New York Journal of Mathematics

New York J. Math. 12 (2006) 183–191.

The E_{2}-term of the descent spectral sequence for continuous G-spectra

Daniel G. Davis

Abstract. Given a profinite group G with finite virtual cohomological dimension, let {X_{i}} be a tower of discrete G-spectra, each of which is fibrant as a spectrum, so that X = holim_{i} X_{i} is a continuous G-spectrum, with homotopy fixed point spectrum X^{hG}. The E_{2}-term of the descent spectral sequence for π_{∗}(X^{hG}) cannot always be expressed as continuous cohomology. However, we show that the E_{2}-term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in lim_{i} M_{i}, where {M_{i}} is a tower of discrete G-modules.

Contents

1. Introduction 183 2. The pro-discrete cochain complex and continuous cohomology 185 3. Constructing the descent spectral sequence 187 4. The E_{2}-term and the pro-discrete cochain complex 188 5. The failure of other possible descriptions of the E_{2}-term 189 References 190

1. Introduction

In this note, G always denotes a profinite group. Let H_{c}^{∗}(G; M) denote the continuous cohomology of G with coefficients in the discrete G-module M. This cohomology is defined as the right derived functors of G-fixed points. Then we always assume that G has finite virtual cohomological dimension; that is, there exists an open subgroup H and a nonnegative integer m, such that H_{c}^{s}(H; M) = 0, for all discrete H-modules M and all s ≥ m.

All of our spectra are Bousfield–Friedlander spectra of simplicial sets. In particular, a discrete G-spectrum is a G-spectrum such that each simplicial set X_{k} is a

Received February 20, 2006. Mathematics Subject Classification. 55P42, 55T99. Key words and phrases. Homotopy fixed point spectrum, descent spectral sequence, continuous

G-spectrum.

The author was partially supported by an NSF grant. Most of this paper was written during a visit to the Institut Mittag-Leffler (Djursholm, Sweden).

183^{ISSN 1076-9803/06}

simplicial object in the category of discrete G-sets (thus, for any l ≥ 0, the action map on the l-simplices, G × (X_{k})_{l} → (X_{k})_{l}, is continuous when (X_{k})_{l} is regarded as a discrete space). The category of discrete G-spectra, with morphisms being G-equivariant maps of spectra, is denoted by Spt_{G}.

Discrete G-spectra are considered in more detail in [2], which shows (see [2, Theorem 3.6]) that Spt_{G} is a model category, where a morphism f in Spt_{G} is a weak equivalence (cofibration) if and only if f is a weak equivalence (cofibration) in Spt, the category of spectra. Given a discrete G-spectrum X, the homotopy fixed point spectrum X^{hG} is obtained as the total right derived functor of fixed points: X^{hG} = (X_{f,G})^{G}, where X → X_{f,G} is a trivial cofibration and X_{f,G} is fibrant, all in Spt_{G}.

Let X_{0} ← X_{1} ← X_{2} ← ⋯ be a tower of discrete G-spectra, such that each X_{i} is a fibrant spectrum. As explained in [2, Lemma 4.4], there exists a tower {X′_{i}} of discrete G-spectra, such that there are weak equivalences

holim_{i} X_{i} ⟶^{≃} holim_{i} X′_{i} ⟵^{≃} lim_{i} X′_{i}.

(In this paper, holim always denotes the version of the homotopy limit of spectra that is constructed levelwise in the category of simplicial sets, as defined in [1] and [10, 5.6].) Since the inverse limit of a tower of discrete G-sets is a topological G-space, and because holim_{i} X_{i} can be identified with lim_{i} X′_{i}, in the eyes of homotopy, holim_{i} X_{i} is a continuous G-spectrum. Notice that, under this identification, the continuous G-action respects the topology of both G and all the X_{i} together. Continuous G-spectra and examples of such in chromatic stable homotopy theory are considered in [2].

Given the tower {X_{i}} and the continuous G-spectrum holim_{i} X_{i}, its homotopy fixed point spectrum, (holim_{i} X_{i})^{hG}, is defined to be holim_{i}(X_{i})^{hG}. This construction is called homotopy fixed points because it is equivalent to the usual definition when G is a finite group and it is the total right derived functor of fixed points in the appropriate sense (see [2, Remark 8.4]).

By [2, Theorem 8.8], thanks to the assumption of finite virtual cohomological dimension, there is a descent spectral sequence

E_{2}^{s,t} ⇒ π_{t−s}((holim_{i} X_{i})^{hG}), (1.1)

where

E_{2}^{s,t} = π^{s}π_{t}(holim_{i}(Γ_{G}^{•}(X_{i})_{f,G})^{G}) (1.2)

(see the beginning of Section 2 for the meaning of Γ_{G}^{•}), and, if the tower of abelian groups {π_{t}(X_{i})} satisfies the Mittag-Leffler condition for every integer t, then E_{2}^{s,t} ≅ H_{cont}^{s}(G; {π_{t}(X_{i})}), which is continuous cohomology in the sense of Jannsen. (This cohomology is obtained by taking the right derived functors of lim_{i}(−)^{G}, a functor from towers of discrete G-modules to abelian groups; see [7].)

In expression (1.2), since π_{t}(holim_{i}(−)) is not necessarily lim_{i} π_{t}(−), the E_{2}-term of descent spectral sequence (1.1), in general, can not be expressed as continuous cohomology, and, in general, it has no compact algebraic description. For example, as pointed out in [2, Remark 8.10], when G = {e},

H_{cont}^{s}({e}; {π_{t}(X_{i})}) = lim_{i} π_{t}(X_{i})

The E_{2}-term of the descent spectral sequence 185

and E_{2}^{0,t} = π_{t}(holim_{i} X_{i}), and these need not be isomorphic, due to the familiar lim^{1}_{i} π_{t+1}(X_{i}) obstruction. However, in this note, we show that the E_{2}-term (1.2) can always be described in an interesting way.

In more detail, Theorem 2.3 gives a particular cochain complex C^{∗} for computing the continuous cochain cohomology of G for a topological G-module lim_{i} M_{i}, where {M_{i}} is a tower of discrete G-modules. In Corollary 4.4, we show that the E_{2}-term of (1.2) can always be given by taking the cohomology of the homotopy groups of the complex C^{∗}, where lim_{i} M_{i} is replaced by the continuous G-spectrum holim_{i} X_{i}, in an appropriate sense. This presentation of the E_{2}-term shows that E_{2}^{∗,∗} always takes into account the topology of the continuous G-spectrum, even when it cannot be expressed as continuous cohomology.

Section 3 of this note consists of a discussion of the construction of descent spectral sequence (1.1). Section 5 explains why two other potentially plausible interpretations of (1.2) fail to work.

Acknowledgements. I am grateful to the referee for helpful comments and for pointing out that the statement of the main result could be simplified. Also, I thank Paul Goerss and Halvard Fausk for useful remarks.

2. The pro-discrete cochain complex and continuous cohomology

We begin this section with some terminology. If C is a category, then tow(C) is the category of towers

C_{0} ← C_{1} ← C_{2} ← ⋯

in C. The morphisms {f_{i}} are natural transformations such that each f_{i} is a morphism in C. In this note, we will be working with tow(DMod(G)), where DMod(G) is the category of discrete G-modules, and tow(Spt_{G}).

If A is an abelian group with the discrete topology, let Map_{c}(G, A) be the abelian group of continuous maps from G to A. If X is a spectrum, one can also define Map_{c}(G, X), where the l-simplices of the kth simplicial set (Map_{c}(G, X)_{k})_{l} are given by Map_{c}(G, (X_{k})_{l}), where (X_{k})_{l} is given the discrete topology.

Consider the functor

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

where the action of G on Map_{c}(G, X) is induced on the level of sets by (g · f)(g′) = f(g′g), for g, g′ ∈ G and f ∈ Map_{c}(G, (X_{k})_{l}), for each k, l ≥ 0. As explained in [2, Definition 7.1], the functor Γ_{G} forms a triple and there is a cosimplicial discrete G-spectrum Γ_{G}^{•}X. Also, it is clear that Γ_{G}: DMod(G) → DMod(G) can be defined as above, so that, given a discrete G-module M, Γ_{G}^{•}M is a cosimplicial discrete G-module.

We do not claim any originality in the following definition.

Definition 2.1. Let {X_{i}} be an object in tow(DMod(G)) or in tow(Spt_{G}). Then the pro-discrete cochain complex is defined to be the complex

C^{∗}(G; {X_{i}}) = lim_{i}(Γ_{G}^{∗}X_{i})^{G},

where Γ_{G}^{∗}X_{i} is the canonical complex associated to Γ_{G}^{•}X_{i}, where, if {X_{i}} is in tow(Spt_{G}), then the complex lives in the stable homotopy category. The prodiscrete cochain complex is a complex of abelian groups or spectra, respectively, and the limit and colimit are both formed in abelian groups or in spectra (not the stable homotopy category), respectively.

Let M be any topological G-module (that is, an abelian Hausdorff topological group that is a G-module, with a continuous G-action). Then the continuous cochain cohomology of G with coefficients in M, H_{cts}^{∗}(G; M), is the cohomology of a cochain complex that has the form

M → Map_{c}(G, M) → Map_{c}(G^{2}, M) → ⋯ (2.2)

(see [9, p. 106] for details). We note that, by [7, Theorem (2.2)], if {M_{i}} is a tower of discrete G-modules that satisfies the Mittag-Leffler condition, then

H_{cts}^{s}(G; lim_{i} M_{i}) ≅ H_{cont}^{s}(G; {M_{i}}),

for all s ≥ 0, but, in general, these two versions of continuous cohomology need not be isomorphic. Also, if M is a discrete G-module, then H_{cts}^{s}(G; M) = H_{c}^{s}(G; M).

Now we show that the pro-discrete cochain complex can be used to compute continuous cochain cohomology.

Theorem 2.3. If {M_{i}} is a tower of discrete G-modules, then

H_{cts}^{s}(G; lim_{i} M_{i}) ≅ H^{s}[C^{∗}(G; {M_{i}})].

Proof. As explained in [9, p. 106], for a topological G-module M, the chain complex in (2.2) is defined by taking the G-fixed points of the complex

X^{∗}(G; M) = [Map_{c}(G, M) → Map_{c}(G^{2}, M) → ⋯], (2.4)

where X^{n}(G; M) = Map_{c}(G^{n+1}, M) has a G-action that is defined by

(g · f)(g_{1}, ..., g_{n+1}) = g · f(g^{−1}g_{1}, ..., g^{−1}g_{n+1}).

Now let M be a discrete G-module. Then it is a standard fact that the cochain complex (X^{∗}(G, M))^{G} is naturally isomorphic as a complex to the cochain complex (Γ_{G}^{∗}M)^{G}. This isomorphism uses the fact that the abelian group of n-cochains of (Γ_{G}^{∗}M)^{G} is isomorphic to Map_{c}(G^{n+1}, M)^{G}, where Map_{c}(G^{n+1}, M) has a G-action that is given by

(g · f)(g_{1}, g_{2}, g_{3}, ..., g_{n+1}) = f(g_{1}g, g_{2}, g_{3}, ..., g_{n+1}).

Since

(X^{n}(G; lim_{i} M_{i}))^{G} ≅ lim_{i}((X^{n}(G; M_{i}))^{G}) ≅ lim_{i}(Γ_{G}^{n+1}M_{i})^{G},

we have:

H_{cts}^{s}(G; lim_{i} M_{i}) = H^{s}[(X^{∗}(G; lim_{i} M_{i}))^{G}] = H^{s}[lim_{i}(Γ_{G}^{∗}M_{i})^{G}],

where we used the aforementioned fact that (X^{∗}(G, M_{i}))^{G} and (Γ_{G}^{∗}M_{i})^{G} are naturally isomorphic cochain complexes. □

3. Constructing the descent spectral sequence

In this section, we review how descent spectral sequence (1.1) is constructed and we compare it with a spectral sequence whose E_{2}-term is always Jannsen’s continuous cohomology.

Given a tower {X_{i}} of discrete G-spectra, such that each X_{i} is a fibrant spectrum, by [2, Remark 7.8, Definition 8.1],

(holim_{i} X_{i})^{hG} = holim_{i} holim_{Δ}(Γ_{G}^{•}(X_{i})_{f,G})^{G}.

Thus,

(holim_{i} X_{i})^{hG} ≅ holim_{Δ}(holim_{i}(Γ_{G}^{•}(X_{i})_{f,G})^{G}),

and descent spectral sequence (1.1) is the conditionally convergent homotopy spectral sequence

lim^{s}_{Δ} π_{t}(holim_{i}(Γ_{G}^{•}(X_{i})_{f,G})^{G}) ⇒ π_{t−s}(holim_{Δ}(holim_{i}(Γ_{G}^{•}(X_{i})_{f,G})^{G}))

(see [2, Theorem 8.8]).

In the above context, there is another spectral sequence that is natural to consider. Since

(holim_{i} X_{i})^{hG} ≅ holim_{Δ×{i}}(Γ_{G}^{•}(X_{i})_{f,G})^{G},

there is a conditionally convergent homotopy spectral sequence

lim^{s}_{Δ×{i}} π_{t}((Γ_{G}^{•}(X_{i})_{f,G})^{G}) ⇒ π_{t−s}(holim_{Δ×{i}}(Γ_{G}^{•}(X_{i})_{f,G})^{G}) ≅ π_{t−s}((holim_{i} X_{i})^{hG}),

such that

lim^{s}_{Δ×{i}} π_{t}((Γ_{G}^{•}(X_{i})_{f,G})^{G}) ≅ H_{cont}^{s}(G; {π_{t}(X_{i})})

(see [5, Proposition 3.1.2]). This spectral sequence is closely related to the ℓ-adic descent spectral sequence of algebraic K-theory (see [10], [8]).

We see that we have two spectral sequences with abutment π_{∗}((holim_{i} X_{i})^{hG}). As pointed out in the Introduction, if the tower {π_{t}(X_{i})} satisfies the Mittag-Leffler condition for every integer t, then the spectral sequences have isomorphic E_{2}-terms. However, as stated in the Introduction, the case G = {e} shows that these two spectral sequences can have different E_{2}-terms, so that the spectral sequences can be different from each other.

Since the second spectral sequence described above has an E_{2}-term that always has a nice algebraic description, it is natural to ask what is the value of descent spectral sequence (1.1). We will see that, because the descent spectral sequence is the homotopy spectral sequence of a cosimplicial spectrum, in certain cases it can be compared with an Adams-type spectral sequence that is strongly convergent.

Let n ≥ 1 and let p be a prime. Let K(n) be the nth Morava K-theory spectrum with K(n)_{∗} = 𝔽_{p}[v_{n}^{±1}], where the degree of v_{n} is 2(p^{n} − 1). Also, let E_{n} denote the Lubin–Tate spectrum, where E_{n∗} = W (𝔽_{p^{n}})[[u_{1}, ..., u_{n−1}]][u^{±1}], where the degree of u is −2, and the complete power series ring over the Witt vectors is in degree zero.

Let Z be a K(n)-local spectrum and suppose that there is an augmentation Z → holim_{i}(Γ_{G}^{•}(X_{i})_{f,G})^{G}. If the associated complex of spectra

∗ → Z → holim_{i} Map_{c}(G, (X_{i})_{f,G})^{G} → holim_{i} Map_{c}(G, Map_{c}(G, (X_{i})_{f,G}))^{G} → ⋯

is a K(n)-local E_{n}-resolution of Z (for the definition of this, see [4, Appendix A]), then descent spectral sequence (1.1) is isomorphic to the strongly convergent K(n)- local E_{n}-Adams spectral sequence with abutment π_{∗}(Z) (see [4, Proposition A.5, Corollary A.8]). Thus, the descent spectral sequence is strongly convergent and the map Z → holim_{Δ} holim_{i}(Γ_{G}^{•}(X_{i})_{f,G})^{G} ≅ (holim_{i} X_{i})^{hG} is a weak equivalence ([4, Corollary A.8]).

In this way, in [3, Chapter 10], the author showed that, given a finite spectrum X, the descent spectral sequence for π_{∗}((E_{n}∧X)^{hG}) is strongly convergent and isomorphic to the K(n)-local E_{n}-Adams spectral sequence with abutment π_{∗}(E_{n}^{dhG} ∧ X), where G is a closed subgroup of the extended Morava stabilizer group G_{n} and E_{n}^{dhG} is the spectrum constructed by Devinatz and Hopkins in [4] (E_{n}^{dhG} is denoted by E_{n}^{hG} in [4]).

4. The E_{2}-term and the pro-discrete cochain complex

In this section, we show that the E_{2}-term of (1.2) can be built out of the same complex that computes continuous cochain cohomology. More precisely, given a continuous G-spectrum holim_{i} X_{i}, there exists a tower {X′_{i}} of discrete G-spectra, such that

E_{2}^{s,t} ≅ H^{s}[π_{t}(C^{∗}(G; {X′_{i}}))]. (4.1)

We find the expression on the right-hand side in (4.1) interesting for the following reason. The homotopy fixed point spectrum is defined with respect to a continuous action of G on the spectrum. Thus, homotopy fixed points take into account the topology of the spectrum. Similarly, since the E_{2}-term is built out of the prodiscrete cochain complex of spectra, the E_{2}-term is always taking into account the topology of the spectrum.

By [6, VI, Proposition 1.3], tow(Spt_{G}) is a 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}.

Theorem 4.2. The E_{2}-term (1.2) of descent spectral sequence (1.1) has the form

E_{2}^{s,t} ≅ π^{s}π_{t}(lim_{i}(Γ_{G}^{•}X′_{i})^{G}), (4.3)

where {X_{i}} → {X′_{i}} is a trivial cofibration with {X′_{i}} fibrant, all in tow(Spt_{G}).

Proof. Let {X′_{i}} be as stated in the theorem. By [6, VI, Remark 1.5], each X′_{i} is fibrant and each map X′_{i} → X′_{i−1} is a fibration, all in Spt_{G}.

For any k ≥ 0, we consider the expression

holim_{i}((Γ_{G}^{•}X′_{i})^{G})^{k} = holim_{i}(Map_{c}(G, Map_{c}(G, ⋯, Map_{c}(G, X′_{i}) ⋯)))^{G},

where Map_{c}(G, −) appears k + 1 times. By [2, Section 3], the forgetful functor U: Spt_{G} → Spt, Map_{c}(G, −): Spt → Spt_{G}, where Map_{c}(G, X) = Γ_{G}(X), and the functor (−)^{G}: Spt_{G} → Spt all preserve fibrations. Thus, {X′_{i}} is a tower of fibrations of fibrant spectra, all in Spt. This implies that {Map_{c}(G, X′_{i})} is a tower of fibrations of fibrant spectra, in Spt_{G}, and hence, in Spt. By iteration,

{Map_{c}(G, Map_{c}(G, ⋯, Map_{c}(G, X′_{i}) ⋯))}

is a tower of fibrations of fibrant spectra, in Spt_{G}, so that

{(Map_{c}(G, Map_{c}(G, ⋯, Map_{c}(G, X′_{i}) ⋯)))^{G}}

is a tower of fibrations of fibrant spectra in Spt. Therefore, the canonical map

lim_{i}((Γ_{G}^{•}X′_{i})^{G})^{k} → holim_{i}((Γ_{G}^{•}X′_{i})^{G})^{k}

is a weak equivalence.

Since {((Γ_{G}^{•}X_{i})^{G})^{k}} and {((Γ_{G}^{•}(X_{i})_{f,G})^{G})^{k}} are towers of fibrant spectra, there is a zigzag of weak equivalences

lim_{i}((Γ^{•}X′_{i})^{G})^{k} → holim_{i}((Γ^{•}X′_{i})^{G})^{k} ← holim_{i}((Γ^{•}X_{i})^{G})^{k} → holim_{i}((Γ^{•}(X_{i})_{f,G})^{G})^{k},

where Γ = Γ_{G}. This zigzag of weak equivalences implies that

π^{s}π_{t}(lim_{i}((Γ_{G}^{•}X′_{i})^{G})) ≅ π^{s}π_{t}(holim_{i}((Γ_{G}^{•}(X_{i})_{f,G})^{G})).  □

Corollary 4.4. Let {X′_{i}} be as in Theorem 4.2. Then there is an isomorphism

E_{2}^{s,t} ≅ H^{s}[π_{t}(C^{∗}(G; {X′_{i}}))],

where E_{2}^{s,t} is the E_{2}-term of (1.2).

Remark 4.5. By Theorem 2.3, H^{s}[C^{∗}(G; {π_{t}(X′_{i})})] ≅ H_{cts}^{s}(G; lim_{i} π_{t}(X_{i})).

5. The failure of other possible descriptions of the E_{2}-term

After studying the expression in (4.3) further, one recalls that lim_{i}(−)^{G} is the functor used to define H_{cont}^{s}(G; −), and, if M is any discrete G-module, then

0 → M → Γ_{G}^{∗}M

is a (−)^{G}-acyclic resolution of M, so that H^{s}[(Γ_{G}^{∗}M)^{G}] = H_{c}^{s}(G; M).

Let {M_{i}} be a tower of discrete G-modules. If

{0} → {M_{i}} → {Γ_{G}^{∗}M_{i}}

is a lim_{i}(−)^{G}-acyclic resolution of {M_{i}} in tow(DMod(G)), then

H^{s}[(lim_{i}(−)^{G})({Γ_{G}^{∗}M_{i}})] = H_{cont}^{s}(G; {M_{i}}).

This would imply that E_{2}^{s,t} ≅ H^{s}[π_{t}(lim_{i}(Γ_{G}^{∗}X′_{i})^{G})] is computed by taking the cohomology of the homotopy groups of a complex of spectra in the stable homotopy category, that, in the context of abelian groups, computes continuous cohomology. This would be an interesting presentation of the E_{2}-term.

However, it is not hard to show that

{0} → {M_{i}} → {Γ_{G}^{∗}M_{i}}

need not be a lim_{i}(−)^{G}-acyclic resolution of {M_{i}} in tow(DMod(G)), so that the above interpretation of the E_{2}-term does not work out. For example, by [7, (2.1)], there is a short exact sequence

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

190 Daniel G. Davis

for each s ≥ 0, where H_{c}^{−1}(G; −) = 0. Therefore, when s ≥ 1, H_{c}^{s}(G; Γ_{G}M_{i}) = 0, so that, for all s ≥ 2, H_{cont}^{s}(G; {Γ_{G}M_{i}}) = 0. But, the short exact sequence also implies that

H_{cont}^{1}(G; {Γ_{G}M_{i}}) ≅ lim^{1}_{i} M_{i},

which need not vanish. Thus, {Γ_{G}M_{i}}, the first object in the complex {Γ_{G}^{∗}M_{i}}, need not be lim_{i}(−)^{G}-acyclic in tow(DMod(G)).

Upon further consideration of the expression in (4.3), one notices that, for any k, l, m ≥ 0,

((lim_{i}(Γ_{G}^{m+1}X′_{i})^{G})_{k})_{l} = lim_{i}(Γ_{G}^{m+1}((X′_{i})_{k})_{l})^{G} ≅ Map_{c}(G^{m}, lim_{i}((X′_{i})_{k})_{l})

is an isomorphism of sets. If one could promote this isomorphism to

(5.1)  lim_{i}(Γ_{G}^{m+1}X′_{i})^{G} ≅ Map_{c}(G^{m}, lim_{i} X′_{i}),

then one could use this to interpret the expression in (4.3) as being the cohomology of homotopy groups applied to the complex of continuous cochains with target (“coefficients”) the continuous G-spectrum lim_{i} X′_{i}.

But notice that, in this interpretation, the expression Map_{c}(G^{m}, lim_{i} X′_{i}) does not have the desired meaning. For isomorphism (5.1) to hold, lim_{i} X′_{i} must be a spectrum whose simplicial sets have simplices with the pro-discrete topology. But, as a Bousfield–Friedlander spectrum, in the construction Map_{c}(G^{m}, lim_{i} X′_{i}), lim_{i} X′_{i} consists of simplicial sets whose simplices all have the discrete topology, by default. This conflict means that this interpretation also fails to work.

References

[1] Bousfield, A. K.; Kan, D. M. Homotopy limits, completions and localizations. Lecture Notes in Mathematics, 304. Springer-Verlag, Berlin, 1972. MR0365573 (51 #1825), Zbl 0259.55004.

[2] Davis, Daniel G. Homotopy fixed points for L_{K(n)}(E_{n} ∧ X) using the continuous action, J. Pure Appl. Algebra 206 (2006), no. 3, 322–354.

[3] Davis, Daniel G. The Lubin–Tate spectrum and its homotopy fixed point spectra. Thesis. Northwestern University, May 9, 2003.

[4] Devinatz, Ethan S.; Hopkins, Michael J. Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43 (2004), no. 1, 1–47. MR2030586 (2004i:55012), Zbl 1047.55004.

[5] Geisser, Thomas; Hesselholt, Lars. Topological cyclic homology of schemes. Algebraic K-Theory (Seattle, WA, 1997) (W. Raskind and C. Weibel, eds.), 41–87. Proc. Sympos. Pure Math., 67. Amer. Math. Soc., Providence, RI, 1999. MR1743237 (2001g:19003), Zbl 0953.19001.

[6] Goerss, Paul G.; Jardine, John F. Simplicial homotopy theory. Birkhäuser Verlag, Basel, 1999. MR1711612 (2001d:55012), Zbl 0949.55001.

[7] Jannsen, Uwe. Continuous étale cohomology. Math. Ann. 280 (1988), no. 2, 207–245. MR929536 (89a:14022), Zbl 0649.14011.

[8] Mitchell, Stephen A. Hypercohomology spectra and Thomason’s descent theorem. Algebraic K-Theory (Toronto, ON, 1996) (V. Snaith, ed.), 221–277. Fields Inst. Commun., 16. Amer. Math. Soc., Providence, RI, 1997. MR1466977 (99f:19002), Zbl 0888.19003.

[9] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay. Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323. Springer-Verlag, Berlin, 2000. MR1737196 (2000j:11168), Zbl 0948.11001.

[10] Thomason, R. W. Algebraic K-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552. MR826102 (87k:14016), Zbl 0596.14012.

Department of Mathematics, Purdue University, 150 N. University St., W. Lafayette, IN, 47907 dgdavis@math.purdue.edu

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