3. It only remains to show that λ: Hom_{G}(−, X) → holim_{Δ} Hom_{G}(−, Γ^{•}X_{f}) is a weak equivalence.

By hypothesis, G contains an open subgroup H with cd(H) ≤ m. Then by [45, Lem. 0.3.2], H contains a subgroup K that is an open normal subgroup of G. Let {N} be the collection of open normal subgroups of G. Let N′ = N ∩ K. Observe that {N′} is a cofinal subcollection of open normal subgroups of G so that G ≅ lim_{N′} G/N′. Since N′ <_{c} H, cd(N′) ≤ cd(H). Thus, H_{c}^{s}(N′; M) = 0, for all s > m + 1, whenever M is a discrete N′-module. Henceforth, we drop the ′ from N′ to ease the notation: N is really N ∩ K.

Any presheaf of sets F has stalk colim_{N} F(G/N), so that λ is a weak equivalence if λ_{u}: X ≅ colim_{N} X^{N} → colim_{N} holim_{Δ}(Γ^{•}X_{f})^{N} is a weak equivalence. Since Hom_{G}(−, Γ^{•}X_{f}) is a cosimplicial globally fibrant spectrum, the diagram (Γ^{•}X_{f})^{N} is a cosimplicial fibrant spectrum. Then, for each N, there is a conditionally convergent spectral sequence

(7.6) E_{2}^{s,t}(N) = π^{s}π_{t}((Γ^{•}X_{f})^{N}) ⇒ π_{t−s}(holim_{Δ}(Γ^{•}X_{f})^{N}).

Because π_{t}(X) is a discrete G-module, we have

π_{t}(Map_{c}(G, X_{f})^{N}) ≅ π_{t}(Map_{c}(G/N, X_{f})) ≅ ∏_{G/N} π_{t}(X) ≅ Map_{c}(G, π_{t}(X))^{N} and π_{t}(Map_{c}(G, X_{f})) ≅ π_{t}(colim_{N} ∏_{G/N} X_{f}) ≅ Map_{c}(G, π_{t}(X)). By iterating such manipulations, we obtain π^{s}π_{t}((Γ^{•}X_{f})^{N}) ≅ H^{s}((Γ^{∗}π_{t}(X))^{N}). The cochain complex 0 → π_{t}(X) → Γ^{∗}π_{t}(X) of discrete N-modules is exact (see e.g. [11, pp. 210-211]), and, for k ≥ 1 and s > 0,

H_{c}^{s}(N; Γ^{k}π_{t}(X)) ≅ H_{c}^{s}(N; Map_{c}(G, Γ^{k−1}π_{t}(X))) = 0.

Thus, the above cochain complex is a resolution of π_{t}(X) by (−)^{N}-acyclic modules, so that E_{2}^{s,t}(N) ≅ H_{c}^{s}(N; π_{t}(X)). Taking a colimit over {N} of (7.6) gives the spectral sequence

(7.7) E_{2}^{s,t} = colim_{N} H_{c}^{s}(N; π_{t}(X)) ⇒ π_{t−s}(colim_{N} holim_{Δ}(Γ^{•}X_{f})^{N}).

Since E_{2}^{s,∗}(N) = 0 whenever s > m+1, the E_{2}-terms E_{2}(N) are uniformly bounded on the right. Therefore, by [29, Prop. 3.3], the colimit of the spectral sequences does converge to the colimit of the abutments, as asserted in (7.7).

Finally,

E_{2}^{∗,t} ≅ H_{c}^{∗}(lim_{N} N; π_{t}(X))≅H^{∗}({e}; π_{t}(X)),

which is isomorphic to π_{t}(X), concentrated in degree zero. Thus, (7.7) collapses and for all t, π_{t}(colim_{N} holim_{Δ}(Γ^{•}X_{f})^{N}) ≅ π_{t}(X), and hence, λ_{u} is a weak equivalence.

□

Remark 7.8. Because of Theorem 7.4, if vcd(G) < ∞ and X is a discrete Gspectrum, we make the identification

X^{hG} = holim_{Δ}(Γ_{G}^{•}X_{f,G})^{G} = ℍ^{•}(∗, X_{f,G}).

In [43, §2.14], an expression that is basically equivalent to holim_{Δ}(Γ_{G}^{•}X_{f,G})^{G} is defined to be the homotopy fixed point spectrum X^{hG}, even if vcd(G) = ∞. This approach has the disadvantage that (−)^{hG} need not always be the total right derived functor of (−)^{G}. Thus, we only make the identification of Remark 7.8 when vcd(G) < ∞.

Now it is easy to construct the descent spectral sequence. Note that if X is a discrete G-spectrum, the proof of Theorem 7.4 shows that

π^{s}π_{t}((Γ_{G}^{•}X)^{G}) ≅ π^{s}((Γ_{G}^{•}π_{t}(X))^{G}) ≅ H_{c}^{s}(G; π_{t}(X)).

Theorem 7.9. If vcd(G) < ∞ and X is a discrete G-spectrum, then there is a conditionally convergent descent spectral sequence

E_{2}^{s,t} = H_{c}^{s}(G; π_{t}(X)) ⇒ π_{t−s}(X^{hG}). (7.10)

Proof. As in Theorem 7.4, (Γ^{•}X_{f})^{G} is a cosimplicial fibrant spectrum. Thus, we can form the homotopy spectral sequence for π_{∗}(holim_{Δ}(Γ^{•}X_{f})^{G}). □

Remark 7.11. Spectral sequence (7.10) has been constructed in other contexts: for simplicial presheaves, presheaves of spectra, and S_{G}, see [21, Cor. 3.6], [23, §6.1], and [11, §§4, 5], respectively. In several of these examples, a Postnikov tower provides an alternative to the hypercohomology spectrum that we use. In all of these constructions of the descent spectral sequence, some kind of finiteness assumption is required in order to identify the homotopy groups of the abutment as being those of the homotopy fixed point spectrum.

Let X be a discrete G-spectrum. We now develop a second model for X^{hK}, where K is a closed subgroup of G, that is functorial in K.

The map X → X_{f,G} in Spt_{K} gives a weak equivalence X^{hK} → (X_{f,G})^{hK}. Composition with the weak equivalence (X_{f,G})^{hK} → holim_{Δ}(Γ_{K}^{•}((X_{f,G})_{f,K}))^{K} gives a weak equivalence X^{hK} → holim_{Δ}(Γ_{K}^{•}((X_{f,G})_{f,K}))^{K} between fibrant spectra. The inclusion K → G induces a morphism Γ_{G}(X_{f,G}) → Γ_{K}(X_{f,G}), giving a map Γ_{G}^{•}(X_{f,G}) → Γ_{K}^{•}(X_{f,G}) of cosimplicial discrete K-spectra.

Lemma 7.12. There is a weak equivalence

ρ: holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K} → holim_{Δ}(Γ_{K}^{•}(X_{f,G}))^{K} → holim_{Δ}(Γ_{K}^{•}((X_{f,G})_{f,K}))^{K}.

Proof. Recall the conditionally convergent spectral sequence

H_{c}^{s}(K; π_{t}(X)) ≅ H_{c}^{s}(K; π_{t}((X_{f,G})_{f,K})) ⇒ π_{t−s}(holim_{Δ}(Γ_{K}^{•}((X_{f,G})_{f,K}))^{K}).

We compare this spectral sequence with the homotopy spectral sequence for holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K}. Note that if Y is a discrete G-spectrum that is fibrant as a spectrum, then Map_{c}(G, Y) ≅ colim_{N} ∏ _{G/N} Y and

Map_{c}(G, Y)^{K} ≅ Map_{c}(G/K, Y) ≅ colim_{N} ∏_{G/(NK)} Y

are fibrant spectra. Thus, (Γ_{G}^{•}(X_{f,G}))^{K} is a cosimplicial fibrant spectrum, and there is a conditionally convergent spectral sequence

E_{2}^{s,t} = H^{s}((Γ_{G}^{∗}π_{t}(X_{f,G}))^{K}) ⇒ π_{t−s}(holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K}).

As in the proof of Theorem 7.4, 0 → π_{t}(X_{f,G}) → Γ_{G}^{∗}(π_{t}(X_{f,G})) is a (−)^{K}-acyclic resolution of π_{t}(X_{f,G}), and thus, we have E_{2}^{s,t} ≅ H_{c}^{s}(K; π_{t}(X_{f,G})) ≅ H_{c}^{s}(K; π_{t}(X)).

Since ρ is compatible with the isomorphism between the two E_{2}-terms, the spectral sequences are isomorphic and ρ is a weak equivalence. □

Remark 7.13. Lemma 7.12 gives the following weak equivalences between fibrant spectra:

X^{hK} = (X_{f,K})^{K} → holim_{Δ}(Γ_{K}^{•}((X_{f,G})_{f,K}))^{K} ← holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K}.

Thus, if vcd(G) < ∞, X is a discrete G-spectrum, and K is a closed subgroup of G, then holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K} is a model for X^{hK}, so that

X^{hK} = holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K}

is another definition of the homotopy fixed points.

This discussion yields the following result.

Theorem 7.14. If X is a discrete G-spectrum, with vcd(G) < ∞, then there is a presheaf of spectra P (X): (O_{G})^{op} → Spt, defined by

P (X)(G/K) = holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K} = X^{hK}.

Proof. If Y is a discrete G-set, any morphism f: G/H → G/K, in O_{G}, induces a map

Map_{c}(G, Y)^{K} ≅ Map_{c}(G/K, Y) → Map_{c}(G/H, Y) ≅ Map_{c}(G, Y)^{H}.

Thus, if Y ∈ Spt_{G}, f induces a map Map_{c}(G, Y)^{K} → Map_{c}(G, Y)^{H}, so that there is a map P (X)(f): holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{K} → holim_{Δ}(Γ_{G}^{•}(X_{f,G}))^{H}. It is easy to check that P (X) is actually a functor. □

We conclude this section by pointing out a useful fact: smashing with a finite spectrum, with trivial G-action, commutes with taking homotopy fixed points. To state this precisely, we first define the relevant map.

Let X be a discrete G-spectrum and let Y be any spectrum with trivial G-action. Then there is a map

(holim_{Δ}(Γ_{G}^{•}X_{f})^{G}) ∧ Y → holim_{Δ}((Γ_{G}^{•}X_{f})^{G} ∧ Y) → holim_{Δ}((Γ_{G}^{•}X_{f}) ∧ Y)^{G}.

Also, there is a natural G-equivariant map Map_{c}(G, X) ∧ Y → Map_{c}(G, X ∧ Y) that is defined by the composition

(colim_{N} ∏_{G/N} X) ∧ Y ≅ colim_{N}((∏_{G/N} X) ∧ Y) → colim_{N} ∏_{G/N} (X ∧ Y),

by using the isomorphism Map_{c}(G, X) ≅ colim_{N} ∏_{G/N} X. This gives a natural Gequivariant map (Γ_{G} Γ_{G}X)∧Y → Γ_{G} ((Γ_{G}X)∧Y) → Γ_{G} Γ_{G}(X∧Y). Thus, iteration gives a G-equivariant map (Γ^{•}X) ∧ Y → Γ^{•}(X ∧ Y) of cosimplicial spectra. Hence, if vcd(G) < ∞, X^{hG} ∧ Y → (X_{f} ∧ Y)^{hG} is a canonical map that is defined by composing X^{hG} ∧ Y → holim_{Δ}((Γ^{•}X_{f}) ∧ Y)^{G}, from above, with the map

holim_{Δ}((Γ^{•}X_{f}) ∧ Y)^{G} → holim_{Δ}(Γ^{•}(X_{f} ∧ Y))^{G} → holim_{Δ}(Γ^{•}(X_{f} ∧ Y)_{f})^{G}.

Lemma 7.15 ([29, Prop. 3.10]). If vcd(G) < ∞, X ∈ Spt_{G}, and Y is a finite spectrum with trivial G-action, then X^{hG}∧Y → (X_{f,G}∧Y)^{hG} is a weak equivalence.

Remark 7.16. By Lemma 7.15, when Y is a finite spectrum, there is a zigzag of natural weak equivalences X^{hG} ∧ Y → (X_{f,G} ∧ Y)^{hG} ← (X ∧ Y)^{hG}. We refer to this zigzag by writing X^{hG} ∧ Y ≅ (X ∧ Y)^{hG}.

8. Homotopy fixed points for towers in Spt_{G}

In this section, {Z_{i}} is always in tow(Spt_{G}) (except in Definition 8.7). For {Z_{i}} a tower of fibrant spectra, we define the homotopy fixed point spectrum (holim_{i} Z_{i})^{hG} and construct its descent spectral sequence. Also, recall from §5 that, if G is finite and X ∈ Spt_{G}, then X^{h′G} = holim_{G} X_{f}, where X → X_{f} is a weak equivalence that is G-equivariant, with X_{f} fibrant in Spt.

Definition 8.1. If {Z_{i}} in tow(Spt_{G}) is a tower of fibrant spectra, we define Z = holim_{i} Z_{i}, a continuous G-spectrum. The homotopy fixed point spectrum Z^{hG} is defined to be holim_{i} Z_{i}^{hG}, a fibrant spectrum.

We make some comments about Definition 8.1. Let H be a closed subgroup of G. Then the map holim_{i}((Z_{i})_{f})^{H} → holim_{i} holim_{Δ}(Γ_{H}^{•}(Z_{i})_{f})^{H} and the map holim_{i} holim_{Δ}(Γ_{G}^{•}(Z_{i})_{f,G})^{H} → holim_{i} holim_{Δ}(Γ_{H}^{•}((Z_{i})_{f,G})_{f,H})^{H} are weak equivalences. Thus, in Definition 8.1, each of our three definitions for homotopy fixed points (Definition 5.2, Remarks 7.8, 7.13) can be used for Z_{i}^{hH}.

In Definition 8.1, suppose that not all the Z_{i} are fibrant in Spt. Then the map Z = holim_{i} Z_{i} → holim_{i}(Z_{i})_{f,{e}} = holim_{i} Z_{i}^{h{e}} = Z^{h{e}} need not be a weak equivalence. Thus, for an arbitrary tower in Spt_{G}, Definition 8.1 can fail to have the desired property that Z → Z^{h{e}} is a weak equivalence.

Below, Lemmas 8.2 and 8.3, and Remark 8.4, show that when G is a finite group, Z^{hG} ≃ Z^{h′G}, and, for any G, Z^{hG} can be obtained by using a total right derived functor that comes from fixed points. Thus, Definition 8.1 generalizes the notion of homotopy fixed points to towers of discrete G-spectra.

Lemma 8.2. Let G be a finite group and let {Z_{i}} in tow(Spt_{G}) be a tower of fibrant spectra. Then there is a weak equivalence Z^{hG} → Z^{h′G}.

Proof. It is not hard to see that the map Z^{hG} → Z^{h′G} can be defined to be

holim_{i} lim_{G}(Z_{i})_{f} → holim_{i} holim_{G}(Z_{i})_{f} ≅ holim_{G} holim_{i}(Z_{i})_{f},

which is easily seen to be a weak equivalence. □

In the lemma below, whose elementary proof is omitted, the functor

R(lim_{i}(−)^{G}): Ho(tow(Spt_{G})) → Ho(Spt)

is the total right derived functor of the functor lim_{i}(−)^{G}: tow(Spt_{G}) → Spt.

Lemma 8.3. If {Z_{i}} is an arbitrary tower in Spt_{G}, then

holim_{i}((Z_{i})_{f})^{G} ⟶^{≃} holim_{i}((Z_{i})′_{f})^{G} ⟵^{≃} lim_{i}((Z_{i})′_{f})^{G} = R(lim_{i}(−)^{G})({Z_{i}}).

Remark 8.4. By Lemma 5.3, if X ∈ Spt_{G}, then X^{hG} = (R(−)^{G})(X). Also, by Lemma 8.3, if {Z_{i}} in tow(Spt_{G}) is a tower of fibrant spectra, then

Z^{hG} = holim_{i} Z_{i}^{hG} = holim_{i}((Z_{i})_{f})^{G} ≅ R(lim_{i}(−)^{G})({Z_{i}}).

Thus, the homotopy fixed point spectrum Z^{hG} is again given by the total right derived functor of an appropriately defined functor involving G-fixed points.

Given any tower in Spt_{G} of fibrant spectra, there is a descent spectral sequence whose E_{2}-term is a version of continuous cohomology.

Theorem 8.5. If vcd(G) < ∞ and {Z_{i}} in tow(Spt_{G}) is a tower of fibrant spectra, then there is a conditionally convergent descent spectral sequence

H_{cont}^{s}(G; {π_{t}(Z_{i})}) ⇒ π_{t−s}(Z^{hG}). (8.6)

We omit the proof of Theorem 8.5, since it is a special case of [8, Prop. 3.1.2], and also because (8.6) is not our focus of interest. However, we point out that spectral sequence (8.6), whose construction goes back to the ℓ-adic descent spectral sequence of algebraic K-theory ([41], [29]), is the homotopy spectral sequence

E_{2}^{s,t} = lim^{s}_{Δ×{i}} π_{t}((Γ_{G}^{•}((Z_{i})_{f,G}))^{G}) ⇒ π_{t−s}(holim_{Δ×{i}}(Γ_{G}^{•}((Z_{i})_{f,G}))^{G}).

For our applications, instead of spectral sequence (8.6), we are more interested in descent spectral sequence (8.9) below. Spectral sequence (8.9), a homotopy spectral sequence for a particular cosimplicial spectrum, is more suitable for comparison with the K(n)-local E_{n}-Adams spectral sequence (see [5, Prop. A.5]), when (8.9) has abutment π_{∗}((E_{n} ∧ X)^{hG}), where X is a finite spectrum.

Definition 8.7. If {Z_{i}} is a tower of spectra such that {π_{t}(Z_{i})} satisfies the Mittag-Leffler condition for every t ∈ ℤ, then {Z_{i}} is a Mittag-Leffler tower of spectra.

Theorem 8.8. If vcd(G) < ∞ and {Z_{i}} in tow(Spt_{G}) is a tower of fibrant spectra, then there is a conditionally convergent descent spectral sequence

E_{2}^{s,t} = π^{s}π_{t}(holim_{i}(Γ_{G}^{•}(Z_{i})_{f})^{G}) ⇒ π_{t−s}(Z^{hG}). (8.9)

If {Z_{i}} is a Mittag-Leffler tower, then E_{2}^{s,t} ≅ H_{cont}^{s}(G; {π_{t}(Z_{i})}).

Remark 8.10. In Theorem 8.8, when {Z_{i}} is a Mittag-Leffler tower, spectral sequence (8.9) is identical to (8.6). However, in general, spectral sequences (8.6) and (8.9) are different. For example, if G = {e}, then in (8.6), E_{2}^{0,t} = lim_{i} π_{t}(Z_{i}), whereas in (8.9), E_{2}^{0,t} = π_{t}(holim_{i} Z_{i}).

Proof of Theorem 8.8. Note that Z^{hG} ≅ holim_{Δ} holim_{i}(Γ_{G}^{•}(Z_{i})_{f})^{G}, and the diagram holim_{i}(Γ_{G}^{•}(Z_{i})_{f})^{G} is a cosimplicial fibrant spectrum.

Let {Z_{i}} be a Mittag-Leffler tower. For k ≥ 0, Lemma 2.10 and Remark 2.11 imply that lim^{1}_{i} Map_{c}(G^{k}, π_{t+1}(Z_{i})) = 0. Therefore,

π_{t}(holim_{i}(Map_{c}(G^{k+1}, (Z_{i})_{f}))^{G}) ≅ lim_{i} Map_{c}(G^{k+1}, π_{t}(Z_{i}))^{G},

and hence, π_{t}(holim_{i}(Γ^{•}(Z_{i})_{f})^{G}) ≅ lim_{i}(Γ^{•}π_{t}(Z_{i}))^{G}. Thus,

E_{2}^{s,t} ≅ π^{s}(lim_{i}(Γ^{•}π_{t}(Z_{i}))^{G}) ≅ H^{s}(lim_{i}(−)^{G} {Γ^{∗}π_{t}(Z_{i})}_{i}).

Consider the exact sequence {0} → {π_{t}(Z_{i})} → {Γ^{∗}π_{t}(Z_{i})} in tow(C_{G}). Note that, for s, k > 0, by Theorem 2.16,

H_{cont}^{s}(G; {Γ^{k}π_{t}(Z_{i})}) ≅ lim_{i} H_{c}^{s}(G; Γ^{k}π_{t}(Z_{i})) = 0,

since the tower {Γ^{k}π_{t}(Z_{i})} satisfies the Mittag-Leffler condition, and, for each i, Γ^{k}π_{t}(Z_{i}) ≅ Map_{c}(G, Γ^{k−1}π_{t}(Z_{i})) is (−)^{G}-acyclic. Thus, the above exact sequence is a (lim_{i}(−)^{G})-acyclic resolution of {π_{t}(Z_{i})}, so that we obtain the isomorphism E_{2}^{s,t} ≅ H_{cont}^{s}(G; {π_{t}(Z_{i})}). □

By Remark 8.4, we can rewrite spectral sequence (8.9), when {Z_{i}} is a Mittag-Leffler tower, in a more conceptual way:

R^{s}(lim_{i}(−)^{G}){π_{t}(Z_{i})} ⇒ π_{t−s}(R(lim_{i}(−)^{G})({Z_{i}})).

Spectral sequence (8.6) can always be written in this way.

9. Homotopy fixed point spectra for E^{∨}(X)

Recall that E^{∨}(X) = L̂(E_{n} ∧ X). In this section, for any spectrum X and for G <_{c} G_{n}, we define the homotopy fixed point spectrum (E^{∨}(X))^{hG}, using the continuous action of G.

Let X be an arbitrary spectrum with trivial G_{n}-action. By Corollary 6.5, there is a weak equivalence F_{n} ∧ M_{I} ∧ X → E_{n} ∧ M_{I} ∧ X. Then, by functorial fibrant replacement, there is a map {(F_{n} ∧M_{I} ∧X)_{f}} → {(E_{n} ∧M_{I} ∧X)_{f}} of towers, which yields the weak equivalence

E^{∨}(X) ≅ holim_{I}(E_{n} ∧ M_{I} ∧ X)_{f} ⟵^{≃} holim_{I}(F_{n} ∧ M_{I} ∧ X)_{f}.

As in the proof of Theorem 6.6, this implies the following lemma, since the diagram {(F_{n} ∧ M_{I} ∧ X)_{f,G_{n}}} is a tower of fibrant spectra.

Lemma 9.1. Given any spectrum X with trivial G_{n}-action, the isomorphism

E^{∨}(X) ≅ holim_{I}(F_{n} ∧ M_{I} ∧ X)_{f,G_{n}}

makes E^{∨}(X) a continuous G_{n}-spectrum.

Let G be any closed subgroup of G_{n}. Since {(F_{n} ∧ M_{I} ∧ X)_{f,G_{n}}} is a tower of discrete G-spectra that are fibrant in Spt, Lemma 9.1 also shows that E^{∨}(X) is a continuous G-spectrum. By Corollary 3.7, the composition

(F_{n} ∧ M_{I} ∧ X) → (F_{n} ∧ M_{I} ∧ X)_{f,G_{n}} → ((F_{n} ∧ M_{I} ∧ X)_{f,G_{n}})_{f,G}

is a trivial cofibration in Spt_{G}, with target fibrant in Spt_{G}. Therefore, we have ((F_{n} ∧ M_{I} ∧ X)_{f,G_{n}})^{hG} = (F_{n} ∧ M_{I} ∧ X)^{hG}. Then, by Definition 8.1, we obtain the following.

Definition 9.2. Let G <_{c} G_{n}. Then

E_{n}^{hG} = (holim_{I}(F_{n} ∧ M_{I})_{f,G_{n}})^{hG} = holim_{I}(F_{n} ∧ M_{I})^{hG}.

More generally, for any spectrum X,

(E^{∨}(X))^{hG} = (holim_{I}(F_{n} ∧ M_{I} ∧ X)_{f,G_{n}})^{hG} = holim_{I}(F_{n} ∧ M_{I} ∧ X)^{hG}.

Remark 9.3. When X is a finite spectrum, E_{n} ∧ X ≃ E^{∨}(X). Thus, we have (E_{n} ∧ X)^{hG} ≅ (E^{∨}(X))^{hG}.

Remark 9.4. For any X, E^{∨}(X) ≅ holim_{I}(F_{n} ∧ M_{I} ∧ X)_{f,G} also shows that E^{∨}(X) is a continuous G-spectrum. By definition, ((F_{n} ∧ M_{I} ∧ X)_{f,G})^{hG} and (F_{n} ∧ M_{I} ∧ X)^{hG} are identical. Thus, as before,

(E^{∨}(X))^{hG} = holim_{I}((F_{n} ∧ M_{I} ∧ X)_{f,G})^{hG} = holim_{I}(F_{n} ∧ M_{I} ∧ X)^{hG}.

Note that Definition 9.2 implies the identifications

E_{n}^{hG} = holim_{I} holim_{Δ}(Γ_{G_{n}}^{•}(F_{n} ∧ M_{I})_{f,G_{n}})^{G} and

(E^{∨}(X))^{hG} = holim_{I} holim_{Δ}(Γ_{G_{n}}^{•}(F_{n} ∧ M_{I} ∧ X)_{f,G_{n}})^{G}.

The first identification, coupled with Theorem 7.14, implies the following.

Theorem 9.5. There is a functor P: (O_{G_{n}})^{op} → Spt, defined by P (G_{n}/G) = E_{n}^{hG}, where G is any closed subgroup of G_{n}.

In addition to the above identifications, we also have

E_{n}^{hG} = holim_{I} holim_{Δ}(Γ_{G}^{•}(F_{n} ∧ M_{I})_{f,G})^{G} and
(E^{∨}(X))^{hG} = holim_{I} holim_{Δ}(Γ_{G}^{•}(F_{n} ∧ M_{I} ∧ X)_{f,G})^{G}.

Below we show that, like E_{n}^{dhG}, E_{n}^{hG} is K(n)-local.

Lemma 9.6. Let G <_{c} G_{n} and let X be any spectrum. Then (F_{n} ∧ M_{I} ∧ X)^{hG} and (E^{∨}(X))^{hG} are K(n)-local. Also, (F_{n} ∧ X)^{hG} is E(n)-local.

Proof. Recall that (F_{n} ∧ M_{I} ∧ X)^{hG} = holim_{Δ}(Γ_{G}^{•}(F_{n} ∧ M_{I} ∧ X)_{f,G})^{G}, and

(Γ_{G}^{k}(F_{n} ∧ M_{I} ∧ X)_{f})^{G} ≃ Map_{c}(G^{k−1}, F_{n} ∧ X) ∧ M_{I},

for k ≥ 1. In the isomorphism

Map_{c}(G, F_{n} ∧ X) ≅ colim_{i} ∏_{G/(U_{i}∩G)} (F_{n} ∧ X),

the spectrum F_{n} ∧ X is E(n)-local, the finite product is too, and hence, the direct limit Map_{c}(G, F_{n} ∧ X) is E(n)-local. Iterating this argument shows that

Map_{c}(G^{k−1}, F_{n} ∧ X) ≅ Γ_{G}Γ_{G} ⋯ Γ_{G}(F_{n} ∧ X)

is E(n)-local. Smashing Map_{c}(G^{k−1}, F_{n} ∧ X) with the spectrum M_{I} shows that (Γ_{G}^{k}(F_{n} ∧ M_{I} ∧ X)_{f})^{G} is K(n)-local. Therefore, since the homotopy limit of an arbitrary diagram of E-local spectra is E-local, (F_{n} ∧ M_{I} ∧ X)^{hG} and (E^{∨}(X))^{hG} are K(n)-local. The same argument shows that (F_{n} ∧ X)^{hG} is E(n)-local. □

The following theorem shows that the homotopy fixed points of E^{∨}(X) are obtained by taking the K(n)-localization of the homotopy fixed points of the discrete G_{n}-spectrum (F_{n} ∧ X).

Theorem 9.7. For G <_{c} G_{n} and any spectrum X with trivial G-action, there is an isomorphism (E^{∨}(X))^{hG} ≅ L̂((F_{n} ∧ X)^{hG}) in the stable category. In particular, E_{n}^{hG} ≅ L̂(F_{n}^{hG}).

Proof. After switching M_{I} and X, (E^{∨}(X))^{hG} ≅ holim_{I}(F_{n} ∧ X ∧ M_{I})^{hG}. By Remark 7.16, (F_{n} ∧ X ∧ M_{I})^{hG} ≅ (F_{n} ∧ X)^{hG} ∧ M_{I} ≃ ((F_{n} ∧ X)^{hG} ∧ M_{I})_{f}, where the isomorphism signifies a zigzag of natural weak equivalences. Thus,

(E^{∨}(X))^{hG} ≅ holim_{I}((F_{n} ∧ X)^{hG} ∧ M_{I})_{f} ≅ L̂((F_{n} ∧ X)^{hG}),

since, by Lemma 9.6, (F_{n} ∧ X)^{hG} is E(n)-local. □

Corollary 9.8. If X is a finite spectrum of type n, then there is an isomorphism (F_{n} ∧ X)^{hG} ≅ E_{n}^{hG} ∧ X. In particular, (F_{n} ∧ M_{I})^{hG} ≅ E_{n}^{hG} ∧ M_{I}.

Proof. By Remark 7.16 and the fact that F_{n}^{hG} is E(n)-local (Lemma 9.6),

(F_{n} ∧ X)^{hG} ≅ F_{n}^{hG} ∧ X ≅ L̂(F_{n}^{hG}) ∧ X ≅ E_{n}^{hG} ∧ X.
□

We conclude this section by observing that smashing with a finite spectrum commutes with taking the homotopy fixed points of E_{n}.

Theorem 9.9. Let G be a closed subgroup of G_{n} and let X be a finite spectrum. Then there is an isomorphism (E_{n} ∧ X)^{hG} ≅ E_{n}^{hG} ∧ X.

Proof. Recall that (E_{n} ∧ X)^{hG} = holim_{I}(F_{n} ∧ M_{I} ∧ X)^{hG}. The zigzag of natural weak equivalences between (F_{n} ∧ M_{I} ∧ X)^{hG} and (F_{n} ∧ M_{I})^{hG} ∧ X yields

(E_{n} ∧ X)^{hG} ≅ holim_{I}((F_{n} ∧ M_{I})^{hG} ∧ X)_{f} ≃ (holim_{I}(F_{n} ∧ M_{I})^{hG}) ∧ X,

where the weak equivalence is due to Lemma 2.7. □

10. The descent spectral sequence for (E^{∨}(X))^{hG}

By applying the preceding two sections, it is now an easy matter to build the descent spectral sequence for (E^{∨}(X))^{hG}.

Definition 10.1. Let X be a spectrum. If the tower {π_{t}(E_{n} ∧M_{I} ∧X)}_{I} of abelian groups satisfies the Mittag-Leffler condition for all t ∈ ℤ, then X is an E_{n}-Mittag-Leffler spectrum. If X is an E_{n}-Mittag-Leffler spectrum, then, for convenience, we say that X is E_{n}-ML.

Any finite spectrum X is E_{n}-ML, since {π_{t}(E_{n} ∧ M_{I} ∧ X)}_{I} is a tower of finite abelian groups, by Lemma 2.13. However, an E_{n}-ML spectrum need not be finite. For example, for j ≥ 1, let X = E_{n}^{(j)}. Then π_{t}(E_{n}∧M_{I}∧X) ≅ Map_{c}^{ℓ}(G_{n}^{j}, π_{t}(E_{n})/I). Since {π_{t}(E_{n})/I} is a tower of epimorphisms, the tower {Map_{c}^{ℓ}(G_{n}^{j}, π_{t}(E_{n})/I)} is also, and E_{n}^{(j)} is E_{n}-ML.

Theorem 10.2. Let G be a closed subgroup of G_{n} and let X be any spectrum with trivial G-action. Let E_{2}^{s,t} = π^{s}π_{t}(holim_{I}(Γ_{G}^{•}(F_{n} ∧ M_{I} ∧ X)_{f,G})^{G}). Then there is a conditionally convergent descent spectral sequence

E_{2}^{s,t} ⇒ π_{t−s}((E^{∨}(X))^{hG}). (10.3)

If X is E_{n}-ML, then E_{2}^{s,t} ≅ H_{cont}^{s}(G; {π_{t}(E_{n} ∧ M_{I} ∧ X)}). In particular, if X is a finite spectrum, then descent spectral sequence (10.3) has the form

H_{c}^{s}(G; π_{t}(E_{n} ∧ X)) ⇒ π_{t−s}((E_{n} ∧ X)^{hG}).

Proof. As in Remark 9.4, E^{∨}(X) ≅ holim_{I}(F_{n} ∧ M_{I} ∧ X)_{f,G} is a continuous Gspectrum. Then (10.3) follows from Theorem 8.8 by considering the tower of spectra {(F_{n} ∧ M_{I} ∧ X)_{f}}_{I}. When X is E_{n}-ML, {(F_{n} ∧ M_{I} ∧ X)_{f}} is a Mittag-Leffler tower of spectra, and thus, the simplification of the E_{2}-term in this case follows from Theorem 8.8. By Definition 2.17, when X is finite, there is an isomorphism H_{cont}^{s}(G; {π_{t}(E_{n} ∧ M_{I} ∧ X)}) ≅ H_{c}^{s}(G; π_{t}(E_{n} ∧ X)). □

As discussed in Remark 8.10, Theorem 8.5 gives a spectral sequence with abutment π_{∗}((E^{∨}(X))^{hG}), the same as the abutment of (10.3), but with E_{2}-term given by H_{cont}^{s}(G; {π_{t}(E_{n} ∧ M_{I} ∧ X)}), which is, in general, different from the E_{2}-term of (10.3). We are interested in the descent spectral sequence of Theorem 10.2, not just because it is a second descent spectral sequence with an interesting E_{2}-term, but, as mentioned in §8, it can be compared with the K(n)-local E_{n}-Adams spectral sequence. (This comparison is work in progress.)

We conclude this paper with a computation that uses spectral sequence (10.3) to compute π_{∗}((L̂(E_{n} ∧ E_{n}^{(j)}))^{hG_{n}}), where j ≥ 1 and G_{n} acts only on the leftmost factor. By Theorem 2.4,

π_{t}(L̂(E_{n}^{(j+1)})) ≅ Map_{c}^{ℓ}(G_{n}^{j}, π_{t}(E_{n})) ≅ lim_{I} Map_{c}^{ℓ}(G_{n}, Map_{c}(G_{n}^{j−1}, π_{t}(E_{n} ∧ M_{I})))

≅ lim_{I} Map_{c}(G_{n}, Map_{c}(G_{n}^{j−1}, π_{t}(E_{n} ∧ M_{I}))),

where the tower {Map_{c}(G_{n}, Map_{c}(G_{n}^{j−1}, π_{t}(E_{n} ∧ M_{I})))} satisfies the Mittag-Leffler condition, by Remark 2.11. Thus, in spectral sequence (10.3), Theorem 2.16 implies that E_{2}^{s,t} ≅ lim_{I} H_{c}^{s}(G_{n}; Map_{c}(G_{n}, Map_{c}(G_{n}^{j−1}, π_{t}(E_{n} ∧ M_{I})))), which vanishes for s > 0, and equals Map_{c}(G_{n}^{j−1}, π_{t}(E_{n})), when s = 0. Thus,

π_{∗}((L̂(E_{n}^{(j+1)}))^{hG_{n}}) ≅ Map_{c}(G_{n}^{j−1}, π_{∗}(E_{n})) ≅ π_{∗}(L̂(E_{n}^{(j)})),

as abelian groups. Therefore, for j ≥ 1, there is an isomorphism

(L̂(E_{n}^{(j+1)}))^{hG_{n}} ≅ L̂(E_{n}^{(j)}).

The techniques described in this paper do not allow us to handle the j = 0 case, which would say that E_{n}^{hG_{n}} and L̂(S^{0}) ≃ E_{n}^{dhG_{n}} are isomorphic.

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Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067, USA

   E-mail address: dgdavis@math.purdue.edu