NORTHWESTERN UNIVERSITY

The Lubin-Tate Spectrum and its Homotopy Fixed Point Spectra

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

for the degree DOCTOR OF PHILOSOPHY

Field of Mathematics By Daniel Glen Davis EVANSTON, ILLINOIS June 2003 \begin{matrix}c \\ ◯ \end{matrix} Copyright by Daniel Glen Davis 2003

All Rights Reserved

ABSTRACT

The Lubin-Tate Spectrum and its Homotopy Fixed Point Spectra

Daniel Glen Davis

Using the work of Devinatz and Hopkins, we show that the Lubin-Tate spectrum E_{n} is a continuous G_{n}-spectrum, where G_{n} is the extended Morava stabilizer group. For G closed in G_{n} and any finite spectrum X, we use this continuous action to define the homotopy fixed point spectrum (E_{n}∧X)^{hG} with the associated descent spectral sequence

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

We show that this spectral sequence is isomorphic to the strongly convergent K(n)_{∗}-local E_{n}-Adams spectral sequence abutting to π_{∗}(E_{n}^{hG} ∧ X). We also have a descent spectral sequence for (L_{K(n)}(E_{n} ∧ X))^{hG}, where X satisfies a particular finiteness condition.

Acknowledgements

First of all, many thanks to my thesis advisor Paul Goerss for teaching me so much through my coursework and my research. I had a plethora of helpful conversations with him and received many suggestions that improved the organization and writing of this thesis. His fingerprints are present throughout this work and its proofs.

I want to express my gratitude to Mark Mahowald and Stewart Priddy for the variety of ways that they helped me during the course of my Ph.D. I really appreciate their careful examination of my thesis and their comments about various drafts of it. Most of all, I have been regularly inspired by the wonderful example Professors Goerss, Mahowald and Priddy have set as mathematicians.

I want to point out that [9], a paper by Ethan Devinatz and Mike Hopkins, was very inspirational for this work, and I thank Ethan Devinatz for helpful conversations and e-mails.

When I first suspected that E_{n} ∧ M_{I} ≃ F_{n} ∧ M_{I}, I asked Charles Rezk about this, and he saw its validity and sketched a proof. I thank Charles for his insight on this point and related conversations.

I thank Rick Jardine for answering several questions about homotopy fixed points, and he and Jeff Smith for answering questions about the smash product. Also, I had many helpful conversations with Christian Haesemeyer during the course of my research, and I thank him for his time.

I had several helpful conversations with Halvard Fausk, and, in seminars about this work, Sharon Hollander and Hal Sadofsky asked “do I know if?” questions that helped me in formulating Theorems 10.12 and 10.6, respectively.

I thank Mark Pearson for his camaraderie as we worked on our dissertations concurrently, and Paul Pearson for his answers to tricky L^{A}TEX questions.

I am grateful to my family, especially my parents, for their encouragement during my time in graduate school. Thanks Joy, for your friendship and all the fun chats we’ve had.

Most of all, I thank Lisa, my wonderful wife, for her support, encouragement, sense of humor, and many other things; she made an already pleasureful endeavor so much more satisfying.

Contents

ABSTRACT iii Acknowledgements iv Chapter 1. Introduction 1 Chapter 2. Preliminaries 11 Chapter 3. An Adams-Novikov spectral sequence for L_{K(n)}(X) 18 Chapter 4. Continuous G-spectra of simplicial sets 23

4.1. The model category of discrete G-spectra 23 4.2. Building discrete G-spectra 29 4.3. Towers of discrete G-spectra 30 4.4. Continuous G-spectra 31

Chapter 5. A model for E_{n} as a continuous G_{n}-spectrum 34 5.1. The spectrum F_{n} 34 5.2. Constructing the continuous action 38

Chapter 6. Homotopy fixed point spectra for discrete G-spectra 41 Chapter 7. Homotopy fixed point spectra for E_{n} 53 Chapter 8. Auxiliary results for building the descent spectral sequence 71 Chapter 9. Homotopy fixed points for L_{K(n)}(E_{n} ∧ X) and its descent spectral

sequence 83

9.1. Towers of spectra and homotopy fixed points 83 9.2. The continuous G_{n}-spectrum L_{K(n)}(E_{n} ∧ X) 87

Chapter 10. The descent spectral sequence is an Adams spectral sequence 94 References 103

CHAPTER 1 Introduction

Let E_{n} be the Lubin-Tate spectrum with 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 G_{n} = S_{n} ⋊ Gal(𝔽_{p^{n}}/𝔽_{p}), where S_{n} is the nth Morava stabilizer group, and let G be a closed subgroup of G_{n}. Morava’s change of rings theorem yields a spectral sequence

(1.1)  H_{c}^{∗}(G_{n}; π_{∗}(E_{n} ∧ X)) ⟹ π_{∗}L_{K(n)}(X),

where the E_{2}-term is continuous cohomology and X is a finite spectrum. Using the G_{n}- action on E_{n} by maps of commutative S-algebras (work of Goerss and Hopkins ([15], [16]), and Hopkins and Miller [39]), Devinatz and Hopkins [9] constructed spectra E_{n}^{hG} with strongly convergent spectral sequences

(1.2)  H_{c}^{∗}(G; π_{∗}(E_{n} ∧ X)) ⟹ π_{∗}(E_{n}^{hG} ∧ X).

Also, they showed that E_{n}^{hG_{n}} ∧ X ≃ L_{K(n)}(X).

When K is a discrete group and Y is a K-spectrum, 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 conditionally convergent 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 [34, §1.1]. Such spectral sequences are called descent spectral sequences.

Thus, by comparison, E_{n} appears to be a continuous G_{n}-spectrum with “descent” spectral sequences for “homotopy fixed point” spectra E_{n}^{hG} ∧ X. Indeed, we explain how their work implies that E_{n} is a continuous G_{n}-spectrum - E_{n} is a homotopy inverse limit of discrete G_{n}-spectra. Using this continuous action, we define homotopy fixed point spectra (E_{n} ∧ X)^{hG} that are weakly equivalent to E_{n}^{hG} ∧ X, and we construct descent spectral sequences having the form of 1.2.

In more detail, the K(n)_{∗}-local spectrum E_{n} has an action by G_{n} as a commutative S-algebra. The K(n)_{∗}-local commutative S-algebra E_{n}^{hG} has an associated strongly convergent spectral sequence

E_{2}^{∗,∗} ≅ H_{c}^{∗}(G; E_{n}^{∗}(Z)) ⟹ (E_{n}^{hG})^{∗}(Z),

where Z is any CW-spectrum. Also, Devinatz and Hopkins prove the remarkable formula E_{n} ≃ L_{K(n)}(hocolim_{i E_{∞}} E_{n}^{hU_{i}}), where {U_{i}} is a cofinal descending chain of open normal subgroups of G_{n}, and the homotopy colimit, as the notation indicates, occurs in the category of commutative S-algebras.

Devinatz and Hopkins prove that the homotopy fixed point spectra E_{n}^{hG} have the expected properties and, when one sets Z equal to the Spanier-Whitehead dual of any finite spectrum X, one obtains a spectral sequence

E_{2}^{∗,∗} ≅ H_{c}^{∗}(G; π_{∗}(E_{n} ∧ X)) ⟹ π_{∗}(E_{n}^{hG} ∧ X)

that has the form of a classical descent spectral sequence. Thus, their constructions strongly suggest that G_{n} acts on E_{n} in a continuous sense. However, their highly structured action is not proven to be continuous, their homotopy fixed point spectra are not defined with respect to a continuous action, and their homotopy fixed point spectral sequences are not constructed as descent spectral sequences, but are K(n)_{∗}-local E_{n}-Adams spectral sequences.

Let F_{n} = colim_{i} E_{n}^{hU_{i}} and let M_{I} (when it exists) be the generalized Moore spectrum with BP_{∗}(M_{I}) ≅ BP_{∗}/I. The key for 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 a discrete G_{n}-spectrum, a spectrum that levelwise consists of simplicial discrete G_{n}-sets. This result is almost immediate due to the work of Devinatz and Hopkins.

Theorem 1.3. As the homotopy inverse limit of a pro-object in the category of discrete G_{n}-spectra, E_{n} ≃ holim_{I} (F_{n} ∧ M_{I}) is a continuous G_{n}-spectrum. Also, if the tower of abelian groups {π_{t}(E_{n} ∧ M_{I} ∧ X)}_{I} is Mittag-Leffler for a fixed spectrum X and every integer t (e.g. X is finite), then L_{K(n)}(E_{n} ∧ X) ≃ holim_{I} (F_{n} ∧ M_{I} ∧ X) is a continuous G_{n}-spectrum.

Using this continuous action and for X as in Theorem 1.3, we define homotopy fixed point spectra (L_{K(n)}(E_{n}∧X))^{hG} and construct their descent spectral sequences. To avoid the pile-up of parentheses, instead of (L_{K(n)}(E_{n} ∧ X))^{hG}, we write L_{K(n)}(E_{n} ∧ X)^{hG}.

Theorem 1.4. Let G be a closed subgroup of G_{n} and let X satisfy the hypotheses of the previous theorem. Then there is a conditionally convergent descent spectral sequence

H_{cts}^{s}(G; π_{t}(L_{K(n)}(E_{n} ∧ X))) ⟹ π_{t−s}(L_{K(n)}(E_{n} ∧ X)^{hG}),

where the E_{2}-term is the cohomology of continuous cochains. In particular, if X is a finite spectrum, this descent spectral sequence is strongly convergent and has the form

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

where the E_{2}-term coincides with the continuous cohomology of [9, Rk. 0.3].

We also show that computing with the descent spectral sequence when X is finite is the same as computing with the Adams spectral sequence of Devinatz and Hopkins: the differentials are identical.

Theorem 1.6. When X is a finite spectrum, descent spectral sequence 1.5 is isomorphic to the K(n)_{∗}-local E_{n}-Adams spectral sequence with abutment π_{∗}(E_{n}^{hG} ∧ X). In particular, in the stable homotopy category, the morphism E_{n}^{hG} ∧ X → (E_{n} ∧ X)^{hG} is an isomorphism.

Finally, we prove a theorem that chromatic stable homotopy theorists have suspected is true, at least since the advent of Morava’s change of rings theorem [36] and the Annals work of Miller, Ravenel and Wilson in 1977 [33]: the K(n)_{∗}-localization of any finite complex is a homotopy fixed points spectrum.

Theorem 1.7. Let X be a finite complex. Then L_{K(n)}(X) ≅ (E_{n} ∧ X)^{hG_{n}}, in the stable homotopy category. In particular, if X is also of type n, then L_{n}X ≅ (F_{n} ∧X)^{hG_{n}}.

We want to stress at the outset how indebted we are to the beautiful work of Devinatz and Hopkins, and to the work of Paul Goerss and Mike Hopkins upon which [9] is reliant ([15], [16]), and the earlier work of Hopkins and Haynes Miller [39].

It is interesting to note that the key weak equivalence E_{n}∧M_{I} ≃ F_{n}∧M_{I} is expected, in the following sense. Since π_{∗}(E_{n} ∧ M_{I}) is a discrete G_{n}-module, one might hope for a spectrum E_{n}/I ≃ E_{n} ∧ M_{I} that is a discrete G_{n}-spectrum. Given a profinite group H, the work of Jardine gives a model category structure on the category of discrete Hspectra (see §4.1). Given a discrete H-spectrum Y, one defines Y^{hH} = (Y_{f,H})^{H}, where Y →^{≃} Y_{f,H} is a fibrant replacement in the category of discrete H-spectra. Also, if K is an open normal subgroup of H, then a fibrant discrete H-spectrum is also fibrant as a discrete K-spectrum [27, Rk. 6.26].

Thus,

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

where in the last two steps we applied the ideas that (a) E_{n} ∧ M_{I} ought to have wellbehaved homotopy fixed point spectra (E_{n} ∧ M_{I})^{hU_{i}}; and (b) the work of Devinatz and Hopkins suggests that E_{n}^{hU_{i}} ∧ M_{I} and (E_{n} ∧ M_{I})^{hU_{i}} ought to have the same homotopy type.

It is important to note that the model given here for E_{n} as a continuous G_{n}-spectrum is not completely satisfactory. For example, the continuous action is by morphisms that are just maps of spectra. However, it is known that there are models for E_{n} where the G_{n}-action is by A_{∞}- and E_{∞}-automorphisms of ring spectra, and therefore we would like to know that such enriched actions are actually continuous.

Another deficiency is that I do not know if holim_{I} (F_{n} ∧ M_{I}) is a twisted E_{n}-module spectrum (that is, the module structure map (if it exists), with diagonal action on the domain, is G_{n}-equivariant). Clearly, one expects any fully functional model to be a twisted E_{n}-module spectrum, as E_{n} itself is.

Outline of Contents. In Chapter 2, we give some background material, recall useful facts, and handle several technical issues. In Chapter 3, we show how the Adams-Novikov spectral sequence for π_{∗}(L_{K(n)}X), when X is finite, gives 1.1. In Chapter 4, we define “continuous G-spectrum,” and discuss the model categories of discrete G-spectra and their towers. Chapter 5 shows that E_{n} is a continuous G_{n}-spectrum. Homotopy fixed points for discrete G-spectra are discussed in Chapter 6, while Chapter 7 constructs homotopy fixed point spectra E_{n}^{h′G} (using the continuous action) that are weakly equivalent to E_{n}^{hG}. Chapter 8 contains various results about continuous cohomology and towers of abelian groups that are needed for later work. Chapter 9 builds the descent spectral sequence for L_{K(n)}(E_{n} ∧ X)^{hG}, when X is E_{n}-Mittag-Leffler, and Chapter 10 proves that, when X is a finite spectrum, the descent spectral sequence for (E_{n} ∧ X)^{hG} is strongly convergent and is isomorphic to the K(n)_{∗}-local E_{n}-Adams spectral sequence for E_{n}^{hG} ∧ X. Notation and Terminology. Gal is the Galois group Gal(𝔽_{p^{n}}/𝔽_{p}). For a profinite group G, we write G ≅ \underleftarrow{lim} _{N} G/N, an inverse limit over the open normal subgroups.

The nth Morava stabilizer group S_{n} is a profinite group, defined to be the automorphism group of the Honda formal group law Γ_{n} of height n over 𝔽_{p^{n}}. G_{n} = S_{n} ⋊ Gal is a profinite group with a descending chain of open normal subgroups G_{n} = U_{0} ⪈ U_{1} ⪈ ... ⪈ U_{i} ⪈ ..., such that ⋂_{i=0}^{∞} U_{i} = \{e\}. The canonical map G_{n} → \underleftarrow{lim} _{i} G_{n}/U_{i} is a group isomorphism and a homeomorphism. The notation N <_{o} G means that N is an open subgroup of G; “<_{c}” refers to closed subgroups.

We use G to denote arbitrary profinite groups and, specifically, closed subgroups of G_{n}, sometimes in the same section. The context should always make it clear which we mean.

Let BP denote the Brown-Peterson spectrum with BP_{∗} = ℤ_{(p)}[v_{1}, v_{2}, ...], where |v_{i}| = 2(p^{i} − 1). The ideal (p^{i_{0}}, v_{1}^{i_{1}}, ..., v_{n−1}^{i_{n−1}}) in 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}}). Given an ideal I, M_{I} need not exist; however, enough exist for our constructions (e.g. see 2.3). The spectrum M_{I} is a finite type n spectrum with the property that BP_{∗}(M_{I}) ≅ BP_{∗}/I. (See [5, §4], [23, §4], and [30, Prop. 3.7] for details.) 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 - sends I to the ideal (p^{i_{0}}, u_{1}^{i_{1}}, ..., u_{n−1}^{i_{n−1}}) in E_{n∗}, also denoted by I. Landweber exactness gives π_{∗}(E_{n} ∧ M_{I}) ≅ E_{n∗}/I. The set {i_{0}, ..., i_{n−1}} of superscripts of positive integers varies so that there is a family of ideals {I}. Also, let I_{n} = (p, u_{1}, ..., u_{n−1}) ⊂ E_{n∗}.

Since we will frequently use (homotopy) limits indexed by these ideals, we make the following remarks. Given I_{1}, I_{2} in {I}, I_{1} ≤ I_{2} if and only if I_{2} ⊂ I_{1}. Even though {I} is not the inverse system {0 ← 1 ← 2 ← ...}, since {I} is a countable directed poset, it contains a cofinal directed poset J = {I_{0} ← I_{1} ← I_{2} ← ...}. Therefore, in what follows, the functors \underleftarrow{lim} _{I} (−) and holim_{I} (−) are always over a cofinal subset with the shape of J. We always write the subscript “I” even though we really mean the cofinal subset.

The homotopy limit of (Bousfield-Friedlander) spectra is constructed levelwise in S, the category of simplicial sets. The homotopy limit in S of pointed simplicial sets is again a member of S_{∗}, the category of pointed simplicial sets. We use S^{n} ∈ S to denote the n-sphere.

Given a spectrum X, DX is its Spanier-Whitehead dual. Given a spectrum E, E^{j} = E ∧ E ∧ ⋯ ∧ E, with j factors. K(n) is the nth Morava K-theory spectrum with K(n)_{∗} = 𝔽_{p}[v_{n}, v_{n}^{−1}]. We use L_{n} to denote the Bousfield localization functor with respect to E(n), where E(n)_{∗} = ℤ_{(p)}[v_{1}, ..., v_{n−1}][v_{n}, v_{n}^{−1}]. The category of commutative S-algebras is denoted by E_{∞}. If C is a model category, then Ho(C) is its homotopy category and [−, −]_{C} denotes morphisms in Ho(C). Spt is the category of Bousfield-Friedlander spectra, and “the stable category” always refers to Ho(Spt).

Given a site C, C^{∼} is the Grothendieck topos of sheaves of sets on the site. DMod(G) is the category of discrete G-modules. A discrete G-set X is a G-set such that the action map G × X → X is continuous when X is given the discrete topology. The map r_{g}: G_{n} → G_{n} denotes right multiplication by g ∈ G_{n}.

We use Map_{c}(G, A) to denote the set of continuous maps out of the profinite group G into the object A, which can be a topological abelian group, a discrete G-spectrum, etc. We often write Γ_{G}(A) for Map_{c}(G, A), and Γ_{G}^{k}(A) = Map_{c}(G^{k}, A), where G^{k} is the k-fold product of G and G^{0} = ∗.

If M ≅ \underleftarrow{lim} _{α} M_{α} is an inverse system of discrete G-modules, then we use H_{c}^{∗}(G; M) to denote \underleftarrow{lim} _{α} H_{c}^{∗}(G; M_{α}).

Given a topological abelian group M ≅ \underleftarrow{lim} _{α} M_{α}, where M_{α} is a discrete abelian group, Map_{c}(G_{n}^{j}, M) is the topological G_{n}-module of continuous maps with action defined by (g′ · f)(g_{1}, ..., g_{j}) = f(g_{1}g′, g_{2}, ..., g_{j}). Also, Map_{c}^{ℓ}(G_{n}^{j}, M) is the topological G_{n}- module of continuous maps with action denoted and defined by (g′ · f)^{ℓ}(g_{1}, ..., g_{j}) = f((g′)^{−1}g_{1}, g_{2}, ..., g_{j}). There is a G_{n}-equivariant isomorphism of topological G_{n}-modules

p: Map_{c}^{ℓ}(G_{n}^{j}, M) → Map_{c}(G_{n}^{j}, M), p(f)(g_{1}, g_{2}, ..., g_{j}) = f(g_{1}^{−1}, g_{2}, ..., g_{j}).

We use k̄ to denote the separable closure of the field k, and Ab is the category of abelian groups.

CHAPTER 2 Preliminaries

In this chapter, we recall some frequently used facts, discuss background material, and handle several technical issues to help get our work started.

In [9], Devinatz and Hopkins, using work by Goerss and Hopkins ([15], [16]), and Hopkins and Miller [39], show that the action of G_{n} on E_{n} is by maps of commutative S-algebras. Previously, Hopkins and Miller showed that G_{n} acts on E_{n} by maps of A_{∞}- ring spectra. However, the continuous action presented here is not so structured. As will be seen later, the starting point for the continuous action is the spectrum F_{n} ∧ M_{I} and this spectrum is not known to even be an A_{∞}-ring spectrum. Thus, we work in an unstructured category of spectra of simplicial sets and the continuous action is not known to have any further structure.

As alluded to in the Introduction, the key to our constructions is the homotopy colimit (see [9, Def. 0.5]) hocolim_{i E_{∞}} E_{n}^{hU_{i}}, where the homotopy colimit is taken in the category of commutative S-algebras. As described in [9, Rk. 0.2, Lem. 5.2], each E_{n}^{hU_{i}} is functorially replaced with a weakly equivalent cofibrant cell commutative S-algebra, also labeled E_{n}^{hU_{i}}. Our construction of the continuous action requires that this cofibrant E_{n}^{hU_{i}} be a G_{n}/U_{i}-spectrum. Note that, as will be seen later, since the G_{n}/U_{i}-action is given by functoriality, this E_{n}^{hU_{i}} is indeed a G_{n}/U_{i}-spectrum in the desired sense.

Since E_{n}^{hU_{i}} is assumed to be cell commutative, [9, Lem. 5.2] shows that

hocolim_{i E_{∞}} E_{n}^{hU_{i}} ≃ hocolim_{i} E_{n}^{hU_{i}},

where the second homotopy colimit is in the category of spectra. Notice that by [9, Def. 5.1, Pf. of Lem. 5.2], π_{∗}(hocolim_{i E_{∞}} E_{n}^{hU_{i}}) ≅ π_{∗}(colim_{i} E_{n}^{hU_{i}}), where the undecorated colim is in the category of spectra. Therefore, the canonical map hocolim_{i E_{∞}} E_{n}^{hU_{i}} → colim_{i} E_{n}^{hU_{i}} is a weak equivalence that is G_{n}-equivariant.

Since our action does not involve additional structure, but is just by maps of spectra, we work in the technically simpler model category Spt of Bousfield-Friedlander spectra. We state precisely how we will work in Spt while using [9].

We begin with some recollections from [9]. Let R_{G_{n}}^{+} be the category whose objects are finite discrete left G_{n}-sets together with the continuous profinite G_{n}-space G_{n} itself. The morphisms are continuous G_{n}-equivariant maps. Devinatz and Hopkins construct a presheaf of spectra F: (R_{G_{n}}^{+})^{op} → E_{∞}, such that (a) for each S ∈ R_{G_{n}}^{+}, F(S) is K(n)_{∗}-local; (b) F(G_{n}) = E_{n}; (c) for U <_{o} G_{n}, E_{n}^{hU} := F(G_{n}/U); and (d) remarkably, F(∗) = E_{n}^{hG_{n}} ≃ L_{K(n)}S^{0}.

We need some terminology to simplify our discussion.

Definition 2.1. [21, Def. 1.3.1] Suppose C and D are 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. In addition, if F and P are adjoint, (F, P) is said to be a Quillen pair for the model categories (C, D).

Recall [21, Lem. 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).

Let Sp_{T_{∗}}^{Σ+} be the category of symmetric spectra of topological spaces with the positive stable model category structure [41, pg. 529], and let M_{S} be the model category of Smodules [11]. Then [41, pg. 530] defines functors Λ and Φ such that (Λ, Φ) is a Quillen pair for (Sp_{T_{∗}}^{Σ+}, M_{S}). We recall that Φ preserves homotopy groups.

Let Sp_{T_{∗}}^{Σ} be the category of symmetric spectra of topological spaces with the stable model category structure. Then the identity functor id goes in both directions between Sp_{T_{∗}}^{Σ+} and Sp_{T_{∗}}^{Σ}, and (id, id) is a Quillen pair for (Sp_{T_{∗}}^{Σ+}, Sp_{T_{∗}}^{Σ}) [31, Prop. 14.6].

Recall from [31, §19] that the singular functor 𝕊: T_{∗} → S_{∗} from the category of pointed compactly generated topological spaces extends, by levelwise application, to a functor 𝕊: Sp_{T_{∗}}^{Σ} → Sp^{Σ} to the category of symmetric spectra of simplicial sets. Also, it is easy to see that 𝕊 preserves homotopy groups. Levelwise application of geometric realization gives a functor 𝕋: Sp^{Σ} → Sp_{T_{∗}}^{Σ}, and once again, (𝕋, 𝕊) is a Quillen pair for (Sp^{Σ}, Sp_{T_{∗}}^{Σ}) [31, Thm. 19.3].

Finally, let U: Sp^{Σ} → Spt be the forgetful functor with left adjoint V. Then (V, U) is a Quillen pair for (Spt, Sp^{Σ}) [22, Prop. 4.2.4]. Let (−)_{c} : Spt → Spt and (−)_{f} : M_{S} → M_{S} denote the cofibrant and fibrant replacement functors, respectively, for the stated model categories; e.g. given any S-module M, M_{c} → M is a trivial fibration with M_{c} cofibrant.

Notice that π_{∗}((U𝕊Φ(−)_{f}F)(S)) ≅ π_{∗}(F(S)). Thus, we can regard (U𝕊Φ(−)_{f}F)(S) as a model for F(S), and we redefine F to be the presheaf of spectra U𝕊Φ(−)_{f}F: (R_{G_{n}}^{+})^{op} → Spt. Also, E_{n} and E_{n}^{hU} are redefined to be (U𝕊Φ(−)_{f}F)(G_{n}) and (U𝕊Φ(−)_{f}F)(G_{n}/U), respectively.

Since left Quillen functors preserve cofibrant objects and right Quillen functors preserve fibrant objects, we have the following isomorphisms, for an arbitrary Bousfield-Friedlander spectrum X (we use both meanings of F):

[Λ𝕋V (X)_{c}, F(S)]_{M_{S}}^{t} ≅ [𝕋V (X)_{c}, (Φ(−)_{f}F)(S)]_{Sp_{T∗}^{Σ+}}^{t} ≅ [V (X)_{c}, (𝕊Φ(−)_{f}F)(S)]_{Sp^{Σ}}^{t}
≅ [X, (U𝕊Φ(−)_{f}F)(S)]_{Spt}^{t} = [X, F(S)]_{Spt}^{t}.

This isomorphism will be needed later.

Now we can define a spectrum that is critical for the constructions in this work.

Definition 2.2. Let F_{n} = colim_{i} E_{n}^{hU_{i}}.

Given a filtered system {X_{i}} of spectra, the canonical map hocolim_{i} X_{i} → colim_{i} X_{i} is a weak equivalence in Spt. Therefore, since Φ preserves homotopy groups, we point out that the four spectra (note the two different meanings of E_{n}^{hU_{i}})

hocolim_{i E_{∞}} E_{n}^{hU_{i}} → colim_{i S} E_{n}^{hU_{i}}, hocolim_{i Spt} E_{n}^{hU_{i}} ← colim_{i Spt} E_{n}^{hU_{i}}

all have the same homotopy groups, and by functoriality, the same G_{n}-action, and thus, are models for the same G_{n}-spectrum. For the remainder of this paper, we will use the model colim_{i} E_{n}^{hU_{i}}, where the Spt is omitted. Since colim X_{i} ≃ hocolim X_{i}, we can interchange these two as needed, so that, for example, we can assume that E(n)_{∗}- localization commutes with colim (see below).

We review some results that will be used frequently in our work. First of all, as implied by the properties of F listed earlier, E_{n} and E_{n}^{hU_{i}} are K(n)_{∗}-local, and hence, E(n)_{∗}-local. Since E(n)_{∗}-localization is smashing [38, Thm. 7.5.6], for any spectrum X, L_{n}X ≃ X ∧ L_{n}S^{0}. Also, E(n)_{∗}-localization commutes with homotopy direct limits [38, Thm. 8.2.2], so that L_{n}(hocolim X_{α}) ≃ hocolim L_{n}(X_{α}), and if each X_{α} is E(n)_{∗}-local, then so is hocolim X_{α}. Note that if X is E(n)_{∗}-local, then for any spectrum Y, X ∧ Y is also E(n)_{∗}-local.

Recall ([6, Remark 3.6], [20, §2], [23, Prop. 7.10]) that if X is any E(n)_{∗}-local spectrum, L_{K(n)}X ≃ holim_{I} (X ∧ M_{I}). Since L_{K(n)}L_{n} ≃ L_{K(n)}, for any spectrum X,

(2.3)  L_{K(n)}X ≃ holim_{I} (L_{n}X ∧ M_{I}).

We also need a result by Hovey and Strickland [23, Lem. 7.2]:

Lemma 2.4. If X is any spectrum, and Y is a finite spectrum of type at least n (so Y is p-local), then L_{K(n)}(X ∧ Y) ≃ L_{K(n)}(X) ∧ Y ≃ L_{n}(X) ∧ Y.

We make a few remarks about smash products in Spt. We require that the smash product be functorial, particularly since our actions are given by functors. Though the smash product is well-known to be cumbersome, [27, Chps. 1,2] gives a functorial definition of it, which we now state.

Definition 2.5. Given spectra X and Y, their smash product X ∧ Y is given by

(X ∧ Y)_{n} = \begin{cases}X_{k} ∧ Y_{k}, & n = 2k; \\ X_{k} ∧ Y_{k+1}, & n = 2k + 1. \end{cases}

Since K∧(−): S_{∗} → S_{∗} is a left adjoint, for any K in S_{∗}, smashing with any spectrum in either variable commutes with colimits in Spt.

Note that in general, in Spt, there are no point-set level maps between X and X ∧S^{0}. We recall some useful facts about compact p-adic analytic groups. Our starting

point is that S_{n} is compact p-adic analytic. Since G_{n} is an extension of S_{n} by the Galois group, [42, Cor. of Thm. 2] implies that G_{n} is a compact p-adic analytic group. By [10, Thm. 10.7], any closed subgroup G of G_{n} is also compact p-adic analytic. Using (a) the characterization of compact p-adic analytic groups in [40, pg. 76]; and (b) the fact that G_{n} has a finitely generated pro-p subgroup S_{n} (the unique p-Sylow open normal subgroup of S_{n} of strict automorphisms) that is strongly complete [40, pg. 124], G_{n} is also strongly complete. Thus, all subgroups in G_{n} of finite index are open.

Recall that virtual cohomological dimension (vcd) refers to the cohomological dimension of an open subgroup. Since vcd(S_{n}) = n^{2} ([28, 2.4.9], [36, pg. 30]), if G is a closed subgroup of G_{n}, then G has finite virtual cohomological dimension. Thus, we write vcd(G_{n}) = m so that any closed subgroup G contains an open subgroup H with cd(H) ≤ m, and hence, H_{c}^{s}(H; M) = 0 for all discrete H-modules M, whenever s > m + 1.

CHAPTER 3 An Adams-Novikov spectral sequence for L_{K(n)}(X)

In the Introduction, we referred to the spectral sequence 1.1 motivated by the Morava change of rings theorem. In this chapter we discuss this spectral sequence, since its existence and form is one of the chief impetuses for this work. Since the spectral sequence is rarely written in this form, we show how it is derived from a certain Adams-Novikov spectral sequence.

Definition 3.1. Let K_{n,∗}(X) = \underleftarrow{lim} _{I} π_{∗}(E_{n} ∧ M_{I} ∧ X), where X is a spectrum.

We recall Proposition 7.4 of [19]: If K_{n,∗}(X) is finitely generated over E_{n∗}, then there is a spectral sequence

(3.2)  E_{2}^{∗,∗} = H_{c}^{∗∗}(S_{n}; K_{n,∗}(X)) ⟹ W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} π_{∗}(L_{K(n)}X).

Remark 3.3. The second “∗” in the superscript of the continuous cohomology above has the following meaning: if N_{∗} is a graded G_{n}-module, H_{c}^{∗t}(S_{n}; N_{∗}) = H_{c}^{∗}(S_{n}; N_{t}).

Spectral sequence 3.2, due to Hopkins and Ravenel (see [18, §2.2], [20, pg. 241], [44, pg. 46], and the proof in [19, pg. 116]), is constructed by taking an inverse limit over {I} of Adams-Novikov spectral sequences of the form

E_{2}^{∗,∗} = W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} Ext_{BP_{∗}BP}^{∗,∗} (BP_{∗}, BP_{∗}(L_{n}X ∧ M_{I})) ⟹ W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} π_{∗}(L_{n}X ∧ M_{I}),

and applying the Morava change of rings isomorphism

W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} Ext_{BP_{∗}BP}^{∗,∗} (BP_{∗}, BP_{∗}(L_{n}X ∧ M_{I})) ≅ H_{c}^{∗∗}(S_{n}; π_{t}(E_{n} ∧ M_{I} ∧ X)).

Let X be a finite spectrum. Then K_{n,∗}(X) ≅ π_{∗}(E_{n} ∧ X) is a finitely generated E_{n∗}-module, by 8.10. We proceed to show that 3.2 gives the spectral sequence

H_{c}^{∗}(G_{n}; π_{∗}(E_{n} ∧ X)) ⟹ π_{∗}(L_{K(n)}X).

Notice that Gal acts on the abutment of (3.2) by acting only on the Witt vectors. There is also a canonical action of Gal on the E_{2}-term, which induces the action of Gal on H_{c}^{s}(S_{n}; K_{n,∗}(X)) = \underleftarrow{lim} _{I} H_{c}^{∗}(S_{n}; π_{∗}(E_{n} ∧ M_{I} ∧ X)) [4, pg. 112]. We recall the following definitions from [4, Lem. 5.4], [8, Def. 3.21, proof of Lem. 3.22].

Definition 3.4. Let M be a W (𝔽_{p^{n}})-module and a Gal-module. If Gal acts by ℤ_{p}-module automorphisms and if σ(cm) = (σc)(σm), whenever σ ∈ Gal, c ∈ W (𝔽_{p^{n}}), and m ∈ M, then M is called a twisted Gal-W (𝔽_{p^{n}}) module. A discrete G_{n}-module N is a discrete twisted G_{n}-W (𝔽_{p^{n}}) module if it is also a torsion W (𝔽_{p^{n}})-module with σ(cn) = (σc)(σn), for n ∈ N, and σ and c as above.

We will make use of the following, a version of [4, Lem. 5.4].

Lemma 3.5. If N is a twisted Gal-W (𝔽_{p^{n}}) module, then the inclusion N^{Gal} ↪ N extends to a Gal-equivariant isomorphism W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} N^{Gal} → N of W (𝔽_{p^{n}})-modules.

Definition 3.6. Given a discrete twisted G_{n}-W (𝔽_{p^{n}}) module N, Map_{c}^{ℓ}(S_{n}^{k}, N) denotes the discrete G_{n}-module of continuous maps into N. Given a continuous map h: S_{n}^{k} → N and (g, σ) ∈ S_{n} ⋊ Gal = G_{n}, the action is given by ((g, σ)h)(s_{1}, ..., s_{k}) = σ · h(σ^{−1}(g^{−1}s_{1}), σ^{−1}s_{2}, ..., σ^{−1}s_{k}). (Note that σ^{−1}s_{i} refers to the action of Gal on S_{n}, instead of being the product of two elements in G_{n}.)

Lemma 3.7. If N is a twisted Gal-W (𝔽_{p^{n}}) module with the discrete topology, then, for any j ≥ 0, Map_{c}^{ℓ}(S_{n}^{j}, N) is a twisted Gal-W (𝔽_{p^{n}}) module.

Proof. It is easy to show that the module structure is twisted and, since Gal acts trivially on ℤ_{p}, it acts by ℤ_{p}-homomorphisms. □

Lemma 3.8. If N is a finite twisted Gal-W (𝔽_{p^{n}}) module with the discrete topology, then H^{s}(Gal; N) = 0, for s > 0.

Proof. Since S_{n} ◁_{o} G_{n}, the continuous projection G_{n} → G_{n}/S_{n} ≅ Gal makes N a discrete G_{n}-module, and the hypotheses imply that N is a discrete twisted G_{n}-W (𝔽_{p^{n}}) module.

Recall [8, pg. 23] that given a discrete twisted G_{n}-W (𝔽_{p^{n}}) module N and a closed subgroup H of G_{n}, there is a cochain complex D_{H}^{∗}(N) = Map_{c}^{ℓ}(S_{n}^{∗+1}, N)^{H}, such that H^{∗}(D_{H}^{∗}(−)) is equivalent to H_{c}^{∗}(H; −) on the category of discrete twisted G_{n}-W (𝔽_{p^{n}}) modules. A tedious computation shows that the Gal-action on the cochain complex Map_{c}^{ℓ}(S_{n}^{∗+1}, N) is by maps of cochain complexes.

Since Map_{c}^{ℓ}(S_{n}^{j}, N) is a twisted Gal-W (𝔽_{p^{n}}) module, there is a Gal-equivariant isomorphism W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} Map_{c}^{ℓ}(S_{n}^{j}, N)^{Gal} → Map_{c}^{ℓ}(S_{n}^{j}, N) of W (𝔽_{p^{n}})-modules that is induced by the inclusion. As in [4, proof of Lem. 5.15], since Gal acts by maps of cochain complexes, there is an isomorphism W (𝔽_{p^{n}}) ⊗_{ℤ_{p}} Map_{c}^{ℓ}(S_{n}^{∗+1}, N)^{Gal} → Map_{c}^{ℓ}(S_{n}^{∗+1}, N) of cochain complexes. Since W (𝔽_{p^{n}}) ≅ ⨁_{i=1}^{n} ℤ_{p} is a free ℤ_{p}-module, there is an isomorphism

⨁_{i=1}^{n}H^{∗}(Gal; N) ≅ ⨁_{i=1}^{n}H^{∗}(Map_{c}^{ℓ}(S_{n}^{∗+1}, N)^{Gal}) ≅ H^{∗}(Map_{c}^{ℓ}(S_{n}^{∗+1}, N)) ≅ H^{∗}({e}; N),

and the conclusion of the theorem follows. □

Theorem 3.9. If X is a finite spectrum, then for all s and t,

H_{c}^{s}(S_{n}; π_{t}(E_{n} ∧ M_{I} ∧ X))^{Gal} ≅ H_{c}^{s}(G_{n}; π_{t}(E_{n} ∧ M_{I} ∧ X)).

Proof. Let M = π_{t}(E_{n} ∧ M_{I} ∧ X), and consider the restriction map H_{c}^{s}(G_{n}; M) → H_{c}^{s}(S_{n}; M)^{Gal} and the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{p,q} = H^{p}(Gal; H_{c}^{q}(S_{n}; M)) ⟹ H_{c}^{p+q}(G_{n}; M).

It is enough to prove that E_{2}^{p,q} = 0 for all p ≥ 1, q ≥ 0, since this implies that

H_{c}^{q}(G_{n}; M) ≅ H^{0}(Gal; H_{c}^{q}(S_{n}; M)) ≅ H_{c}^{q}(S_{n}; M)^{Gal}.

As a finite G_{n}-module, M is also an E_{n,0}-module with the E_{n,0}-module structure map G_{n}-equivariant (since G_{n} acts on E_{n∗} by ring automorphisms). Then W (𝔽_{p^{n}})⊂ E_{n,0} implies that M is a twisted Gal-W (𝔽_{p^{n}}) module.

By the preceding result, we only have to prove that H_{c}^{q}(S_{n}; M) is a finite discrete twisted Gal-W (𝔽_{p^{n}}) module, which follows from applying the following: (a) since S_{n} is a compact p-adic analytic group, the proofs of 8.12 and 8.13 show that H_{c}^{q}(S_{n}; M) is a finite abelian group; and (b) the complex of continuous cochains used to compute H_{c}^{q}(S_{n}; M) consists of the discrete abelian groups Map_{c}(S_{n}^{k}, M), so that the induced topology on the cohomology is also discrete. □ Since taking fixed points is a limit, we get

Corollary 3.10. If X is a finite spectrum, then, for all s and t,

H_{c}^{s}(S_{n}; K_{n,t}(X))^{Gal} ≅ H_{c}^{s}(G_{n}; π_{t}(E_{n} ∧ X)).

Therefore, after taking Gal-fixed points [44, pg. 46], the spectral sequence in 3.2 does indeed take on the form of 1.1.

CHAPTER 4 Continuous G-spectra of simplicial sets

In this chapter, we consider the model categories of discrete G-spectra and towers of discrete G-spectra. We give a definition of continuous G-spectrum and see how in certain cases these can be obtained by taking the holim of a tower.

4.1. The model category of discrete G-spectra Let Spt be the category of Bousfield-Friedlander spectra of simplicial sets, and let

G be a profinite group. A discrete G-spectrum X is a (naive) G-spectrum of simplicial sets X_{k}, for k ≥ 0, such that each simplicial set 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 G-action). Thus, a discrete G-spectrum is a continuous G-spectrum in the sense that each set constituting the spectrum is a continuous G-space with the discrete topology and all face and degeneracy maps are (trivially) continuous. We let Spt_{G} denote the category of discrete G-spectra.

Example 4.1.1. Let k be a field with separable closure k̄. The simplicial set BGL_{∞}(k̄) is functorially a simplicial discrete G-set, where G = Gal(k̄/k). Let K(k̄) denote the corresponding algebraic K-theory spectrum. Since Ω^{∞}K(k̄) ≃ ℤ × BGL_{∞}(k̄)^{+}, K(k̄) is a discrete G-spectrum [34, §1.1].

Let G−Sets_{df} be the canonical site of finite discrete G-sets. This site has a unique point u: Sets → (G−Sets_{df})^{∼}, where (G−Sets_{df})^{∼} is the category of sheaves of sets on the site. (“Unique” means that {u} is a sufficient set of points for the Grothendieck topos.) The left adjoint of the topos morphism is

u^{∗}: (G−Sets_{df})^{∼} → Sets, F ↦ colim_{N} F(G/N),

with right adjoint

u_{∗}: Sets → (G−Sets_{df})^{∼}, X ↦ Hom_{G}(−, Map_{c}(G, X)) ≅ Sets(−, X)

[27, Rk. 6.25]. The action on 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.

Let ShvSpt(G−Sets_{df}) 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 each surjective covering family {f_{i}: W_{i} → U}, the diagram of spectra

\xymatrix{F(U) \ar[r] & ∏ _{i} F(W_{i}) \ar[r] \ar[r] & ∏ _{j,k} F(W_{j} ×_{U} W_{k})}

is an equalizer.

Alternatively, 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.

The category PreSpt(G−Sets_{df}) of presheaves of spectra has the following “stable” model category structure ([26], [27, §2.3]). A map h: F → G of presheaves of spectra is a weak equivalence if and only if the canonical map colim_{N} F(G/N) → colim_{N} G(G/N) is a weak equivalence of spectra. Recall that a map k of simplicial presheaves is a cofibration if it is an objectwise monomorphism in the sense that, for each S ∈ G−Sets_{df}, the map k(S) is a monomorphism of simplicial sets. The map h is a cofibration of presheaves of spectra if the following two conditions hold:

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

(2) for each n ≥ 0, the canonical map (S^{1} ∧ G^{n}) ∪_{(S^{1}∧F^{n})} F^{n+1} ↪ G^{n+1} is a

cofibration of simplicial presheaves. Fibrations are those maps with the right lifting property with respect to trivial cofibrations.

In this stable model category structure, fibrant presheaves of spectra are often referred to as globally fibrant, and if F → GF is a weak equivalence of presheaves of spectra, with GF globally fibrant, then we say that GF is a globally fibrant model for F.

Remark 4.1.2. Henceforth, we use PreSpt and ShvSpt to denote the categories PreSpt(G−Sets_{df}) and ShvSpt(G−Sets_{df}), respectively.

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

Lemma 4.1.3. 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, which preserves cofibrations and weak equivalences. See [27, pg. 60] and [34, Cor. 3.16] for the details. □

Now we consider sheaves of spectra. Let i: ShvSpt → PreSpt denote the inclusion functor, which is right adjoint to L^{2}, the sheafification functor. In [13, Rk. 3.11], ShvSpt is given 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 of spectra. A map k of simplicial sheaves is a cofibration if it is a cofibration of simplicial presheaves. Then h is a cofibration of sheaves of spectra if the following two conditions hold:

(1) the map h^{0} : F^{0} → G^{0} is a cofibration of simplicial sheaves, 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 sheaves.

Since i preserves all weak equivalences and fibrations, L^{2} preserves cofibrations, trivial cofibrations, and weak equivalences between cofibrant objects, so that (L^{2}, i) is a Quillen pair. Given a presheaf of spectra F, F → L^{2}F is a weak equivalence (and a strict weak equivalence), so that LL^{2} and Ri give an equivalence of categories

Ho(ShvSpt) ≅ Ho(PreSpt).

Corresponding to ShvSpt is the category Spt_{G} of discrete G-spectra. Building on the equivalence between the categories of discrete G-sets and sheaves of sets on G−Sets_{df}, the categories ShvSpt and Spt_{G} are equivalent to each other, via the functors L: ShvSpt → Spt_{G}, where L(F) = colim_{N} F(G/N), and R: Spt_{G} → ShvSpt, where R(X) = Hom_{G}(−, X).

We make Spt_{G} a model category in the following way. Define a map h of discrete G-spectra to be a weak equivalence (fibration) if and only if Hom_{G}(−, h) is a weak equivalence (fibration) of sheaves of spectra. Also, define h to be a cofibration if and only if h has the left lifting property with respect to all trivial fibrations. Then it is easy to show that h is a cofibration if and only if Hom_{G}(−, h) is a cofibration in ShvSpt. Using this, it is easy to show that Spt_{G} is a model category.

Note that h is a weak equivalence in Spt_{G} if and only if h is a weak equivalence in Spt. This setup immediately implies the following.

Theorem 4.1.4. The Quillen pair (L, R) induces an equivalence of homotopy categories: Ho(ShvSpt) ≅ Ho(Spt_{G}).

Lemma 4.1.5. Let e: Spt → Spt_{G} give a spectrum trivial G-action, so that e(X) = X. The right adjoint of e is the fixed points functor (−)^{G}, and (e, (−)^{G}) is a Quillen pair for (Spt, Spt_{G}).

Proof. It is clear that e preserves weak equivalences; it suffices to show that e preserves cofibrations. Let X ∈ Spt, Y ∈ Spt_{G}, and let cX denote the constant presheaf on X. In Spt_{G}, X ≅ colim_{N} (L^{2}(cX))(G/N), due to the adjunction isomorphisms

Spt_{G}(X, Y) ≅ Spt(X, Y^{G}) ≅ PreSpt(cX, Hom_{G}(−, Y))

≅ ShvSpt(L^{2}(cX), Hom_{G}(−, Y)) ≅ Spt_{G}(colim_{N} (L^{2}(cX))(G/N), Y).

Let f: X → Z be a cofibration in Spt. By 4.1.3, c(f) is a cofibration in PreSpt, and thus, L^{2}(c(f)) is a cofibration in ShvSpt. Since e(f) factors as

X ≅ colim_{N} (L^{2}(cX))(G/N) → colim_{N} (L^{2}(cZ))(G/N) ≅ Z,

e(f) is a cofibration in Spt_{G}. □ We need the following observation.

Lemma 4.1.6. If h: X → Y is a fibration in Spt_{G}, then it is a fibration in Spt. In particular, if X is fibrant as a discrete G-spectrum, then X is fibrant as a spectrum.

Proof. Since Hom_{G}(−, h) is a fibration of presheaves of spectra, Hom_{G}(G/N, h) is a fibration of spectra for each open normal subgroup N, by 4.1.3. Thus, the map colim_{N} Hom_{G}(G/N, h) is a fibration of spectra. Since h factors as X ≅ colim_{N} X^{N} → colim_{N} Y^{N} ≅ Y, the result follows. □

4.2. Building discrete G-spectra In this section, we give several elementary lemmas showing that certain colimits and

smash products yield discrete G-spectra.

Lemma 4.2.1. Given a profinite group G ≅ \underleftarrow{lim} _{N} G/N, let \{X_{N}\} be a direct system of (discrete) G/N-spectra. Then colim_{N} X_{N} is a discrete G-spectrum.

Proof. This follows from the fact that given a direct system {Z_{N}} of G/N-sets, the projection π: G → G/N makes {Z_{N}} a direct system of discrete G-sets, and the colimit of discrete G-sets is a discrete G-set. We remark that this lemma was used implicitly in defining L: ShvSpt → Spt_{G}. (See [27, Prop. 6.20], [29, III-9, Thm. 1].) □

Lemma 4.2.2. Given the hypotheses of the previous lemma, let Y be a trivial Gspectrum. Then there is a G-equivariant isomorphism of discrete G-spectra

(colim_{N} X_{N}) ∧ Y ≅ colim_{N} (X_{N} ∧ Y).

Proof. By the functoriality of (−) ∧ Y, (colim X_{N}) ∧ Y is a G-spectrum, and X_{N} ∧ Y is a G/N-spectrum, making colim (X_{N} ∧ Y) a discrete G-spectrum. The isomorphism is G-equivariant, since it is so on the level of simplicial sets. The isomorphism makes (colim X_{N}) ∧ Y a discrete G-spectrum. □

Lemma 4.2.3. Let G and Y be as in the two previous lemmas. If X is a discrete G- spectrum, then X ∧ Y is a discrete G-spectrum.

Proof. We have X ∧Y ≅ (colim X^{N})∧Y ≅ colim (X^{N} ∧Y), a discrete G-spectrum. □ This lemma is used repeatedly in upcoming chapters.

4.3. Towers of discrete 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, [17, VI, Prop. 1.3] shows that tow(Spt_{G}) is 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 [17, Rk. 1.5], if {Z_{i}} is fibrant in tow(Spt_{G}), then each map Z_{i} → Z_{i−1} in the tower is a fibration and each Z_{i} is fibrant in Spt_{G}.

Example 4.3.1. Let M(p^{i}) be the mod p^{i} Moore spectrum. Then the p-adic completion KU_{p}^{∧} = L_{M(p)}KU = holim_{i≥1} KU ∧ M(p^{i}) ≃ E_{1}, of the complex K-theory spectrum gives {KU ∧ M(p^{i})} in tow(Spt_{G_{1}}).

Example 4.3.2. Let ℓ be a prime not equal to char(k̄). Then the ℓ-adic completion K(k̄)_{ℓ}^{∧} of K(k̄) gives {K(k̄) ∧ M(ℓ^{i})} ∈ tow(Spt_{G}), where G is the absolute Galois group.

The functor \underleftarrow{lim} _{i} (−)^{G}: tow(Spt_{G}) → Spt, given by \{X_{i}\} ↦ \underleftarrow{lim} _{i} X_{i}^{G}, is right adjoint to the functor \underline{e}: Spt → tow(Spt_{G}) that sends a spectrum X to the constant diagram {X}, where X has trivial G-action.

Remark 4.3.3. It is important to note that the inverse limit above is formed in Spt. Limits in Spt_{G} are not formed in Spt, and lim_{Spt_{G}} X_{i} ≅ colim_{N} (lim_{Spt} X_{i})^{N}. To prevent any confusion, \underleftarrow{lim} and holim are always formed in Spt.

Since \underline{e} preserves all weak equivalences and cofibrations, by 4.1.5, we have:

Lemma 4.3.4. The pair (\underline{e}, \underleftarrow{lim} _{i} (−)^{G}) is a Quillen pair for (Spt, tow(Spt_{G})).

This implies the existence of the total right derived functor

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

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

4.4. Continuous G-spectra Definition 4.4.1. Let \{X_{i}\} be in tow(Spt_{G}). Then the inverse limit \underleftarrow{lim} _{i} X_{i},

formed in Spt, is a continuous G-spectrum. Let {Y_{i}} be a tower in tow(Spt_{G}). If X → holim_{i} Y_{i} or holim_{i} Y_{i} → X is a G-equivariant weak equivalence in Spt, where X is a continuous G-spectrum, then holim_{i} Y_{i} is a continuous G-spectrum.

Remark 4.4.2. Definition 4.4.1 can be generalized to more complicated diagrams of discrete G-spectra, but our definition is sufficient for the applications considered here. Also, we will use the term “continuous G-spectrum” more loosely. Suppose that X is a continuous G-spectrum, Z is a G-spectrum, and X → Z or Z → X is a weak equivalence of spectra that is G-equivariant. Then X is a model for Z as a continuous G-spectrum and we call Z a continuous G-spectrum. Also, if a G-spectrum Z is isomorphic to a continuous G-spectrum in the stable category, then Z is a continuous G-spectrum.

Let {X_{i}} be in tow(Spt_{G}). Given X in Spt_{G}, let X → X_{f} be a trivial cofibration with X_{f} fibrant in Spt_{G}, so that there is a weak equivalence {X_{i}} → {(X_{i})_{f}} of towers. Let {(X_{i})_{f}} → {(X_{i})′_{f}} be a trivial cofibration with {(X_{i})′_{f}} fibrant in tow(Spt_{G}). Then there is a commutative diagram, all of whose maps are G-equivariant:

(4.4.3)  \xymatrix{\underleftarrow{lim} _{i} X_{i} \ar[r] \ar[d] & holim_{i} X_{i} \ar[d]^{a} \\ \underleftarrow{lim} _{i} (X_{i})_{f} \ar[r]^{b} \ar[d]_{c} & holim_{i} (X_{i})_{f} \ar[d]^{d} \\ \underleftarrow{lim} _{i} (X_{i})′_{f} \ar[r]^{e} & holim_{i} (X_{i})′_{f}.}

Remark 4.4.4. We recall the definition of the Mittag-Leffler condition. Let {A_{i}} be a tower ⋯ → A_{2} → A_{1} → A_{0} of abelian groups. If for each k there exists a j ≥ k such that the image of A_{i} → A_{k} equals the image of A_{j} → A_{k} for all i ≥ j, then the tower {A_{i}} is said to be Mittag-Leffler. For example, if all the maps A_{i} → A_{i−1} are onto, then {A_{i}} is Mittag-Leffler. Also, a tower of finite abelian groups is Mittag-Leffler.

Lemma 4.4.5. In diagram 4.4.3, holim_{i} (X_{i})_{f} and holim_{i} (X_{i})′_{f} are continuous Gspectra. If the tower {π_{t}(X_{i})} is Mittag-Leffler for every integer t, then holim_{i} X_{i} is a continuous G-spectrum.

Proof. Since d is the homotopy limit of a diagram that consists of weak equivalences between fibrant spectra, it is a weak equivalence. If {π_{t}(X_{i})} is always Mittag-Leffler, then π_{∗}(holim_{i} X_{i}) ≅ \underleftarrow{lim} _{i} π_{∗}((X_{i})_{f}) ≅ π_{∗}(holim_{i} (X_{i})_{f}), and a is a weak equivalence.

Each (X_{i})′_{f} is a fibrant spectrum and, as a fibration of discrete G-spectra, the map (X_{i})′_{f} → (X_{i−1})′_{f} is a fibration of spectra, for i ≥ 1. Thus, for any j ≥ 0, ((X_{i})′_{f})_{j} → ((X_{i−1})′_{f})_{j} is a fibration between Kan complexes, and \underleftarrow{lim} _{i} ((X_{i})′_{f})_{j} → holim_{i} ((X_{i})′_{f})_{j} is a weak equivalence in S. This implies that e is a weak equivalence of spectra. □ Lemma 4.4.5 will be used frequently in later chapters to obtain continuous G-spectra.

CHAPTER 5 A model for E_{n} as a continuous G_{n}-spectrum

5.1. The spectrum F_{n} We consider the spectrum F_{n} = colim_{i} E_{n}^{hU_{i}} and its relationship to E_{n} in more detail.

Note that F_{n} is the stalk of the presheaf of spectra F|_{(G_{n}−Sets_{df})^{op}}, and L_{K(n)}S^{0} ≃ F(∗) is the “spectrum of global sections” Γ_{∗}F = \underleftarrow{lim} _{S∈R_{Gn}^{+}} F(S) of the presheaf F.

After the author realized that holim_{I} (F_{n} ∧ M_{I}) might provide a model of E_{n} with a continuous action, Devinatz pointed out to him that by applying the results of [9], E_{n} ≃ L_{K(n)}F_{n} is immediate. We pause to explain this and temporarily return to the notation of [9].

Theorem 5.1.1 (Devinatz and Hopkins). There is a weak equivalence E_{n} ≃ L_{K(n)}F_{n}.

Proof. As in [9], let E_{n}^{h′G} denote the ordinary homotopy fixed point spectrum for the action of G on E_{n}, when G is a finite subgroup of G_{n}. By applying [9, Thm. 3, Def. 0.5, Rk. 0.6], we have: E_{n} ≃ E_{n}^{h′{e}} ≃ E_{n}^{h{e}} = L_{K(n)}((hocolim_{i E_{∞}} E_{n}^{hU_{i}})_{cc}), where hocolim_{i E_{∞}} E_{n}^{hU_{i}} is functorially replaced with (hocolim_{i E_{∞}} E_{n}^{hU_{i}})_{cc}, a weakly equivalent cell commutative S-algebra. Since (hocolim_{i E_{∞}} E_{n}^{hU_{i}})_{cc} ≃ F_{n}, we obtain the result. □

We return to our earlier notation and F_{n}, as defined in Definition 2.2. Without relying on the results of [9] used in proving Theorem 5.1.1, we give a different proof of the key weak equivalence that underlies this work. The proof follows an outline given by Charles Rezk.

Theorem 5.1.2. There is a weak equivalence E_{n} ≃ L_{K(n)}F_{n}.

Proof. Let f: F_{n} → E_{n} be the canonical map out of the colimit, determined by the obvious maps F(G_{n}/U_{i}) → F(G_{n}). Let K_{n} be the two-periodic version of K(n) satisfying K_{n∗} ≅ 𝔽_{p^{n}}[u^{±1}], where |u| = −2. Let η: F_{n} → L_{K_{n}}F_{n} be the K_{n∗}-localization of F_{n}. Since E_{n} is K(n)_{∗}-local, it is K_{n∗}-local, so if f is a K_{n∗}-equivalence, then f and η are canonically equivalent, implying that E_{n} ≃ L_{K_{n}}F_{n} ≃ L_{K(n)}F_{n}, since K_{n} and K(n) have weakly equivalent localization functors. Thus, it suffices to prove that K_{n∗}(f) is an isomorphism.

Observe that K_{n∗}(F_{n}) ≅ colim_{i} K_{n∗}(E_{n}^{hU_{i}}). Let E be the version of “Morava Etheory” considered in [23]. Since E_{n} is a finite wedge of suspended copies of E [45, pg. 1022], E_{n∗} is the direct sum of various shifted copies of E_{∗}, with the consequence that the results of Proposition 8.4 in [23] are valid with E replaced with E_{n} and K = K(n) replaced with K_{n}. By [9, Theorem 2], as E_{n∗}-modules, π_{∗}(L_{K(n)}(E_{n} ∧ E_{n}^{hU_{i}})) ≅ Map_{c}^{ℓ}(G_{n}, E_{n∗})^{U_{i}} ≅ ∏_{G_{n}/U_{i}} E_{n∗} is a pro-free E_{n∗}-module, by [23, Corollary A.11], since E_{n∗} is pro-free (because E_{n∗} is a complete ring). Therefore, Proposition 8.4(e) implies that K_{n∗}(E_{n}^{hU_{i}}) ≅ (∏_{G_{n}/U_{i}} E_{n∗})/I_{n} ≅ (⨁_{G_{n}/U_{i}} E_{n∗})⊗_{E_{n∗}}E_{n∗}/I_{n} ≅ Map_{c}(G_{n}/U_{i}, K_{n∗}). Thus, K_{n∗}(F_{n}) ≅ Map_{c}(G_{n}, K_{n∗}), since K_{n∗} is a discrete G_{n}-module.

By [9, pg. 8], π_{∗}(L_{K(n)}(E_{n} ∧ E_{n})) ≅ Map_{c}(G_{n}, E_{n∗}) is a pro-free E_{n∗}-module. Thus, Proposition 8.4(e) implies that K_{n∗}(E_{n}) ≅ Map_{c}(G_{n}, E_{n∗})/I_{n}. This isomorphism indicates that K_{n∗}(E_{n}) is two-periodic, since E_{n∗} is two-periodic. As in [45, Thm. 12], (K_{n})_{0}(E_{n}) = Map_{c}(G_{n}, E_{n,0})/I_{n} = Map_{c}(G_{n}, E_{n,0}/I_{n}), so that two-periodicity gives K_{n∗}(E_{n}) ≅ Map_{c}(G_{n}, E_{n∗}/I_{n}) ≅ Map_{c}(G_{n}, K_{n∗}).

Since K_{n∗}(f) = K_{n∗}(colim_{i} (E_{n}^{hU_{i}} → E_{n})) = colim_{i} (K_{n∗}(E_{n}^{hU_{i}}) → K_{n∗}(E_{n})), it is clear that K_{n∗}(f) is an isomorphism. □

Corollary 5.1.3. There is a weak equivalence of spectra

E_{n} ≃ holim_{I} (F_{n} ∧ M_{I}) ≃ holim_{I} (E_{n} ∧ M_{I}).

Proof. We have E_{n} ≃ holim_{I} L_{n}(F_{n}) ∧ M_{I}. Since each E_{n}^{hU_{i}} is K(n)_{∗}-local and hence, E(n)_{∗}-local, the colimit F_{n} is also E(n)_{∗}-local so that L_{n}F_{n} ≃ F_{n}. □

The following result, whose proof was suggested by Rezk, shows that E_{n} ∧ M_{I} ≃ F_{n} ∧ M_{I}. As will be seen later, this weak equivalence is the main fact that makes it possible to construct the homotopy fixed point spectra of E_{n}.

Corollary 5.1.4. 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 ≃ L_{K(n)}(F_{n}) ∧ Y ≃ L_{n}(F_{n}) ∧ Y ≃ F_{n} ∧ Y. □

We point out some simple facts that are necessary for building the model in the next section.

Lemma 5.1.5. Let N be an open normal subgroup of G_{n}. The homotopy fixed point spectrum E_{n}^{hN} is a G_{n}/N-spectrum and the projection π: G_{n} → G_{n}/N gives a G_{n}-equivariant map F(π): E_{n}^{hN} → E_{n}. Thus, the canonical map f: F_{n} → E_{n} is G_{n}- equivariant.

Proof. Since Hom_{G_{n}}(G_{n}/N, G_{n}/N) ≅ G_{n}/N, F gives a G_{n}/N-action on F(G_{n}/N). □ The following is an example of a useful manipulation that can be performed with F_{n}.

Lemma 5.1.6. The maps L_{K(n)}(F_{n} ∧ F_{n}) → L_{K(n)}(E_{n} ∧ F_{n}) → L_{K(n)}(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} ∧ 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. □

We conclude this section with the following observation.

Theorem 5.1.7. The spectrum F_{n} is not K(n)_{∗}-local, and F_{n} and E_{n} are not weakly equivalent.

Proof. Suppose F_{n} and E_{n} are weakly equivalent. Then F_{n} is K(n)_{∗}-local and since f: F_{n} → E_{n} is a K_{n∗}-equivalence, the G_{n}-equivariant map f is a weak equivalence.

Then π_{∗}(f): π_{∗}(F_{n}) → π_{∗}(E_{n}) is a G_{n}-equivariant isomorphism from a discrete module to a graded profinite module, which is impossible, so that F_{n} and E_{n} are not weakly equivalent. If F_{n} were K(n)_{∗}-local, we obtain the same contradiction. □

5.2. Constructing the continuous action We explain how the continuous action is constructed. Recall that E_{n} = F(G_{n}). Also,

in Spt, there is no map between E_{n} ∧ S^{0} and E_{n}, but only an isomorphism in the stable category. There is a G_{n}-equivariant map holim_{I} (F_{n} ∧ M_{I}) → holim_{I} (E_{n} ∧ M_{I}), where M_{I} has trivial G_{n}-action. By the unit map S → M_{I} [23, Prop. 4.22], there is a map E_{n} ∧ S^{0} → lim_{I} (E_{n} ∧ M_{I}) → holim_{I} (E_{n} ∧ M_{I}). Notice that there is no map between holim_{I} (F_{n} ∧ M_{I}) and E_{n} ∧ S^{0}, since, for example, there is no map E_{n} → F_{n}.

Consider the following zig-zag of maps:

holim_{I} (F_{n} ∧ M_{I}) ⟶^{α} holim_{I} (E_{n} ∧ M_{I}) ⟵^{β} E_{n} ∧ S^{0} ≅^{γ} E_{n}.

The maps α and β are G_{n}-equivariant and weak equivalences, and γ is an isomorphism in the stable category. Since E_{n} ∧ S^{0} is a model for E_{n} as a G_{n}-spectrum, the zig-zag presents holim_{I} (F_{n} ∧ M_{I}) as a model for E_{n} as a G_{n}-spectrum.

Theorem 5.2.1. The spectrum 𝔼_{n} = holim_{I} (F_{n}∧M_{I}) is a continuous G_{n}-spectrum, and thus, E_{n} is a continuous G_{n}-spectrum.

Proof. Since E_{n}^{hU_{i}} is a (discrete) G_{n}/U_{i}-spectrum, F_{n} and F_{n} ∧ M_{I} are discrete G_{n}- spectra. Since {π_{∗}(F_{n} ∧ M_{I})} is a Mittag-Leffler tower of epimorphisms, 𝔼_{n} is a continuous G_{n}-spectrum, by 4.4.5. □

Now that we have a model for E_{n} as a continuous G_{n}-spectrum, one might wonder if there is a simpler model. We believe that there is not a simpler model, and we explain why one natural proposal does not work. Notice that π_{∗}(E_{n} ∧ M_{I}) = E_{n∗}/I is twoperiodic and finite in degree zero. This implies that the G_{n}-action on E_{n,0}/I factors through some finite quotient G_{n}/U_{I}, so that π_{0}(E_{n} ∧ M_{I}) is a G_{n}/U_{I}-module. Thus, one might conjecture that E_{n} ∧ M_{I} is a G_{n}/U_{I} spectrum and hence, a particularly simple discrete G_{n}-spectrum, providing E_{n} with a simpler continuous action. However, we will show that this is false: E_{n} ∧ M_{I} is not a G_{n}/U_{i}-spectrum for any i. As far as the author knows, Mike Hopkins first made this observation and the author learned the justification given below from Hal Sadofsky.

First of all, note that it is morally clear that E_{n}∧M_{I} should not be a G_{n}/U_{I}-spectrum since it has the “same action” as the weakly equivalent F_{n} ∧ M_{I}, which by construction “is not” a G_{n}/U_{i}-spectrum for any U_{i}. However, the author does not know rigorous proofs of these assertions without using the kind of argument given below.

Theorem 5.2.2. 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; E_{n} ∧ M_{I} is not a G_{n}/U-spectrum for any U ◁_{o} G_{n}.

Proof. Suppose the G_{n}-spectrum E_{n} ∧ M_{I} is a G_{n}/U-spectrum. This implies that 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 naturally a G_{n}/U-module. Also, E_{n}∧E_{n}∧M_{I} ≃ L_{n}(E_{n}∧E_{n}∧M_{I}) ≃ L_{K(n)}(E_{n} ∧ E_{n}) ∧ M_{I}, so that π_{∗}(E_{n} ∧ E_{n} ∧ M_{I}) ≅ Map_{c}^{ℓ}(G_{n}, E_{n∗}/I) by [9, pg. 10].

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. □

As mentioned in the Introduction, one of the deficiencies of the model given in this paper for E_{n} as a continuous G_{n}-spectrum is that even though E_{n} is a twisted E_{n}- module spectrum (“twisted” means that the module structure map is G_{n}-equivariant, where E_{n}∧E_{n} has the diagonal action), we can not prove that holim_{I} (F_{n}∧M_{I}) is an E_{n}- module spectrum. Clearly, any satisfactory model of E_{n} as a continuous G_{n}-spectrum would also be a twisted E_{n}-module continuous G_{n}-spectrum. Similarly, since E_{n} ∧ M_{I} is an E_{n}-module, one would hope that F_{n} ∧ M_{I} is an E_{n}-module. However, the primary cause of this deficiency in our model is that it appears that F_{n} is not an E_{n}-module spectrum.

The apparent failure of F_{n} to be an E_{n}-module is a topological version of the fact that if R is a non-discrete continuous G-ring, then the discrete G-module colim_{N} R^{N} need not be an R-module.

CHAPTER 6 Homotopy fixed point spectra for discrete G-spectra

Let G be a closed subgroup of G_{n}. Then G is a profinite group and E_{n} is a continuous G-spectrum by the inclusion of G in G_{n}. Recall that [9, Def. 0.5] defines homotopy fixed point spectra

E_{n}^{hG} = L_{K(n)}(hocolim_{i E_{∞}} E_{n}^{hU_{i}G}),

where U_{i}G is an open subgroup of G_{n}, with associated spectral sequences

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

The spectra E_{n}^{hG} are called homotopy fixed point spectra because these associated spectral sequences have the form of descent spectral sequences. However, E_{n}^{hG} is not defined with respect to any continuous action, and the associated spectral sequences are not actually descent spectral sequences, but are instead K(n)_{∗}-local E_{n}-Adams spectral sequences.

In this chapter, we define homotopy fixed point spectra for discrete G-spectra. Also, when G has finite virtual cohomological dimension, we develop a specific model of the homotopy fixed points that is better suited for building the descent spectral sequence. In subsequent chapters, we use this model to define homotopy fixed point spectra E_{n}^{h′G} and construct descent spectral sequences, using the continuous action. Furthermore, we partially recover the constructions of [9], by using the continuous action, in the following sense: our homotopy fixed point spectra are weakly equivalent to theirs, and our descent spectral sequences are isomorphic to theirs.

Definition 6.1. Given a discrete G-spectrum X, X^{hG} denotes the homotopy fixed point spectrum of X with respect to the continuous action of G. We define X^{hG} = (X_{f})^{G}, where X → X_{f,G} ≡ X_{f} is a trivial cofibration, and X_{f} is a fibrant discrete G-spectrum. (This definition has been developed in other categories: see [14] for simplicial discrete G-sets, [25] for simplicial (pre)sheaves, and, for example, [27, Chp. 6] for presheaves of spectra.)

We make explicit several properties of homotopy fixed points.

Lemma 6.2. 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.

Proof. Lemma 4.1.5 implies R(−)^{G} = (−)^{hG}. The map Hom_{G}(−, X_{f}) → Hom_{G}(−, Y_{f}) is a weak equivalence between globally fibrant presheaves, and evaluation at ∗ gives a weak equivalence, by 4.1.3. □

Remark 6.3. 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 Gspectrum, and let X → X′ be a weak equivalence that is G-equivariant, with X′ a fibrant spectrum. Then the definition of the homotopy fixed point spectrum X^{h′G} for a finite group G is holim_{G} X′. Note that by Remark 6.9 and [3, Ch. XI, 6.3], 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^{hG}).

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 the canonical map X^{hG} = lim_{G} X_{f} → holim_{G} X_{f} = X^{h′G} is a weak equivalence, as desired, by [27, Prop. 6.39].

Unlike the simplicial presheaves setting, not every presheaf of spectra is cofibrant. Thus, given two globally fibrant models GF and G′F for the presheaf of spectra F, there need not be a weak equivalence GF → G′F. Also, a cofibration of sheaves of spectra can fail to be a cofibration of presheaves. Given these facts, we state the following relationship between homotopy fixed points and the global sections of globally fibrant models.

Lemma 6.4. Let X be a discrete G-spectrum. Then

X^{hG} = (X_{f})^{G} ⟵^{≃} F(∗) ⟶^{≃} GHom_{G}(−, X)(∗),

where Hom_{G}(−, X) → F is a trivial cofibration of presheaves of spectra and F is globally fibrant. Thus, X^{hG} and GHom_{G}(−, X)(∗) have the same stable homotopy type.

Proof. Weak equivalences from F to the globally fibrant models Hom_{G}(−, X_{f}) and GHom_{G}(−, X) exist by the left lifting property of trivial cofibrations with respect to fibrations. Evaluation at the point gives the desired weak equivalences. □

We ease our construction of the descent spectral sequence by using a different model for X^{hG}. We do this by discussing some tools that utilize the fact that the closed subgroups of G_{n} have uniformly bounded finite virtual cohomological dimension m.

Definition 6.5. The functor Γ_{G}: Spt_{G} → Spt_{G} is given by Γ_{G}(X) = Map_{c}(G, X), where Γ_{G}(X) has the G-action defined in §4.1. We write Γ instead of Γ_{G}, when G is understood from the context. There is a G-equivariant monomorphism ι: X → ΓX defined by ι(x)(g) = g · x. Since Γ forms a triple, there is a cosimplicial discrete Gspectrum Γ^{·}X.

We recall the construction of Thomason’s hypercohomology spectrum (see [48, 1.31-1.33] and [34, §3.2] for more details). Let F be a sheaf of fibrant spectra so that for each S ∈ G−Sets_{df}, F(S) is a fibrant spectrum. Then the functor

T = u_{∗}u^{∗}: ShvSpt → ShvSpt, F ↦ Hom_{G}(−, Map_{c}(G, colim_{N} F(G/N))),

gives a cosimplicial sheaf T.F of fibrant spectra. This last fact is verified in the following lemma.

Lemma 6.6. If F is a sheaf of fibrant spectra, then T.F is a cosimplicial sheaf of fibrant spectra.

Proof. It suffices to show that T F is a sheaf of fibrant spectra. Any S ∈ G−Sets_{df} is isomorphic to a finite disjoint union ∐ _{i} G/H_{i}, where each H_{i} is an open subgroup of G. Thus,

(T F)(S) ≅ ∏_{i} Hom_{G}(G/H_{i}, Map_{c}(G, colim_{N} F(G/N))) ≅ ∏_{i} ∏_{G/H_{i}} colim_{N} F(G/N).

Since filtered colimits and products of fibrant spectra are fibrant, (T F)(S) is a fibrant spectrum. □

Definition 6.7. The presheaf of hypercohomology spectra of Φ = G−Sets_{df} with coefficients in F is the presheaf of spectra

ℍ._{Φ} (−; F): (G−Sets_{df})^{op} → Spt, U ↦ ℍ._{Φ} (U; F) = holim_{Δ} (T.F)(U),

and ℍ._{Φ} (U; F) is the hypercohomology spectrum of U with coefficients in F.

Lemma 6.8. Let X → X_{f} in Spt_{G} be a trivial cofibration with X_{f} a fibrant discrete G-spectrum. Then T.Hom_{G}(−, X_{f}) ≅ Hom_{G}(−, Γ.X_{f}).

Proof. Note that the fibrant presheaf Hom_{G}(−, X_{f}) is a sheaf of fibrant spectra, as required. The result is then obtained by induction. By definition,

T (Hom_{G}(−, X_{f})) ≅ Hom_{G}(−, Map_{c}(G, colim_{N} (X_{f})^{N})) ≅ Hom_{G}(−, Map_{c}(G, X_{f})). Similarly, the inductive hypothesis gives

T^{k+1}(Hom_{G}(−, X_{f})) ≅ T (Hom_{G}(−, Map_{c}(G^{k}, X_{f})))
≅ Hom_{G}(−, Map_{c}(G, colim_{N} Map_{c}(G^{k}, X_{f})^{N}))
≅ Hom_{G}(−, Map_{c}(G^{k+1}, X_{f})).
□

The map ι induces the map X_{f} → Γ.X_{f} out of the constant cosimplicial diagram, which induces the canonical map Hom_{G}(−, X_{f}) → lim Hom_{G}(−, X_{f}) → holim Hom_{G}(−, X_{f}) → holim Hom_{G}(−, Γ.X_{f}), where each (ho)lim is over Δ. Then the preceding lemma gives a map

Hom_{G}(−, X_{f}) ⟶ holim_{Δ} Hom_{G}(−, Γ.X_{f}) ⟶^{≅} ℍ._{Φ} (−; Hom_{G}(−, X_{f})).

Remark 6.9. We recall the Bousfield-Kan spectral sequence, an important tool for our work. [3, Chp. XI, §7] constructs a homotopy spectral sequence

π^{s}π_{t}(F.) ⟹ π_{t−s}(TotF.),

where F. is a fibrant cosimplicial pointed simplicial set. This spectral sequence extends to cosimplicial spectra: if F. is a cosimplicial diagram of fibrant spectra (a “cosimplicial fibrant spectrum”), then there is a spectral sequence

(6.10)  E_{2}^{s,t} = π^{s}π_{t}(F.) ⟹ π_{t−s}(holim_{Δ} F.)

[48, Prop. 5.13, Lem. 5.31].

Here, as elsewhere, the holim of spectra is constructed levelwise in S and is defined as in [3]. Thus, associated to F. is the cosimplicial spectrum ∏^{∗} F., with ∏^{n}F. = ∏_{(BΔ)_{n}} F^{j_{n}}, where the n-simplices of the classifying space BΔ consist of all strings [j_{0}] → ⋯ → [j_{n}] of n morphisms in Δ. For any k ≥ 0, (∏^{∗}F.)_{k} is a fibrant cosimplicial pointed simplicial set, and holim_{Δ} F. = Tot(∏^{∗}F.).

More generally, if J is a small category and F: J → Spt is a diagram of fibrant spectra, then there is a spectral sequence

(6.11)  E_{2}^{s,t} = lim^{s} _{J} [Z, F]_{t} ⟹ [Z, holim_{J} F]_{t−s},

where Z is a finite spectrum equivalent to a CW-spectrum, and lim^{s} is the sth right derived functor of lim_{J}: Ab^{J} → Ab. All of the above spectral sequences are conditionally convergent [34, pg. 246].

Now we prove the result that gives us a more useful model of X^{hG}. This result is not original: a version of it is in [25, Prop. 3.3], and more general forms of it appear in [34, Prop. 3.20] and [35, Prop. 6.1].

Theorem 6.12. 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}) ⟶^{≃} ℍ._{Φ} (−; Hom_{G}(−, X_{f})),

where ℍ._{Φ} (−; Hom_{G}(−, X_{f})) is a globally fibrant model for Hom_{G}(−, X). Then the canonical map

X^{hG} = Hom_{G}(∗, X_{f}) ⟶ holim_{Δ} (Γ._{G}X_{f})^{G} ≅ ℍ._{Φ} (∗; Hom_{G}(−, X_{f}))

is a weak equivalence.

Remark 6.13. Since (X_{f})^{G} is already X^{hG} and given Lemma 6.17 below, the reader might wonder if X_{f} can be replaced by a weaker model for X. This can be done, but after trying several alternatives, X_{f} was decided upon as the best choice, in part because of the above canonical map.

Proof. We refer to the references above, especially [25, Prop. 3.3], for the proof that the hypercohomology presheaf ℍ._{Φ} (−; Hom_{G}(−, X_{f})) is globally fibrant. It only remains to show that η: Hom_{G}(−, X) → ℍ._{Φ} (−; Hom_{G}(−, X_{f})) is a weak equivalence.

By hypothesis, G contains an open subgroup H with cd(H) ≤ m. Then by [50, 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. Set N′ = N ∩ K. Observe that {N′} is a cofinal subcollection of open normal subgroups of G so that G ≅ \underleftarrow{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 sheaf of sets F has stalk colim_{N} F(G/N) and so η is a weak equivalence of presheaves of spectra if η̂: X ≅ colim_{N} X^{N} → colim_{N} ℍ._{Φ} (G/N; Hom_{G}(−, X_{f})) is a weak equivalence, where colim_{N} ℍ._{Φ} (G/N; Hom_{G}(−, X_{f})) ≅ colim_{N} holim_{Δ} (Γ._{G}X_{f})^{N}. For each N, apply 6.10 to obtain the conditionally convergent spectral sequence

(6.14)  E_{2}^{s,t}(N) = π^{s}π_{t}((Γ._{G}X_{f})^{N}) ⟹ π_{t−s}(holim_{Δ} (Γ._{G}X_{f})^{N}).

Since T.Hom_{G}(−, X_{f}) ≅ Hom_{G}(−, Γ.X_{f}) is a cosimplicial sheaf of fibrant spectra, the diagram (Γ.X_{f})^{N} is a cosimplicial fibrant spectrum, as required. Note that

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}.

Thus, π^{s}π_{t}((Γ._{G}X_{f})^{N}) ≅ H^{s}((Γ^{∗}π_{t}(X))^{N}). The cochain complex 0 → π_{t}(X) → Γ^{∗}π_{t}(X) is exact and H_{c}^{s}(N; Γ^{k}π_{t}(X)) ≅ H_{c}^{s}(N; Map_{c}(G, Γ^{k−1}π_{t}(X))) = 0, for s > 0, by 8.16. Thus, Γ^{∗}π_{t}(X) is a resolution of π_{t}(X) by (−)^{N}-acyclics and thus,

E_{2}^{s,t}(N) ≅ H_{c}^{s}(N; π_{t}(X)).

Taking a colimit over {N} of 6.14 gives the spectral sequence

(6.15)  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 [34, Prop. 3.3], the colimit of the spectral sequences does converge to the colimit of the abutments, as stated in 6.15. Finally,

colim_{N} H_{c}^{∗}(N; π_{t}(X))≅H_{c}^{∗}(\underleftarrow{lim} N; π_{t}(X))≅H^{∗}(\{e\}; π_{t}(X)),

which is isomorphic to π_{t}(X), concentrated in degree 0. Thus, 6.15 collapses and for all t, π_{t}(colim_{N} holim_{Δ} (Γ._{G}X_{f})^{N}) ≅ π_{t}(X), showing that η̂ is a weak equivalence. □

Remark 6.16. Because of Theorem 6.12, whenever the profinite group G has finite virtual cohomological dimension, and X is a discrete G-spectrum, we will always make the identification X^{hG} = holim_{Δ} (Γ._{G}(X_{f}))^{G}. (There are a few exceptions to this, where it is clear from the context.)

The following lemma is obtained from 6.6 and 6.8 by evaluating at ∗ ∈ G−Sets_{df}. It is used repeatedly so we make it explicit.

Lemma 6.17. If X is fibrant in Spt_{G}, then (Γ._{G}X)^{G} is a cosimplicial fibrant spectrum.

Given a profinite group G of finite virtual cohomological dimension and a discrete G-spectrum X, the following theorem shows that there is always a descent spectral sequence.

Remark 6.18. Given a discrete G-module M, there are canonical maps

η(M): M → Γ_{G}(M) = Map_{c}(G, M) and ε: Map_{c}(G, Map_{c}(G, M)) → Map_{c}(G, M),

so that Γ gives a triple and there is an associated cosimplicial discrete G-module Γ._{G}(M). Recall that H_{c}^{s}(G, M) can be computed by taking the cohomology of the cochain complex

M → Map_{c}(G, M) → Map_{c}(G^{2}, M) → · · · .

As pointed out in [14, Lem. 5.4 and its proof], the maps in this cochain complex are identical to the maps in the cochain complex associated to the cosimplicial abelian group (Γ.(M))^{G}. Thus, H_{c}^{s}(G; M) ≅ π^{s}((Γ.(M))^{G}).

Theorem 6.19. If the profinite group G has finite virtual cohomological dimension 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}).

Proof. The preceding lemma and 6.10 give a spectral sequence

π^{s}π_{t}((Γ.X_{f})^{G}) ⟹ π_{t−s}(X^{hG}).

By the above remark, E_{2}^{s,t} ≅ H_{c}^{s}(G; π_{t}(X)), since π_{t}(X_{f}) ≅ π_{t}(X) is an isomorphism of discrete G-modules. □

CHAPTER 7 Homotopy fixed point spectra for E_{n}

Using the machinery of the previous chapter, it is easy to define homotopy fixed point spectra E_{n}^{h′G} for G <_{c} G_{n}, using the continuous action. In this chapter, we also show that E_{n}^{h′G} matches the homotopy fixed point spectrum E_{n}^{hG} of Devinatz and Hopkins.

Definition 7.1. Let E_{n}^{h′G} = holim_{I} (F_{n} ∧ M_{I})^{hG} be the homotopy fixed point spectrum of E_{n} with respect to the continuous action of G. Since F_{n} ∧ M_{I} is a discrete G-spectrum and vcd(G) < ∞, (F_{n} ∧ M_{I})^{hG} is defined as in Remark 6.16.

Remark 7.2. Note that the map ((F_{n} ∧ M_{I})_{f})^{G} → holim_{Δ} (Γ_{G}^{·}(F_{n} ∧ M_{I})_{f})^{G} is a weak equivalence between fibrant spectra, since the map is obtained by taking the global sections of a weak equivalence between fibrant presheaves of spectra. Thus, there is a weak equivalence holim_{I} ((F_{n} ∧ M_{I})_{f})^{G} → holim_{I} holim_{Δ} (Γ_{G}^{·}(F_{n} ∧ M_{I})_{f})^{G} between the “strict” definition of E_{n}^{h′G} and our definition.

Remark 7.3. We make explicit the following elementary fact, which will be used several times later. If {X_{j}} is a finite set of spectra and Y is any spectrum, then there is a weak equivalence (∏ X_{j}) ∧ Y → ∏ (X_{j} ∧ Y).

It is not hard to see that E_{n}^{h′G} is K(n)_{∗}-local, as expected, since E_{n}^{hG} is.

Theorem 7.4. The homotopy fixed point spectra (F_{n} ∧ M_{I})^{hG} and E_{n}^{h′G} are K(n)_{∗}- local. Also, F_{n}^{hG} is E(n)_{∗}-local.

Proof. Note that

(Γ^{k}(F_{n} ∧ M_{I})_{f})^{G} ≅ Map_{c}(G^{k−1}, (F_{n} ∧ M_{I})_{f}) ≃ colim_{N} ∏_{(G/N)^{k−1}} (F_{n} ∧ M_{I})
≃ (colim_{N} ∏_{(G/N)^{k−1}} F_{n}) ∧ M_{I}.

Since F_{n} is E(n)_{∗}-local, the finite product is also. Since L_{n} commutes with direct limits, colim_{N} ∏_{(G/N)^{k−1}} F_{n} is E(n)_{∗}-local, and thus, (Γ^{k}(F_{n} ∧ M_{I})_{f})^{G} is K(n)_{∗}-local. Given a CW-spectrum E, the homotopy limit of an arbitrary diagram of E_{∗}-local spectra is E_{∗}-local [2, pg. 259], making (F_{n} ∧ M_{I})^{hG} = holim_{Δ} (Γ.(F_{n} ∧ M_{I})_{f})^{G} K(n)_{∗}-local. Hence, the homotopy limit E_{n}^{h′G} is K(n)_{∗}-local. The same arguments show that F_{n}^{hG} is E(n)_{∗}-local. □

After seeing how E_{n}^{h′G} is defined, we now examine the construction of E_{n}^{hU} by Devinatz and Hopkins, for U <_{o} G_{n}. For this discussion, we use the notation of [9] and work, as needed, in the category of S-modules. At times, we will not be completely rigorous, since we are trying to understand from the perspective of a continuous action how the construction of Devinatz and Hopkins works.

Construction 3.11 of [9] defines a diagram

C: (R_{G_{n}}^{+})^{op} × Δ → Ho(E_{∞}), C(S, [j]) = L_{K(n)}(X(S) ∧ E_{n}^{j}),

where X: (R_{G_{n}}^{+})^{op} → Ho(E_{∞}) is a certain diagram with X(G_{n}/U) = ∏ _{G_{n}/U} E_{n}. Then [9] shows that C lifts to a diagram \overline{C}: (R_{G_{n}}^{+})^{op} × Δ → E_{∞}, such that there is a natural transformation \overline{C} → C in Ho(E_{∞}), with \overline{C}(S, [j]) → C(S, [j]) a weak equivalence for every S and [j]. Henceforth, we write C._{G_{n}/U} instead of \overline{C}(G_{n}/U, −) for the cosimplicial diagram Δ → E_{∞}, so that C_{G_{n}/U}^{j} ≡ \overline{C}(G_{n}/U, [j]). Then, by [9, Def. 3.12],

E_{n}^{hU} = Tot(∏^{∗}C._{G_{n}/U}) = holim_{Δ} C._{G_{n}/U}.

Since C._{G_{n}/U} ∈ (Ho(E_{∞}))^{Δ}, one cannot form holim_{Δ} C._{G_{n}/U}. However, since C._{G_{n}/U} is a strict diagram of spectra, holim_{Δ} C._{G_{n}/U} is a model for the undefined holim_{Δ} C._{G_{n}/U}.

We work to understand C._{G_{n}/U} better. We ignore the fact that it is only a diagram in the homotopy category; we assume that E_{n}^{j} ∧ M_{I} is a discrete G_{n}-spectrum (as is its weakly equivalent model F_{n}^{j} ∧ M_{I}); we assume that holim preserves natural transformations consisting of weak equivalences; and we ignore the fact that the G_{n}-action we should have below is that of Map_{c}^{ℓ}(G_{n}^{j}, −) instead of Map_{c}(G_{n}^{j}, −) [9, 3.19]. Applying [9, pg. 10], we have

C_{G_{n}/U}^{j} ≃ C_{G_{n}/U}^{j} = L_{K(n)}((∏_{G_{n}/U} E_{n}) ∧ E_{n}^{j}) ≃ holim_{I} (∏_{G_{n}/U} (E_{n}^{j+1} ∧ M_{I}))

≃ holim_{I} Map_{c}(G_{n}, E_{n}^{j+1} ∧ M_{I})^{U} ≃ holim_{I} Map_{c}(G_{n}, Map_{c}(G_{n}^{j}, E_{n} ∧ M_{I}))^{U}

≃ holim_{I} Map_{c}(G_{n}^{j+1}, E_{n} ∧ M_{I})^{U}.

Thus, holim_{Δ} C._{G_{n}/U} ≃ holim_{I} holim_{Δ} (Γ._{G_{n}}(E_{n} ∧ M_{I}))^{U}. We examine this last expression further.

Consider the functor (U − Sets_{df})^{op} → (G_{n} − Sets_{df})^{op}, defined by S ↦ G_{n}/U × S, where S has trivial G_{n}-action. Composition with this functor gives the restriction functor res: PreSpt(G_{n} − Sets_{df}) → PreSpt(U − Sets_{df}), which sends the cosimplicial sheaf Hom_{G_{n}}(−, Γ._{G_{n}}X_{f}) to the cosimplicial sheaf

Hom_{G_{n}}(G_{n}/U × (−), Γ._{G_{n}}X_{f}) ≅ Hom_{G_{n}}(G_{n}/U, Sets(−, Γ._{G_{n}}X_{f})) ≅ Sets(−, Γ._{G_{n}}X_{f})^{U}

≅ Hom_{U}(−, Γ._{G_{n}}X_{f}).

Since U is an open subgroup of G_{n}, the restriction of the globally fibrant presheaf holim_{Δ} Hom_{G_{n}}(−, Γ._{G_{n}}X_{f}) is the globally fibrant presheaf holim_{Δ} Hom_{U}(−, Γ._{G_{n}}X_{f}). (When U is a normal subgroup, [27, Rk. 6.26] shows that res preserves globally fibrant presheaves.)

Note that Hom_{U}(S, Γ._{G_{n}}X_{f}) ≅ Hom_{G_{n}}(G_{n}/U × S, Γ._{G_{n}}X_{f}) is a cosimplicial fibrant spectrum since Hom_{G_{n}}(−, Γ._{G_{n}}X_{f}) is a cosimplicial sheaf of fibrant spectra. Any subgroup H that is open in U, is open in G_{n}. These two observations imply that the proof of Theorem 6.12 goes through to show that Hom_{U}(−, X) → holim_{Δ} Hom_{U}(−, Γ._{G_{n}}X_{f}) is a weak equivalence of presheaves.

Since Hom_{G_{n}}(−, X_{f,G_{n}}) is globally fibrant on (G_{n} − Sets_{df})^{op}, Hom_{U}(−, X_{f,G_{n}}) is globally fibrant on (U − Sets_{df})^{op}, and X_{f,G_{n}} is a fibrant discrete U-spectrum. Thus, there is a weak equivalence Hom_{U}(−, X_{f,G_{n}}) → holim_{Δ} Hom_{U}(−, Γ._{G_{n}}X_{f,G_{n}}) between globally fibrant presheaves, which gives a weak equivalence

X^{hU} ≅ Hom_{U}(∗, X_{f,G_{n}}) → holim_{Δ} Hom_{U}(∗, Γ._{G_{n}}X_{f,G_{n}}) = holim_{Δ} (Γ._{G_{n}}X_{f,G_{n}})^{U},

where the isomorphism is in Ho(Spt). Therefore, given any discrete G_{n}-spectrum X, holim_{Δ} (Γ._{G_{n}}(X_{f,G_{n}}))^{U} is a model for X^{hU}.

Let us return to our assumptions about C._{G_{n}/U} and allow ourselves to move freely between S-modules and Bousfield-Friedlander spectra. Observe that

(Γ._{G_{n}}(E_{n} ∧ M_{I}))^{U} → (Γ._{G_{n}}(E_{n} ∧ M_{I})_{f,G_{n}})^{U}

is a weak equivalence in each cosimplicial degree. Thus,

holim_{Δ} C._{G_{n}/U} ≃ holim_{I} holim_{Δ} (Γ._{G_{n}}(E_{n} ∧ M_{I}))^{U}

≃ holim_{I} holim_{Δ} (Γ._{G_{n}}(E_{n} ∧ M_{I})_{f,G_{n}})^{U}.

We highlight our conclusion in

Remark 7.5. The above discussion shows that, in some sense,

E_{n}^{hU} = Tot(∏^{∗}C._{G_{n}/U}) ≃ holim_{I} (E_{n} ∧ M_{I})^{hU},

and thus, E_{n}^{hU} is the continuous homotopy fixed point spectrum. Also, conceptually, we see that E_{n}^{hU} and E_{n}^{h′U} are the same.

We make this remark precise by showing in what sense the continuous homotopy fixed point spectra E_{n}^{h′G} match the homotopy fixed point spectra E_{n}^{hG} of Devinatz and Hopkins.

Lemma 7.6 ([7, proof of Lem. 3.5]). For any G <_{c} G_{n} and any integer t, the homotopy group π_{t}(E_{n}^{hG} ∧ M_{I}) of the S-module E_{n}^{hG} ∧ M_{I} is a finite abelian group.

Remark 7.7. By the discussion about homotopy colimits in Chapter 2, in Spt, L_{K(n)}(colim_{i} E_{n}^{hU_{i}G}) is a model for the S-module E_{n}^{hG} defined by Devinatz and Hopkins. Thus, in the context of Spt, we set E_{n}^{hG} ≡ L_{K(n)}(colim_{i} E_{n}^{hU_{i}G}).

However, clarification is needed if G = U is an open subgroup. In S-modules, the spectra E_{n}^{hU} and L_{K(n)}(colim_{i} E_{n}^{hU_{i}U}) have the same homotopy type. But it is not obvious that this is the case in Spt, where E_{n}^{hU} = (U𝕊Φ(−)_{f}F)(G_{n}/U). Let X_{C} and X∧_{C} Y refer to the spectra X and X∧Y, respectively, where X and Y are in C, a category of spectra. As in the proof of 7.9, there are strongly convergent spectral sequences

H_{c}^{s}(U; π_{t}(E_{n} ∧_{Spt} M_{I})) ⟹ π_{t−s}(E_{n}^{hU} ∧_{Spt} M_{I}),

and

H_{c}^{s}(U; π_{t}(E_{n} ∧_{M_{S}} M_{I})) ⟹ π_{t−s}(E_{n}^{hU} ∧_{M_{S}} M_{I}).

By Landweber exactness, π_{∗}(E_{n}∧_{Spt}M_{I}) ≅ E_{n∗}/I ≅ π_{∗}(E_{n}∧_{M_{S}} M_{I}). Thus, the spectral sequences have isomorphic E_{2}-terms and π_{∗}(E_{n}^{hU} ∧_{Spt} M_{I}) and π_{∗}(E_{n}^{hU} ∧_{M_{S}} M_{I}) are the same up to group extensions. Furthermore, since (E_{n}^{hU})_{Spt} is a model for (E_{n}^{hU})_{M_{S}} and M_{I} has the same characterization in either category of spectra, we conclude that π_{∗}(E_{n}^{hU} ∧_{Spt}M_{I}) ≅ π_{∗}(E_{n}^{hU} ∧_{M_{S}} M_{I}). Thus, since E_{n}^{hU} ∧_{M_{S}} M_{I} ≃ colim_{i} (E_{n}^{hU_{i}U} ∧_{M_{S}} M_{I}) and π_{∗}(E_{n}^{hU} ∧ M_{I}) is degree-wise finite,

π_{∗}((E_{n}^{hU})_{Spt}) ≅ π_{∗}((E_{n}^{hU})_{M_{S}}) ≅ π_{∗}(L_{K(n)}(colim_{i} (E_{n}^{hU_{i}U})_{M_{S}}))

≅ π_{∗}(holim_{I} colim_{i} (E_{n}^{hU_{i}U} ∧_{M_{S}} M_{I})) ≅ \underleftarrow{lim} _{I} colim_{i} π_{∗}(E_{n}^{hU_{i}U} ∧_{M_{S}} M_{I}) ≅ \underleftarrow{lim} _{I} colim_{i} π_{∗}(E_{n}^{hU_{i}U} ∧_{Spt} M_{I}) ≅ π_{∗}(holim_{I} colim_{i} (E_{n}^{hU_{i}U} ∧_{Spt} M_{I}))

≅ π_{∗}(L_{K(n)}(colim_{i} (E_{n}^{hU_{i}U})_{Spt})).

Therefore, the canonical map L_{K(n)}(colim_{i} (E_{n}^{hU_{i}U})_{Spt}) → (E_{n}^{hU})_{Spt} in Ho(Spt) is an isomorphism. Thus, in Spt, just as in M_{S}, we can move freely between the two different meanings of E_{n}^{hU}, L_{K(n)}(colim_{i} (E_{n}^{hU_{i}U})_{Spt}) and (U𝕊Φ(−)_{f}F)(G_{n}/U) in the stable category. In any particular instance, the context will make it clear which meaning we intend.

Given a spectrum X, though there is no point-set level map between X and X ∧ S^{0}, there is an isomorphism E_{n}^{hG} = L_{K(n)}(colim_{i} E_{n}^{hU_{i}G}) ≅ L_{K(n)}((colim_{i} E_{n}^{hU_{i}G}) ∧ S^{0}) in Ho(Spt). Also, since there is no point-set level map between L_{K(n)}(X ∧ S^{0}) and holim_{I} (X ∧ M_{I}), for an E(n)_{∗}-local spectrum X, the best we can do in our case is the following zigzag:

\xymatrix{L_{K(n)}((colim_{i} E_{n}^{hU_{i}G}) ∧ S^{0}) \ar[d]^{≃} & holim_{I} ((colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I}) \ar@{-->}[dl]_{≃} \\ L_{K(n)}(holim_{I} ((colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I})).}

Now we specify the map ψ_{G}: holim_{I} ((colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I}) → E_{n}^{h′G}. When G is finite, [9, pg. 5] shows that there is a canonical map into the homotopy fixed points E_{n}^{hG} → holim_{G} E_{n}, defined by the composition E_{n}^{hG} → lim_{G} E_{n} → holim_{G} E_{n}. Similarly, we would like to construct ψ by composing with a map

ϕ_{G}: lim_{G} (F(G_{n}) ∧ S^{0}) → E_{n}^{h′G} = holim_{I} (F_{n} ∧ M_{I})^{hG}.

However, ϕ_{{e}} forces the existence of a map F(G_{n})∧S^{0} → holim_{I} (F_{n} ∧M_{I}). But we saw earlier that there is no such canonical G_{n}-map, and thus, there is no ϕ_{{e}}. We conclude that there is no canonical map ϕ_{G}, and we proceed to construct ψ another way.

Since U_{i}G/U_{i} is a subgroup of G_{n}/U_{i}, the canonical map E_{n}^{hU_{i}G} → E_{n}^{hU_{i}} gives a map

E_{n}^{hU_{i}G} → lim_{U_{i}G/U_{i}} E_{n}^{hU_{i}} →^{≅} (E_{n}^{hU_{i}})^{U_{i}G/U_{i}}. Letting (E_{n}^{hU_{i}})^{U_{i}G/U_{i}} also denote the constant

cosimplicial spectrum, there are canonical maps

(E_{n}^{hU_{i}})^{U_{i}G/U_{i}} → lim (E_{n}^{hU_{i}})^{U_{i}G/U_{i}} → holim (E_{n}^{hU_{i}})^{U_{i}G/U_{i}} → holim (Γ._{U_{i}G/U_{i}}(E_{n}^{hU_{i}}))^{U_{i}G/U_{i}},

where all (ho)limits are over Δ. Also, the projection U_{i}G → U_{i}G/U_{i} induces canonical maps

holim_{Δ} (Γ._{U_{i}G/U_{i}}(E_{n}^{hU_{i}}))^{U_{i}G/U_{i}} → holim_{Δ} (Γ._{U_{i}G}E_{n}^{hU_{i}})^{U_{i}G/U_{i}} → holim_{Δ} (Γ._{U_{i}G}E_{n}^{hU_{i}})^{U_{i}G}.

The inclusion G → U_{i}G gives maps

holim_{Δ} (Γ._{U_{i}G}E_{n}^{hU_{i}})^{U_{i}G} → holim_{Δ} (Γ._{U_{i}G}E_{n}^{hU_{i}})^{G} → holim_{Δ} (Γ._{G}E_{n}^{hU_{i}})^{G}.

Putting all these maps together gives a canonical map E_{n}^{hU_{i}G} → holim_{Δ} (Γ._{G}E_{n}^{hU_{i}})^{G}. The G_{n}-equivariant map E_{n}^{hU_{i}} → F_{n} induces the map

holim_{Δ} (Γ._{G}E_{n}^{hU_{i}})^{G} → holim_{Δ} (Γ._{G}F_{n})^{G},

and thus, there are maps

colim_{i} E_{n}^{hU_{i}G} → colim_{i} holim_{Δ} (Γ._{G}E_{n}^{hU_{i}})^{G} → holim_{Δ} (Γ._{G}F_{n})^{G}.

This composition defines the map colim_{i} E_{n}^{hU_{i}G} → holim_{Δ} (Γ._{G}F_{n})^{G}.

Definition 7.8. Since there is a map

(holim_{Δ} (Γ.F_{n})^{G}) ∧ M_{I} → holim_{Δ} (Γ.(F_{n} ∧ M_{I}))^{G},

we define ϕ_{I,G}: (colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I} → (F_{n} ∧ M_{I})^{hG} to be the composition

(colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I} → holim_{Δ} (Γ._{G}(F_{n} ∧ M_{I}))^{G} → holim_{Δ} (Γ._{G}(F_{n} ∧ M_{I})_{f})^{G}

Thus, the map ψ_{G}: holim_{I} ((colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I}) → E_{n}^{h′G} is defined to be holim_{I} ϕ_{I,G}.

The following sequence of results shows that ψ_{G} is a weak equivalence for all G <_{c} G_{n}. We first do this for G open, then for G closed. Though we will prove this again in Chapter 10 with a stronger theorem, we include these proofs because the technique is simple and useful.

Lemma 7.9. For any open subgroup U of G_{n}, the map

ϕ_{I,U}: (colim_{i} E_{n}^{hU_{i}U}) ∧ M_{I} → (F_{n} ∧ M_{I})^{hU}

is a weak equivalence. Also, there is an isomorphism in the stable category:

E_{n}^{hU} ∧ M_{I} ≅ (F_{n} ∧ M_{I})^{hU}.

Proof. By [9, (3.13), Prop. 4.16], there is a conditionally convergent (actually, strongly convergent) spectral sequence with

E_{2}^{s,t} ≅ H_{c}^{s}(U; E_{n}^{−t}(Λ𝕋V (DM_{I})_{c})) ≅ π^{s}([Λ𝕋V (DM_{I})_{c}, C_{G_{n}/U}^{∗}]_{M_{S}}^{−t}

and abutment

[Λ𝕋V (DM_{I})_{c}, Tot(∏^{∗}C._{G_{n}/U})]_{M_{S}}^{−t+s} ≅ [DM_{I}, E_{n}^{hU}]_{Spt}^{−t+s} ≅ π_{t−s}(E_{n}^{hU} ∧ M_{I})
≅ π_{t−s}((colim_{i} E_{n}^{hU_{i}U}) ∧ M_{I}),

by Remark 7.7. Also, by 6.19, there is a conditionally convergent spectral sequence

H_{c}^{s}(U; π_{t}(F_{n} ∧ M_{I})) ≅ π^{s}(π_{t}((Γ_{U}^{∗}(F_{n} ∧ M_{I})_{f})^{U})) ⟹ π_{t−s}((F_{n} ∧ M_{I})^{hU}).

Since

E_{n}^{−t}(Λ𝕋V (DM_{I})_{c}) = [Λ𝕋V (DM_{I})_{c}, F(G_{n})]_{M_{S}}^{−t} ≅ [DM_{I}, E_{n}]_{Spt}^{−t} ≅ π_{t}(E_{n} ∧ M_{I}),

there is an isomorphism between the E_{2}-terms of the spectral sequences that is compatible with the map ϕ_{I,U} between the abutments. Thus, by [1, Thm 7.2], comparison of spectral sequences gives an isomorphism of abutments. □

Corollary 7.10. For any open subgroup U of G_{n}, there is a weak equivalence

ψ_{U}: holim_{I} ((colim_{i} E_{n}^{hU_{i}U}) ∧ M_{I}) → E_{n}^{h′U},

and an isomorphism E_{n}^{hU} ≅ E_{n}^{h′U} in the stable category.

Proof. This follows from the preceding lemma by changing π_{∗}(holim (−)) into \underleftarrow{lim} π_{∗}(−) by a \underleftarrow{lim}^{1} argument. □

Theorem 7.11. Let G be closed in G_{n}. Then

ϕ_{I,G}: (colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I} ⟶ (F_{n} ∧ M_{I})^{hG}

and ψ_{G}: holim_{I} ((colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I}) ⟶ E_{n}^{h′G} are weak equivalences. Also, there are isomorphisms E_{n}^{hG} ∧ M_{I} ≅ (F_{n} ∧ M_{I})^{hG} and E_{n}^{hG} ≅ E_{n}^{h′G} in the stable category.

Proof. As above, it suffices to verify the first weak equivalence, which we do by a comparison of spectral sequences. There is a conditionally convergent descent spectral sequence

H_{c}^{s}(G; π_{t}(F_{n} ∧ M_{I})) ⟹ π_{t−s}((F_{n} ∧ M_{I})^{hG}).

Let C._{G_{n}/G} be the cosimplicial S-module with C_{G_{n}/G}^{j} = L_{K(n)}(E_{n}^{hG} ∧_{M_{S}} E_{n}^{j+1}) [9, Rk. AI.9]. Then, by remarks from the proof of Lemma 4.11, pg. 35, and Prop. AI.5 in [9], there is a conditionally convergent spectral sequence with E_{2}-term

H_{c}^{s}(G; π_{t}(E_{n} ∧_{M_{S}} M_{I})) ≅ π^{s}([DM_{I}, C_{G_{n}/G}^{∗}]_{M_{S}}^{−t}

and abutment

[DM_{I}, Tot(∏^{∗}C._{G_{n}/G})]_{M_{S}}^{−t+s} ≅ π_{t−s}(E_{n}^{hG} ∧_{M_{S}} M_{I}) ≅ colim_{i} π_{t−s}(E_{n}^{hU_{i}G} ∧_{M_{S}} M_{I})
≅ π_{t−s}((colim_{i} E_{n}^{hU_{i}G}) ∧_{Spt} M_{I}).

The proof is finished by the same argument used for Lemma 7.9. □

We want to show that E_{n}^{h′G} can be viewed as the K(n)_{∗}-localization of F_{n}^{hG}. To do this, we give some results about commuting finite spectra with holim that will also be helpful later.

If J is a small category and {X_{j}} is a diagram in (S_{∗})^{J}, then for any i ∈ J, the functor {i} → J induces a canonical map holim_{J} X_{j} → holim_{{i}} X_{i} ≅ X_{i}, where the holim is taken in S. If K is any pointed simplicial set, then there is a canonical map

(holim_{J} X_{j}) ∧ K → lim_{J} (X_{j} ∧ K) → holim_{J} (X_{j} ∧ K).

Thus, if {X_{j}} is a diagram in Spt^{J} and Y is any spectrum, there is a canonical map

(holim_{J} X_{j}) ∧ Y → holim_{J} (X_{j} ∧ Y).

The following result gives conditions for when smashing with a finite spectrum commutes with homotopy limits (see also [34, pg. 251], [49, pg. 96]).

Lemma 7.12. Let J be a small category and let {X_{j}} be a J-shaped diagram of spectra. Let Y be any finite spectrum, and given a spectrum Z, let Z → Z_{f} be a weak equivalence with Z_{f} a fibrant spectrum. Then there is a weak equivalence

(holim (X_{j})_{f}) ∧ Y → holim ((X_{j})_{f} ∧ Y)_{f}.

Furthermore, if J is the category \{0 ← 1 ← 2 ← ⋯ \} and if \underleftarrow{lim}^{1} π_{∗}(X_{j}) = 0 = \underleftarrow{lim}^{1} π_{∗}(X_{j} ∧ Y), then the canonical map (holim X_{j}) ∧ Y → holim (X_{j} ∧ Y) is a weak equivalence.

Remark 7.13. The last statement of the lemma is used several times in Chapter 10 and Corollary 8.10. In Lemma 8.18, we verify the \underleftarrow{lim}^{1} hypotheses for the cases we need.

Proof. Setting Z equal to DY and S^{0} in 6.11 successively gives the spectral sequences

\underleftarrow{lim}^{s} _{j} π_{t}(X_{j} ∧ Y) ⟹ π_{t−s}((holim (X_{j})_{f}) ∧ Y)

and

\underleftarrow{lim}^{s} _{j} π_{t}(X_{j} ∧ Y) ⟹ π_{t−s}(holim ((X_{j})_{f} ∧ Y)_{f}),

respectively.

Comparison of these two spectral sequences, using the map

(holim (X_{j})_{f}) ∧ Y → holim ((X_{j})_{f} ∧ Y) → holim ((X_{j})_{f} ∧ Y)_{f},

implies that the desired map is a weak equivalence ([1, Thm. 7.2], [34, pf. of Prop. 3.3]).

The last statement of the theorem, given the hypotheses, follows from

π_{t}(holim (X_{j} ∧ Y)) ≅ \underleftarrow{lim} π_{t}(X_{j} ∧ Y) ≅ \underleftarrow{lim} π_{t}(((X_{j})_{f} ∧ Y)_{f}) ≅ π_{t}(holim ((X_{j})_{f} ∧ Y)_{f}),

and the fact that π_{t}(holim X_{j}) ≅ \underleftarrow{lim} π_{t}((X_{j})_{f}) ≅ π_{t}(holim (X_{j})_{f}). □

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

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

Consider the isomorphism ((Γ._{G}X_{f})∧Y)^{k} ≅ colim_{N} ((∏_{(G/N)^{k+1}} X_{f})∧Y) and the canonical weak equivalence

colim_{N} ((∏_{(G/N)^{k+1}} X_{f}) ∧ Y) → colim_{N} ∏_{(G/N)^{k+1}} (X_{f} ∧ Y) ≅ (Γ._{G}(X_{f} ∧ Y))^{k}.

Putting these maps together gives a G-equivariant map of cosimplicial spectra

(Γ.X_{f}) ∧ Y → Γ.(X_{f} ∧ Y).

Thus, there is a canonical map X^{hG} ∧ Y → (X_{f} ∧ Y)^{hG} defined by

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

A version of the following result appears in [34, Prop. 3.10].

Lemma 7.14. Suppose G has finite virtual cohomological dimension, X is a discrete G-spectrum, and Y is a finite spectrum with trivial G-action. Then the canonical maps

X^{hG} ∧ Y ⟶ (X_{f} ∧ Y)^{hG} ⟵ (X ∧ Y)^{hG}

are weak equivalences, where X → X_{f}, in Spt_{G}, is a trivial cofibration with X_{f} fibrant.

Remark 7.15. Because there is no canonical G_{n}-equivariant map (F_{n})_{f} ∧ M_{I} → (F_{n} ∧M_{I})_{f}, as explained in Remark 9.2.7, there is no map F_{n}^{hG} ∧M_{I} → (F_{n} ∧M_{I})^{hG} that can be used to define a map holim_{I} (F_{n}^{hG} ∧ M_{I}) → holim_{I} (F_{n} ∧ M_{I})^{hG}. Thus, we do not concern ourselves in the theorem with the existence of a map X^{hG} ∧ Y → (X ∧ Y)^{hG}.

Proof. By 6.19, there is a conditionally convergent spectral sequence

H_{c}^{s}(G; π_{t}(X ∧ Y)) ⟹ π_{t−s}((X_{f} ∧ Y)^{hG}).

Also, by 6.11 with Z = DY, there is a conditionally convergent spectral sequence

H_{c}^{s}(G; π_{t}(X∧Y)) ≅ H^{s}[π_{t}((Γ_{G}^{∗}(X_{f}∧Y))^{G})] ≅ H^{s}([DY, (Γ_{G}^{∗}X_{f})^{G}]^{−t})⟹ π_{t−s}(X^{hG}∧Y).

The map X^{hG} ∧ Y → (X_{f} ∧ Y)^{hG}, comparison of spectral sequences, and [1, Thm. 7.2] give the desired conclusion. □

Corollary 7.16. For any G <_{c} G_{n}, there is an isomorphism E_{n}^{h′G} ≅ L_{K(n)}(F_{n}^{hG}) in the stable category.

Proof. This follows from a \underleftarrow{lim}^{1} argument applied to π_{∗}(holim (−)) using the fact that π_{∗}(F_{n}^{hG} ∧ M_{I}) ≅ π_{∗}((F_{n} ∧ M_{I})^{hG}) ≅ π_{∗}(E_{n}^{hG} ∧ M_{I}) is finite. □

The next result is an example of homotopy fixed points commuting with a non-finite spectrum, which is not true in general.

Lemma 7.17. For any G <_{c} G_{n}, the spectra F_{n}^{hG} ∧ F_{n} ∧ M_{I} and (F_{n} ∧ F_{n} ∧ M_{I})^{hG} have the same homotopy type. (The second F_{n} in each expression is regarded as a trivial G_{n}-spectrum.)

Proof. By [9, proof of Prop. 5.3], π_{∗}(F_{n}^{hG} ∧ F_{n} ∧ M_{I}) ≅ π_{∗}(E_{n}^{hG} ∧ E_{n} ∧ M_{I}) ≅ Map_{c}^{ℓ}(G_{n}, E_{n∗}/I)^{G}. Also, there is the descent spectral sequence

H_{c}^{s}(G; π_{t}(F_{n} ∧ F_{n} ∧ M_{I})) ⟹ π_{t−s}((F_{n} ∧ F_{n} ∧ M_{I})^{hG}),

and because π_{∗}(F_{n} ∧ F_{n} ∧ M_{I}) ≅ Map_{c}^{ℓ}(G_{n}, E_{n∗}/I) ≅ Map_{c}(G_{n}, E_{n∗}/I) is G-acyclic, by 8.16, π_{∗}((F_{n} ∧ F_{n} ∧ M_{I})^{hG}) ≅ Map_{c}^{ℓ}(G_{n}, E_{n∗}/I)^{G}. □

Theorem 7.18. If G <_{c} G_{n} and X is any finite spectrum of type n, then there are isomorphisms (F_{n} ∧ X)^{hG} ≅ F_{n}^{hG} ∧ X ≅ E_{n}^{h′G} ∧ X ≅ E_{n}^{hG} ∧ X in Ho(Spt).

Proof. We have F_{n}^{hG} ∧ X ≅ L_{n}(F_{n}^{hG}) ∧ X ≅ L_{K(n)}(F_{n}^{hG}) ∧ X ≅ E_{n}^{h′G} ∧ X. □

Corollary 7.19. If G <_{c} G_{n}, then (F_{n} ∧M_{I})^{hG} ≅ colim_{i} (F_{n} ∧M_{I})^{hU_{i}G} in the stable category.

Proof. This follows from (F_{n}∧M_{I})^{hG} ≅ colim_{i} (E_{n}^{hU_{i}G}∧M_{I}) ≅ colim_{i} (F_{n}∧M_{I})^{hU_{i}G}. □

Recall that if G ◁_{c} G_{n} does not have finite index, then G_{n}/G ≅ \underleftarrow{lim} _{i} G_{n}/U_{i}G is profinite and uncountable.

Corollary 7.20. If the normal subgroup G is closed in G_{n}, then (F_{n} ∧ M_{I})^{hG} is a discrete G_{n}/G-spectrum. Thus, E_{n}^{hG} is a continuous G_{n}/G-spectrum.

Proof. Note that (F_{n}∧M_{I})^{hG} ≃ colim_{i} (E_{n}^{hU_{i}G}∧M_{I}). Since U_{i}G is normal and has finite index in G_{n}, E_{n}^{hU_{i}G} ∧ M_{I} is automatically a discrete G_{n}/U_{i}G-spectrum. The canonical continuous map G_{n}/G → G_{n}/U_{i}G makes E_{n}^{hU_{i}G} ∧ M_{I} a discrete G_{n}/G-spectrum. Since the category of discrete G_{n}/G-spectra is equivalent to the category of sheaves of spectra on the site G_{n}/G−Sets_{df}, it is easy to see that Spt_{G_{n}/G} is closed under colimits, so that colim_{i} (E_{n}^{hU_{i}G} ∧ M_{I}) is a discrete G_{n}/G-spectrum. □

Example 7.21. At the prime 2, G_{1} = ℤ_{2} × ℤ/2 and E_{1}^{hℤ/2} ≃ KO_{2}^{∧} is a continuous ℤ_{2}-spectrum [19, pg. 101].

CHAPTER 8 Auxiliary results for building the descent spectral sequence

This chapter collects together various results about continuous cohomology, towers of abelian groups, and inverse limits that are required for the work of later chapters.

Definition 8.1. Let DMod(G)^{ℕ} denote the category of diagrams in discrete Gmodules of the form ⋯ → M_{2} → M_{1} → M_{0}. Define H_{cont}^{s}(G; {M_{i}}), the continuous cohomology of G with coefficients in the tower {M_{i}}, to be the sth right derived functor of the left exact functor

\underleftarrow{lim} _{i} (−)^{G}: DMod(G)^{ℕ} → Ab, \{M_{i}\} ↦ \underleftarrow{lim} _{i} M_{i}^{G}.

(This version of continuous cohomology is developed in [24].)

The next two results are special cases of statements in [12, Lem. 3.1.3] and its proof, for the site G−Sets_{df}. Recall that lim^{s} _{Δ×ℕ} is the sth right derived functor of lim_{Δ×ℕ}: Ab^{Δ×ℕ} → Ab.

Lemma 8.2. If I = {I_{i}} is an injective object in DMod(G)^{ℕ}, then, for s > 0,

lim^{s} _{Δ×ℕ} (Γ._{G}I_{i})^{G} = 0.

Proof. Let (j, i) be a typical element of Δ × ℕ. The functors \underleftarrow{lim} _{i}: Ab^{ℕ} → Ab and

lim_{Δ} : Ab^{Δ×ℕ} → Ab^{ℕ}, {A_{j,i}}_{j,i} ↦ {lim_{Δ} A_{j,i}}_{i}

are left exact functors. Also, lim_{Δ} is right adjoint to the diagonal functor c: Ab^{ℕ} → Ab^{Δ×N} that sends {A_{i}} in Ab^{ℕ} to the diagram with A_{j,i} = A_{i}, for every j ∈ Δ. Since c is exact, lim_{Δ} preserves injectives. Thus, there is a Grothendieck spectral sequence

E_{2}^{s,t} = \underleftarrow{lim}^{s} _{i} (lim^{t} _{Δ} (Γ._{G}I_{i})^{G}) ⟹ lim^{s+t} _{Δ×ℕ} (Γ._{G}I_{i})^{G}.

Since I is an injective object, each I_{i} is injective in DMod(G), and every map d_{i}: I_{i+1} → I_{i} is split surjective, with section r_{i} [24, Prop. 1.1]. This implies that d_{i}: I_{i+1}^{G} → I_{i}^{G} is surjective for each i: if m ∈ I_{i}^{G}, then m = d_{i}(r_{i}(m)), and g · r_{i}(m) = r_{i}(g · m) = r_{i}(m), for all g ∈ G, so that r_{i}(m) ∈ I_{i+1}^{G}. Thus, {I_{i}^{G}} is a Mittag-Leffler tower of epimorphisms.

Then lim^{t} _{Δ} (Γ._{G}I_{i})^{G} ≅ H^{t}((Γ_{G}^{∗}I_{i})^{G}) ≅ H_{c}^{t}(G; I_{i}), which equals 0 whenever t > 0, and is I_{i}^{G} when t = 0. Thus, for t > 0, E_{2}^{s,t} = 0, so that, for s > 0, lim^{s} _{Δ×ℕ} (Γ._{G}I_{i})^{G} ≅ E_{2}^{s,0} = \underleftarrow{lim}^{s} _{i} I_{i}^{G} = 0. □

Theorem 8.3. Let {M_{i}} be in DMod(G)^{ℕ}. Then for all s ≥ 0,

lim^{s} _{Δ×ℕ} (Γ._{G}M_{i})^{G} ≅ H_{cont}^{s}(G; {M_{i}}).

Proof. By definition, H_{cont}^{s}(G; \{M_{i}\}) ≅ H^{s}(\underleftarrow{lim} _{i} (−)^{G}(I^{0} → I^{1} → ⋯)), where 0 → {M_{i}} → I^{0} → I^{1} → ⋯ is an injective resolution in DMod(G)^{ℕ}. If {N_{i}} is in DMod(G)^{ℕ}, then

lim_{Δ×ℕ} (Γ._{G}N_{i})^{G} ≅ \underleftarrow{lim} _{i} lim_{Δ} (Γ._{G}N_{i})^{G} ≅ \underleftarrow{lim} _{i} H^{0}((Γ_{G}^{∗}N_{i})^{G})
≅ \underleftarrow{lim} _{i} H_{c}^{0}(G; N_{i}) ≅ \underleftarrow{lim} _{i} N_{i}^{G}.

Thus, H_{cont}^{s}(G; {M_{i}}) ≅ H^{s}(lim_{Δ×ℕ}((Γ._{G}I^{0})^{G} → (Γ._{G}I^{1})^{G} → ⋯)), which is associated to the sequence

(8.4)  0 → (Γ.{M_{i}})^{G} → (Γ.I^{0})^{G} → (Γ.I^{1})^{G} → ⋯,

in Ab^{Δ×ℕ}. Since Ab^{Δ×ℕ} is an abelian category with enough injectives, and the functor lim_{Δ×ℕ}: Ab^{Δ×ℕ} → Ab is left exact, if 8.4 is a (lim_{Δ×ℕ})-acyclic resolution of (Γ.{M_{i}})^{G}, then H_{cont}^{s}(G; {M_{i}}) ≅ lim^{s} _{Δ×ℕ} (Γ.M_{i})^{G}.

Fix any ([j], k) ∈ Δ × ℕ and consider the sequence

(8.5)  0 → (Γ^{j+1}M_{k})^{G} → (Γ^{j+1}I_{k}^{0})^{G} → (Γ^{j+1}I_{k}^{1})^{G} → · · · .

Since 0 → M_{k} → I_{k}^{0} → I_{k}^{1} → ⋯ is exact in DMod(G),

(8.6)  0 → Γ^{j+1}M_{k} → Γ^{j+1}I_{k}^{0} → Γ^{j+1}I_{k}^{1} → ⋯

is exact. Since H_{c}^{s}(G; Γ^{j+1}I_{k}^{m}) = 0, for any m ≥ 0 and s > 0, 8.6 is a resolution of Γ^{j+1}M_{k} by (−)^{G}-acyclics, and thus, H^{s}((Γ^{j+1}I_{k}^{0})^{G} → (Γ^{j+1}I_{k}^{1})^{G} → ⋯) ≅ H_{c}^{s}(G; Γ^{j+1}M_{k}) = 0, for s > 0. Since (−)^{G} is left exact, it follows that 8.5 is an exact sequence. Thus, 8.4 is an exact sequence in Ab^{Δ×ℕ}.

The proof is finished by applying Lemma 8.2. □ The following lemma is used, for example, with the exact functors Γ_{G}^{k}: DMod(G) →

DMod(G), for any k ≥ 1.

Lemma 8.7. Let ⋯ ⟶^{f_{2}} A_{2} ⟶^{f_{1}} A_{1} ⟶^{f_{0}} A_{0} be a tower of abelian groups, and let F: Ab → Ab be an additive exact functor. If {A_{i}} satisfies the Mittag-Leffler condition, then so does {F (A_{i})}.

Proof. For each i ≥ 0, there exists some j > i, such that, for all k ≥ j, im(f_{k}^{i}) = im(f_{j}^{i}), where f_{k}^{i}: A_{k} → A_{i}. (The definition of Mittag-Leffler provides for “j ≥ i,” but this implies our statement, and “j > i” simplifies the argument.) We use f̄_{k}^{i} to denote the obvious map A_{k} → im(f_{k}^{i}). Since F is exact, there is an isomorphism h: im(F (f_{k}^{i})) → im(F (f_{j}^{i})) given by the composite im(F (f_{k}^{i})) →^{≅} F (im(f_{k}^{i})) = F (im(f_{j}^{i})) →^{≅} im(F (f_{j}^{i})). If f ∈ F (A_{k}), h is defined by F (f_{k}^{i})(f) ↦ F (f̄_{k}^{i})(f) = F (f̄_{j}^{i})(f′) ↦ F (f_{j}^{i})(f′), where F (f̄_{k}^{i})(f) = F (f̄_{j}^{i})(f′) for some f′ ∈ F (A_{j}). Note that im(F (f_{k}^{i})) ⊂ im(F (f_{j}^{i})). Consider the commutative diagram

\xymatrix{A_{k} \ar[r]_{f_{k}^{j}} \ar[rr]^{f̄_{k}^{i}} & A_{j} \ar[r]^{p} \ar[rr]_{f̄_{j}^{i}} & im(f_{k}^{i}) \ar[r]^{i_{1}} & im(f_{j}^{i}) \ar[r]^{i_{2}} & A_{i},}

where i_{1} and i_{2} are inclusions, i_{1} is the identity map in this case, and i_{2}i_{1}pf_{k}^{j} = f_{k}^{i}. Functoriality gives the commutative diagram

\xymatrix{F (A_{k}) \ar[r]_{F (f_{k}^{j})} \ar[rr]^{F (f̄_{k}^{i})} & F (A_{j}) \ar[r]^{F (p)} \ar[rr]_{F (f̄_{j}^{i})} & F (im(f_{k}^{i})) \ar[r]^{F (i_{1})} & F (im(f_{j}^{i})) \ar[r]^{F (i_{2})} & F (A_{i}),}

where F (i_{1}) is the identity. Therefore,

h(F (f_{k}^{i})(f)) = F (f_{j}^{i})(f′) = F (i_{2})(F (f̄_{j}^{i})(f′)) = F (i_{2})(F (i_{1})(F (f̄_{k}^{i})(f))) = F (f_{k}^{i})(f).

Thus, h is the inclusion map and im(F (f_{k}^{i})) = im(F (f_{j}^{i})). This proves that {F (A_{i})} is Mittag-Leffler. □

Lemma 8.8. Let M = \underleftarrow{lim} _{α} M_{α} be an inverse limit of topological abelian groups. Let H be any profinite group. Then the canonical map κ: Map_{c}(H, M) → \underleftarrow{lim}_{α} Map_{c}(H, M_{α}) is an isomorphism of groups.

Proof. Inverse limits in Ab and in topological spaces are created in Sets: Map_{c}(H, M) = Top(H, \underleftarrow{lim} _{α} M_{α}) ≅ \underleftarrow{lim} _{Sets} _{α} Top(H, M_{α}) ≅ \underleftarrow{lim} _{Ab} _{α} Map_{c}(H, M_{α}). □

Lemma 8.9. If X is a finite spectrum, then for any G <_{c} G_{n} and any integer t, the abelian group π_{t}(E_{n}^{hG} ∧ M_{I} ∧ X) is finite. In particular, π_{t}(E_{n} ∧ M_{I} ∧ X) is finite.

Proof. We use the fact that π_{t}(E_{n}^{hG} ∧ M_{I}) is finite, and for convenience, we work in the stable homotopy category of CW-spectra. Since X is a finite spectrum, there exists some m such that X_{n} = Σ^{n−m}X_{m} whenever n ≥ m, and X_{m} is a finite complex. Since π_{t}(E_{n} ∧ M_{I} ∧ X) ≅ π_{t−m}(E_{n} ∧ M_{I} ∧ X_{m}) and X_{m} can be built out of a finite number of cofiber sequences, the result follows. □

Corollary 8.10 ([19, pg. 116]). If X is a finite spectrum, then π_{t}(E_{n} ∧ X) ≅ \underleftarrow{lim} _{I} π_{t}(E_{n} ∧ M_{I} ∧ X).

Proof. Since π_{t}(E_{n} ∧ X) ≅ π_{t}((holim_{I} E_{n} ∧ M_{I}) ∧ X) ≅ π_{t}(holim_{I} (E_{n} ∧ M_{I} ∧ X)), the result follows from the preceding lemma. □

We need to be precise about the definition of continuous cohomology that we use. We let H_{cts}^{s}(G; M) denote the cohomology of continuous cochains for a profinite group G with coefficients in the topological G-module M, in the sense of [47, §2]. If M is a discrete G-module, this is the usual continuous (Galois) cohomology H_{c}^{s}(G; M) of [43, §2.2]. Also, for an inverse system of discrete G-modules, there is the following result.

Theorem 8.11 ([24, (2.1), Thm. 2.2]). Let {M_{n}}_{n≥0} be an inverse system of discrete G-modules satisfying the Mittag-Leffler condition and let M = \underleftarrow{lim} _{n} M_{n} as a topological G-module. Then, for each s ≥ 0, there is a short exact sequence

0 → \underleftarrow{lim}^{1} _{n} H_{c}^{s−1}(G; M_{n}) → H_{cts}^{s}(G; M) → \underleftarrow{lim} _{n} H_{c}^{s}(G; M_{n}) → 0,

where H_{c}^{−1}(G; −) = 0.

Corollary 8.12. If the profinite group G is a compact p-adic analytic group and \{M_{n}\}_{n≥0} is an inverse system of finite discrete ℤ_{p}[[G]]-modules, with M = \underleftarrow{lim} _{n} M_{n}, a topological G-module, then H_{cts}^{s}(G; M) ≅ \underleftarrow{lim} _{n} H_{c}^{s}(G; M_{n}), for s ≥ 0.

Proof. When s = 0, the statement is immediate. For s ≥ 1, it suffices to prove that for t ≥ 0, H_{c}^{t}(G, N) is a finite abelian group whenever N is a finite discrete ℤ_{p}[[G]]-module.

Since G is a compact p-adic analytic group, [46, Thm. 5.1.2] implies that the trivial ℤ_{p}[[G]]-module ℤ_{p} (that is, ℤ_{p} has trivial action by G) has a resolution by free objects of the form ⋯ → P_{2} → P_{1} → P_{0} → ℤ_{p} → 0, in the category ℭ_{p}(G), whose objects are topological ℤ_{p}[[G]]-modules that are inverse limits of finite discrete left ℤ_{p}[[G]]-modules, and whose morphisms are continuous ℤ_{p}[[G]]-module homomorphisms. Furthermore, we can assume that each P_{i} is topologically finitely generated as a ℤ_{p}[[G]]-module. Then [46, (3.2.4)] implies that

H_{c}^{s}(G; N) = Ext_{ℤ_{p}[[G]]}^{s}(ℤ_{p}, N) = H^{s}(Hom_{ℤ_{p}[[G]]}^{c}(P_{0}, N) → Hom_{ℤ_{p}[[G]]}^{c}(P_{1}, N) → ...),

where Hom_{ℤ_{p}[[G]]}^{c}(−, −) denotes continuous ℤ_{p}[[G]]-module homomorphisms.

Let P_{i} be topologically generated by the finite subset {p_{i1}, ..., p_{ik}}. This set generates the submodule ⨁ _{j} ℤ_{p}[[G]]·p_{ij}. Since this submodule is closed in P_{i}, by [50, Lem. 7.2.2], P_{i} = ⨁ _{j} ℤ_{p}[[G]] · p_{ij}. Thus, each P_{i} is isomorphic to a finite number of copies of ℤ_{p}[[G]]. Then any ℤ_{p}[[G]]-module homomorphism from P_{i} to N is determined by a finite number of values and since N is finite, Hom_{ℤ_{p}[[G]]}^{c}(P_{i}, N) is a finite abelian group, so that H_{c}^{s}(G; N) is itself finite. □

Corollary 8.13. If G is closed in G_{n} and X is a finite spectrum, then

H_{cts}^{s}(G; π_{t}(E_{n} ∧ X)) ≅ \underleftarrow{lim} _{k} H_{c}^{s}(G; E_{n,t}(X)/I_{n}^{k}), s ≥ 0,

where I_{n}^{k} is the submodule I_{n}^{k}E_{n,t}(X).

Proof. Note that G is a compact p-adic analytic group, and since X is a finite spectrum, E_{n,t}(X) ≅ \underleftarrow{lim} _{k} E_{n,t}(X)/I_{n}^{k} [5, pg. 767]. Also, π_{t}(E_{n} ∧ X) ≅ \underleftarrow{lim} _{I} π_{t}(E_{n} ∧ M_{I} ∧ X), and the profinite and I_{n}-adic topologies on π_{t}(E_{n} ∧ X) are identical [23, pf. of Prop. 11.9]. Since I_{n}^{k} is an open subgroup, E_{n,t}(X)/I_{n}^{k} is a finite discrete G-module.

By the previous result, our proof is finished by showing that E_{n,t}(X)/I_{n}^{k} is a discrete ℤ_{p}[[G]]-module. By [4, Def. 5.8], E_{n,t}(X)/I_{n}^{k} is a discrete twisted E_{n,0}-G_{n}-module such that g · (rm) = (g · r)(g · m), for g ∈ G_{n}, r ∈ E_{n,0}, and m ∈ E_{n,t}(X)/I_{n}^{k}. Since ℤ_{p} ⊂ E_{n,0}, if the G_{n}-action on ℤ_{p} is trivial, then E_{n,t}(X)/I_{n}^{k} is a discrete ℤ_{p}[[G]]-module [40, Prop. 5.3.6(d)].

Gal acts trivially on ℤ_{p}, and since S_{n} acts on E_{n,t}(X) by W (𝔽_{p^{n}})-module homomorphisms, the S_{n}-action on ℤ_{p} ⊂ W (𝔽_{p^{n}}) is trivial. Thus, G_{n} acts trivially on ℤ_{p}, and E_{n,t}(X)/I_{n}^{k} is a finite discrete ℤ_{p}[[G]]-module. □

Remark 8.14. This result shows that H_{cts}^{s}(G; π_{t}(E_{n} ∧X)) agrees with the definition of continuous cohomology used in [9, Remark 0.3].

The next result extends the conclusion of Remark 6.18 to pro-discrete G-modules, and is useful in the proof of Theorem 10.6.

Theorem 8.15. If M = \underleftarrow{lim} _{α} M_{α} is, as a topological G-module, an inverse limit of discrete G-modules, then H_{cts}^{s}(G; M) ≅ π^{s}((Γ._{G}M)^{G}), for s ≥ 0.

Proof. We have:

H_{cts}^{s}(G; M) = H^{s}(\underleftarrow{lim} M_{α} → Map_{c}(G, \underleftarrow{lim} M_{α}) → Map_{c}(G^{2}, \underleftarrow{lim} M_{α}) → ⋯)
≅ H^{s}(\underleftarrow{lim} (M_{α} → Map_{c}(G, M_{α}) → Map_{c}(G^{2}, M_{α}) → ⋯))
≅ H^{s}(\underleftarrow{lim} (Γ^{∗}M_{α})^{G}) ≅ π^{s}(\underleftarrow{lim} (Γ.M_{α})^{G}) ≅ π^{s}((Γ.M)^{G}).
□

The proof of the following result closely follows [37, pp. 30-1].

Theorem 8.16. Let K be a closed subgroup of a profinite group H. Let A = \underleftarrow{lim} _{α} A_{α}, as a topological abelian group, be an inverse limit of discrete abelian groups. Then H_{cts}^{s}(K; Map_{c}(H, A)) = 0, for s > 0.

Proof. The space H/K is profinite and there is a K-equivariant homeomorphism H → K × H/K, where K × H/K has the diagonal action. Thus, there is an isomorphism Map_{c}(H, A) ≅ Map_{c}(K, Map_{c}(H/K, A)) of topological K-modules.

As described in [37, pg. 106], for any topological K-module M, H_{cts}^{∗}(K; M) can be computed by taking the cohomology of the cochain complex obtained by taking the K-fixed points of the acyclic complex X(K, M) given by

Map_{c}(K, M) ⟶^{∂^{1}} Map_{c}(K^{2}, M) ⟶^{∂^{2}} Map_{c}(K^{3}, M) ⟶^{∂^{3}} ⋯,

where Map_{c}(K^{n}, M) has a K-action given by (k · f)(k_{1}, ..., k_{n}) = k · f(k^{−1}k_{1}, ..., k^{−1}k_{n}), and ∂^{n} is defined by ∂^{n}(f)(k_{1}, ..., k_{n+1}) = ∑_{i=1}^{n+1} (−1)^{i+1}f(k_{1}, ..., k̂_{i}, ..., k_{n+1}), where k̂_{i} means that the ith entry is omitted. Note that this acyclic complex is defined when M is just a topological abelian group.

Let M ≅ \underleftarrow{lim} _{β} M_{β} be a topological abelian group, where each M_{β} is a discrete abelian group. Consider the map δ: Map_{c}(K^{n+1}, Map_{c}(K, M))^{K} → Map_{c}(K^{n+1}, M), defined by f ↦ [(k_{1}, ..., k_{n+1}) ↦^{f̂} f(k_{1}, ..., k_{n+1})(1)], where f̂ is continuous, since δ is induced by the similarly-defined map Map_{c}(K^{n+1}, Map_{c}(K, M_{β}))^{K} → Map_{c}(K^{n+1}, M_{β}). Because δ is an isomorphism that commutes with ∂^{n+1}, there is an isomorphism of cochain complexes X(K, Map_{c}(K, M))^{K} ≅ X(K, M). Thus,

H_{cts}^{s}(K; Map_{c}(H, A)) ≅ H^{s}(X(K, Map_{c}(K, Map_{c}(H/K, A)))^{K})
≅ H^{s}(X(K, Map_{c}(H/K, A))) = 0,

for s > 0, since X(K, Map_{c}(H/K, A)) is acyclic. □ The following useful result is Proposition 2.2 in [23].

Theorem 8.17. Given a spectrum Z, define Λ(Z) to be the filtered category of pairs (Y, u), where Y is a finite spectrum and u: Y → Z. The morphisms are the obvious commutative triangles and all maps are in the stable category. Then, for any integer t, the canonical map E_{n}^{t}(Z) → \underleftarrow{lim} _{(Y,u)∈Λ(Z)} E_{n}^{t}(Y) is an isomorphism of abelian groups and a homeomorphism of profinite spaces.

In Lemma 7.12, we showed that when certain \underleftarrow{lim}^{1} terms vanish, smashing with a finite spectrum commutes with holim for a tower of spectra. As mentioned in Remark 7.13, we now verify that the appropriate \underleftarrow{lim}^{1} terms vanish for the cases where Lemma 7.12 is applied in Corollary 8.10 and Chapter 10. All of these cases follow from the lemma below, by: setting G and/or K equal to {e}, letting X = S^{0}, using the weak equivalence E_{n}^{hG} ∧ M_{I} ≃ (colim_{i} E_{n}^{hU_{i}G}) ∧ M_{I}, and/or by applying the fact that the Spanier-Whitehead dual of a finite spectrum is again finite.

Lemma 8.18. Let G and K be closed subgroups of G_{n}, and let X be a finite spectrum. When j ≥ 0 and m is equal to 0 or 1, then the tower of abelian groups

{π_{t}((colim_{N} ∏_{G^{j}/N} X) ∧ E_{n}^{hK} ∧ E_{n}^{m} ∧ M_{I})}_{I}

is Mittag-Leffler for every integer t, where we write G^{j} ≅ \underleftarrow{lim} _{N◁_{o}G^{j}} G^{j}/N. We let G^{0} = {e} and E_{n}^{0} = S^{0}.

Proof. Observe that π_{t}((colim_{N} ∏_{G^{j}/N} X) ∧ E_{n}^{hK} ∧ E_{n}^{m} ∧ M_{I}) is isomorphic to the expression colim_{N} ∏_{G^{j}/N} π_{t}(E_{n}^{hK} ∧M_{I} ∧E_{n}^{m}∧X). Since filtered colimits and taking finite products are exact functors, it is enough to show that {π_{t}(E_{n}^{hK} ∧M_{I} ∧E_{n}^{m}∧X)} is Mittag-Leffler. Since π_{t}(E_{n}^{hK} ∧M_{I} ∧E_{n}^{m}∧X) ≅ colim_{i} π_{t}(E_{n}^{hU_{i}K} ∧E_{n}^{m}∧M_{I} ∧X), we only need to show that {π_{t}(E_{n}^{hU_{i}K} ∧E_{n}^{m}∧M_{I} ∧X)} is Mittag-Leffler. If m = 0, then we are done, since π_{t}(E_{n}^{hU_{i}K} ∧ M_{I} ∧ X) is a finite abelian group. When m = 1, π_{t}(E_{n}^{hU_{i}K} ∧ E_{n} ∧ M_{I} ∧ X) ≅ π_{t}(L_{K(n)}(E_{n}^{hU_{i}K} ∧ E_{n}) ∧ M_{I} ∧ X). By [9, Cor. 4.5], L_{K(n)}(E_{n}^{hU_{i}K} ∧ E_{n}) ≃ ∏_{G_{n}/U_{i}K} E_{n}. Thus, π_{t}(L_{K(n)}(E_{n}^{hU_{i}K} ∧ E_{n}) ∧ M_{I} ∧ X) ≅ ∏_{G_{n}/U_{i}K} π_{t}(E_{n} ∧ M_{I} ∧ X), which gives a Mittag-Leffler tower since π_{t}(E_{n} ∧ M_{I} ∧ X) is a finite abelian group. □

CHAPTER 9 Homotopy fixed points for L_{K(n)}(E_{n} ∧ X) and its descent

spectral sequence

Let {Z_{i}} be a tower of discrete G-spectra, where G has finite virtual cohomological dimension. By Lemma 4.4.5, if each Z_{i} is fibrant in Spt_{G} or if {π_{t}(Z_{i})} is Mittag-Leffler for every t, then Z = holim_{i} Z_{i} is a continuous G-spectrum. In this chapter, we construct a descent spectral sequence for such continuous G-spectra (e.g. holim_{I} (F_{n}∧M_{I} ∧X)_{f,G}), and use it to obtain the descent spectral sequence for the homotopy fixed points of L_{K(n)}(E_{n} ∧ X), when X satisfies a particular finiteness condition.

9.1. Towers of spectra and homotopy fixed points Theorem 9.1.1. Let G be a profinite group and let {Z_{i}} be any tower in tow(Spt_{G}).

Then there is a conditionally convergent spectral sequence

(9.1.2)  E_{2}^{s,t} ≅ H_{cont}^{s}(G; {π_{t}(Z_{i})}) ⟹ π_{t−s}(holim_{i} holim_{Δ} (Γ._{G}((Z_{i})_{f,G}))^{G}).

Remark 9.1.3. This theorem is a special case of [12, Prop. 3.1.2] for G−Sets_{df}. Also related to 9.1.2 is the ℓ-adic descent spectral sequence of algebraic K-theory ([48], [34, pg. 266]). For a “nice scheme” U, the K-theory presheaf K, and ℍ._{ét}(U; K)_{ℓ}^{∧} = holim_{n} ℍ._{ét}(U; K) ∧ M(ℓ^{n}), this spectral sequence has the form

H_{ét}^{s}(U; ℤ_{ℓ}(t/2)) ⟹ π_{t−s}(ℍ._{ét}(U; K)_{ℓ}^{∧}).

Proof. Note that holim_{i} holim_{Δ} (Γ._{G}(Z_{i})_{f})^{G} ≅ holim_{Δ×ℕ} (Γ._{G}(Z_{i})_{f})^{G}, and (Γ._{G}(Z_{i})_{f})^{G} is a diagram of fibrant spectra. Then by Remark 6.9, there is a conditionally convergent spectral sequence

E_{2}^{s,t} = lim^{s} _{Δ×ℕ} π_{t}((Γ._{G}(Z_{i})_{f})^{G}) ⟹ π_{t−s}(holim_{i} holim_{Δ} (Γ._{G}(Z_{i})_{f})^{G}).

We only need to identify the E_{2}-term:

E_{2}^{s,t} = lim^{s} _{Δ×ℕ} π_{t}((Γ._{G}(Z_{i})_{f})^{G}) ≅ lim^{s} _{Δ×ℕ} (Γ._{G}π_{t}(Z_{i}))^{G} ≅ H_{cont}^{s}(G; {π_{t}(Z_{i})}),

by Lemma 8.3. □ Now we focus on two types of towers that give continuous G-spectra.

Definition 9.1.4. Let {Z_{i}} be in tow(Spt_{G}). Suppose that one of the following conditions is satisfied: (a) Z_{i} is fibrant in Spt_{G} for every i; or (b) {π_{t}(Z_{i})} is Mittag-Leffler for every integer t. Then we let Z = holim_{i} Z_{i} be the continuous G-spectrum associated to the tower. Also, we define the homotopy fixed point spectrum Z^{hG} = holim_{i} Z_{i}^{hG}.

The following result shows that when G is a finite group and the tower {Z_{i}} satisfies one of the conditions given in the definition, Z^{hG} agrees with the usual homotopy fixed points Z^{h′G} (see Remark 6.3), so that Definition 9.1.4 generalizes the notion of homotopy fixed points for a particular kind of continuous G-spectrum.

Theorem 9.1.5. Let {Z_{i}} be as in Definition 9.1.4. Suppose that G is a finite group. Then the canonical map Z^{hG} → Z^{h′G} is a weak equivalence.

Proof. Note that if each Z_{i} is fibrant in Spt_{G}, we can let (Z_{i})_{f} = Z_{i}. The map is defined in the following way:

Z^{hG} = holim_{i} lim_{G} (Z_{i})_{f} → holim_{i} holim_{G} (Z_{i})_{f} ≅ holim_{G} (holim_{i} (Z_{i})_{f}) = Z^{h′G}.

By hypothesis, Z = holim_{i} Z_{i} → holim_{i} (Z_{i})_{f} is a weak equivalence and a G-equivariant map, and holim_{i} (Z_{i})_{f} is fibrant in Spt. Thus, Z^{h′G} is indeed the target of the above map. For a fixed i, the map Hom_{G}(∗, (Z_{i})_{f}) ≅ lim_{G} (Z_{i})_{f} → holim_{G} (Z_{i})_{f} is a weak equivalence between fibrant spectra, so that the above map is a weak equivalence. □

By applying Remark 6.16 and Theorem 9.1.1, we obtain the following.

Corollary 9.1.6. Let G be a profinite group with vcd(G) < ∞, and let {Z_{i}} be an object in tow(Spt_{G}) that satisfies one of the conditions in Definition 9.1.4. Let Z = holim_{i} Z_{i} be a continuous G-spectrum and let Z^{hG} = holim_{i} Z_{i}^{hG} be its homotopy fixed point spectrum. Then there is a conditionally convergent descent spectral sequence

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

Remark 9.1.8. As in Remark 7.2, there is a weak equivalence between the two possible interpretations of Z^{hG}: holim_{i} ((Z_{i})_{f})^{G} ⟶^{≃} holim_{i} holim_{Δ} (Γ_{G}^{·}(Z_{i})_{f})^{G}.

In case (b) of Definition 9.1.4, we can simplify the E_{2}-term of spectral sequences 9.1.2 and 9.1.7.

Theorem 9.1.9. Let {Z_{i}} be a tower in tow(Spt_{G}) such that {π_{t}(Z_{i})} is Mittag-Leffler for every integer t, so that Z = holim_{i} Z_{i} is a continuous G-spectrum. Then in spectral sequences 9.1.2 and 9.1.7, E_{2}^{s,t} ≅ H_{cts}^{s}(G; π_{t}(Z)), the cohomology of continuous cochains.

Proof. By [24, Thm. 2.2], H_{cont}^{s}(G; \{π_{t}(Z_{i})\}) ≅ H_{cts}^{s}(G; \underleftarrow{lim} _{i} π_{t}(Z_{i})) ≅ H_{cts}^{s}(G; π_{t}(Z)).

□

Because of the finite vcd hypothesis in Corollary 9.1.6, spectral sequence 9.1.7 has abutment π_{∗}(holim_{i} Z_{i}^{hG}). We point out that the functor

holim_{i} (−)^{hG}: Ho(tow(Spt_{G})) → Ho(Spt), {Z_{i}} ↦ holim_{i} Z_{i}^{hG}

is the total right derived functor R( \underleftarrow{lim} _{i} (−)^{G}), in the following sense, generalizing the homotopy fixed points X^{hG} = (R(−)^{G})(X), for a discrete G-spectrum X.

Lemma 9.1.10. Let {Z_{i}} be a tower of discrete G-spectra. Let {(Z_{i})_{f}} → {(Z_{i})′_{f}} be a trivial cofibration with {(Z_{i})′_{f}} fibrant in tow(Spt_{G}). Then

holim_{i} ((Z_{i})_{f})^{G} ⟶^{≃} holim_{i} ((Z_{i})′_{f})^{G} ⟵^{≃} \underleftarrow{lim} _{i} ((Z_{i})′_{f})^{G} = R(\underleftarrow{lim} _{i} (−)^{G})(\{Z_{i}\}),

where holim_{i} Z_{i}^{hG} = holim_{i} ((Z_{i})_{f})^{G}.

Proof. The first weak equivalence follows from the fact that for each i, ((Z_{i})_{f})^{G} → ((Z_{i})′_{f})^{G} is a weak equivalence between fibrant spectra. The second weak equivalence is verified as in the proof of 4.4.5, since {((Z_{i})′_{f})^{G}} is a tower in Spt of fibrations between fibrant spectra. Finally, the equality comes from the fact that the composition {Z_{i}} → {(Z_{i})′_{f}} is a trivial cofibration in tow(Spt_{G}). □

Remark 9.1.11. By the above observation, we can rewrite spectral sequence 9.1.7 in a more conceptual way:

R^{s}(\underleftarrow{lim} _{i} (−)^{G})\{π_{t}(Z_{i})\} ⟹ π_{t−s}(R(\underleftarrow{lim} _{i} (−)^{G})(\{Z_{i}\})).
9.2. The continuous G_{n}-spectrum L_{K(n)}(E_{n} ∧ X)

We apply the results of the previous section to continuous G-spectra associated with E_{n}, where G is a closed subgroup of G_{n}. Corollary 9.1.6 immediately gives

Theorem 9.2.1. Let G be a closed subgroup of G_{n} and let X be an arbitrary spectrum with trivial G-action. Then holim_{I} (F_{n} ∧M_{I} ∧X)_{f} is a continuous G-spectrum and there is a conditionally convergent descent spectral sequence

H_{cont}^{s}(G; {π_{t}(E_{n} ∧ M_{I} ∧ X)}) ⟹ π_{t−s}(holim_{I} (F_{n} ∧ M_{I} ∧ X)^{hG}).

Remark 9.2.2. For arbitrary X, we do not know that holim_{I} (F_{n} ∧ M_{I} ∧ X) is a continuous G_{n}-spectrum, so that, in general, holim_{I} (F_{n} ∧ M_{I} ∧ X)^{hG} is the homotopy fixed point spectrum (holim_{I} (F_{n} ∧ M_{I} ∧ X)_{f})^{hG} and not (holim_{I} (F_{n} ∧ M_{I} ∧ X))^{hG}.

Let X be a spectrum with trivial G-action. When X satisfies a certain condition, we define the homotopy fixed point spectrum L_{K(n)}(E_{n} ∧ X)^{hG} and construct its descent spectral sequence.

Definition 9.2.3. If the tower {π_{t}(E_{n} ∧ M_{I} ∧ X)}_{I} is Mittag-Leffler for all t ∈ ℤ, then we say that the spectrum X is E_{n}-Mittag-Leffler.

Remark 9.2.4. An arbitrary finite spectrum X is E_{n}-Mittag-Leffler, since the tower {π_{t}(E_{n} ∧ M_{I} ∧ X)}_{I} consists of finite abelian groups, by Lemma 8.9. However, an E_{n}- Mittag-Leffler spectrum need not be finite. For example, for j ≥ 1, let X = E_{n}^{j}. Then

π_{t}(E_{n} ∧ M_{I} ∧ X) ≅ π_{t}(E_{n}^{j+1} ∧ M_{I}) ≅ π_{t}(L_{K(n)}(E_{n}^{j+1}) ∧ M_{I}) ≅ Map_{c}^{ℓ}(G_{n}^{j}, π_{t}(E_{n})/I),

by [9, pg. 10]. Since {π_{t}(E_{n})/I}_{I} is Mittag-Leffler, 8.7 implies that E_{n}^{j} is E_{n}-Mittag-Leffler.

Let X be E_{n}-Mittag-Leffler. Then there is an isomorphism L_{K(n)}(E_{n} ∧ X) ≅ holim_{I} (E_{n} ∧ M_{I} ∧ X) in Ho(Spt), and a weak equivalence holim_{I} (F_{n} ∧ M_{I} ∧ X) → holim_{I} (E_{n} ∧ M_{I} ∧ X), since π_{t}(holim_{I} (E_{n} ∧ M_{I} ∧ X)) ≅ \underleftarrow{lim} _{I} π_{t}(E_{n} ∧ M_{I} ∧ X) ≅ π_{t}(holim_{I} (F_{n} ∧ M_{I} ∧ X)). Similarly, if X is a finite spectrum, E_{n} ∧ X ≃ L_{K(n)}(E_{n} ∧ X), by 8.10. Since F_{n} ∧ M_{I} ∧ X is a discrete G_{n}-spectrum, we have

Theorem 9.2.5. If X is an E_{n}-Mittag-Leffler spectrum with trivial G-action, then L_{K(n)}(E_{n} ∧ X) ≅ holim_{I} (F_{n} ∧ M_{I} ∧ X) is a continuous G-spectrum. If X is a finite spectrum, then E_{n} ∧ X is a continuous G-spectrum.

Definition 9.2.6. If X is an E_{n}-Mittag-Leffler spectrum, then

(L_{K(n)}(E_{n} ∧ X))^{hG} = holim_{I} ((F_{n} ∧ M_{I})_{f} ∧ X)^{hG}.

We write L_{K(n)}(E_{n}∧X)^{hG}, in place of (L_{K(n)}(E_{n}∧X))^{hG}, to avoid excessive parentheses. If X is a finite spectrum, then we can write

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

in place of L_{K(n)}(E_{n} ∧ X)^{hG}. As usual, F_{n} ∧ M_{I} → (F_{n} ∧ M_{I})_{f} is a trivial cofibration with (F_{n} ∧ M_{I})_{f} fibrant in Spt_{G}.

Remark 9.2.7. We explain why we use ((F_{n} ∧ M_{I})_{f} ∧ X)^{hG}, in the above definition, instead of (F_{n}∧M_{I}∧X)^{hG}. We want to be able to define the map E_{n}^{h′G}∧X → (E_{n}∧X)^{h′G}.

If we use (F_{n} ∧ M_{I} ∧ X)^{hG}, then this map requires a map of diagrams

{(F_{n} ∧ M_{I})_{f} ∧ X}_{I} ⟶ {(F_{n} ∧ M_{I} ∧ X)_{f}}_{I}.

This requires, for example, a commutative diagram

\xymatrix{(F_{n} ∧ M_{I})_{f} ∧ X \ar[r] \ar[d] & (F_{n} ∧ M_{I} ∧ X)_{f} \ar[d] \\ (F_{n} ∧ M_{J})_{f} ∧ X \ar[r] & (F_{n} ∧ M_{J} ∧ X)_{f}.}

However, there is no natural map (F_{n} ∧ M_{I})_{f} ∧ X → (F_{n} ∧ M_{I} ∧ X)_{f}, and thus, no such commutative diagram. Thus, we cannot use (F_{n} ∧M_{I} ∧X)^{hG} above and have the desired map. But since (F_{n} ∧ M_{I} ∧ X)^{hG} → ((F_{n} ∧ M_{I})_{f} ∧ X)^{hG} is a weak equivalence between fibrant spectra, the two definitions are equivalent.

Note that by definition, L_{K(n)}(E_{n} ∧ X)^{hG} ≅ holim_{Δ} holim_{I} (Γ._{G}((F_{n} ∧ M_{I})_{f} ∧ X)_{f})^{G}.

Theorem 9.2.8. Let G be a closed subgroup of G_{n}. If X is an E_{n}-Mittag-Leffler spectrum with trivial G-action, then there is a conditionally convergent descent spectral sequence

(9.2.9)  H_{cts}^{s}(G; π_{t}(L_{K(n)}(E_{n} ∧ X))) ⟹ π_{t−s}(L_{K(n)}(E_{n} ∧ X)^{hG}).

In particular, if X is a finite spectrum, then there is a descent spectral sequence

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

Proof. This follows from Corollary 9.1.6 and Theorem 9.1.9 by considering the tower {(F_{n}∧M_{I})_{f} ∧X}_{I}. The G-equivariant isomorphism π_{t}((F_{n}∧M_{I})_{f} ∧X) ≅ π_{t}(F_{n}∧M_{I} ∧X) gives the desired E_{2}-term. When X is finite, 8.13 showed that H_{cts}^{s}(G; π_{t}(E_{n} ∧ X)) ≅ H_{c}^{s}(G; π_{t}(E_{n} ∧ X)). □

The above descent spectral sequences for L_{K(n)}(E_{n} ∧ X)^{hG} and (E_{n} ∧ X)^{hG} can be obtained in a way that is slightly different from the above. This second method of derivation of the spectral sequence is required for Theorem 10.6, because of the spectral sequence comparison argument used in its proof. Thus, we give the needed alternative proof. Proof of Theorem 9.2.8. Consider the cosimplicial fibrant spectrum

C._{X} = holim_{I} (Γ._{G}((F_{n} ∧ M_{I})_{f} ∧ X)_{f})^{G}.

Let Z_{I} = (F_{n} ∧ M_{I})_{f} ∧ X and note that L_{K(n)}(E_{n} ∧ X)^{hG} ≅ holim_{Δ} holim_{I} (Γ._{G}(Z_{I})_{f})^{G}. Then spectral sequence 6.10 takes the form

E_{2}^{s,t} = π^{s}π_{t}(holim_{I} (Γ._{G}(Z_{I})_{f})^{G}) ⟹ π_{t−s}(L_{K(n)}(E_{n} ∧ X)^{hG}).

Note that, for k ≥ 0, \underleftarrow{lim}^{1} _{I} Map_{c}(G^{k}, π_{t+1}(Z_{I})) = 0, since \{π_{t+1}(Z_{I})\}_{I} is Mittag-Leffler and Map_{c}(G^{k}, −) is an exact functor. Therefore,

π_{t}(holim_{I} Map_{c}(G^{k}, (Z_{I})_{f})) ≅ \underleftarrow{lim} _{I} Map_{c}(G^{k}, π_{t}(Z_{I})).

Thus, π_{t}(holim_{I} (Map_{c}(G^{k+1}, (Z_{I})_{f}))^{G}) ≅ \underleftarrow{lim} _{I} Map_{c}(G^{k+1}, π_{t}(Z_{I}))^{G}, and

π_{t}(holim_{I} (Γ.(Z_{I})_{f})^{G}) ≅ \underleftarrow{lim} _{I} (Γ.π_{t}(Z_{I}))^{G}.

Now we identify the E_{2} term:

E_{2}^{s,t} ≅ π^{s}(\underleftarrow{lim} _{I} (Γ.π_{t}(Z_{I}))^{G}) ≅ π^{s}(lim_{G} \underleftarrow{lim} _{I} Γ.π_{t}(Z_{I})) ≅ π^{s}((Γ.(\underleftarrow{lim} _{I} π_{t}(Z_{I})))^{G})
≅ π^{s}((Γ.π_{t}(L_{K(n)}(E_{n} ∧ X)))^{G}) ≅ H_{cts}^{s}(G; π_{t}(L_{K(n)}(E_{n} ∧ X))),

by applying 8.15 and the fact that, since \underleftarrow{lim}^{1} _{I} π_{t+1}(Z_{I}) = 0, π_{t}(L_{K(n)}(E_{n} ∧ X)) ≅ \underleftarrow{lim} _{I} π_{t}(Z_{I}) is an inverse limit of discrete G_{n}-modules. □

Remark 9.2.11. In the above proof, note that E_{2}^{s,t} ≅ H^{s}(\underleftarrow{lim} _{I} (−)^{G} \{Γ^{∗}π_{t}(Z_{I})\}_{I}). One can also show that E_{2}^{s,t} ≅ H_{cts}^{s}(G; π_{t}(L_{K(n)}(E_{n} ∧ X))) by showing that the exact sequence

0 → {π_{t}(Z_{I})} → {Γ^{∗}π_{t}(Z_{I})} → ⋯

in DMod(G)^{ℕ} is a (\underleftarrow{lim} _{I} (−)^{G})-acyclic resolution of \{π_{t}(Z_{I})\}, by applying [24, Thm. 2.2].

Remark 9.2.12. For j ≥ 1, consider L_{K(n)}(E_{n} ∧ E_{n}^{j})^{hG}, where G acts nontrivially only on the leftmost factor. By [9, 3.19, 3.24], π_{t}(L_{K(n)}(E_{n}^{j+1})) ≅ Map_{c}^{ℓ}(G_{n}^{j}, π_{t}(E_{n})) ≅ Map_{c}^{ℓ}(G_{n}, Map_{c}(G_{n}^{j−1}, π_{t}(E_{n}))) ≅ Map_{c}(G_{n}, Map_{c}(G_{n}^{j−1}, π_{t}(E_{n}))). Thus, in 9.2.9, E_{2}^{s,t} ≅ H_{cts}^{s}(G; π_{t}(L_{K(n)}(E_{n}^{j+1}))) vanishes for s > 0 by 8.16, and equals Map_{c}(G_{n}^{j}, π_{t}(E_{n}))^{G} when s = 0. Therefore, when G = G_{n}, π_{∗}(L_{K(n)}(E_{n} ∧ E_{n}^{j})^{hG_{n}}) ≅ Map_{c}(G_{n}^{j−1}, π_{∗}(E_{n})), and hence, L_{K(n)}(E_{n} ∧ E_{n}^{j})^{hG_{n}} ≅ L_{K(n)}(E_{n}^{j}), in the stable category, although we do not concern ourselves with defining the map that induces the given isomorphism. This generalizes the formula E_{n}^{h′G_{n}} ≅ L_{K(n)}(S^{0}).

CHAPTER 10 The descent spectral sequence is an Adams spectral

sequence

Let X be a finite spectrum. We compare (E_{n} ∧ X)^{hG} with the spectrum E_{n}^{hG} ∧ X of Devinatz and Hopkins, and show that, in this case, the descent spectral sequence is strongly convergent. First of all, we make explicit the relationship between these two homotopy fixed point spectra.

We start by noting that there is an isomorphism

E_{n}^{hG} ∧ X ≅ (holim_{I} ((colim_{i}E_{n}^{hU_{i}G}) ∧ M_{I})) ∧ X

in Ho(Spt). Consider the canonical map E_{n}^{h′G} ∧ X → (E_{n} ∧ X)^{hG} defined by

(holim_{I} (F_{n} ∧ M_{I})^{hG}) ∧ X → holim_{I} ((F_{n} ∧ M_{I})^{hG} ∧ X) → holim_{I} ((F_{n} ∧ M_{I})_{f} ∧ X)^{hG}.

Definition 10.1. The map ψ̂: (holim_{I} ((colim_{i}E_{n}^{hU_{i}G}) ∧ M_{I})) ∧ X → (E_{n} ∧ X)^{hG} is defined to be the composition (holim_{I} ((colim_{i}E_{n}^{hU_{i}G}) ∧ M_{I})) ∧ X → E_{n}^{h′G} ∧ X → (E_{n} ∧ X)^{hG}.

To go further, we need some results from [9, AI]. We begin by briefly discussing the K(n)_{∗}-local E_{n}-Adams spectral sequence. For more details, the interested reader is referred to [9, AI] and [32].

Definition 10.2. Let X be a K(n)_{∗}-local spectrum and suppose that the sequence ∗ → X → I^{0} → I^{1} → ⋯, in Ho(Spt), is E_{n}-exact with each I^{s} a K(n)_{∗}-local E_{n}- injective spectrum. Then this sequence is said to be a K(n)_{∗}-local E_{n}-resolution of X; such a resolution always exists. Then the K(n)_{∗}-local E_{n}-Adams spectral sequence is the conditionally and strongly convergent spectral sequence E_{1}^{s,t} = [Z, I^{s}]^{t} ⇒ [Z, X]^{t+s}, where Z in Spt is equivalent to a CW-spectrum [9, Prop. AI.3].

Theorem 10.3 ([9, Prop. AI.5]). Let X be a K(n)_{∗}-local spectrum and let C. be a cosimplicial fibrant spectrum in Spt with an augmentation X → C. such that

(10.4)  ∗ → X → C^{0} → C^{1} → C^{2} → ⋯

is a K(n)_{∗}-local E_{n}-resolution of X. Then for any spectrum Z (equivalent to a CWspectrum), the K(n)_{∗}-local E_{n}-Adams spectral sequence

E_{2}^{s,t} ≅ H^{s}([Z, C^{∗}]^{t}) ⟹ [Z, X]^{t+s}

is isomorphic to the Bousfield-Kan spectral sequence E_{r}^{∗,∗}(Z, C.) given by

E_{2}^{s,t}(Z, C.) ≅ H^{s}([Z, C^{∗}]^{t}) ⟹ [Z, holim_{Δ} C.]^{t+s}.

Corollary 10.5 ([9, Cor. AI.8]). Let X and C. be as in Theorem 10.3. Then the spectral sequence E_{r}^{∗,−∗}(S^{0}, C.) of the form

E_{2}^{s,−t} = π^{s}π_{t}(C.) ⟹ π_{t−s}(holim_{Δ} C.)

is strongly convergent and is isomorphic to the K(n)_{∗}-local E_{n}-Adams spectral sequence with abutment π_{∗}(X). Thus, X → holim_{Δ} C. is a weak equivalence.

We use these two results to obtain

Theorem 10.6. Let X be a finite spectrum. Then descent spectral sequence 9.2.10 for (E_{n} ∧ X)^{hG} is strongly convergent and is isomorphic to the K(n)_{∗}-local E_{n}-Adams spectral sequence with abutment π_{∗}(E_{n}^{hG} ∧ X). Furthermore, the map

ψ̂: (holim_{I} ((colim_{i}E_{n}^{hU_{i}G}) ∧ M_{I})) ∧ X ⟶ (E_{n} ∧ X)^{hG}

is a weak equivalence and E_{n}^{hG} ∧ X ≅ (E_{n} ∧ X)^{hG} in Ho(Spt).

Proof. Let W = (holim_{I} ((colim_{i}E_{n}^{hU_{i}G}) ∧ M_{I})) ∧ X. Since

W ≃ holim_{I} (((colim_{i}E_{n}^{hU_{i}G}) ∧ M_{I}) ∧ X) ≃ L_{K(n)}((colim_{i}E_{n}^{hU_{i}G}) ∧ X),

W is K(n)_{∗}-local. We write C. for the cosimplicial spectrum

C._{X} = holim_{I} (Γ._{G}((F_{n} ∧ M_{I})_{f} ∧ X)_{f})^{G},

when doing so causes no confusion. The canonical map Tot(∏^{∗}C.) → Tot_{0}(∏^{∗}C.) ≅ ∏_{j} C^{j} → C^{0} gives an augmentation α: W → C. defined by the map W → (E_{n}∧X)^{hG} ≅ holim_{Δ} C._{X} = Tot(∏^{∗}C.) → C^{0}.

Consider the sequence

(10.7)  ∗ → W →^{α} C^{0} → C^{1} → C^{2} → · · · .

We claim that this sequence is a K(n)_{∗}-local E_{n}-resolution of W.

Suppose the claim is true. Then Theorems 10.3 and 10.5 give the conclusions of the theorem, and we only need to verify the claim.

First of all, we show that each C^{i} is E_{n}-injective. Note that by 8.7, the tower {π_{t}(Map_{c}(G^{i}, F_{n} ∧ M_{I} ∧ X))}_{I} is Mittag-Leffler. Then

C^{i} ≅ holim_{I} Map_{c}(G^{i}, ((F_{n} ∧ M_{I})_{f} ∧ X)_{f}) ≃ holim_{I} colim_{N} ∏_{G^{i}/N} (F_{n} ∧ M_{I} ∧ X)

≃ holim_{I} ((colim_{N} ∏_{G^{i}/N} X) ∧ E_{n} ∧ M_{I}) ≃ L_{K(n)}((colim_{N} ∏_{G^{i}/N} X) ∧ E_{n}),

so that C^{i} is K(n)_{∗}-local and a retract of L_{K(n)}((colim_{N} ∏_{G^{i}/N} X) ∧ E_{n}). Thus, C^{i} is K(n)_{∗}-local E_{n}-injective.

It only remains to show that 10.7 is E_{n}-exact. As in [9, pp. 37-38], it suffices to show that

(10.8) 0 → [Z, L_{K(n)}(W ∧ E_{n})]^{t} → [Z, L_{K(n)}(C^{0} ∧ E_{n})]^{t} → [Z, L_{K(n)}(C^{1} ∧ E_{n})]^{t} → ⋯

is exact for any Z equivalent to a CW-spectrum.

Note that L_{K(n)}(C^{i} ∧E_{n}) ≃ holim_{I} ((colim_{N} ∏_{G^{i}/N} X)∧E_{n}^{2} ∧M_{I}), and since X is a finite spectrum, this is weakly equivalent to (holim_{I} (colim_{N} ∏_{G^{i}/N} S^{0})∧E_{n}^{2}∧M_{I})∧X ≃ L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n}) ∧ X. Thus, [Z, L_{K(n)}(C^{i} ∧ E_{n})]^{t} ≅ [Z ∧ DX, L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})]^{t}.

Recall the category Λ = Λ(Z ∧ DX) described in Theorem 8.17. Let

τ_{i}: [Z ∧ DX, L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})]^{t} → \underleftarrow{lim}_{(Y,u)∈Λ} [Y, L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})]^{t}

be the canonical map, where each Y is a finite spectrum. Since τ_{i} is a natural map and cosimplicial abelian groups give associated cochain complexes,

τ: [Z ∧ DX, L_{K(n)}(C_{S^{0}}^{∗} ∧ E_{n})]^{t} → \underleftarrow{lim}_{(Y,u)∈Λ} [Y, L_{K(n)}(C_{S^{0}}^{∗} ∧ E_{n})]^{t}

is a map of cochain complexes. Note that

[Y, L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})]^{t} ≅ π_{−t}(holim_{I} ((colim_{N} ∏_{G^{i}/N} S^{0}) ∧ E_{n}^{2} ∧ M_{I} ∧ DY)).

Also, since π_{−t}(E_{n}^{2} ∧ M_{I} ∧ DY) ≅ π_{−t}(F_{n}^{2} ∧ M_{I} ∧ DY) is a discrete G_{n}-module,

π_{−t}((colim_{N} ∏_{G^{i}/N} S^{0}) ∧ E_{n}^{2} ∧ M_{I} ∧ DY) ≅ Γ_{G}^{i}(Map_{c}^{ℓ}(G_{n}, π_{−t}(E_{n} ∧ M_{I} ∧ DY)))

≅ Γ_{G}^{i}(Map_{c}(G_{n}, π_{−t}(E_{n} ∧ M_{I} ∧ DY))).

By 8.9 and 8.7, \underleftarrow{lim}^{1} _{I} Γ_{G}^{i}(Map_{c}(G_{n}, π_{−t+1}(E_{n} ∧ M_{I} ∧ DY))) = 0. This implies that π_{−t}(holim_{I} ((colim_{N} ∏_{G^{i}/N} S^{0})∧E_{n}^{2} ∧M_{I} ∧DY)) ≅ Γ_{G}^{i}Map_{c}(G_{n}, π_{−t}(E_{n} ∧DY)). Thus, there is a natural isomorphism

[Y, L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})]^{t} ≅ Map_{c}(G^{i}, Map_{c}(G_{n}, π_{−t}(E_{n} ∧ DY))).

Therefore, τ is the map of cochain complexes

[Z ∧ DX, L_{K(n)}(C_{S^{0}}^{∗} ∧ E_{n})]^{t} → Map_{c}(G^{∗}, Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX))),

and Map_{c}(G^{i}, Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX))) ≅ Map_{c}(G^{i} × G_{n}, E_{n}^{t}(Z ∧ DX)). Now consider the canonical map (obtained as above)

τ̂: [V, L_{K(n)}(C_{S^{0}}^{∗} ∧ E_{n})]^{t} → Map_{c}(G^{∗} × G_{n}, E_{n}^{t}(V))

of cochain complexes, for an arbitrary spectrum V (equivalent to a CW-spectrum). Since L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})^{t}(−) and Map_{c}(G^{i} × G_{n}, E_{n}^{t}(−)) are cohomology theories satisfying the product axiom, to show that τ̂ is an isomorphism for all V, it suffices to check this for V = S^{0} [9, pg. 25]. Setting V = S^{0} yields

[S^{0}, L_{K(n)}(C_{S^{0}}^{i} ∧ E_{n})]^{t} ≅ Map_{c}(G^{i}, Map_{c}(G_{n}, π_{−t}(E_{n}))),

showing that τ̂ is an isomorphism for any V. In particular, V = Z ∧ DX implies that τ is an isomorphism of cochain complexes for any Z, and

[Z, L_{K(n)}(C^{i} ∧E_{n})]^{t} ≅ [Z ∧DX, L_{K(n)}(C_{S^{0}}^{i} ∧E_{n})]^{t} ≅ Map_{c}(G^{i}, Map_{c}(G_{n}, E_{n}^{t}(Z ∧DX))).

Therefore, since π_{−t}(C^{∗}) gives continuous cochains, the sequence [Z, L_{K(n)}(C^{∗} ∧ E_{n})]^{t} from 10.8 is the cochain complex of continuous cochains

Γ_{G_{n}}(E_{n}^{t}(Z ∧ DX)) → Map_{c}(G, Γ_{G_{n}}E_{n}^{t}(Z ∧ DX)) → Map_{c}(G^{2}, Γ_{G_{n}}E_{n}^{t}(Z ∧ DX)) → · · · .

Thus, since Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX)) ≅ \underleftarrow{lim} _{(Y,u),I} Map_{c}(G_{n}, π_{−t}(E_{n} ∧ M_{I} ∧ DY)) is an inverse limit of discrete G_{n}-modules, 8.15 implies that

H^{s}([Z, L_{K(n)}(C_{X}^{∗} ∧ E_{n})]^{t}) ≅ H_{cts}^{s}(G; Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX))),

which vanishes for s > 0 by 8.16, and equals Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX))^{G}, when s = 0. Now consider the portion of 10.8

(10.9)  0 → [Z, L_{K(n)}(W ∧ E_{n})]^{t} → [Z, L_{K(n)}(C^{0} ∧ E_{n})]^{t}

that is induced by α. This can be rewritten as

0 → [Z ∧ DX, L_{K(n)}(E_{n}^{hG} ∧ E_{n})]^{t} → [Z ∧ DX, L_{K(n)}(C_{S^{0}}^{0} ∧ E_{n})]^{t}.

As in [9, proof of Prop. 5.4], the map

π_{∗}L_{K(n)}(E_{n}^{hG} ∧ E_{n}) → π_{∗}L_{K(n)}(C_{S^{0}}^{0} ∧ E_{n}) ≅ π_{∗}L_{K(n)}(E_{n} ∧ E_{n})

is an injection with image Map_{c}(G_{n}, E_{n∗})^{G} ⊂Map_{c}(G_{n}, E_{n∗}). Therefore, as in [9, pp. 37-39], 10.9 is an injection with image Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX))^{G} ⊂Map_{c}(G_{n}, E_{n}^{t}(Z ∧ DX)). This completes the proof that 10.8 is an exact sequence for any Z. □

The following corollary shows that the K(n)_{∗}-localization of any finite spectrum is a homotopy fixed point spectrum associated to E_{n}.

Corollary 10.10. If X is a finite spectrum, then L_{K(n)}(X) ≅ (E_{n} ∧ X)^{hG_{n}}, in the stable category.

Proof. We have: (E_{n} ∧ X)^{hG_{n}} ≅ E_{n}^{hG_{n}} ∧ X ≅ L_{K(n)}(S^{0}) ∧ X ≅ L_{K(n)}(X). □ Putting these results together gives

Corollary 10.11. Let X be a finite spectrum. Using the continuous G_{n}-action on E_{n} ∧ X, there is a strongly convergent descent spectral sequence

H_{c}^{s}(G_{n}; π_{t}(E_{n} ∧ X)) ⟹ π_{t−s}(L_{K(n)}X)

that is isomorphic to the K(n)_{∗}-local E_{n}-Adams spectral sequence with π_{∗}(E_{n}^{hG_{n}} ∧ X) as abutment.

This Corollary shows that spectral sequence 1.1 can indeed be realized as a descent spectral sequence for E_{n} ∧ X with actual homotopy fixed point spectrum L_{K(n)}(X).

One can also view the E(n)_{∗}-localization of any finite type n complex as a homotopy fixed point spectrum:

Theorem 10.12. If X is a finite type n spectrum, then L_{n}X ≅ (F_{n} ∧ X)^{hG_{n}}.

Proof. We have: L_{n}X ≅ L_{n}S^{0} ∧ X ≅ L_{K(n)}(X) ≅ (E_{n} ∧ X)^{hG_{n}} ≅ E_{n}^{hG_{n}} ∧ X ≅ (F_{n} ∧ X)^{hG_{n}}. □

Remark 10.13. Recall that by [9, Prop. 5.7], if X is a CW-spectrum such that, for each E(n)-module spectrum M, there exists a k with I_{n}^{k}M_{∗}(X) = 0, then there is a K(n)_{∗}-local E_{n}-Adams spectral sequence of the form

H_{c}^{∗}(G; π_{∗}(E_{n} ∧ X)) ⟹ π_{∗}(L_{K(n)}(E_{n}^{hG} ∧ X)).

Also, as discussed in Chapter 3, if \underleftarrow{lim} _{I} π_{∗}(E_{n} ∧ M_{I} ∧ X) is finitely generated over E_{n∗}, then there is a spectral sequence

H_{c}^{∗}(S_{n}; \underleftarrow{lim} _{I} π_{∗}(E_{n} ∧ M_{I} ∧ X))^{Gal} ⟹ π_{∗}(L_{K(n)}X).

Our descent spectral sequence was motivated by these earlier spectral sequences. However, we do not know precisely the relationships among the X’s that appear in the three spectral sequences.

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