THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS

DANIEL G. DAVIS^{1}

Abstract. Let G be a profinite group. We define an S[[G]]-module to be a G-spectrum X that satisfies certain conditions, and, given an S[[G]]-module X, we define the homotopy orbit spectrum X_{hG}. When G is countably based and X satisfies a certain finiteness condition, we construct a homotopy orbit spectral sequence whose E_{2}-term is the continuous homology of G with coefficients in the graded profinite ℤ̂[[G]]-module π_{∗}(X). Let G_{n} be the extended Morava stabilizer group and let E_{n} be the Lubin-Tate spectrum. As an application of our theory, we show that the function spectrum F (E_{n}, L_{K(n)}(S^{0})) is an S[[G_{n}]]-module with an associated homotopy orbit spectral sequence.

1. Introduction

Let G be a finite group and let X be a (left) G-spectrum. Then the homotopy orbit spectrum X_{hG} is defined to be hocolim_{G} X, the homotopy colimit of the Gaction on X (see, for example, [16, pg. 42]). Furthermore, there is a homotopy orbit spectral sequence

H_{p}(G, π_{q}(X)) ⇒ π_{p+q}(X_{hG}),

where the E_{2}-term is the group homology of G, with coefficients in the graded Gmodule π_{∗}(X) ([17, 5.1]). In this paper, under certain hypotheses, we extend these constructions to the case where G is a profinite group.

After making a few comments about notation, we summarize the contents of this paper. We follow the convention that all of our spectra are in Spt, the category of Bousfield-Friedlander spectra of simplicial sets. We use (−)_{f} to denote functorial fibrant replacement in the category of spectra; for any spectrum Z, Z → Z_{f} is a weak equivalence with Z_{f} fibrant. Also, “holim” always denotes the version of the homotopy limit of spectra that is constructed levelwise in the category of simplicial sets, as defined in [2] and [22, 5.6].

In Section 2, for a finite group G, we use the fact that the homotopy colimit of a diagram of pointed simplicial sets is the diagonal of the simplicial replacement of the diagram, to obtain an alternative model for homotopy orbits. Then, in Section 3, we use this model to define the homotopy orbit spectrum for a profinite group G and a certain type of G-spectrum X, which we now define. We point out that the following definition was essentially first formulated by Mark Behrens.

Definition 1.1. Given a profinite group G, let {N_{i}} be a collection of open normal subgroups of G such that G ≅ lim_{i} G/N_{i}. Then we let S[[G]] be the spectrum holim_{i}(S[G/N_{i}])_{f}. Also, let {X_{i}} be an inverse system of G-spectra and G-equivariant maps indexed over {i}, such that, for each i, the G-action on X_{i} factors through G/N_{i} (so that X_{i} is a G/N_{i}-spectrum) and X_{i} is a fibrant spectrum. Then we call the associated G-spectrum X = holim_{i} X_{i} an S[[G]]-module.

1The author was partially supported by an NSF grant.

After defining the homotopy orbit spectrum X_{hG} for an S[[G]]-module X, we show that this construction agrees with the classical definition when G is finite. Now let us recall a definition (see [4, Section 3] for more detail).

Definition 1.2. A discrete right 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 right G-set, and each bonding map S^{1} ∧X_{k} → X_{k+1} is G-equivariant (S^{1} has trivial G-action).

In Section 4, given a discrete right G-spectrum X and any spectrum Z, we show that the function spectrum F (X, Z) is an S[[G]]-module, so that one can form F (X, Z)_{hG}. Interesting examples of F (X, Z) are the Brown-Comenetz dual cX of X and the Spanier-Whitehead dual DX.

Now we summarize the main result of Section 5, in which we build a homotopy orbit spectral sequence.

Recall that a topological space Y is countably based if there is a countable family B of open sets, such that each open set of Y is a union of members of B. In particular, one can consider countably based profinite groups. Then, if G is a countably based profinite group, by [24, Proposition 4.1.3], G has a chain

N_{0} ≥ N_{1} ≥ N_{2} ≥ ⋯

of open normal subgroups, such that G ≅ lim_{i≥0} G/N_{i}. For example, a compact p-adic analytic group is a countably based profinite group. (If a topological group G is compact p-adic analytic, then G is a profinite group with an open subgroup H of finite rank, by [7, Corollary 8.34]. Then, by [7, Exercise 1, pg. 58], G has finite rank, and thus, G is finitely generated ([7, pg. 51]), so that G is countably based, by [24, pg. 55].)

Let G be a countably based profinite group, with a chain {N_{i}}_{i≥0} of open normal subgroups, such that G ≅ lim_{i} G/N_{i}, and let {X_{i}} be a diagram of G-spectra, such that holim_{i} X_{i} is an S[[G]]-module. Recall from [3, pg. 5] that a spectrum X is an f-spectrum, if, for each integer q, the abelian group π_{q}(X) is finite. Let ℤ̂ denote lim_{n≥1} ℤ/nℤ. Then, if each X_{i} is an f-spectrum, we show that there is a homotopy orbit spectral sequence

E_{2}^{p,q} ≅ H_{p}^{c}(G, π_{q}(holim_{i} X_{i})) ⇒ π_{p+q}((holim_{i} X_{i})_{hG}),

where the E_{2}-term is the continuous homology of G with coefficients in the graded profinite ℤ̂[[G]]-module π_{∗}(holim_{i} X_{i}).

We continue to assume that G is a countably based profinite group in Section 6, where we prove a result about the homotopy orbits of a tower of Eilenberg-Mac Lane spectra that was suggested to the author by Mark Behrens. To be precise, let A_{0} ← A_{1} ← A_{2} ← ⋯ be a tower of G-modules and G-equivariant maps, such that, for each i, the G-action on A_{i} factors through G/N_{i}, so that A_{i} is a G/N_{i}-module. If each abelian group A_{i} is finite, then we call {A_{i}} a nice tower of G-modules. Given a nice tower {A_{i}} of G-modules, we show that holim_{i} H(A_{i}) is an S[[G]]-module, with

π_{∗}((holim_{i} H(A_{i}))_{hG}) ≅ H_{∗}^{c}(G, lim_{i} A_{i}),

where H(A_{i}) is an Eilenberg-Mac Lane spectrum.

In Section 7, we consider an example that arises in chromatic homotopy theory. Let ℓ be a prime and let n ≥ 1. Let E_{n} denote the Lubin-Tate spectrum, so that π_{∗}(E_{n}) = W (𝔽_{ℓ^{n}})[[u_{1}, ..., u_{n−1}]][u^{±1}], where the degree of u is −2 and the power series ring over the Witt vectors has degree zero. Also, let G_{n} = S_{n} ⋊ Gal(𝔽_{ℓ^{n}}/𝔽_{ℓ}) be the extended Morava stabilizer group, where S_{n} is the nth Morava stabilizer group, and recall that the profinite group G_{n} is compact ℓ-adic analytic. By work of Paul Goerss, Mike Hopkins, and Haynes Miller (see [18], [10], and [6]), the group G_{n} acts on E_{n}.

Now let K(n) be the nth Morava K-theory, with K(n)_{∗} = 𝔽_{ℓ}[v_{n}^{±1}], where the degree of v_{n} is 2(ℓ^{n}−1). Also, let L_{K(n)}(−) denote the Bousfield localization functor with respect to K(n). Then we show that the K(n)-local Spanier-Whitehead dual of E_{n}, the function spectrum F (E_{n}, L_{K(n)}(S^{0})), is an S[[G_{n}]]-module and there is a homotopy orbit spectral sequence

H_{p}^{c}(G_{n}, π_{q}(F (E_{n}, L_{K(n)}(S^{0})))) ⇒ π_{p+q}(F (E_{n}, L_{K(n)}(S^{0}))_{hG_{n}}).

When (ℓ−1) ∤ n, the use of the above spectral sequence to identify the spectrum F (E_{n}, L_{K(n)}(S^{0}))_{hG_{n}} more concretely is work in progress.

We want to point out that others have investigated homotopy orbits for profinite groups. In [8], Halvard Fausk studies homotopy orbits in the setting of proorthogonal G-spectra, where G is profinite. As pointed out in various places in this paper, Mark Behrens has worked on homotopy orbits for a profinite group. Also, Dan Isaksen has thought about homotopy orbits for profinite groups in the context of pro-spectra. Finally, [12], by Mike Hopkins and Hal Sadofsky, contains some explorations of homotopy orbits involving E_{n} and G_{n}. It was the author’s study of [12] that motivated him to begin work on the project represented by this paper.

Acknowledgements. I thank Hal Sadofsky for sparking my interest in homotopy orbits for profinite groups. I thank Mark Behrens for helpful and encouraging discussions about this work. His positive influence on this work is seen in the many places where his name is mentioned. Also, I thank Paul Goerss and Mark Hovey for their encouragement.

2. Homotopy orbits when G is finite

In this section, let G be a finite group and let X be a (left) G-spectrum. We use simplicial replacement to obtain an alternative model for the homotopy orbit spectrum X_{hG}, which will be useful for defining the homotopy orbit spectrum when G is a profinite group. Given a spectrum Z, we let Z_{k} denote the kth pointed simplicial set of Z and we use Z_{k,l} to signify the set of l-simplices of Z_{k}.

By [2, XII, 5.2],

(X_{hG})_{k} = hocolim_{G} X_{k} = diag( ∐_{∗} (G → X_{k})),

where diag(−) is the functor that takes the diagonal of a bisimplicial set, and G → X_{k} is the diagram out of the groupoid G that is defined by the action of G on X_{k}. The simplicial replacement is defined as follows:

∐_{l} (G → X_{k}) = ⋁_{u_{l}∈I_{l}} X_{k},

where I_{l} is the indexing set that consists of all compositions of the form

u_{l} = i_{0} ⟵^{α_{1}} ⋯ ⟵^{α_{l}} i_{l},

such that each α_{j} is a morphism in the groupoid G.

The face and degeneracy maps of the pointed simplicial set ∐_{l} (G → X_{k}) are defined in the following way. Let 0 ≤ j < l and let d_{j}(u_{l}) be the map that, given a morphism u_{l} ∈ I_{l}, is defined by the composition

X_{k} ⟶^{id} X_{k} ⟶ ⋁_{I_{l−1}} X_{k},

where the target of id is the copy of X_{k} that is indexed by the morphism

u_{l−1} = i_{1} ⟵^{α_{2}} ⋯ ⟵^{α_{l}} i_{l} ∈ I_{l−1}.

Then the jth face map d_{j}: ⋁_{I_{l}} X_{k} → ⋁_{I_{l−1}} X_{k} is the unique map that is induced by all the d_{j}(u_{l}), where d_{j}(u_{l}) is the canonical map out of the copy of X_{k} that is indexed by u_{l}. Also, let d_{l}(u_{l}) be the map that, given a morphism u_{l} ∈ I_{l}, is defined by the composition

X_{k} ⟶^{α_{l}} X_{k} ⟶ ⋁_{I_{l−1}} X_{k},

where the target of α_{l} is the copy of X_{k} that is indexed by the morphism

u_{l−1} = i_{0} ⟵^{α_{1}} ⋯ ⟵^{α_{l−1}} i_{l−1} ∈ I_{l−1}.

Then the lth face map d_{l}: ⋁_{I_{l}} X_{k} → ⋁_{I_{l−1}} X_{k} is the unique map that is induced by all the d_{l}(u_{l}), where d_{l}(u_{l}) is the canonical map out of the copy of X_{k} that is indexed by u_{l}.

For 0 ≤ j ≤ l, let s_{j}(u_{l}) be the map that, given a morphism u_{l} ∈ I_{l}, is defined by the composition

X_{k} ⟶^{id} X_{k} ⟶ ⋁_{I_{l+1}} X_{k},

where the target of id is the copy of X_{k} that is indexed by the morphism

u_{l+1} = i_{0} ⟵^{α_{1}} ⋯ ⟵^{α_{l}} i_{l} ⟵^{id} i_{l} ∈ I_{l+1}.

Then the jth degeneracy s_{j}: ⋁_{I_{l}} X_{k} → ⋁_{I_{l+1}} X_{k} is the unique map that is induced by all the s_{j}(u_{l}), where s_{j}(u_{l}) is the canonical map out of the copy of X_{k} that is indexed by u_{l}.

Definition 2.1. Let K be a pointed simplicial set and let L be a set. Then

K[L] = K ∧ L_{+},

where L_{+} is the constant simplicial set on L, together with a disjoint basepoint. Similarly, if Z is a spectrum, then Z[L] is the spectrum with each (Z[L])_{k} equal to Z_{k} ∧ L_{+}.

Since each u_{l} can be identified with an element of G^{l}, the l-fold product of G, I_{l} = G^{l} and

∐_{l}(G → X_{k}) = ⋁_{G^{l}} X_{k} ≅ X_{k} ∧ (G^{l})_{+} = X_{k}[G^{l}].

Thus,

(X_{hG})_{k} ≅ diag(X_{k}[G^{•}]),

where the l-simplices of the simplicial pointed simplicial set X_{k}[G^{•}] are the pointed simplicial set X_{k}[G^{l}].

We introduce some notation that organizes the above, allowing us to summarize it in a theorem.

Definition 2.2. Let X be a G-spectrum. Then X[G^{•}] is the simplicial spectrum that is defined above, with (X[G^{•}])_{k} = X_{k}[G^{•}] and l-simplices equal to the spectrum X[G^{l}]. Thus, X[G^{•}] is the simplicial spectrum

\xymatrix{X ≅ X[∗] & X[G] \ar[l] \ar[l] & X[G^{2}] \ar[l] \ar[l] \ar[l] & · · · . \ar[l] \ar[l] \ar[l] \ar[l]}

We let π_{q}(X[G^{•}]) denote the simplicial abelian group associated to X[G^{•}].

Definition 2.3. Let d(Z_{•}) denote the spectrum that is the diagonal of the simplicial spectrum Z_{•} (see [14, pg. 100]). Thus, for all k ≥ 0,

(d(Z_{•}))_{k} = diag((Z_{•})_{k})

and, for all l ≥ 0,

(d(Z_{•}))_{k,l} = (Z_{l})_{k,l}.

Theorem 2.4. If G is a finite group and X is a G-spectrum, then

X_{hG} ≅ d(X[G^{•}]).

3. The homotopy orbit spectrum X_{hG}

In this section, we use the model for the homotopy orbit spectrum that is given in Theorem 2.4 to help us define homotopy orbits for G a profinite group. We begin with a comment about Definition 1.1.

Remark 3.1. Let X = holim_{i} X_{i} be an S[[G]]-module. The choice of the term “S[[G]]-module” is motivated by the fact that, for each i, the action of G/N_{i} on X_{i} yields a function G/N_{i} → Hom_{Spt}(X_{i}, X_{i}) in

Hom_{Sets}(G/N_{i}, Hom_{Spt}(X_{i}, X_{i})) ≅ ∏_{G/N_{i}}Hom_{Spt}(X_{i}, X_{i})
≅ Hom_{Spt}(X_{i}[G/N_{i}], X_{i})

that corresponds to a map X_{i} ∧ (G/N_{i})_{+} → X_{i}. The analogy with actual modules over the spectrum S[[G]] could be pursued further (as Mark Behrens has done); however, for simplicity, we do not do this.

Using the author’s preliminary work on homotopy orbits as a reference point, the following definition was basically given by Behrens, and then later independently formulated by the author.

Definition 3.2. Let X = holim_{i} X_{i} be an S[[G]]-module. For each l ≥ 0, the diagrams {X_{i}} and {G/N_{i}} induce the diagram {X_{i}[(G/N_{i})^{l}]}, so that one can form holim_{i}(X_{i}[(G/N_{i})^{l}])_{f}, which gives the simplicial spectrum holim_{i}(X_{i}[(G/N_{i})^{•}])_{f}. Then we define X_{hG}, the homotopy orbit spectrum of X with respect to the Gaction, to be the spectrum

X_{hG} = d(holim_{i}(X_{i}[(G/N_{i})^{•}])_{f}).

Henceforth, we use the notation X_{h′G} to denote the classical homotopy orbit spectrum hocolim_{G} X, when G is finite.

The definition of the homotopy orbit spectrum comes from imitating the model in Theorem 2.4 and from the demands of the homotopy orbit spectral sequence (see the proof of Theorem 5.3). For example, the reader might expect that the definition involve a homotopy colimit of G-spectra, instead of the homotopy limit of such; however, the homotopy limit is necessitated by the nature of continuous group homology, which is the E_{2}-term of the homotopy orbit spectral sequence.

Remark 3.3. The functor (−)_{f} appears in Definition 3.2 so that the homotopy limit is well-behaved. Let us suppose that the functor (−)_{f} does not appear in the definition of X_{hG}, so that it reads instead as X_{hG} = d(holim_{i} X_{i}[(G/N_{i})^{•}]). Recall that X_{i}[(G/N_{i})^{•}] is involved in the construction of (X_{i})_{hG/N_{i}} ≅ d(X_{i}[(G/N_{i})^{•}]), by Theorem 2.4. Thus, one might wonder if there is an isomorphism

d(holim_{i} X_{i}[(G/N_{i})^{•}]) ≅ holim_{i} d(X_{i}[(G/N_{i})^{•}]),

so that, morally, X_{hG} = holim_{i}(X_{i})_{hG/N_{i}}. However, it is not hard to see that this is not the case, since

(d(holim_{i} X_{i}[(G/N_{i})^{•}]))_{k,l} = (diag(holim_{i}(X_{i})_{k}[(G/N_{i})^{•}]))_{l}

= (holim_{i}(X_{i})_{k}[(G/N_{i})^{l}])_{l}

= Hom_{S^{({i}^{op})}}(Δ[l] × ({i}^{op})/−, {(X_{i})_{k}[(G/N_{i})^{l}]}_{i})

(see [2, Chapter XI, 2.2, 3.2]), where S is the category of simplicial sets, is not equivalent to

(holim_{i} d(X_{i}[(G/N_{i})^{•}]))_{k,l} = (holim_{i} diag((X_{i})_{k}[(G/N_{i})^{•}]))_{l}
= Hom_{S^{({i}^{op})}}(Δ[l] × ({i}^{op})/−, {diag((X_{i})_{k}[(G/N_{i})^{•}])}_{i}).

We recall the following useful result.

Theorem 3.4 ([14, Corollary 4.22]). If X_{•} is a simplicial spectrum, then there is a spectral sequence

E_{2}^{p,q} = H_{p}(π_{q}(X_{∗})) ⇒ π_{p+q}(d(X_{•})),

where H_{∗}(π_{q}(X_{∗})) is the homology of the Moore complex of the simplicial abelian group π_{q}(X_{•}).

Lemma 3.5. Let X_{•} → Y_{•} be a map between simplicial spectra, such that, for each n ≥ 0, the map X_{n} → Y_{n} is a weak equivalence between the n-simplices. Then the induced map d(X_{•}) → d(Y_{•}) is a weak equivalence of spectra.

Proof. There is a spectral sequence

H_{p}(π_{q}(X_{∗})) ⇒ π_{p+q}(d(X_{•})),

and a map to the spectral sequence

H_{p}(π_{q}(Y_{∗})) ⇒ π_{p+q}(d(Y_{•})).

Since π_{q}(X_{n}) ≅ π_{q}(Y_{n}), for each n ≥ 0, and π_{q}(X_{∗}) and π_{q}(Y_{∗}) are chain complexes, there is an isomorphism H_{p}(π_{q}(X_{∗})) →^{≅} H_{p}(π_{q}(Y_{∗})) of E_{2}-terms. Therefore, the abutments of the above two spectral sequences are isomorphic, giving the conclusion of the lemma. □

Remark 3.6. Let G be finite and let X be any G-spectrum. Let {i} = {0}, N_{0} = {e}, and X_{0} = X_{f}. Then holim_{{0}} X_{0} ≅ X_{f} is an S[[G]]-module and

(holim_{{0}} X_{0})_{hG} = d(holim_{{0}}(X_{f}[G^{•}])_{f}) ≅ d((X_{f}[G^{•}])_{f}).

Thus, there is a weak equivalence

X_{h′G} ≅ d(X[G^{•}]) ⟶^{≃} d((X_{f}[G^{•}])_{f}) ≅ (holim_{{0}} X_{0})_{hG},

so that Definition 3.2 recovers the classical definition of homotopy orbits.

Now we show that (−)_{hG} preserves weak equivalences of S[[G]]-modules, as defined below.

Definition 3.7. Let {X_{i}} → {Y_{i}} be a natural transformation of diagrams of Gspectra, such that X_{i} → Y_{i} is a weak equivalence, for each i, and the induced map X = holim_{i} X_{i} → holim_{i} Y_{i} = Y is a map between S[[G]]-modules. Then we say that the weak equivalence X → Y is a weak equivalence of S[[G]]-modules.

Theorem 3.8. If X → Y, as in Definition 3.7, is a weak equivalence of S[[G]]- modules, then X_{hG} → Y_{hG} is a weak equivalence.

Proof. Let i ≥ 0. Since X_{i} → Y_{i} is a weak equivalence, the induced map

(X_{i}[(G/N_{i})^{l}])_{f} → (Y_{i}[(G/N_{i})^{l}])_{f}

is a weak equivalence between fibrant spectra, for each l ≥ 0. Thus,

holim_{i}(X_{i}[(G/N_{i})^{l}])_{f} → holim_{i}(Y_{i}[(G/N_{i})^{l}])_{f}

is a weak equivalence, so that, by Lemma 3.5, X_{hG} → Y_{hG} is a weak equivalence. □

4. Examples of S[[G]]-modules

In this section, we consider a way that S[[G]]-modules arise naturally from discrete right G-spectra.

The following recollection will be helpful. There is a functor (−)_{c}: Spt → Spt, such that, given Y in Spt, Y_{c} is a cofibrant spectrum, and there is a natural transformation (−)_{c} → id_{Spt}, such that, for any Y, the map Y_{c} → Y is a trivial fibration. For example, if Y is a right K-spectrum for some group K, then Y_{c} is also a right K-spectrum, and the map Y_{c} → Y is K-equivariant.

Let Z be any spectrum and let X be a discrete right G-spectrum. Then

F (X, Z) ≃ F (colim_{i} X^{N_{i}}, Z_{f}) ≃ F (hocolim_{i} X^{N_{i}}, Z_{f}) ≃ holim_{i} F ((X^{N_{i}})_{c}, Z_{f}).

Thus, when X is a discrete right G-spectrum, we identify the left G-spectrum F (X, Z) with the G-spectrum holim_{i} F ((X^{N_{i}})_{c}, Z_{f}). Under this identification, we make the following observation.

Theorem 4.1. If X is a discrete right G-spectrum, then F (X, Z) is an S[[G]]- module.

Proof. The spectrum (X^{N_{i}})_{c} is a right G/N_{i}-spectrum, because X^{N_{i}} is a right G/N_{i}-spectrum. Thus, F ((X^{N_{i}})_{c}, Z_{f}) is a G/N_{i}-spectrum. Also, since the source and target of the function spectrum F ((X^{N_{i}})_{c}, Z_{f}) are cofibrant and fibrant, respectively, the function spectrum itself is fibrant. These facts imply that the spectrum holim_{i} F ((X^{N_{i}})_{c}, Z_{f}) is an S[[G]]-module. □

We give two examples of F (X, Z) as an S[[G]]-module.

Example 4.2. Recall from [3] that, if cS^{0} is the Brown-Comenetz dual of S^{0}, then cX, the Brown-Comenetz dual of X, is F (X, cS^{0}). Thus, if X is a discrete right G-spectrum, then

cX = holim_{i} F ((X^{N_{i}})_{c}, (cS^{0})_{f})

is an S[[G]]-module.

The following example is due to Mark Behrens.

Example 4.3. If X is a discrete right G-spectrum, then the Spanier-Whitehead dual of X,

DX = F (X, S^{0}) = holim_{i} F ((X^{N_{i}})_{c}, (S^{0})_{f}),

is an S[[G]]-module.

5. The homotopy orbit spectral sequence for an S[[G]]-module

In this section, we construct a homotopy orbit spectral sequence for a certain type of S[[G]]-module. We attempted to construct such a spectral sequence for any profinite group G. However, there were difficulties that we were able to get around only by putting an additional hypothesis on the group G, that is, by assuming that G is countably based.

Now we recall the construction of the classical homotopy orbit spectral sequence for X_{h′G}; the brief analysis below of its E_{2}-term will be useful. Thus, for the duration of this paragraph, let G be a finite group and let X be a G-spectrum. By Theorem 3.4, there is a spectral sequence

E_{2}^{p,q} ⇒ π_{p+q}(d(X[G^{•}])) = π_{p+q}(X_{h′G}),

where

E_{2}^{p,q} = H_{p}(π_{q}(X[G^{∗}])) ≅ H_{p}(G, π_{q}(X)), (5.1)

the pth group homology of G, with coefficients in π_{q}(X) (see, for example, [14, (7.9)]). Note that, by (5.1),

H_{p}(G, π_{q}(X)) ≅ H_{p}(π_{q}(X)[G^{∗}]), (5.2)

where, if A is an abelian group and K is a group, we use A[K] to denote ⨁_{K} A.

Now we are ready to construct the homotopy orbit spectral sequence for a countably based profinite group.

Theorem 5.3. Let G be a countably based profinite group, and let

N_{0} ≥ N_{1} ≥ N_{2} ≥ ⋯

be a chain of open normal subgroups of G, such that G ≅ lim_{i} G/N_{i}. Also, let

X_{0} ← X_{1} ← X_{2} ← ⋯

be a tower of G-spectra and G-equivariant maps such that X = holim_{i} X_{i} is an S[[G]]-module. If, for all i, X_{i} is an f-spectrum, then there is a spectral sequence of the form

E_{2}^{p,q} ≅ H_{p}^{c}(G, π_{q}(X)) ⇒ π_{p+q}(X_{hG}),

where the E_{2}-term is the continuous homology of G with coefficients in the graded profinite ℤ̂[[G]]-module π_{∗}(X).

Proof. By Theorem 3.4, there is a spectral sequence

E_{2}^{p,q} ⇒ π_{p+q}(d(holim_{i}(X_{i}[(G/N_{i})^{•}])_{f})) = π_{p+q}(X_{hG}),

where

E_{2}^{p,q} = H_{p}(π_{q}(holim_{i}(X_{i}[(G/N_{i})^{∗}])_{f})).

We spend the rest of the proof identifying the E_{2}-term as continuous group homology.

Let l ≥ 0. Since {π_{q+1}(X_{i})[(G/N_{i})^{l}]} is a tower of finite abelian groups,

lim^{1}_{i} π_{q+1}(X_{i})[(G/N_{i})^{l}] = 0.

Thus,

π_{q}(holim_{i}(X_{i}[(G/N_{i})^{l}])_{f}) ≅ lim_{i} π_{q}(X_{i})[(G/N_{i})^{l}],

which implies that

E_{2}^{p,q} ≅ H_{p}(lim_{i}(π_{q}(X_{i})[(G/N_{i})^{∗}])).

Since, for any l ≥ 0, the tower of abelian groups {π_{q}(X_{i})[(G/N_{i})^{l}]}_{i} satisfies the Mittag-Leffler condition, by [23, Theorem 3.5.8], there is a short exact sequence

0 → lim^{1}_{i} H_{p+1}(π_{q}(X_{i})[(G/N_{i})^{∗}]) → E_{2}^{p,q} → lim_{i} H_{p}(π_{q}(X_{i})[(G/N_{i})^{∗}]) → 0.

By (5.2), this short exact sequence can be written as

0 → lim^{1}_{i} H_{p+1}(G/N_{i}, π_{q}(X_{i})) → E_{2}^{p,q} → lim_{i} H_{p}(G/N_{i}, π_{q}(X_{i})) → 0.

Since G/N_{i} and π_{q}(X_{i}) are finite, H_{p+1}(G/N_{i}, π_{q}(X_{i})) is a finite abelian group, for each p ≥ 0 ([23, Corollary 6.5.10]). Thus, lim^{1}_{i} H_{p+1}(G/N_{i}, π_{q}(X_{i})) = 0. Therefore, we obtain that

E_{2}^{p,q} ≅ lim_{i} H_{p}(G/N_{i}, π_{q}(X_{i})) ≅ H_{p}^{c}(G, π_{q}(X)),

where the last isomorphism uses [19, Proposition 6.5.7]. □

6. Eilenberg-Mac Lane spectra and their homotopy orbits

Let the profinite group G be countably based, with a chain {N_{i}} of open normal subgroups, such that G ≅ lim_{i} G/N_{i}, and, as defined in the Introduction, let {A_{i}} be a nice tower of G-modules. Let Γ: Ch_{+} → sAb be the functor in the Dold-Kan correspondence from Ch_{+}, the category of chain complexes C_{∗} with C_{n} = 0 for n < 0, to sAb, the category of simplicial abelian groups (see, for example, [11, Chapter III, Corollary 2.3]). Also, if A is an abelian group, let A[−n] be the chain complex that is A in degree n and zero elsewhere.

Given the tower {A_{i}}, we explain how to form {H(A_{i})}, a tower of Eilenberg-Mac Lane spectra (we follow the construction given in [15]), so that holim_{i} H(A_{i}) is an S[[G]]-module. By functoriality, for each k ≥ 0, {Γ(A_{i}[−k])} is a tower of simplicial G-modules and G-equivariant maps, such that, for each i, Γ(A_{i}[−k]) is the Eilenberg-Mac Lane space K(A_{i}, k) and Γ(A_{i}[−k]) is a simplicial G/N_{i}-module. Furthermore, by taking 0 as the basepoint, each Γ(A_{i}[−k]) is a pointed simplicial set.

For each i, we define the Eilenberg-Mac Lane spectrum H(A_{i}) by (H(A_{i}))_{k} = Γ(A_{i}[−k]), so that π_{0}(H(A_{i})) = A_{i} and π_{n}(H(A_{i})) = 0, when n ≠ 0. Then, by functoriality, {H(A_{i})} is a tower of G-spectra and G-equivariant maps, such that each H(A_{i}) is a G/N_{i}-spectrum. Since each (H(A_{i}))_{k} is a fibrant simplicial set and each H(A_{i}) is an Ω-spectrum (see, for example, [15, Example 21]), H(A_{i}) is a fibrant spectrum. These facts imply the following result.

Lemma 6.1. The spectrum holim_{i} H(A_{i}) is an S[[G]]-module.

Now we show that the homotopy orbit spectral sequence can be used to compute π_{∗}((holim_{i} H(A_{i}))_{hG}).

Theorem 6.2. If G is a countably based profinite group with a chain {N_{i}} of open normal subgroups, such that G ≅ lim_{i} G/N_{i}, and, if {A_{i}} is a nice tower of G-modules, then

π_{∗}((holim_{i} H(A_{i}))_{hG}) ≅ H_{∗}^{c}(G, lim_{i} A_{i}).

Proof. By hypothesis, each H(A_{i}) is an f-spectrum. Then, by Theorem 5.3, there is a homotopy orbit spectral sequence

E_{2}^{p,q} ≅ H_{p}^{c}(G, π_{q}(holim_{i} H(A_{i}))) ⇒ π_{p+q}((holim_{i} H(A_{i}))_{hG}).

Since lim^{1}_{i} A_{i} = 0, we have:

E_{2}^{p,q} ≅ H_{p}^{c}(G, lim_{i} π_{q}(H(A_{i}))) = \begin{cases}H_{p}^{c}(G, 0) & if q ≠ 0 \\ H_{p}^{c}(G, lim_{i} A_{i}) & if q = 0 \end{cases}

= \begin{cases}0 & if q ≠ 0 \\ H_{p}^{c}(G, lim_{i} A_{i}) & if q = 0. \end{cases}

Thus, the spectral sequence collapses, giving the conclusion of the theorem. □

7. The G_{n}-homotopy orbits of F (E_{n}, L_{K(n)}(S^{0}))

Our main motivation for constructing homotopy orbits for a profinite group was that we were interested in constructing the G_{n}-homotopy orbit spectrum of E_{n}, as part of our effort to understand [12]. Though we have not been able to construct such a spectrum, we are able to construct the G_{n}-homotopy orbits of the closely related spectrum

F (E_{n}, L_{K(n)}(S^{0})) ≃ Σ^{−n^{2}}E_{n},

where the weak equivalence applies [20, Proposition 16] and the function spectrum F (E_{n}, L_{K(n)}(S^{0})) is the K(n)-local Spanier-Whitehead dual of E_{n}. After explaining why we have not been able to construct a homotopy orbit spectrum for E_{n}, we consider F (E_{n}, L_{K(n)}(S^{0}))_{hG_{n}}.

Recall from the Introduction that, since G_{n} is a compact ℓ-adic analytic group, it is a countably based profinite group. Then, following [6, (1.4)], we fix a chain G_{n} = N_{0} ⪈ N_{1} ⪈ N_{2} ⪈ ⋯ of open normal subgroups of G_{n}, such that ⋂_{i} N_{i} = {e}, so that G_{n} ≅ lim_{i} G_{n}/N_{i}. Thus, an S[[G_{n}]]-module comes from a tower X_{0} ← X_{1} ← X_{2} ← ⋯ of G_{n}-spectra and G_{n}-equivariant maps.

To form the G_{n}-homotopy orbit spectrum of E_{n}, we have to show that E_{n} is an S[[G_{n}]]-module. However, we do not know how to do this and we explain how a natural way to try to do this fails. Let E_{n}^{dhN_{i}} be the spectrum constructed by Devinatz and Hopkins (see [6]) that behaves like the N_{i}-homotopy fixed points of E_{n} with respect to a continuous N_{i}-action (see [4]). Let

M_{I_{0}} ← M_{I_{1}} ← M_{I_{2}} ← ⋯

be a tower of generalized Moore spectra such that

holim_{j}(L_{K(n)}(M_{I_{j}}))_{f} ≃ L_{K(n)}(S^{0})

([13, Proposition 7.10 (e)]). Also, recall from [4, Theorem 6.6] that

E_{n} ≃ holim_{j}(colim_{i}(E_{n}^{dhN_{i}} ∧ M_{I_{j}})_{f}).

Thus, one might attempt to present E_{n} as an S[[G_{n}]]-module by considering the tower {colim_{i}(E_{n}^{dhN_{i}}∧M_{I_{j}})_{f}}_{j}. However, by [4, Lemma 6.2], there is no open normal subgroup N of G_{n} such that the G_{n}-action on colim_{i}(E_{n}^{dhN_{i}} ∧M_{I_{j}})_{f} factors through G_{n}/N. Thus, this attempt fails to present E_{n} as an S[[G_{n}]]-module.

We noted above that F (E_{n}, L_{K(n)}(S^{0})) ≃ Σ^{−n^{2}}E_{n}. Thus, one might like to assert that, since we can construct the G_{n}-homotopy orbit spectrum of F (E_{n}, L_{K(n)}(S^{0})), as done below, then we can also construct the homotopy orbit spectrum of E_{n}, by considering the homotopy orbits of Σ^{n^{2}}F (E_{n}, L_{K(n)}(S^{0})). However, we have been informed by Mark Behrens that, in general, this can not be done, because the equivalence F (E_{n}, L_{K(n)}(S^{0})) ≃ Σ^{−n^{2}}E_{n} need not be G_{n}-equivariant. According to Behrens, the S^{−n^{2}} in Σ^{−n^{2}}E_{n}, when properly interpreted, can have a non-trivial G_{n}-action on it.

To form F (E_{n}, L_{K(n)}(S^{0}))_{hG_{n}}, we only need to show that F (E_{n}, L_{K(n)}(S^{0})) is an S[[G_{n}]]-module. As in [4], let F_{n} = colim_{i} E_{n}^{dhN_{i}}. Then

F (E_{n}, L_{K(n)}(S^{0})) ≃ F (L_{K(n)}(F_{n}), holim_{j}(L_{K(n)}(M_{I_{j}}))_{f})

≃ F (F_{n}, holim_{j}(L_{K(n)}(M_{I_{j}}))_{f})

≃ F (hocolim_{i}(E_{n}^{dhN_{i}})_{c}, holim_{j}(L_{K(n)}(M_{I_{j}}))_{f})

≅ holim_{i} holim_{j} F ((E_{n}^{dhN_{i}})_{c}, (L_{K(n)}(M_{I_{j}}))_{f})

≃ holim_{i} F ((E_{n}^{dhN_{i}})_{c}, (L_{K(n)}(M_{I_{i}}))_{f}),

where the first weak equivalence applies [6, Definition 1.5] and [4, Theorem 6.3], and the last weak equivalence uses the fact that {(i, i)}_{i} is cofinal in {(i, j)}_{i,j}. Thus, we make the identification

F (E_{n}, L_{K(n)}(S^{0})) = holim_{i} F ((E_{n}^{dhN_{i}})_{c}, (L_{K(n)}(M_{I_{i}}))_{f}).

Since E_{n}^{dhN_{i}} is a right G_{n}/N_{i}-spectrum, as in Section 4, the above identification implies the following result.

Lemma 7.1. The spectrum F (E_{n}, L_{K(n)}(S^{0})) is an S[[G_{n}]]-module.

The next result allows us to build the homotopy orbit spectral sequence for F (E_{n}, L_{K(n)}(S^{0}))_{hG_{n}}.

Lemma 7.2. For each i ≥ 0, the spectrum F (E_{n}^{dhN_{i}}, L_{K(n)}(M_{I_{i}})) is an f-spectrum.

Proof. Our proof follows [5, proof of Lemma 3.5]. Since

L_{K(n)}(M_{I_{i}}) ≃ L_{K(n)}(S^{0}) ∧ M_{I_{i}} ≃ E_{n}^{dhG_{n}} ∧ M_{I_{i}}

(see [6, Theorem 1(iii)]), [6, Theorem 2(ii)] gives a strongly convergent K(n)-local E_{n}-Adams spectral sequence

H_{c}^{s}(G_{n}, (E_{n} ∧ M_{I_{i}})^{−t}(E_{n}^{dhN_{i}})) ⇒ π_{t−s}(F (E_{n}^{dhN_{i}}, L_{K(n)}(M_{I_{i}}))),

where the E_{2}-term is continuous cohomology for profinite continuous ℤ_{ℓ}[[G_{n}]]- modules (see [6, Remark 1.3]). By [9, Proposition 2.5],

(E_{n} ∧ M_{I_{i}})^{−t}(E_{n}^{dhN_{i}}) ≅ ⨁_{G_{n}/N_{i}} π_{t}(E_{n} ∧ M_{I_{i}})

is a finite abelian group. Thus, by [21, Proposition 4.2.2], the abelian group H_{c}^{s}(G_{n}, (E_{n} ∧ M_{I_{i}})^{−t}(E_{n}^{dhN_{i}})) is finite. The fact that E_{∞}^{∗,∗} has a horizontal vanishing line (see [1, Theorem 6.10], [5, Proposition 2.3, proof of Lemma 3.5], and [6, Proposition A.3]) implies that π_{∗}(F (E_{n}^{dhN_{i}}, L_{K(n)}(M_{I_{i}}))) is finite in each degree. □

Theorem 5.3 and Lemmas 7.1 and 7.2 give the following.

Theorem 7.3. There is a spectral sequence

H_{p}^{c}(G_{n}, π_{q}(F (E_{n}, L_{K(n)}(S^{0})))) ⇒ π_{p+q}(F (E_{n}, L_{K(n)}(S^{0}))_{hG_{n}}).

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