RATIONAL S^{1}-EQUIVARIANT STABLE HOMOTOPY THEORY.

J.P.C.Greenlees

The author is grateful to the Nuffield Foundation for its support.

Author addresses:

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: j.greenlees@sheffield.ac.uk

Abstract. We make a systematic study of rational S^{1}-equivariant cohomology theories, or rather of their representing objects, rational S^{1}-spectra.

In Part I we construct a complete algebraic model for the homotopy category of S^{1}- spectra, reminiscent of the localization theorem. The model is of homological dimension one, and simple enough to allow practical calculations; in particular we obtain a classification of rational S^{1}-equivariant cohomology theories.

In Part II we identify the algebraic counterparts of all the usual S^{1}-spectra and constructions on S^{1}-spectra. This enables us in Part III to give a rational analysis of a number of interesting phenomena, such as the Atiyah-Hirzebruch spectral sequence, the Segal conjecture, K-theory and topological cyclic cohomology.

Contents

Chapter 0. General Introduction. 1 0.1. Motivation 1 0.2. Overview 2

Part I The algebraic model of rational 𝕋-spectra.

Chapter 1. Introduction to Part I. 13 1.1. Outline of the algebraic models. 13 1.2. Reading Guide. 17 1.3. Haeberly’s example. 18 1.4. McClure’s Chern character isomorphism for F-spaces. 18 Chapter 2. First steps. 23 2.1. Natural cells and basic cells 23 2.2. Separating isotropy types. 26 2.3. The single strand spectra E⟨H⟩. 29 2.4. Operations. 32 Chapter 3. The Adams spectral sequence. 35 3.1. The Adams short exact sequence. 35 3.2. Construction of the Adams resolution. 36 3.3. Convergence of the Adams resolution. 38 3.4. Torsion ℚ[c_{H}]-modules and geometric implications. 39 Chapter 4. Categorical reprocessing. 43 4.1. Recollections about derived categories. 43 4.2. Split linear triangulated categories. 46

iv CONTENTS

4.3. The algebraicization of the category of 𝕋-spectra over H. 51 Chapter 5. Assembly and the standard model. 55 5.1. Assembly. 55 5.2. Global assembly. 57 5.3. The standard model category. 59 5.4. The algebraicization of rational 𝕋-spectra. 63 5.5. Notation. 66 Chapter 6. The torsion model. 71 6.1. Practical calculations. 71 6.2. The torsion model 72 6.3. Equivalence of derived categories of standard and torsion models. 75

Part II Algebraic counterparts to standard constructions.

Chapter 7. Introduction to Part II 81 7.1. General outline. 81 7.2. Modelling of the smash product and the function spectrum. 84 7.3. Modelling functors changing equivariance. 86 7.4. Modelling Eilenberg-MacLane spectra and related objects. 88 7.5. Functors between split triangulated categories. 88 Chapter 8. Basic algebra for models and their derived categories. 91 8.1. Euler classes and F-finite torsion O_{F}-modules. 91 8.2. Products in the torsion model. 94 8.3. The tensor hom adjunction. 97 8.4. Hom, tensor and torsion functors in standard models. 101 8.5. Hom, tensor and torsion functors on derived categories. 105 8.6. Products in the standard model. 107 Chapter 9. Products, smash products and function 𝕋-spectra. 109 9.1. Maps between injective spectra. 109 9.2. Models of smash products. 110 9.3. Models of function spectra. 112 Chapter 10. Induction, coinduction and geometric fixed points. 117 10.1. Forgetful, induction and coinduction functors. 117

CONTENTS v

10.2. Geometric fixed points. 119 Chapter 11. Algebraic inflation and deflation. 125 11.1. Algebraic inflation and Hausdorff O_{F}-modules. 125 11.2. Algebraic inflation and deflation of O_{F}-modules. 128 11.3. Inflation and its right adjoint on the torsion model category. 130 Chapter 12. Inflation, Lewis-May fixed points and quotients. 135 12.1. The topological inflation and Lewis-May fixed point functors. 135 12.2. Correspondence of Algebraic and geometric inflation maps. 139 12.3. A direct approach to the Lewis-May fixed point functor. 141 12.4. Lewis-May fixed points on objects in the standard model. 143 12.5. Quotient functors. 146 Chapter 13. Homotopy Mackey functors and related constructions. 149 13.1. The homotopy Mackey functor on A. 149 13.2. Eilenberg-MacLane spectra. 152 13.3. coMackey functors and spectra representing ordinary homology. 155 13.4. Brown-Comenetz spectra. 157

Part III Applications.

Chapter 14. Introduction to Part III. 163 14.1. General Outline 163 14.2. Prospects and problems. 165 Chapter 15. Classical miscellany. 167 15.1. The collapse of the Atiyah-Hirzebruch spectral sequence. 167 15.2. Orbit category resolutions. 169 15.3. Suspension spectra. 171 15.4. K-theory revisited. 172 15.5. The geometric equivariant rational Segal conjecture for 𝕋. 175 Chapter 16. Cyclic and Tate cohomology. 179 16.1. Cyclic cohomology. 179 16.2. Rational Tate spectra. 180 16.3. The integral 𝕋-equivariant Tate spectrum for complex K-theory. 181 Chapter 17. Cyclotomic spectra and topological cyclic cohomology. 185

vi CONTENTS

17.1. Cyclotomic spectra. 185 17.2. Free loop spaces. 188 17.3. Topological cyclic cohomology of cyclotomic spectra. 190 Appendix A. Mackey functors. 195 Appendix B. Closed model categories. 201 Appendix C. Conventions. 207 C.1. Conventions for spaces and spectra. 207 C.2. Standing conventions. 209 Appendix D. Indices. 211 D.1. Index of definitions and terminology. 211 D.2. Index of notation. 215 Bibliography 225

CHAPTER 0 General Introduction.

0.1. Motivation

Spaces with actions of the circle group 𝕋 are of particular interest. Loops occur in many constructions, and it is often appropriate to take into account the action of the circle by rotation; in particular the free loop space has been the object of much study. This in turn leads towards the use of the circle group in cyclic cohomology; the refinements of topological Hochschild homology and topological cyclic constructions are also important in algebraic K-theory. More prosaically the circle is simply the first infinite compact Lie group, and it plays a fundamental role in the understanding of all positive dimensional groups. For any one of these reasons it is important to understand equivariant cohomology theories for spaces with circle action.

To obtain a reasonably broad and simple picture, we consider the case of rational cohomology theories; these have been considered before for special classes of spaces (see for example [5]), but this appears to be the first attempt to obtain a complete algebraic picture. In any case, the understanding of the rational case is a necessary first step towards a general understanding of 𝕋-equivariant cohomology theories. It is well known [14] that, for finite groups, all cohomology theories are products of ordinary cohomology theories, but this is false for the circle group. A test case of particular interest is rational topological K-theory. The example of J.-P.Haeberly [16] shows that, by contrast with the case of finite groups of equivariance, there is no Chern character isomorphism. It follows that 𝕋-equivariantly some topological structure remains, even after rationalization. The author began the present work to understand the 𝕋-equivariant Chern character, the 𝕋equivariant Segal conjecture, the Tate construction on 𝕋-equivariant K-theory and several other 𝕋-equivariant rational objects that had come to light. The list of contents contains a list of examples treated here.

From now on we let 𝕋 denote the circle group. We only consider closed subgroups, and the letters H, K and L will denote finite subgroups. The family of all finite subgroups will be denoted by F. We work rationally throughout, without displaying this in the notation; for example S^{n} denotes the rationalized n-sphere.

0.2. Overview

Equivariant cohomology theories are represented by equivariant spectra, and we shall conduct most of the investigation at the represented level. This gives more precise information both about individual theories and about natural transformations between them; indeed, the only loss is any geometric interpretation of the cohomology theory concerned, which is inevitable in any general study. It is important to be explicit that we only consider cohomology theories which admit suspension isomorphisms for arbitrary representations; these are sometimes known as ‘genuine’ or ‘RO(G)-graded’ cohomology theories. The corresponding representing objects are G-spectra. For these too there are adjectives to emphasize the type of spectra concerned: they are ‘genuine’ G-spectra or G-spectra ‘indexed on a complete G-universe’. Since these cohomology theories and these G-spectra form the most natural classes to consider, we shall not use these adjectives unless required for emphasis. As made clear by the title, we consider the circle group G = 𝕋.

Before summarizing our results we begin by putting the circle group into context. In fact the circle stands at a watershed: for finite groups of equivariance rational cohomology theories may be analysed completely, and any group more complicated than the circle is substantially harder to understand.

The main problem in analyzing spectra is to choose basic objects which are easy to work with and which give theorems of practical use. It is natural to be guided by one’s favourite algebraic invariant, and this suggests analysis in terms of Moore spectra or Eilenberg-MacLane spectra. For finite groups of equivariance both approaches work well, and one may analyse rational spectra completely. There are two reasons for this: firstly the group has no topology, and secondly the classifying space has no rational cohomology. The first fact means the category of Mackey functors is very simple, and the second means that the classes of Eilenberg-MacLane spectra, of Moore spectra and of Brown-Comenetz spectra coincide, so that all their characteristic properties can be used at once. Both simplifying factors fail for infinite groups, and the three basic classes are distinct. This means that different methods must be used: in essence we base our analysis for the circle group on a slightly embellished version of equivariant homotopy with its primary operations. The reason such a simple invariant suffices is that the rank of the circle group is one. In general the injective dimension of the category of rational Mackey functors and the Krull dimension of the cohomology of its classifying space are both equal to the rank of the group. When the rank is one there is no room for extension problems, and some hope of a simple answer. However, even for the group O(2), it is necessary to take into account a topology on the space of subgroups, and to work with sheaves: it is no longer possible to treat different conjugacy classes of subgroups entirely separately. This explains why it is worthwhile to treat the single case of the circle in such detail.

The work is broken into three parts: Part I in which we construct the algebraic models for various classes of 𝕋-spectra, Part II in which we identify the algebraic counterparts of various general constructions, and Part III in which we consider several classes of examples of particular interest. Each part has a detailed introduction of its own, but we give a general outline here.

Part I begins by discussing K-theory. On the one hand, we give Haeberly’s example showing that K-theory cannot be described simply using ordinary cohomology. On the other hand, we give a generalization of McClure’s result that the K-theory Atiyah-Hirzebruch spectral sequence collapses for F-free spaces. This suggests the necessity of the present work and that it is practical. We then turn to the main business of constructing a model: in this introduction we describe the model in an aesthetically satisfying way, but do not attempt to explain the proof that it is a model. The introduction to Part I gives a different approach to the model which does suggest the proof. We would prefer to achieve these two ideals simultaneously.

To motivate the form of the model, one should recall the classical localization theorem for semifree 𝕋-spaces. This states that if X is a finite space which only has isotropy groups 𝕋 and 1, then the inclusion of the fixed point space X^{𝕋} ⟶ X induces an isomorphism in Borel cohomology once the Euler classes E = {1, c_{1}, c_{1}^{2}, … } are inverted:

E^{−1}H^{∗}(E𝕋_{+} ∧_{𝕋} X) ⟶^{≅} E^{−1}H^{∗}(E𝕋_{+} ∧_{𝕋} X^{𝕋}) = E^{−1}H^{∗}(B𝕋_{+}) ⊗ H^{∗}(X^{𝕋}).

We conclude that N = H^{∗}(E𝕋_{+} ∧_{𝕋} X), regarded as a module over ℚ[c_{1}] = H^{∗}(B𝕋_{+}) is very nearly enough to identify the homology of the fixed point space X^{𝕋}, but we need to pick out a vector subspace V = H_{∗}(X^{𝕋}) of E^{−1}N which is a basis in the sense that E^{−1}N ≅ E^{−1}H^{∗}(B𝕋_{+}) ⊗ V. In particular, if X is free then N is E-torsion.

Now 𝕋-equivariant cohomology theories are represented by 𝕋-spectra, and the localization theorem suggests a model which turns out to be a complete invariant. To describe it, we first note that there is a natural homotopy-level analogue of the set of isotropy groups which occur. This uses the geometric K-fixed point functor X ⟼ Φ^{K}X, which is the functor extending the K-fixed point functor on spaces, in the sense that Φ^{K}(Σ^{∞}Y) = Σ^{∞}(Y^{K}); it also enjoys similar formal properties to the space-level functor. We then define the set of isotropy groups of a spectrum X to be the set of subgroups K for which the geometric fixed point spectra Φ^{K}X are non-equivariantly essential. This gives the notion of a free 𝕋-spectrum (alternatively characterized as a 𝕋-spectrum X for which E𝕋_{+} ∧ X ⟶ X is an equivalence). We therefore suppose given a collection H of finite subgroups of 𝕋, and we may consider the class of H-free spectra (i.e. those with isotropy in H), and the class of H-semifree spectra (i.e. those with isotropy in H ∪ {𝕋}). The reader should concentrate on the case H = {1}, which gives the usual classes of free and semifree spectra, and on the case H = F: the class of F-semifree spectra is the class of all 𝕋-spectra. However the additional generality makes the picture clearer, and the two special cases are representative of the two classes of examples: those with H finite, and those with H infinite. Analagous to the ring H^{∗}(B𝕋_{+}) we have the ring of operations

O_{H} = C(H, ℚ)[c],

where C(H, ℚ) denotes the ℚ-valued functions on the discrete set H, and c is of degree −2. The notation is chosen to suggest that O_{H} is a ring of functions on H. This ring is Noetherian if H is finite and not otherwise. We let e_{H} ∈ C(H, ℚ) = (O_{H})_{0} denote the idempotent with support H ∈ H, and we let c_{H} = e_{H}c. Next we need the set E = E_{H} of Euler classes. If H = {1} this is simply the multiplicative subset {1, c_{1}, c_{1}^{2}, … } of O_{H} used for the localization theorem above, but in general it needs a little more explanation. For any finite subset ϕ ⊆ H we have an associated idempotent e_{ϕ} ∈ O_{H}, and we have an Euler class c_{ϕ} = e_{ϕ}c + (1 − e_{ϕ}), which is not a homogeneous element of O_{H}. The effect of c_{ϕ} on an O_{H}-module M = e_{ϕ}M ⊕ (1 − e_{ϕ})M is to multiply by c on the first factor and do nothing to the second: thus the result of inverting c_{ϕ} on M is again a graded module: e_{ϕ}M[c^{−1}] ⊕ (1 − e_{ϕ})M. Thus our second ingredient is the set

E_{H} = {c_{ϕ}^{k} | ϕ ⊆ H finite, k ≥ 0}

of Euler classes. The category modelling semifree H-spectra is then the category A_{H} of O_{H}-modules N with a specified graded vector space V to act as a basis of E^{−1}N. It is convenient to package this as saying that we are given a basing map

β : N ⟶ (E^{−1}O_{H}) ⊗ V

which becomes an isomorphism when E is inverted. This makes clear that a morphism in A_{H} is a diagram

M ⟶^{θ} N
α ↓  ↓ β
(E^{−1}O_{H}) ⊗ U ⟶^{1⊗ϕ} (E^{−1}O_{H}) ⊗ V.

We refer to N as the nub and V as the vertex. We also refer to an O_{H}-module with specified basing map as a based O_{H}-module, and to a morphism θ : M ⟶ N for which there is a compatible map ϕ as a based map. Note that if H is a singleton the existence of a basing isomorphism E^{−1}N ≅ E^{−1}O_{H} ⊗ V for some V is automatic, but in general it puts a restriction on the modules N.

The connection with topology arises since O_{F} = [EF_{+}, EF_{+}]_{∗}^{𝕋}, and hence this acts on both π_{∗}^{𝕋}(EF_{+} ∧ X) and π_{∗}^{𝕋}(DEF_{+} ∧ ẼF ∧ X) for any X; if X is H-semifree this action factors through the projection O_{F} ⟶ O_{H}. Furthermore, since c is of negative degree and any element of π_{∗}^{𝕋}(EF_{+} ∧ X) is supported on a finite subspectrum, one sees that E^{−1}π_{∗}^{𝕋}(EF_{+} ∧ X) = 0. Next, we have a map

DEF_{+} ∧ X ⟶ DEF_{+} ∧ ẼF ∧ X ≃ DEF_{+} ∧ ẼF ∧ Φ^{𝕋}X

with cofibre DEF_{+} ∧ ΣEF_{+} ∧ X ≃ ΣEF_{+} ∧ X. Since the homotopy of the cofibre is Euler-torsion, its homotopy

π_{∗}^{A}(X) := (π_{∗}^{𝕋}(DEF_{+} ∧ X) ⟶ π_{∗}^{𝕋}(DEF_{+} ∧ ẼF ∧ Φ^{𝕋}X) = π_{∗}^{𝕋}(DEF_{+} ∧ ẼF) ⊗ π_{∗}(Φ^{𝕋}X))

is therefore an object of A_{H}.

Now we may state our main classification theorem.

Classification Theorem: For any collection H of finite subgroups of the circle 𝕋, the above invariant induces bijections (i)

{H-free rational spectra} / ≃ ⟷ {Euler-torsion O_{H}-modules} / ≅ where ≃ denotes homotopy equivalence, and ≅ denotes isomorphism, and (ii)

{H-semifree rational spectra} / ≃ ⟷ {based O_{H}-modules} / ≅

where ≃ denotes homotopy equivalence, and ≅ denotes isomorphism. In particular, rational 𝕋-equivariant cohomology theories are in bijective correspondence to isomorphism classes of based O_{F}-modules.

In practice this is derived as a corollary of a theorem identifying the categories of spectra in algebraic terms. More precisely, recall that the derived category of a graded abelian category is the category of differential graded objects with homology isomorphisms inverted, although for practical purposes a more concrete construction is essential. The theorem identifies the categories of spectra as the derived category of the associated algebraic category:

H-free 𝕋-spectra ≃ D(Euler torsion O_{H}-modules)

and

H-semifree 𝕋-spectra ≃ D(based O_{H}-modules).

Furthermore, cofibre sequences of spectra correspond to triangles under these equivalences. The point here is that both algebraic categories turn out to be abelian and one dimensional, so that morphisms in the derived category can be calculated from a short exact sequence involving Hom and Ext in the abelian category.

It is sometimes more practical to identify the place of a spectrum X in the classification by a different route. This amounts to identifying first EF_{+} ∧ X and Φ^{𝕋}X, and then the map

q_{X} : ẼF ∧ Φ^{𝕋}X = ẼF ∧ X ⟶ ΣEF_{+} ∧ X

of which X is the fibre. It is not enough to identify the effect of q_{X} in homotopy: one must also take into account the twisting given by representations, and in general this requires both primary and secondary information. Nonetheless, there is a second model for semifree H-spectra based on this approach, which we call the torsion model. We show it is equivalent to the standard model described above, and it is often the easiest route to placing a spectrum in the classification.

There are really three stages to the proof of these theorems. Firstly one shows, using idempotents in the Burnside rings of finite subgroups, that for F-free spectra it is essentially enough to deal with the case of free spectra. Next, one constructs an Adams spectral sequence for free spectra, which collapses to a short exact sequence and gives a means of calculation. Because of the particularly simple algebraic behaviour of O_{1} = ℚ[c_{1}] this is enough to identify the entire triangulated category. The final stage is to take this work and process it: this stage is essentially formal.

Once we have algebraic models for various categories of spectra we naturally want to understand familiar topological constructions in algebraic terms. This is the business of Part II. We have followed the order suggested by logic, and therefore begin by studying the smash product and function spectrum constructions, and then go on to functors changing equivariance. Unfortunately the smash product and function spectrum are by far the most complicated examples, and require more algebraic machinery than any of the other examples we consider. Furthermore, their complexity means that we are not able to show that our description is functorial, and our approach is necessarily indirect. This highlights a shortcoming of our method: the correct proof of our results would follow that used by Quillen in modelling rational homotopy of simply connected spaces. The functorial identification of smash products and function spectra would then be automatic. At present, such a proof is not accessible, but the present results strongly suggest that such a proof exists. In any case, the model of the smash product is essentially the left derived tensor product, and the model of function spectra is its right adjoint. There are two warnings here: in the categories of H-free spectra, there are not enough flat objects, so the left derived tensor product must be calculated in a larger category; it results in an Euler-torsion object since it coincides with the suspension of the right derived torsion product. With this caveat, if the spectra X and Y are modelled by M and N respectively then

X ∧ Y is modelled by M ⊗^{L} N.

There is also a caveat for function objects, which we now explain. It is convenient in both cases to consider the larger algebraic category in which no condition is placed on the behaviour of Euler classes. For H-free spectra this is the category of all O_{H}-modules, and for H-semifree spectra it is the category of all maps N ⟶ E^{−1}O_{H} ⊗ V. It turns out that the internal Hom functor in the abelian category is the composite functor Γ Hom(M, N), where Hom(M, N) is an object in the category with no condition on behaviour under inversion of Euler classes, and where Γ is the right adjoint to the inclusion of the smaller category. For example, in the case of H-free spectra Hom(M, N) is simply the O_{H}-module of O_{H}-morphisms, and for an arbitrary O_{H}-module M′, the Euler-torsion module ΓM is defined to be the kernel of M ⟶ E^{−1}M. In the semifree case both functors are harder to describe, and we refer the reader to Chapter 8. It turns out that the right adjoint of M ⟼ M ⊗^{L} N is not the right derived functor of P ⟼ Γ Hom(N, P), but rather it is P ⟼ RΓ RHom(N, P). Thus if the spectra Y and Z are modelled by N and P, then

The internal function spectrum of maps from Y to Z is modelled by RΓ RHom(N, P).

An essential step in identifying the function spectrum on objects is to give a functorial identification of the product. In these terms we may say that if X_{i} is modelled by M_{i} then

The internal product of the spectra X_{i} is modelled by RΓ ∏_{i} M_{i},

and this model is functorial.

The other topological functors we consider can be modelled functorially, and we shall discuss only the full category of 𝕋-spectra. The forgetful functor and its left and right adjoints, induction and coinduction, are straightforward. Similarly the geometric fixed point functor X ⟼ Φ^{𝕋}X is the passage-to-vertex functor given as part of the structure. The first interesting functor is the geometric fixed point functor Φ^{K} : 𝕋 − spectra ⟶ 𝕋/K − spectra for a finite subgroup K. This turns out to be easy to describe: we simply let e ∈ C(F, ℚ) denote the idempotent supported on the set [⊇ K] of subgroups containing K. The algebraic model of Φ^{K} is multiplication by e; this make sense since eO_{F} is naturally identified with the ring \overline{O}\overline{F} of operations for \overline{𝕋} = 𝕋/K. As usual, the Lewis-May fixed point functor Ψ^{K} : 𝕋 − spectra ⟶ 𝕋/K − spectra (the spectrum Ψ^{K}X is written X^{K} in [18]) is much harder to understand, and we only describe its behaviour here for F-free and F-contractible spectra, referring the reader to Chapters 11 and 12 for details of how these are spliced. On F-contractible spectra X ≃ ẼF ∧ Φ^{𝕋}X, we have Ψ^{K}(X) = Ẽ\overline{F} ∧ Φ^{𝕋}X, so this is easy. We have seen that an F-spectrum X is modelled by an Euler-torsion O_{F}- module N; from the form of Euler classes it follows that this is equivalent to specifying the function

\begin{matrix}[N] : F & ⟶ & torsionℚ[c] − modules \\ H & ⟼ & e_{H}N. \end{matrix}

The Lewis-May fixed point functor groups these modules together according to the behaviour of the subgroup on passage to quotient. More precisely, we observe that passage to quotient q : 𝕋 ⟶ 𝕋/K = \overline{𝕋} defines a map q_{∗} : F ⟶ \overline{F} on finite subgroups. If the function [N] models the F-free spectrum X then the function [Ψ^{K}N] modelling Ψ^{K}X is the map

\begin{matrix}\overline{F} & ⟶ & torsionℚ[c] − modules \\ \overline{H} & ⟼ & ⨁_{q_{∗}(H)=\overline{H}}[N](H). \end{matrix}

A little thought shows that it is not a trivial matter to see how the F-free and Fcontractible parts should be spliced together. Because the Lewis-May fixed point functor is so complicated, we actually approach it via its left adjoint, the inflation map inf_{𝕋/K}^{𝕋} : 𝕋/K − spectra ⟶ 𝕋 − spectra. This is the functor given by regarding a 𝕋/K spectrum as a 𝕋-spectrum by pullback along the quotient, and then building in representations (it is written q^{#} in [18], but more commonly i_{∗} by abuse of notation; we shall stick to the more descriptive notation). From our description of Lewis-May fixed points it is easy to deduce inflation on \overline{F}-contractible and \overline{F}-free specta. On \overline{F}-contractible spectra Y ≃ Ẽ\overline{F} ∧ Φ\overline{𝕋}Y we have inf_{𝕋/K}^{𝕋}Y = ẼF ∧ Φ\overline{𝕋}Y. If [P] is the model of the \overline{F}-spectrum Y then the model [inf_{𝕋/K}^{𝕋}P] of inf_{𝕋/K}^{𝕋}Y is the composite

F ⟶^{q_{∗}} \overline{F} ⟶^{[P]} torsionℚ[c] − modules.

In cases where N is Euler-torsion, the right adjoint of the inflation map is also its left adjoint; it therefore also gives a model for the topological quotient when X is K-free.

The final chapter of Part II turns to ordinary cohomology and its variants. After Eilenberg and Steenrod we define a cohomology theory to be ordinary if its coefficients are nonzero only in degree 0, and similarly in homology. For each integer q, an equivariant cohomology theory F_{G}^{∗}(·) specifies a contravariant additive functor G/H_{+} ⟼ F_{G}^{q}(G/H_{+}) = F_{H}^{q} on the stable category of orbits; such a functor is called a Mackey functor. As in the classical case, ordinary cohomology theories are classified by their non-zero Mackey functor M in degree 0, and we write H_{G}^{∗}(·; M) for this theory and HM for its representing spectrum. Similarly, for each integer q a homology theory F_{∗}^{G}(·) defines a covariant additive functor G/H_{+} ⟼ F_{q}^{G}(G/H_{+}) on the stable category of orbits; such a functor is called a coMackey functor. Ordinary homology theories are classified by their associated coMackey functors N, and we write H_{∗}^{G}(·; N) for this functor and JN for the representing spectrum. For finite groups G the stable orbit category is self-dual, so that a coMackey functor can also be viewed as a Mackey functor; in this case the ordinary homology theory classified by a Mackey functor M is also represented by HM. However, for positive dimensional groups such as the circle, the functor given by a homology theory cannot usually be viewed as a Mackey functor.

Our first task is to identify objects of the form HM and JN in our model; we find that they are well behaved but by no means trivial. Finally, whenever one has an injective Mackey functor I one may consider the cohomology theory defined by Brown-Comenentz I-duality

hI_{G}^{q}(X) = Hom(\underline{π}_{q}^{G}(X), I),

and its representing spectrum hI. Again, in the case of a finite group all rational Mackey functors are injective, and HM = JM = hM. Indeed, this is the basis of a simple proof that all rational cohomology theories are ordinary for finite groups. However, for the circle group the spectrum hI is rather complicated, and in particular it is unbounded; we identify it exactly in our model.

In Part III we apply the general theory of Parts I and II to several examples of particular interest. First we answer a number of obvious general questions. To begin with, we relate the model we have used to the use of Postnikov towers and the use of cells. In fact, we can understand the Atiyah-Hirzebruch spectral sequence H_{𝕋}^{∗}(X; \underline{K}_{𝕋}^{∗}) ⟹ K_{𝕋}^{∗}(X) for F-free spectra X completely, in terms of our model. It collapses at the E_{2} page if and only if K_{∗}^{𝕋}(EF_{+}) is injective over O_{F}. The latter condition holds for complex K-theory, so we recover McClure’s theorem that the Atiyah-Hirzebruch spectral sequence for the rational K-theory of an F-space collapses at E_{2}. However, in general there are arbitrarily long differentials. The contrast with the simplicity of the one dimensional nature of the category of Euler-torsion O_{F}-modules suggests that the Postnikov tower is a poor way to study 𝕋spectra. On the other hand, because of the simplicity of the graded maps between cells, we can contemplate homological algebra over it, and it is easy to construct a convergent spectral sequence based on cellular resolutions with a calculable E_{2} term. Unfortunately the spectral sequence does not appear to be useful in general.

We do not have the means to detect purely unstable phenomena, but the splitting theorem of Segal and tom Dieck shows that suspension spectra of 𝕋-spaces are very special, and we briefly comment on the implications of this for their algebraic model.

Finally we return to complex K-theory and identify its algebraic model. It is simple to describe in terms of representation theory, and is well behaved algebraically (‘formal’ in the torsion model). However there remain many interesting questions that we have not treated. Firstly, a qualitative comparison of the F-spectrum Euler classes and the Ktheory Euler classes is sufficient for our purpose, but an exact comparison using the Chern character, along the lines of Crabb’s work [5], would be illuminating. Secondly, it would be interesting to compare our model with that of Brylinski [3]. Presumably these questions would be useful preparation for the more substantial project of modelling 𝕋-equivariant elliptic cohomology as constructed by Grojnowski [8] and Ginzburg-Kapranov-Vaserrot [6].

The other motivating problem was that of understanding the 𝕋-equivariant analogue of the Segal conjecture. We had the ironic situation that we understood the harder profinite part by virtue of work on the Segal conjecture for finite groups, whilst we could not understand the rational part. Using the model described here, it is now an easy exercise to identify DE𝕋_{+} in the torsion model as the composite

E^{−1}O_{F} ⊗_{ℚ} E^{−1}O_{F} ⟶ E^{−1}O_{F} ⟶ E^{−1}O_{F}/O_{F} ⟶ ℚ[c_{1}, c_{1}^{−1}]/ℚ[c_{1}]

where the first map is the product. It is quite instructive to view this as a special case of the identification of the function spectrum.

Turning to more specialised examples, we reach Tate cohomology theories in the sense of [14]. This construction on 𝕋-spectra corresponds precisely to Tate cohomology in commutative algebra in the sense of [10]. Perhaps more interesting is our study of the integral Tate spectrum of complex equivariant K-theory. We are able to identify the exact homotopy types of both t(Kℤ) ∧ EF_{+} and t(Kℤ) ∧ ẼF and the map q of which t(Kℤ) is the fibre: the first is rational, and identified using our general theory, and the second is formed from K-theory with suitable coefficients by inflating and smashing with ẼF.

Finally we turn to examples gaining their importance from algebraic K-theory. The motivation for the notion of a cyclotomic spectrum comes from the free loop space ΛX = map(𝕋, X) on a 𝕋-fixed space X. This has the property that if we take K-fixed points we obtain the 𝕋/K-space map(𝕋/K, X), and if we identify the circle 𝕋 with the circle 𝕋/K by the |K|th root isomorphism we recover ΛX. For spectra one also needs to worry about the indexing universe, but a cyclotomic spectrum is basically one whose geometric fixed point spectrum Φ^{K}X, regarded as a 𝕋-spectrum, is the original 𝕋-spectrum X. After the suspension spectrum of a free loop space, the principal example comes from the topological Hochschild homology of T HH(F) of a functor F with smash products. Given such a cyclotomic spectrum X one may construct the topological cyclic spectrum T C(X) of Bökstedt-Hsiang-Madsen [2], which is a non-equivariant spectrum. An intermediate construction of some interest is the 𝕋-spectrum T R(X). Although these constructions are principally of interest profinitely, it is instructive to identify the cyclotomic spectra in our model and follow the constructions through. In fact we show that cyclotomic spectra, are those spectra X so that the function [N] : F ⟶ torsionℚ[c] − modules modelling EF_{+} ∧ X is constant, and so that the structure map E^{−1}O_{F} ⊗ V ⟶ ΣN commutes with any translation of the finite subgroups. It therefore factors through E^{−1}O_{F} ⊗ V ⟶ (E^{−1}O_{F})/O_{F} ⊗ V, and the map (E^{−1}O_{F})/O_{F} ⊗ V ⟶ ΣN is a direct sum of copies of ℚ[c, c^{−1}]/ℚ[c] ⊗ V ⟶ Σ[N](1). Furthermore, we may recover Goodwillie’s theorem that for any cyclotomic spectrum X we have T C(X) = X^{h𝕋}: topological cyclic cohomology coincides with cyclic cohomology in the rational setting.

This summarises the contents of the body. There are also a number of appendices. Appendix A gives the structure of rational Mackey functors, and is of independent interest: in particular the category is of projective and injective dimension 1. Appendix B gives Quillen closed model category structure on the algebraic categories. Finally we suggest the reader glance at Appendix C summarising our conventions. There are also a number of indices.

It is appropriate to comment briefly on reading this document. Formally, Part I is the basis of all that follows, and is cumulative. Part II consists of an introductory chapter, followed by the treatment of four classes of examples. Since it gives algebraic models of topological constructions it must therefore develop the relevant algebra before comparing it to topology. Thus Chapters 8 and 11 are purely algebraic, and are prerequisites for Chapters 9 and 12 respectively. Otherwise the chapters are independent of each other, but the geometric results depend on Part I. Finally, the chapters of Part III are again independent, and depend only on Part I and the appropriate results from Part II. We have made some effort to ensure it is possible for the trusting reader to read a part without previously reading its predecessors.

We expect there will be those only interested in Chapters 1 to 3. There may also be those wanting to gain a feel for the behaviour of certain functors, who may find Part II worthwhile, even without reading Part I. Finally, there may be those who want to begin with Part III and read earlier chapters as necessary.

The author is grateful to the Nuffield Foundation for its support, to the Universities of Georgia (Athens) and Chicago for their hospitality, and to the towns of Karlsruhe and Worms. The author also thanks L.Hesselholt, J.P.May and N.P.Strickland for useful comments and conversations.

Part I

The algebraic model of rational 𝕋-spectra.

12

CHAPTER 1 Introduction to Part I.

This chapter motivates Part I and provides a map for it. In Section 1.1 we explain the strategy used in Part I to analyse the category of rational 𝕋-spectra, and in Section 1.2 give a brief guide to help readers with particular interests. This is followed in Sections 1.3 and 1.4 by accounts of Haeberly’s example and a generalization of McClure’s theorem: this is designed to show there is a need for analysis and some hope of achieving it.

1.1. Outline of the algebraic models.

The main business of Part I is to construct a complete algebraic model of the category of rational 𝕋-spectra. Since spectra represent cohomology theories, this gives a complete algebraic classification of rational 𝕋-equivariant cohomology theories. Having given the overview in the General Introduction, we concentrate here on the practical approach. In fact, we lead the reader through the investigative process to the algebraic model of 𝕋spectra. This should help explain the how geometric information is packaged in the model, and how the algebraic model can be used.

The main problem in analyzing 𝕋-spectra is to choose basic objects which are easy to work with and which give theorems of practical use. We explained in the introduction that the building blocks familiar from finite groups of equivariance are not suitable: Eilenberg-MacLane spectra, Moore spectra and Brown-Comenetz spectra form distinct classes. This means that different methods must be used.

The redeeming feature is that there is no complication at all from representation theory since the Weyl groups are all connected. This means we can return to geometric intuition and concentrate on isotropy groups. It is appropriate for our present purpose to think of 𝕋-spectra as generalized stable spaces. It is standard practice in transformation groups to consider various fixed point spaces X^{H} of a space X. In particular, spaces with a free action are especially approachable. One reason for this is that only one subgroup occurs as an isotropy group. In the rational case the behaviour at each finite subgroup is reasonably similar and reasonably simple. Therefore it is common to consider spaces X all of whose isotropy groups are finite. These are variously called F-spaces, F-free spaces, almost free spaces, or spaces without fixed points. We shall call them F-spaces, and concentrate on the fact that they are equivalent to spaces constructed from cells G/H × E^{n} with H finite.

In any case, our analysis follows this time-honoured pattern, by breaking any object X into into F-free and F-contractible parts by the isotropy separation cofibering

X ⟶ X ∧ ẼF ⟶^{q_{X}} X ∧ ΣEF_{+}.

We thus consider X in two parts: the F-contractible object X(𝕋) = X ∧ ẼF and the Ffree object X(F) = X ∧ EF_{+}. The object X(𝕋) is determined by its 𝕋-homotopy groups as rational vector spaces. The main content of the analysis is therefore in understanding F-objects such as X(F), and how they may be stuck to F-contractible objects X(𝕋). By use of idempotents in Burnside rings it is easy to see that X(F) splits as a wedge of objects X(H), one for each finite subgroup H, where only the isotropy group H is relevant to X(H). The category of these will be called the category of 𝕋-spectra over H and denoted 𝕋−Spec/H; the mathematical core of the whole enterprise is the analysis of this category of objects X(H). It turns out that π_{∗}^{𝕋}(X(H)) is a torsion module over the ring O_{H} = ℚ[c_{H}], in which c_{H} is an Euler class, and of degree −2, and that the category 𝕋−Spec/H of objects X(H) is equivalent to the derived category of differential graded torsion ℚ[c_{H}]-modules. The object X(F) is thus determined by the torsion module π_{∗}^{𝕋}(X(F)) over O_{F} = ∏_{H} ℚ[c_{H}]. Because we are working rationally it is not difficult to calculate homotopy groups of any precisely described spectrum, so this description is of practical use.

Finally we must determine the assembly map q_{X} : X(𝕋) ⟶ ΣX(F). Note first that π_{∗}^{𝕋}(X(𝕋)) is not naturally a module over O_{F}, and also that π_{∗}^{𝕋}(q_{X}) may be zero without q_{X} being zero. The answer is to take into account the twisting available from representations of 𝕋. This twisting is measured by Euler classes, and since there are Thom isomorphisms for arbitrary F-spectra we may consider the ring E^{−1}O_{F} formed from O_{F} by inverting all Euler classes. We denote this ring t_{∗}^{F}, since it is in fact the F-Tate cohomology of S^{0} in the sense of [14]. It turns out that t_{∗}^{F} is ⨁_{H} ℚ in positive even degrees and ∏_{H} ℚ in even degrees ≤ 0. By construction, t_{∗}^{F} is a O_{F}-module, and q_{X} determines a map

q̂_{X} : t_{∗}^{F} ⊗ π_{∗}^{𝕋}(X(𝕋)) ⟶ π_{∗}^{𝕋}(X(F))

in the derived category of differential graded O_{F}-modules. It transpires that q̂_{X} is a complete invariant of q_{X}, so that X is determined by the rational vector space π_{∗}^{𝕋}(X(𝕋)), the torsion ℚ[c_{H}]-modules π_{∗}^{𝕋}(X(H)), and the derived O_{F}-map q̂_{X}. Continuing from this stage, it is not hard to identify which triples (π_{∗}^{𝕋}(X(𝕋)), π_{∗}^{𝕋}(X(F)), q̂_{X}) occur, and to identify the relevant algebraic triangulated category.

In fact we may consider the torsion model category A_{t} whose objects are maps t_{∗}^{F} ⊗V ⟶ T of O_{F}-modules, T being a sum ⨁_{H} T (H) with T (H) a torsion ℚ[c_{H}]-module. It turns out that this category is abelian and of injective dimension 2. One may therefore consider differential graded objects in A_{t}, and invert homology isomorphisms to form the derived category DA_{t} . This category is equivalent to the category of rational 𝕋-spectra, and provides the complete algebraic model we seek. However we prefer not to emphasize this model: the analysis is only possible by introducing a second model, which we call the standard model. This proves to be more convenient for most purposes. The real difficulty is that, since A_{t} is of dimension 2, it is rather hard to get a precise hold on morphisms in the derived category. On the other hand the standard model is of dimension 1. The identification of the standard model is the most important result of the analysis.

It will help to explain the construction of algebraic models for four triangulated categories of 𝕋-spectra in increasing order of complexity. They are (i) the category of free 𝕋-spectra, or more generally the category 𝕋−Spec/H of 𝕋-spectra in which only the isotropy group H is important, (ii) the category of 𝕋−Spec/F of F-spectra, (iii) the category 𝕋−Spec_{sf} of semifree 𝕋-spectra and (iv) the category of all rational 𝕋-spectra. For each of these categories ℂ, we find an abelian category A = A_{ℂ} of dimension 1, and a linearization functor π_{∗}^{A} : ℂ ⟶ A_{ℂ}. Because the abelian category A_{ℂ} is so simple in each case, it is possible to reconstruct the original triangulated category ℂ from it. Recall that the derived category of an abelian category A is the category formed from the category of differential graded objects by inverting homology isomorphisms; if A is finite dimensional, the derived category may be constructed explicitly.

Theorem 1.1.1. If ℂ is one of the above four categories of rational 𝕋-spectra, there is a category A = A_{ℂ} which is abelian and one dimensional so that there is an equivalence of triangulated categories

ℂ ≃ DA,

where DA is the derived category of A. Hence in particular, for any objects X and Y of ℂ, there is a natural short exact sequence

0 ⟶ Ext_{A}(π_{∗}^{A}(ΣX), π_{∗}^{A}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{A}(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ⟶ 0,

which splits unnaturally.

Before making the theorem explicit for the four categories we make some general remarks about the levels at which the theorem is useful. Firstly, every geometric object X of ℂ has an algebraic model π_{∗}^{A}(X) and there is a bijection between isomorphism classes in ℂ and isomorphism classes in A. Next, if we know the algebraic models of two objects X and Y, the short exact sequence allows us to use the algebra of the abelian category to calculate the group [X, Y]_{∗}^{𝕋} of maps between them. Finally, we may model all primary constructions (such as formation of cofibres, smash products, function spectra, composition of functions and calculation of Toda brackets) in the algebraic category. This much is internal to the category, but in addition, all homotopy functors of 𝕋-spectra have their algebraic counterparts. It is very illuminating to identify the algebraic behaviour of various well known functors.

We now make Theorem 1.1.1 explicit in the four cases.

Theorem 1.1.2. If ℂ = 𝕋−Spec/H is the category of 𝕋-spectra over H, then A is the category of torsion ℚ[c_{H}]-modules. The functor π_{∗}^{A} is simply 𝕋-equivariant homotopy π_{∗}^{𝕋}. This category A is abelian and one dimensional. Accordingly, for two 𝕋-spectra X and Y over H there is a split short exact sequence

0 ⟶ Ext_{ℚ[c_{H}]}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{ℚ[c_{H}]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) ⟶ 0.

The proof of this will be completed in Section 4.3. The short exact sequence is Theorem 3.1.1, and it is the central result of the analysis of Part I.

Theorem 1.1.3. If ℂ = 𝕋−Spec/F is the category of F-spectra, then A is the full subcategory of O_{F}-modules M of the form M = ⨁_{H} M(H) for torsion ℚ[c_{H}]-modules M(H). We refer to these as F-finite torsion modules, and they may also be described as the O_{F}-modules annihilated by inverting all Euler classes. The functor π_{∗}^{A} is simply 𝕋-equivariant homotopy π_{∗}^{𝕋}. The category of F-finite torsion modules is abelian and one dimensional. Accordingly, for two F-spectra X and Y there is a split short exact sequence

0 ⟶ Ext_{O_{F}} (π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{O_{F}} (π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) ⟶ 0.

The proof of this will also be completed in Section 4.3.

Theorem 1.1.4. If ℂ = 𝕋−Spec_{sf} is the category of semi-free spectra, then A is the category whose objects are morphisms M ⟶ ℚ[c, c^{−1}] ⊗ V of ℚ[c]-modules (for some graded vector space V) which become isomorphisms when c is inverted. This category A is abelian and one dimensional. The functor π_{∗}^{A} is defined by

π_{∗}^{A}(X) := (π_{∗}^{𝕋}(X ∧ DE𝕋_{+}) ⟶ π_{∗}^{𝕋}(X ∧ DE𝕋_{+} ∧ ẼF)) .

Accordingly, for two semifree 𝕋-spectra there is a split short exact sequence

0 ⟶ Ext_{A}(π_{∗}^{A}(ΣX), π_{∗}^{A}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{A}(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ⟶ 0.

Finally the model of all rational 𝕋-spectra is as follows.

Theorem 1.1.5. If ℂ = 𝕋−Spec then A is the category whose objects are morphisms M ⟶ t_{∗}^{F} ⊗ V of O_{F}-modules (for some graded vector space V) which become isomorphisms when all Euler classes are inverted (i.e. the kernel and cokernel are F-finite torsion modules). This category A is abelian and one dimensional. The functor π_{∗}^{A} is defined by

π_{∗}^{A}(X) := (π_{∗}^{𝕋}(X ∧ DEF_{+}) ⟶ π_{∗}^{𝕋}(X ∧ DEF_{+} ∧ ẼF)) .

Accordingly, for two 𝕋-spectra there is a split short exact sequence

0 ⟶ Ext_{A}(π_{∗}^{A}(ΣX), π_{∗}^{A}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{A}(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ⟶ 0.

The proof of this is given in Section 5.4. It should be emphasized that Hom_{A}(M, N) and Ext_{A}(M, N) are routinely computable, and that, because we are working rationally, there is usually no serious trouble in calculating π_{∗}^{A}(X).

Part I begins with the concrete and moves towards the abstract in two steps. Thus we begin with the cohomology theories, move on to homotopy theory, pass to algebra by an Adams spectral sequence, and finally package this in categorical terms. Here is a more detailed outline of contents.

We begin with two sections which can be expressed in classical terms. These give evidence that there is some complexity in rational 𝕋-equivariant cohomology theories, but not too much. In particular they give some evidence for the simplicity of F-objects.

After this, the discussion is conducted in the Lewis-May [18] stable category of 𝕋-spectra. The first step is to introduce the basic building blocks and the methods for breaking general objects up. This gives us the setting to construct an Adams spectral sequence, which provides the connection between topology and algebra. Once the Adams spectral sequence for 𝕋−Spec/H has been constructed we need only do some algebra and certain formal manipulations to obtain and exploit all the algebraic models. We have taken the view that an abstract machine should only be introduced when there is a particular case on which its operation can be illustrated. Accordingly we have not described the transition from an Adams spectral sequence to an algebraic model (in Section 4.2) until we have constructed the simplest instance to which it applies. On the other hand Section 4.2 may be relevant in quite different settings, and it is written axiomatically so that it can be read and applied independently of the preceding sections.

Once the general analysis is completed we consider standard 𝕋-spectra and constructions on 𝕋-spectra in Part II. In Part III we consider in more detail certain examples of established interest. More detailed accounts of the contents of Parts II and III may be found in their introductions.

1.2. Reading Guide.

Some readers may not wish to read all of the material in Part I, so we provide further guidance here.

Those only interested in the Atiyah-Hirzebruch spectral sequence for the K-theory of an F-space will only need to read Sections 1.3, 1.4, 2.1, referring to Appendix A for the necessary facts about Mackey functors. Sections 1.3 and 1.4 are not used elsewhere in Part I. We shall return to the Atiyah-Hirzebruch spectral sequence in Section 15.1 of Part III, where we give more complete results.

Those interested in Mackey functors should read Section 2.1 and then refer to Appendix A. Mackey functors are not used until we consider ordinary cohomology theories in Chapter 13 from Part II.

The central material constructing the main Adams spectral sequence for the categories of F-spectra and 𝕋-spectra over H is to be found in Chapters 2 and 3. Maps from Fcontractible spectra to F-free spectra are deduced in Sections 5.1 and 5.2. This is sufficient to answer most direct questions about particular 𝕋-spectra, and may satisfy some readers. On the other hand readers wishing to understand the shape of the algebraic models without reading these chapters.

In Chapter 4, we explain the abstract process of reaching an algebraic model from an Adams spectral sequence and we illustrate it for 𝕋-spectra over H. However the goal of a full algebraic model is fulfilled in Chapter 5. We deduce the remaining topological input from the Adams spectral sequence in Sections 5.1 and 5.2, and construct the algebraic model in Section 5.3. It is then a simple matter to show in Section 5.4 that the algebra does indeed model the topology. Chapter 6 completes the circle by introducing the torsion model, closely following geometric intuition, and by showing that it gives a model equivalent to the standard model.

1.3. Haeberly’s example.

We give Haeberly’s example [16] showing there is no Chern character isomorphism, for 𝕋equivariant K-theory. This simply involves constructing a 𝕋-space X whose equivariant Ktheory is concentrated in even degrees, but whose ordinary cohomology with coefficients in the rationalized representation ring functor is nonzero in odd degrees. Since the homotopy functors of the K-theory spectrum are in even degrees the K-theory cannot be a product of copies of ordinary cohomology. In the next section we give a proof of a generalization of McClure’s result that there is a Chern isomorphism for 𝕋-spaces X with X^{𝕋} trivial.

To explain Haeberly’s example it is convenient to consider the group Γ = 𝕋 × 𝕋′ where both 𝕋 and 𝕋′ are copies of the circle group. The group Γ has a 3-dimensional complex representation V = (1 ⊕ t ⊕ t^{2}) ⊗ t′, where t is the natural representation of 𝕋 on ℂ, and similarly for 𝕋′. We may consider the unit sphere S(V) as a Γ-space, give it a disjoint basepoint and then form the 𝕋-space X = S(V)_{+}/𝕋′. We could equally well describe X as a copy of ℂP_{+}^{2} on which 𝕋 acts via s(z_{0} : z_{1} : z_{2}) = (z_{0} : sz_{1} : s^{2}z_{2}). From the first description it is easy to calculate the K-theory since we have K_{𝕋}^{∗}(X) = K_{Γ}^{∗}(S(V)_{+}), because S(V) is free as a 𝕋′-space. Indeed, the cofibre sequence S(V)_{+} ⟶ S^{0} ⟶ S^{V} of Γ-spaces gives an exact sequence

⋯ ⟶ K_{Γ}^{i}(S^{V}) ⟶^{λ(V)} K_{Γ}^{i}(S^{0}) ⟶ K_{Γ}^{i}(S(V)_{+}) ⟶ K_{Γ}^{i+1}(S^{V}) ⟶ · · · .

Now by Bott periodicity K_{Γ}^{i}(S^{V}) is R(Γ) if i is even and 0 if i is odd, and because the degree 0 Euler class λ(V) = (1−t′)(1−tt′)(1−t^{2}t′) is not a zero divisor in R(Γ) = ℤ[t, t^{−1}, t′, (t′)^{−1}] we find

K_{𝕋}^{0}(X) = R(Γ)/λ(V) and K_{𝕋}^{1}(X) = 0.

In particular the K-theory of X is entirely in even degrees.

On the other hand from the second description it is not hard to see that X has isotropy groups 𝕋, C_{2} and 1. Furthermore X^{C_{2}} = (S^{t^{2}} ∨S^{0})_{+} and X may be given a 𝕋-CW structure with two free 1-cells, one free 2-cell and one free 3-cell. Hence for any Mackey functor M we see that H_{𝕋}^{∗}(X; M) is the cohomology of a complex

3M(𝕋) ⟶^{d^{0}} M(C_{2}) ⊕ 2M(1) ⟶^{d^{1}} M(1) ⟶^{d^{2}} M(1),

and it is easy to see that d^{1} is surjective. Thus H^{3}(X; M) = M(1), and in particular if M is the rationalized representation ring Mackey functor this is the non-zero group ℚ.

1.4. McClure’s Chern character isomorphism for F-spaces.

McClure has observed that the if X is an F-space then the Atiyah-Hirzebruch spectral sequence for the K-cohomology of X does collapse at E_{2}. His proof involves appealing to unstable results and the work of Slominska. We shall give a proof of the corresponding statement for any cohomology theory whose homotopy functors are concentrated entirely in even degrees, and of the corresponding statement for homology theories. Of course this applies in particular to K theory, by the Bott periodicity theorem. In Section 15.1 of Part 3 we shall give a necessary and sufficient condition for the collapse of the Atiyah-Hirzebruch spectral sequence for F-spaces, which will give an alternative to the proof of this section.

1.4. MCCLURE’S CHERN CHARACTER ISOMORPHISM FOR F-SPACES. 19

Before stating the theorem, we recall that for each integer k it is appropriate to consider the entire system of homotopy groups π_{k}^{H}(X) = [G/H_{+} ∧ S^{k}, X]^{𝕋} as H runs through all subgroups of 𝕋. It is appropriate to regard this as a functor \underline{π}_{k}^{𝕋}(X) : G/H_{+} ⟼ [G/H_{+} ∧S^{k}, X]^{𝕋}, on the category of stable orbits. An additive functor of this form is called a Mackey functor; we examine the algebraic structure of the category of rational Mackey functors in Appendix A, but for the present we only need the basic terminology. In line with the usual abbreviation we write the coefficient functor \underline{π}_{k}^{𝕋}(K) as \underline{K}_{k}^{𝕋}.

Since the orbits are the equivariant analogues of points, an ordinary cohomology theory is one for which the cohomology of each orbit is concentrated in degree zero. Thus ordinary cohomology theories correspend to Mackey functors M, and they are represented by Eilenberg-MacLane spectra HM.

Theorem 1.4.1. If K is any rational 𝕋-spectrum with homotopy functors \underline{K}_{m}^{𝕋} = 0 for all odd integers m then for any F-space X there are isomorphisms (a)

K_{𝕋}^{∗}(X) ≅ ∏_{n∈ℤ}H_{𝕋}^{∗}(Σ^{2n}X; \underline{K}_{−2n}^{𝕋})

and (b)

K_{∗}^{𝕋}(X) ≅ ⨁_{n∈ℤ}H_{∗}^{𝕋}(Σ^{2n}X; \underline{K}_{2n}^{𝕋}).

This follows from a geometric statement.

Theorem 1.4.2. If K is any rational 𝕋-spectrum with homotopy functors \underline{K}_{m}^{𝕋} = 0 for all odd integers m then (a)

F (EF_{+}, K) ≃ F (EF_{+}, ∏_{n∈ℤ}Σ^{2n}H(\underline{K}_{2n}^{𝕋}))

and (b)

K ∧ EF_{+} ≃ ⋁_{n∈ℤ}EF_{+} ∧ Σ^{2n}H(\underline{K}_{2n}^{𝕋}).

To see how Theorem 1.4.1 follows from 1.4.2 we use a lemma which is immediate from the definition of EF_{+} and its unreduced suspension ẼF.

Lemma 1.4.3. For any F-spectrum X,

(a) X ∧ ẼF ≃ ∗ and hence X ≃ EF_{+} ∧ X; also

(b) for any 𝕋-spectrum Y we have F (X, Y ∧ ẼF) ≃ ∗ and hence F (X, Y ∧ EF_{+}) ≃ F (X, Y).

By 1.4.3 (a), Theorem 1.4.1 follows by applying F (X, ) to Part (a) of 1.4.2 and X∧ to Part (b) of 1.4.2 and taking homotopy groups.

Proof: We turn to the proof of 1.4.2. Note first that it is enough to prove Part (b); indeed, by 1.4.3 (b), Part (a) follows by applying F (EF_{+}, ·) to the equivalence of Part (b).

It is enough to construct a 𝕋-map θ : K ∧ EF_{+} ⟶ EF_{+} ∧ ⋁_{n∈ℤ} Σ^{2n}H(\underline{K}_{2n}) which is an H-equivalence for all finite subgroups H. By the Whitehead theorem it is sufficient that θ induces an isomorphism of π_{∗}^{H} for all finite subgroups H. By 1.4.3 (b) again, it is equivalent to give the composite

θ′ : K ∧ EF_{+} ⟶ ⋁_{n∈ℤ}Σ^{2n}H(\underline{K}_{2n}),

and since this wedge is equivalent to the product we may specify θ′ by giving its components. These are elements of the cohomology groups [K ∧ EF_{+}, HM]_{𝕋}^{∗} = H_{𝕋}^{∗}(K ∧ EF_{+}; M) for various Mackey functors M. Accordingly we set about calculating the cohomology of K ∧ EF_{+}.

The idea is to filter EF_{+} so that the subquotients are analogues of cells, but with all elements of finite order as isotropy groups. This extends the idea of [9]. Thus we note that if H ⊆ L we have a projection 𝕋/H ⟶ 𝕋/L, and that the subgroups of finite order form a directed set. We may therefore let 𝕋//F_{+} := holim_{→}_{H} 𝕋/H_{+} where the limit is over all finite subgroups H (or over a cofinal sequence if that appears more comfortable). Analogously, if H is a finite subgroup of order n we may let V (H) denote the representation t^{n} with kernel H, and there are maps mV (H) ⟶ mV (L) (of degree |L/H|^{m}) for all m. We let S(mV (F))_{+} := holim_{→}_{H} S(mV (H))_{+} for 0 ≤ m ≤ ∞. The usefulness of these constructions is summarized in a lemma.

Lemma 1.4.4. The infinite sphere S(∞V (F))_{+} is a model for EF_{+}. We thus have a filtration

∗ = S(0V (F))_{+} ⊆ S(1V (F))_{+} ⊆ S(2V (F))_{+} ⊆ ⋯ ⊆ S(∞V (F))_{+} = EF_{+}

and the subquotients are generalized cells

S(mV (F))_{+}/S((m − 1)V (F))_{+} ≃ S^{2m−2} ∧ 𝕋//F_{+}

for 1 ≤ m < ∞.

Proof: Since (S(mV (H)))^{L} = ∅ if L ⊈ H or S(mV (H)) if L ⊆ H the fact that S(∞V (F))_{+} is a universal space is clear. To identify the quotients we use the fact that the cofibre sequences

S((m − 1)V (H))_{+} ⟶ S(mV (H))_{+} ⟶ S^{2m−2} ∧ 𝕋/H_{+}

fit into a direct system.

In other words we have

EF_{+} = 𝕋//F_{+} ∪ 𝕋//F_{+} ∧ e^{2} ∪ 𝕋//F_{+} ∧ e^{4} ∪ 𝕋//F_{+} ∧ e^{6} ∪ · · · .

Thus, for any spectrum K, we may form the spectral sequence of the filtered spectrum K ∧ EF_{+} which will have the form

E_{1}^{s,t} = H_{𝕋}^{s+t}(K ∧ (EF_{+}^{(s)}/EF_{+}^{(s−1)}); M) ⇒ H_{𝕋}^{s+t}(K ∧ EF_{+}; M).

Indeed, from the form of the filtration, we find the spectral sequence is concentrated in the first quadrant in terms with even s where we have

E_{1}^{2m,t} = H_{𝕋}^{t}(K ∧ 𝕋//F_{+}; M).

Of course, using the change of groups isomorphism H_{𝕋}^{∗}(K ∧ 𝕋/H_{+}; M) = H_{H}^{∗}(K; M), we have a Milnor exact sequence

0 ⟶ lim_{←}_{H}^{1} H_{H}^{t−1}(K; M) ⟶ H_{𝕋}^{t}(K ∧ 𝕋//F_{+}; M) ⟶ lim_{←}_{H} H_{H}^{t}(K; M) ⟶ 0.

It is in the analysis of this exact sequence that it is essential we are working rationally. Indeed, because H is finite, every rational H-spectrum is a product of Eilenberg-MacLane spectra and these are necessarily also Moore spectra. It now follows that, provided K has its homotopy functors in even degrees, the groups H_{H}^{t}(K; M) are only nonzero for even t. The collapse of the spectral sequence is thus ensured once we show the lim_{←}^{1} terms vanish. In fact the restriction maps

H_{L}^{t}(K; M) ⟶ H_{H}^{t}(K; M)

are surjective. Perhaps the quickest way to see this is to note that H_{H}^{∗}(HM′; M) = [HM′, HM]_{H}^{∗} = Hom_{H}(M′, M), for any Mackey functors M′ and M. We may then use the corresponding fact for Mackey functors, that

Hom_{L}(M′, M) ⟶ Hom_{H}(M′, M)

is surjective. This surjectivity is due to the fact that all Weyl groups are connected, and it is easily deduced from Appendix A.

We conclude that if K has all its homotopy functors in even degrees then

H_{𝕋}^{∗}(K ∧ EF_{+}; M) = ∏_{m∈ℤ} lim_{←}_{H} H_{H}^{∗}(Σ^{2m}K; M),

and in particular we can find a map

θ′_{2m} : K ∧ EF_{+} ⟶ Σ^{2m}H(\underline{K}_{2m}^{𝕋})

inducing the identity in π_{2m}^{H}(•) for all finite subgroups H. The map

θ′ : K ∧ EF_{+} ⟶ ⋁_{n∈ℤ}Σ^{2n}H(\underline{K}_{2n}^{𝕋})

is thus an F-equivalence and hence θ is a homotopy equivalence as required.

In Section 15.1 of Part III we shall complete the picture of Atitiyah-Hirzebruch spectral sequences for F-spaces by giving an analysis without hypothesis on the rational cohomology theory. We characterize those theories K_{𝕋}^{∗}(·) for which the spectral sequence always collapses at E_{2}, show that arbitrarily high differentials occur, and give a geometric explanation of them in terms of universal examples. The behaviour of the spectral sequence for arbitrary spaces X is much more complicated.

22 1. INTRODUCTION TO PART I.

CHAPTER 2 First steps.

This chapter introduces all the basic ingredients for our analysis. In Section 2.1 we summarise what is known about the homotopy groups of the cells 𝕋/H_{+}, and use idempotents of the Burnside rings to break them up; in the following section we show this extends to a complete splitting of the category of F-spectra as a product of categories, one for each finite subgroup. In Section 2.3 we analyse the summands E⟨H⟩ of EF_{+}; their simplicity is fundamental to the success of our programme. Finally, in Section 2.4 we identify the ring of self-maps of E⟨H⟩, which acts as operations on the homotopy groups of all 𝕋-spectra over H, as the principal ideal domain ℚ[c_{H}].

2.1. Natural cells and basic cells

The idea is that in any context one should understand complicated objects by first understanding the building blocks and the way they can be stuck together. In practice this involves a complete description of the full subcategory on the basic building blocks. There are inevitably many different choices for the set of basic objects, and for practical purposes it is useful to choose those for which the basic subcategory is as simple as possible. The ideal is typified by representation theory over ℂ where where every morphism between simple modules is a scalar multiplication.

The natural building blocks for locally nice G-spaces are the G-cells G/H_{+} ∧S^{n}. The full category on these is complicated because of the group theory (which enters via the fixed point spaces (G/H)^{K}) and because of the topology (since we need to know the homotopy groups of spheres). Working over the rationals brings the topology under reasonable control, but the group theoretic complication is still considerable. However, if we work stably, we obtain a reasonable supply of idempotents with which to chop up the cells into simpler pieces, for which the residual group theory and topology is much simpler.

It is time to be quite definite, and we return to the case G = 𝕋. This case is particularly favourable for several reasons. Firstly 𝕋 is abelian, secondly there is only one infinite subgroup, and finally all Weyl groups are connected so that there is hardly any group theoretic complication at all. There are two types of natural cells: the 𝕋-fixed cells S^{n} and the cells 𝕋/H_{+} ∧ S^{n} with finite isotropy group. The stable rational maps between them are not hard to calculate, using the standard change of groups isomorphisms and tom Dieck splitting. Here and elsewhere we shall refer always to homology grading unless otherwise specified, and we use the homology suspension (Σ^{t}M)_{n} = M_{n−t}.

Lemma 2.1.1. (a) The graded self maps of the fixed cell S^{0} are

[S^{0}, S^{0}]_{∗}^{𝕋} = ℚ ⊕ Σ(ℚF [c^{−1}])

where ℚF denotes a rational vector space with basis F, and c^{−1} is an element of degree 2.

(b) The other maps involving the fixed cell are

[S^{0}, 𝕋/H_{+}]_{∗}^{𝕋} = ΣA(H)

and

[𝕋/H_{+}, S^{0}]_{∗}^{𝕋} = A(H)

where A(H) is the rational Burnside ring of H.

(c) Maps between two cells with finite isotropy groups are

[𝕋/H_{+}, 𝕋/K_{+}]_{∗}^{𝕋} = A(H ∩ K) ⊕ ΣA(H ∩ K)

For the present c^{−1} is simply a notational device, but we shall eventually introduce an operation c of degree −2 which will act as the notation suggests. Proof: The tom Dieck splitting theorem [18, V.9.1] states that for any 𝕋-space X

π_{∗}^{𝕋}(X) = π_{∗}(X) ⊕ ⨁_{H} π_{∗}(E(𝕋/H)_{+} ∧_{𝕋/H} X^{H}).

Part (a) follows, as does Part (b).

For Parts (b) and (c) we use the change of groups isomorphism

[𝕋 ∧_{H} X, Y]_{∗}^{𝕋} ≅ [X, Y]_{∗}^{H}

and the Wirthmüller isomorphism [18, II.6.5]

[Y, 𝕋 ∧_{H} X]_{∗}^{𝕋} ≅ [Y, S^{1} ∧ X]_{∗}^{H},

where X is an H-spectrum and Y is a 𝕋-spectrum. To complete the proof of (c) we note that as an H-spectrum 𝕋/K_{+} is S(U)_{+} where U is a representation with kernel L = H ∩K. Now there is a cofibre sequence of H-spaces

H/L_{+} ⟶ H/L_{+} ⟶ S(U)_{+}

where the first map is 1 − g for some generator g of the cyclic group H/L. Since the group is abelian g induces the identity in π_{0}^{H}, and Part (c) follows.

The point to notice is that there is vertical complication (i.e. there are nontrivial maps of positive degree) and there is horizontal complication (i.e. the maps of degree zero do not form a simple ring). The vertical complexity is topological in origin, coming from the facts that 𝕋 is of dimension 1 and that B𝕋 has positive dimensional rational cohomology. This is unavoidable, and indeed it is the source of the interesting mathematics. On the other hand we may eliminate the horizontal complexity simply by using simpler building blocks.

For this we need idempotents from the Burnside rings. Recall that for any finite subgroup H we have A(H) = [S^{0}, S^{0}]^{H} and the ring isomorphism ϕ : A(H) ⟶^{≅} ∏_{K⊆H} ℚ whose Kth coordinate takes f : S^{0} ⟶ S^{0} to the degree of the fixed point map Φ^{H}(f) : S^{0} ⟶ S^{0}, where Φ^{H} is the geometric H-fixed point functor. We let e_{K} denote the idempotent which is 1 on the Kth factor and 0 on the others. This notation is consistent with restriction in the sense that if K ⊆ H′ ⊆ H then res_{H′}^{H}e_{K} = e_{K}. Notice that for any H-spectrum X we have e_{K}X = (e_{K}S^{0}) ∧ X and hence we have non-equivariant equivalences Φ^{L}(e_{K}X) ≃ ∗ if L ≠ K, whilst Φ^{K}e_{K}X = Φ^{K}X.

Definition 2.1.2. The basic cells are the 𝕋-spectra

σ_{𝕋}^{n} = S^{n}
σ_{H}^{n} = 𝕋_{+} ∧_{H} e_{H}S^{n}

for various integers n and finite subgroups H of 𝕋.

We shall need a couple of easy observations about basic cells.

Lemma 2.1.3. (i) The duals of the basic cells are as follows: Dσ_{𝕋}^{0} = σ_{𝕋}^{0} and Dσ_{H}^{0} ≃ Σ^{−1}σ_{H}^{0}. (ii) If K ⊆ H there is an equivalence

σ_{K}^{0} ≃ 𝕋_{+} ∧_{H} e_{K}S^{0}.

(iii) The natural cells can be obtained from the basic cells by the formula

𝕋/H_{+} = ⋁_{K⊆H} σ_{K}^{0}.

Note in particular that this means that 𝕋//F_{+} = ⋁_{H} σ_{H}^{0}, which completes the connection with Section 1.4 on McClure’s theorem. Proof: (i) The duality result follows from the fact that any map f : S^{0} ⟶ S^{0} is self-dual. (ii) The projection H/K_{+} ⟶ S^{0} is an equivalence on K fixed points, so that it induces an equivalence e_{K}H/K_{+} ⟶^{≃} e_{K}S^{0}. Now by consistency of e_{K} under restriction,

e_{K}H/K_{+} ≃ e_{K}(H_{+} ∧_{K} S^{0}) ≃ H_{+} ∧_{K} e_{K}S^{0},

so that H_{+} ∧_{K} e_{K}S^{0} ≃ e_{K}S^{0}. Inducing up to 𝕋, we find σ_{K}^{0} ≃ 𝕋_{+} ∧_{K} e_{K}S^{0}, as required. (iii) For the splitting we choose an H-equivalence S^{0} ⟶^{≃} ⋁_{K} e_{K}S^{0} corresponding to the splitting A(H) ⟶^{≅} ∏_{K} ℚ and extend it to a 𝕋-equivalence 𝕋/H_{+} = 𝕋_{+} ∧_{H} S^{0} ⟶^{≃} ⋁_{K} 𝕋_{+} ∧_{H} e_{K}S^{0}. Now use Part (ii).

Obviously the analogue of 2.1.1 (a) for basic cells is identical, but Parts (b) and (c) are much simpler.

Lemma 2.1.4. (a) The graded self maps of the fixed cell σ_{𝕋}^{0} are

[σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{∗}^{𝕋} = ℚ ⊕ Σ(ℚF [c^{−1}])

where ℚF denotes a rational vector space with basis F, and c^{−1} is an element of degree 2.

(b) The other maps involving the fixed cell are

[σ_{𝕋}^{0}, σ_{H}^{0}]_{∗}^{𝕋} = Σℚ

and

[σ_{H}^{0}, σ_{𝕋}^{0}]_{∗}^{𝕋} = ℚ

(c) Maps between two cells with finite isotropy groups are

[σ_{H}^{0}, σ_{H}^{0}]_{∗}^{𝕋} = ℚ ⊕ Σℚ

and

[σ_{H}^{0}, σ_{K}^{0}]_{∗}^{𝕋} = 0

if H ≠ K.

Proof: The result for [σ_{H}^{0}, σ_{𝕋}^{0}]_{∗}^{𝕋} is clear from the definitions, and the result for [σ_{𝕋}^{0}, σ_{H}^{0}]_{∗}^{𝕋} follows by duality. Part (c) follows by applying the idempotent e_{H} to the proof of 2.1.1 (c).

There are certain features worth special attention, beyond the fact that there are no non-trivial maps σ_{H}^{0} ⟶ σ_{K}^{0} if H ≠ K. Firstly notice that except for degree zero the graded ring [σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{∗}^{𝕋} is entirely in odd degrees, and hence its product structure is trivial. The second feature is the fact that every non-zero map out of σ_{𝕋}^{0} is an isomorphism: this is responsible for the finite dimensionality of the category of 𝕋 -Mackey functors.

The curious reader may wish to read Appendix A; this will complete the justification of Section 1.4, but will otherwise not be needed until Part 2. We also pursue the analysis of the graded category of stable maps between cells a little further in Section 15.2.

2.2. Separating isotropy types.

Building on the observations on cells in the Section 2.1, we pull the whole category of rational 𝕋-spectra apart on similar lines. Obviously the fixed cell σ_{𝕋}^{0} behaves quite differently from the others, so it is appropriate to separate 𝕋 -fixed phenomena from Fphenomena. The success of the present method depends on the fact that, in addition, the effects of different finite subgroups are completely decoupled. This idea is firmly based in geometric intuition since we are separating a spectrum X into parts X(H) which only have geometric fixed points at H (i.e. Φ^{K}X(H) is non-equivariantly contractible if K ≠ H).

Let 𝕋−Spec be the stable homotopy category of rational 𝕋 -spectra, and consider the full subcategory 𝕋−Spec/𝕋 of F-contractible 𝕋 -spectra X (i.e. with Φ^{H}X non-equivariantly contractible for all finite subgroups H), and the full subcategory 𝕋−Spec/F of F-spectra. We view the category 𝕋−Spec as being generated by these two subcategories. More formally we consider the isotropy separation cofibering

EF_{+} ∧ X ⟶ X ⟶ ẼF ∧ X :

the last term ẼF ∧ X lies in 𝕋−Spec/𝕋 since Φ^{H}(ẼF ∧ X) ≃ (Φ^{H} ẼF) ∧ (Φ^{H}X), which is non-equivariantly contractible for all finite subgroups H, and the first term EF_{+} ∧ X is visibly an F-spectrum. We thus let X(𝕋) = ẼF ∧ X and X(F) = EF_{+} ∧ X. Our approach is to analyze the categories 𝕋−Spec/𝕋 and 𝕋−Spec/F rather completely and then study how to assemble them to make 𝕋−Spec. The analysis of 𝕋−Spec/F forms the main content of the work, and it begins by observing that the decoupling implies that 𝕋−Spec/F ≃ ∏_{H} 𝕋−Spec/H where 𝕋−Spec/H is the full subcategory on 𝕋 -spectra X with Φ^{K}X non-equivariantly contractible if K ≠ H. In particular X(F) ≃ ⋁_{H} X(H) in 𝕋−Spec/F, for suitable spectra X(H) in 𝕋−Spec/H. In a sense to be made precise below 𝕋−Spec/H has global dimension 1 and is thus easily controlled.

The analysis of 𝕋−Spec/𝕋 is straightforward obstruction theory, together with the classical analysis of the nonequivariant category of rational spectra. The obstruction theory is easily done integrally.

Proposition 2.2.1. (i) Φ^{H}X is non-equivariantly contractible for all finite subgroups H if and only if the inclusion X = X ∧ S^{0} ⟶ X ∧ ẼF is a 𝕋-homotopy equivalence. (ii) Maps between objects of 𝕋−Spec/𝕋 are calculated by

[ẼF ∧ X, ẼF ∧ Y]_{∗}^{𝕋} = [Φ^{𝕋}X, Φ^{𝕋}Y]_{∗}.

For a vector space W of dimension d we let X[W] be a d-fold wedge of copies on X, and if W is a graded vector space we let X[W] = ⋁_{n} Σ^{n}X[W_{n}]. Since the nonequivariant rational spectrum Φ^{𝕋}X is equivalent to a wedge S^{0}[π_{∗}(Φ^{𝕋}X)] of spheres, we find X(𝕋) ≃ ẼF ∧ S^{0}[π_{∗}(Φ^{𝕋}X)].

Corollary 2.2.2. The functor Φ^{𝕋} induces an equivalence of categories 𝕋−Spec/𝕋≃ Graded vector spaces.

If the equivalence is also to preserve cofibre sequences, one must use the derived category of differential graded vector spaces on the right hand side.

Now return to the analysis of F-spectra. The idea is that any F-spectrum may be formed from natural cells with finite isotropy groups and hence from basic cells with finite isotropy groups. Since basic cells with one isotropy type cannot be non-trivially attached to those of another the splitting follows.

Theorem 2.2.3. (Decoupling of finite isotropy groups.)

(i) For any F-spectrum X and each finite subgroup H of 𝕋 there is a spectrum X(H) in 𝕋−Spec/H formed from basic cells with isotropy H so that there is an equivalence

X ⟶^{≃} ⋁_{H} X(H).

(ii) Given two F-spectra X and Y the natural map

∏_{H}[X(H), Y (H)]^{𝕋} ⟶^{≅} [X, Y]^{𝕋}

is an isomorphism.

To obtain a functorial construction we use the universal example EF_{+}. Here we let E⟨H⟩ = EF_{+}(H), and we may describe it more geometrically. Indeed we may let F^{k} denote the family of finite subgroups of order ≤ k, and hence obtain the filtration

E𝕋_{+} = EF_{+}^{1} ⊆ EF_{+}^{2} ⊆ EF_{+}^{3} ⊆ ⋯ ⊆ ⋃_{k} EF_{+}^{k} = EF_{+}.

The cofibre of EF_{+}^{k−1} ⟶ EF_{+}^{k} is the space E⟨H⟩ for H of order k. Note in particular that E⟨1⟩ = E𝕋_{+}, but that in general E⟨H⟩ is not of the form X_{+} for any space X. The content of 2.2.3 is that the filtration splits rationally (see [12] for the generalization to arbitrary compact Lie groups).

Now since any F-spectrum X is equivalent to EF_{+} ∧ X and EF_{+} ≃ ⋁_{H} E⟨H⟩, one then finds X ≃ ⋁_{H} E⟨H⟩ ∧ X. Since Φ^{K} commutes with smash products it follows that X(H) ≃ E⟨H⟩ ∧ X. In any case we obtain the categorical consequence we need.

Corollary 2.2.4. There is an equivalence of triangulated categories

𝕋−Spec/F ≃ ∏_{H} 𝕋−Spec/H

induced by the functors X ⟼ (X(H))_{H} and (X(H))_{H} ⟼ ⋁_{H} X(H).

We turn to the proof of 2.2.3. The basic tool is the following.

Lemma 2.2.5. If H is a finite subgroups of 𝕋 and if X is formed from basic cells σ_{H}^{n} with isotropy H and if Y is a F-spectrum formed using basic cells σ_{K}^{n} with isotropy K ≠ H then

[X, Y]^{𝕋} = 0 and X ∧ Y ≃ ∗.

Proof: The fact that [σ_{H}^{0}, Y]^{𝕋} = 0 follows by induction on the number of cells of Y and passage to direct limits. The fact that [X, Y]^{𝕋} = 0 now follows by wedges cofibres and limits in the X variable.

To see that X ∧ Y ≃ ∗ note that D(σ_{H}^{0}) ≃ Σ^{−1}σ_{H}^{0} so that σ_{H}^{0} ∧ σ_{K}^{0} ≃ ∗ and argue by induction and limits.

Proof of 2.2.3: If 2.2.3 (i) holds for X, we shall say X splits, and we note first that Part (ii) of the 2.2.3 holds for any pair of F-spectra X and Y which both split. Indeed we have

[⋁_{H} X(H), ⋁_{K} Y (K)]^{𝕋} = ∏_{H}[X(H), ⋁_{K} Y (K)]^{𝕋}.

Since [X(H), ⋁_{K≠H} Y (K)]^{𝕋} = 0 by 2.2.5 we conclude

[⋁_{H} X(H), ⋁_{K} Y (K)]^{𝕋} = ∏_{H}[X(H), Y (H)]^{𝕋}

as required.

We now turn to the proof of Part (i), in which the basic ingredients are the splitting of cells as in 2.1.3, and an inductive step as follows.

Lemma 2.2.6. If X ⟶ Y ⟶ Z is a cofibre sequence of F-spectra and both X and Y split, so too does Z.

2.3. THE SINGLE STRAND SPECTRA E⟨H⟩. 29

Before proving 2.2.6, we note how the theorem follows. Indeed any F-spectrum has an inductive filtration so that the subquotients are natural F-cells. In other words X ≃ holim_{→}_{n} X_{n} where X_{0} = ∗ and where there are cofibre sequences C_{n} ⟶ X_{n} ⟶ X_{n+1} where C_{n} is a wedge of natural cells with finite isotropy. The result therefore follows for X_{n} by induction on n, and since there is a cofibre sequence

⋁_{n} X_{n} ⟶ ⋁_{n} X_{n} ⟶ holim_{→}_{n} X_{n}

it also follows for X. It thus remains only to prove 2.2.6. Proof of 2.2.6: By hypothesis X and Y both split, so we may form the diagram

X  ⟶  Y  ⟶ Z
≃↓  ≃↓
⋁_{H} X(H) ⟶^{f} ⋁_{K} Y (K)  .

By Part (ii) of the theorem f ≃ ⋁_{H} f(H) for suitable maps f(H) : X(H) ⟶ Y (H). It follows that Z ≃ ⋁_{H} Z(H) where Z(H) is the cofibre of f(H). This cofibre is of the required form by the cellular approximation theorem for basic cells.

2.3. The single strand spectra E⟨H⟩.

It turns out that E⟨H⟩ and its skeleta are very basic objects of 𝕋−Spec/H. Furthermore since Y ≃ Y ∧ E⟨H⟩ for any object Y of 𝕋−Spec/H, it follows that [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} acts as an algebra of operations on objects of 𝕋−Spec/H. It is therefore fitting to spend a little time considering maps between the skeleta of E⟨H⟩. Topologists may like to think of E⟨H⟩ as a version of ℂP_{+}^{∞} massively simplified by the facts that η^{2} = 0, and James periodicity is trivial.

From the construction of E⟨H⟩ as EF_{+}(H) we find its basic structure.

Lemma 2.3.1. E⟨H⟩ may be constructed with a single basic cell σ_{H}^{2m} in each even dimension:

E⟨H⟩ = σ_{H}^{0} ∪ σ_{H}^{0} ∧ e^{2} ∪ σ_{H}^{0} ∧ e^{4} ∪ ….

Using this cell structure we may consider the skeleta E⟨H⟩^{(2m)}. These are the fundamental objects of 𝕋−Spec/H. Since we only have contributions from 𝕋-isotropy and H-isotropy we should identify the H-homotopy types.

Proposition 2.3.2. (a) E⟨H⟩ is H-equivariantly equivalent to e_{H}S^{0}.

(b) E⟨H⟩^{(2m)} is H-equivariantly equivalent to e_{H}(S^{0} ∨ S^{2m+1})

Proof: For Part (a) we apply e_{H} to the H-equivalences S^{0} ≃ EF_{+}|_{H} ≃ ⋁_{K} E⟨K⟩|_{H}.

Now for Part (b) we argue by induction on m. Indeed the case m = 0 states that σ_{H}^{0} is H-equivariantly e_{H}(S^{0} ∨ S^{1}), which follows by applying e_{H} to the H-equivalence G/H_{+}|_{H} ≃ S^{0} ∨ S^{1} (which is true at the unstable level). The fact that the attaching map is an H-equivariant equivalence on the common cell follows from (a).

For objects X of 𝕋−Spec/H the non-trivial part of the Mackey functor \underline{π_{n}^{𝕋}}(X) comes from π_{n}^{𝕋}(X) = [S^{0}, X]_{n}^{𝕋} and π_{n}^{H}(X). Indeed by 2.1.3

π_{n}^{K}(X) = [G/K_{+}, X]_{n}^{𝕋} = [ ⋁_{L⊆K} σ_{K}^{0}, X]_{n}^{𝕋}

and this is [σ_{H}^{0}, X]_{n}^{𝕋} if H ⊆ K and 0 otherwise. A form of the Whitehead theorem follows.

Lemma 2.3.3. (Naive Whitehead theorem in 𝕋−Spec/H.) If X and Y are F-spectra over H, a map f : X ⟶ Y is an equivalence provided it is a π_{∗}^{H} isomorphism.

Proof: Since X and Y are F-spectra it is enough to verify that f induces an isomorphism in π_{∗}^{K} for all finite subgroups K; by the above discussion this follows from the particular case H = K.

We record the homotopy groups of the skeleta of E⟨H⟩.

Lemma 2.3.4. (a) For any natural number k

π_{∗}^{𝕋}(E⟨H⟩^{(2k)}) = ⨁_{0≤i≤k} Σ^{2i+1}ℚ and π_{∗}^{H}(E⟨H⟩^{(2k)}) ≅ ℚ ⊕ Σ^{2k+1}ℚ.

(b)

π_{∗}^{𝕋}(E⟨H⟩) = ⨁_{i≥0} Σ^{2i+1}ℚ and π_{∗}^{H}(E⟨H⟩) ≅ ℚ.

Proof: The values of π_{∗}^{H} are immediate from 2.3.2. The 𝕋-homotopy groups are easily calculated from the cell structure by induction, using the fact that π_{∗}^{𝕋}(σ_{H}^{0}) = Σℚ.

It follows that each of the attaching maps

σ_{H}^{2m+1} ⟶ E⟨H⟩^{(2m)} ⟶ E⟨H⟩^{(2m)}/E⟨H⟩^{(2m−2)} = σ_{H}^{2m}

is nontrivial in [σ_{H}^{2m+1}, σ_{H}^{2m}]^{𝕋} = ℚ.

The following uniqueness result will have fundamental implications.

Proposition 2.3.5. Let k be a natural number or ∞, and let X be a 𝕋-spectrum of the form X = σ_{H}^{0} ∪ σ_{H}^{0} ∧ e^{2} ∪ ⋯ ∪ σ_{H}^{0} ∧ e^{2k}. If the attaching maps σ_{H}^{2n+1} ⟶ X^{2n} ⟶ X^{2n}/X^{2n−2} = σ_{H}^{2n} are nontrivial for all n then X is equivalent to E⟨H⟩^{(2k)} .

Three particular cases are of special interest.

Corollary 2.3.6. (a)(Thom): If V (H) denotes a one dimensional representation with kernel H, there is an equivalence

E⟨H⟩ ∧ S^{V (H)} ≃ E⟨H⟩ ∧ S^{2}.

(b) (James): For any natural number m, and for 0 ≤ k ≤ ∞ there is an equivalence

E⟨H⟩^{(2m+2k)}/E⟨H⟩^{(2m−2)} ≃ Σ^{2m}E⟨H⟩^{(2k)}.

(c) (Atiyah): For any natural number m there is an equivalence

D(E⟨H⟩^{(2m)}) ≃ Σ^{−2m−1}E⟨H⟩^{(2m)}.

Part (a) is the statement that there are universal Thom isomorphisms for 𝕋-spectra over H, Part (b) is the strong analogue of James periodicity, and Part (c) is the straightforward analogue of Atiyah duality. We warn that DE⟨H⟩ is not an object of 𝕋−Spec/H, and in particular its form is quite different from the pattern suggested in Part (c). Proof: By the naive Whitehead theorem 2.3.3, it suffices to construct a map f : X ⟶ E⟨H⟩^{(2k)} which is an isomorphism in [σ_{H}^{0}, ·]_{∗}^{𝕋}. We do this by obstruction theory, defining f on successive skeleta. In fact define f on X^{(0)} = σ_{H}^{0} to be the inclusion of the 0-cell. By 2.3.2 [σ_{H}^{r}, E⟨H⟩^{(2k)}]^{𝕋} = 0 for 0 < r < 2k + 1 and so this extends uniquely to a map f.

Now it is easy to calculate π_{∗}^{H}(X^{2n}) by induction on n and see that π_{∗}^{H}(X) ≅ π_{∗}^{H}(E⟨H⟩^{(2k)}). The map f is certainly an isomorphism in π_{0}^{H}, and to see it is an isomorphism in π_{2k+1}^{H} we follow through the construction, supposing by induction on n that f_{n} gives an equivalence X^{(2n)} ≃ E⟨H⟩^{(2n)}. One extension to f_{n+1} may be obtained by completing the map of cofibre sequences

σ_{H}^{2n+1} ⟶ X^{(2n)} ⟶  X^{2n+2}
↓  ≃↓
σ_{H}^{2n+1} ⟶ E⟨H⟩^{(2n)} ⟶ E⟨H⟩^{(2n+2)},

in which the left hand vertical is an equivalence by hypothesis on X; by uniqueness this gives the only possibility.

The main use of the Thom isomorphism is to give Euler classes. In fact we shall refer

to the composite E⟨H⟩ = E⟨H⟩ ∧ S^{0} ⟶^{1∧e(V)} E⟨H⟩ ∧ S^{V (H)} ≃ E⟨H⟩ ∧ S^{2} as the Euler class

c_{H}. It induces a map c_{H} : X ⟶ X ∧ S^{2} for any object X of 𝕋−Spec/H. More exactly the operation c_{H} is the part of the Euler class appearing in 𝕋−Spec/H; the Euler class for V (H) makes contributions to 𝕋−Spec/K for all subgroups K ⊆ H. If K is of order d it is perhaps best to view c_{K} as analogous to the dth cyclotomic polynomial Φ_{d}(z). In fact by analogy with 1 − z^{n} = ∏_{d|n} Φ_{d}(z), if H is of order n we have c(V (H)) acting on the category of spectra over K as c_{K} if K ⊆ H, and as 1 otherwise. We shall discuss this global behaviour of Euler classes futher in Section 8.1, but for the present we need only worry about the behaviour of Euler classes over one subgroup, and we now may make this more precise. Indeed, if we define Ẽ⟨H⟩ by the cofibre sequence E⟨H⟩ ⟶ S^{0} ⟶ Ẽ⟨H⟩ it behaves quite analogously to ẼF.

Lemma 2.3.7. There are equivalences

(i) S^{∞V (H)} ∧ E⟨H⟩ ≃ ∗ (ii) S(∞V (H))_{+} ∧ E⟨H⟩ ≃ E⟨H⟩ (iii) Ẽ⟨H⟩ ∧ S^{∞V (H)} ≃ S^{∞V (H)}.

Proof: Part (i) is clear by considering the various Φ-fixed points, and the other parts follow from the defining cofibre sequences.

The other useful fact is that the skeleton Ẽ⟨H⟩^{(2k−1)} behaves on 𝕋−Spec/H rather like the spheres S^{kV (H)}; we therefore use the notation σ^{kV (H)} = Ẽ⟨H⟩^{(2k−1)}.

Corollary 2.3.8. (i) For any object Y of 𝕋−Spec/H there are equivalences Y ∧σ^{kV (H)} ≃ Y ∧ S^{2k}. (ii) The inclusions σ^{kV (H)} ⟶ σ^{(k+1)V (H)} induce multiplication by c_{H} in the homotopy of any object of 𝕋−Spec/H.

Proof: Both parts follow from the corresponding facts for S^{kV (H)}. Indeed we have a cofibre sequence S(kV (H))_{+} ⟶ S^{0} ⟶ S^{kV (H)} and the first term splits as ⋁_{K⊆H} E⟨K⟩^{(2k−1)}. Only the term E⟨H⟩^{(2k−1)} survives smashing with an object of 𝕋−Spec/H.

2.4. Operations.

As remarked above, for any object X of 𝕋−Spec/H, we have an equivalence X ≃ X ∧ E⟨H⟩ and hence [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} acts on X, and on all homology and cohomology theories on 𝕋−Spec/H. The best thing of all is that this ring of operations is the principal ideal domain ℚ[c_{H}], and is thus of global dimension 1. The aim of the next chapter is to construct an Adams spectral sequence based on these operations, which has the uncharacteristically simple form of a short exact sequence.

We begin by calculating the ring of operations.

Theorem 2.4.1. (Homotopy operations in 𝕋−Spec/H.) The ring of self-maps of E⟨H⟩ is

[E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} = ℚ[c_{H}].

where c_{H} (of degree −2) is the Euler class of V (H) in the sense that it is the composite

E⟨H⟩ = E⟨H⟩ ∧ S^{0} ⟶^{1∧e(V)} E⟨H⟩ ∧ S^{V (H)} ≃ E⟨H⟩ ∧ S^{2}.

Proof: We calculate the additive structure of [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} by passage to limits from the case of finite skeleta. Of course we know that [σ_{H}^{0}, E⟨H⟩]_{∗}^{𝕋} = ℚ so that by induction on k we see that [E⟨H⟩^{(2k)}, E⟨H⟩]_{∗}^{𝕋} is nonzero only in degrees 0, −2, −4, …, −2k, in each of which it is ℚ. The additive result follows using the Milnor exact sequence.

Choose a generator c_{H} : E⟨H⟩ ⟶ Σ^{2}E⟨H⟩ of [E⟨H⟩, E⟨H⟩]_{−2}^{𝕋}; for the multiplicative statement it is sufficient to show that all composites c_{H}^{n} for n ≥ 0 are essential. For this it is enough to show that c_{H} induces an isomorphism in π_{2k}^{𝕋} for k ≥ 1. This follows by the argument used in the proof of 2.3.5, since c_{H} is the unique extension of the composite E⟨H⟩^{(2)} ⟶ E⟨H⟩^{(2)}/E⟨H⟩^{(0)} ≃ σ_{H}^{2} ⟶ Σ^{2}E⟨H⟩, in which the first map is the collapse map and the last is the inclusion of the bottom cell.

To see that c_{H} may be chosen to be the Euler class it is enough to observe that the Euler class is nontrivial in π_{2}^{𝕋}. But the cofibre of the Euler class map is ΣE⟨H⟩ ∧ G/H_{+} ≃ σ_{H}^{1}, which has π_{∗}^{𝕋} concentrated in degree 2.

For us the fundamental object will be the ℚ[c_{H}]-module π_{∗}^{𝕋}(X); notice that c_{H} decreases the degree of a homotopy element by 2.

Theorem 2.4.2. For any object X of 𝕋−Spec/H, the ℚ[c_{H}]-module π_{∗}^{𝕋}(X) is torsion in the sense that for any element x ∈ π_{∗}^{𝕋}(X) there is a natural number n so that (c_{H})^{n}x = 0.

Proof: In fact x is supported on a finite 𝕋−Spec/H-subcomplex X′ ⊆ X in the sense that x = i_{∗}x′ for some x′ ∈ π_{∗}^{𝕋}(X′) where i denotes the inclusion. Since π_{∗}^{𝕋}(X′) is bounded below, it is immediate that c_{H}^{n}x′ = 0 for sufficiently large n.

Note in particular that this implies that ℚ[c_{H}] itself does not occur as π_{∗}^{𝕋}(X) for any object X of 𝕋−Spec/H. Accordingly, the function spectrum F (E⟨H⟩, E⟨H⟩) is not an object of 𝕋−Spec/H, so that 𝕋−Spec/H does not have arbitrary products.

We should record the ℚ[c_{H}]-modules for the spectra we have already studied.

Lemma 2.4.3. There are ℚ[c_{H}]-module isomorpisms

π_{∗}^{𝕋}(E⟨H⟩) = Σ𝕀(H),

where 𝕀(H) = Σ^{−2}ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}] and

π_{∗}^{𝕋}(E⟨H⟩^{(2k)}) = Σ(ℚ[c_{H}]/c_{H}^{k+1}).

Proof: The first calculation was part of the proof of 2.4.1, and the second follows since the inclusion E⟨H⟩^{(2k)} ⟶ E⟨H⟩ induces an isomorphism of π_{i}^{𝕋} for i ≤ 2k + 1.

The fundamental fact here is that π_{∗}^{𝕋}(E⟨H⟩) is ℚ[c_{H}]-injective, and this explains the notation. Indeed the ℚ[c_{H}]-module Σ^{2}𝕀(H) = ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}] is c_{H}-divisible and hence injective. We have chosen to name 𝕀(H) rather than its double suspension because 𝕀(H) is concentrated in degrees 0, 2, 4, 6, . . ..

34 2. FIRST STEPS.

CHAPTER 3 The Adams spectral sequence.

This chapter forms the core of Part I. In it we construct an Adams spectral sequence for the category of 𝕋-spectra over H, and observe that it collapses to a short exact sequence: this gives us control over morphisms between F-spectra. In Section 3.1 we state the theorem and outline the strategy. In Section 3.2 we construct an Adams resolution. In Section 3.3 we prove the Hurewicz and Whitehead theorems, and deduce convergence from them, provided the codomain is bounded below. Finally, in Section 3.4 we are able to deduce the general case, because of the simplicity of the algebra.

3.1. The Adams short exact sequence.

Our aim now is to prove the following theorem.

Theorem 3.1.1. (The Adams short exact sequence.) For any spectra X and Y in 𝕋−Spec/H there is a short exact sequence

0 ⟶ Ext_{ℚ[c_{H}]}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{ℚ[c_{H}]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) ⟶ 0.

We note that one immediate consequence is a Whitehead theorem based on π_{∗}^{𝕋} for 𝕋−Spec/H-spectra. Unlike the naive Whitehead theorem 2.3.3 based on π_{∗}^{H}, this has real content, and will be a major step in the proof (see 3.3.2 below). On the other hand, the resulting theorem allows us to be more quantitative and recover π_{∗}^{H} from π_{∗}^{𝕋}. Indeed we know π_{∗}^{𝕋}(σ_{H}^{0}) = Σℚ so that a little homological algebra gives the result.

Corollary 3.1.2. For any spectrum Y in 𝕋−Spec/H there is a short exact sequence

0 ⟶ π_{∗}^{𝕋}(Y)/c_{H} ⟶ π_{∗}^{H}(Y) ⟶ Σ^{−1}ann(c_{H}, π_{∗}^{𝕋}(Y)) ⟶ 0.

The strategy is the usual one. The result is rather straightforward when π_{∗}^{𝕋}(Y) is injective as a ℚ[c_{H}]-module, and the general case follows by resolving Y using spectra with ℚ[c_{H}]- injective π_{∗}^{𝕋}. The procedure is somewhat complicated in the present case by the fact that S^{0} is not an object of 𝕋−Spec/H and by the fact that 𝕋−Spec/H does not admit arbitrary products.

The pattern of proof is to establish the theorem in the following cases in turn.

3.1.3. Y = E⟨H⟩ and arbitrary X.

3.1.4. Y bounded below of finite type and arbitrary X.

3.1.5. Arbitrary Y and finite X.

3.1.6. Any Y with π_{∗}^{𝕋}(Y) injective and arbitrary X.

3.1.7. The general case.

Of course the case when π_{∗}^{𝕋}(Y) is injective is particularly easy.

Lemma 3.1.8. If the ℚ[c_{H}]-module π_{∗}^{𝕋}(Y) is injective and the functor

π_{∗}^{𝕋} : [X, Y]_{∗}^{𝕋} ⟶ Hom_{ℚ[c_{H}]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y))

is an isomorphism for X = σ_{H}^{0} then it is an isomorphism for all objects X of 𝕋−Spec/H.

Proof: Since π_{∗}^{𝕋}(Y) is ℚ[c_{H}]-injective, the codomain is a cohomology theory of X, and π_{∗}^{𝕋} is a natural transformation of cohomology theories on 𝕋−Spec/H. Since any object of 𝕋−Spec/H can be constructed from cells σ_{H}^{n}, the result follows.

This shows that 3.1.5 implies 3.1.6. It also shows that 3.1.3 follows from a lemma.

Lemma 3.1.9. Passage to homotopy induces an isomorphism

[σ_{H}^{0}, E⟨H⟩]_{∗}^{𝕋} ⟶^{≅} Hom_{ℚ[c_{H}]}(π_{∗}^{𝕋}(σ_{H}^{0}), π_{∗}^{𝕋}(E⟨H⟩)).

Proof: Both sides are simply ℚ in degree 0, and the calculation of π_{∗}^{𝕋}(E⟨H⟩) in 2.4.3 showed that the inclusion σ_{H}^{0} ⟶ E⟨H⟩ induces an isomorphism in π_{0}^{𝕋}.

The other easy step is that 3.1.4 implies 3.1.5, which follows because any map from a finite spectrum X factors through a finite subspectrum, and similarly in algebra. More precisely if X is finite, π_{∗}^{𝕋}(X) is finitely generated and hence [X, Y]_{∗}^{𝕋} = lim_{→}_{α}[X, Y_{α}]_{∗}^{𝕋}, and Hom_{ℚ[c_{H}]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) = lim_{→}_{α} Hom_{ℚ[c_{H}]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y_{α})). Since ℚ[c_{H}] is Noetherian the result for Hom implies the corresponding result for Ext, and by naturality of the short exact sequence we obtain the result for [X, Y]_{∗}^{𝕋} by passage to limits from those of [X, Y_{α}]_{∗}^{𝕋}.

The remaining steps are that 3.1.3 implies 3.1.4 and that 3.1.6 implies 3.1.7. Each of these require a substantial amount of work, and we explain them in the three following sections.

3.2. Construction of the Adams resolution.

In this section and the next we prove that 3.1.3 implies 3.1.4 which is the principal step in the proof of Theorem 3.1.1. The method is the usual one for constructing Adams spectral sequences: we show in this section that there are enough injectives for which the theorem is known, so that we may construct an Adams resolution of Y, and in the next we show that the resulting Adams resolution is convergent. The implication then follows as usual by applying [X, ·]^{𝕋} to the Adams resolution: the E_{2} term is identified by 3.1.3, and convergence follows from that of the resolution.

Suppose then that Y is bounded below and of finite type. We begin by constructing an Adams resolution

Y ⟶ I_{0} ⟶ I_{1} ⟶ I_{2} ⟶ ⋯

in 𝕋−Spec/H, where the associated sequence

3.2.1.

π_{∗}^{𝕋}(Y) ⟶ π_{∗}^{𝕋}(I_{0}) ⟶ π_{∗}^{𝕋}(I_{1}) ⟶ π_{∗}^{𝕋}(I_{2}) ⟶ ⋯

is a resolution of π_{∗}^{𝕋}(Y) by injective ℚ[c_{H}]-modules, and where Theorem 3.1.1 is known to hold for [X, I_{j}]_{∗}^{𝕋}. Of course once 3.1.1 is proved we shall know that there is a resolution stopping at I_{1}, but that is not obvious at present.

In fact we show how to construct a π_{∗}^{𝕋}-monomorphism Y ⟶ I_{0} with I_{0} a bounded below locally finite wedge of suspensions of E⟨H⟩. The next step in the resolution is obtained by applying this to ΣY_{1} = cofibre(Y ⟶ I_{0}), and the rest of the resolution is constructed by iterating the procedure.

We shall use some elementary facts about ℚ[c_{H}]-modules, which we list for convenience. Firstly, any finitely generated ℚ[c_{H}]-module is a direct sum of cyclic modules, each of which is a suspension of either ℚ[c_{H}] itself or a module of form ℚ[c_{H}]/c_{H}^{k} for some k ≥ 0. Next recall that a ℚ[c_{H}]-module M is injective if and only if it is c_{H} divisible. In particular all direct limits of injectives are injective, and the module 𝕀(H) = Σ^{−2}ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}] is injective. Anyway, for any ℚ[c_{H}]-module M and any x ∈ M of degree n there is a ℚ[c_{H}]- morphism ϕ : M ⟶ Σ^{n}𝕀(H) with ϕ(x) ≠ 0: indeed, since 𝕀(H) is injective we may extend a homomorphism from the cyclic submodule ℚ[c_{H}]x over M.

Proposition 3.2.2. If Y is a locally finite and bounded below spectrum in 𝕋−Spec/H, there is a sequence of integers n_{i} tending to infinity with i, and a map

k : Y ⟶ ⋁_{i} Σ^{n_{i}}E⟨H⟩,

inducing a monomorphism in π_{∗}^{𝕋}. Furthermore if π_{∗}^{𝕋}(Y) is (c+1)-connected we may arrange that π_{n}^{𝕋}(k) is an isomorphism for n ≤ c + 3. Consequently the fibre Y_{1} of k is also a locally finite and bounded below spectrum in 𝕋−Spec/H, and if π_{∗}^{𝕋}(Y) is (c+1)-connected π_{∗}^{𝕋}(Y_{1}) is (c + 2)-connected.

The apparently strange indexing will be explained below by the fact that the connectivity of π_{∗}^{𝕋}(Y) is one more than that of Y. Proof: By suspending Y as necessary we may suppose π_{∗}^{𝕋}(Y) is 0-connected.

For any nonzero x ∈ π_{n}^{𝕋}(Y), by 3.1.3 we can find a map k(x) : Y ⟶ Σ^{n−1}E⟨H⟩ so that k(x)_{∗}(x) ≠ 0. Accordingly we let x_{i} run over a ℚ-basis of π_{∗}^{𝕋}(Y) with n_{i} being the degree of x_{i}. By hypothesis, the integers n_{i} tend to infinity, and therefore ⋁_{i} Σ^{n_{i}}E⟨H⟩ ≃ ∏_{i} Σ^{n_{i}}E⟨H⟩. We may thus form the map k by giving it components k(x_{i}):

k : Y ⟶ ∏_{i} Σ^{n_{i}}E⟨H⟩ ≃ ⋁_{i} Σ^{n_{i}}E⟨H⟩.

This certainly gives a monomorphism in π_{∗}^{𝕋}. Since π_{∗}^{𝕋}(E⟨H⟩) = Σ𝕀(H) is ℚ in degree 1, and otherwise concentrated in degrees greater than 2, the construction also gives an isomorphism in the bottom two degrees, π_{1}^{𝕋} and π_{2}^{𝕋}.

3.3. Convergence of the Adams resolution.

The convergence statement says that all of Y is ultimately squeezed into the injective spectra I_{j}. It is convenient to convert the Adams resolution 3.2.1 into the Adams tower

⋮
↓
Y_{2} ⟶ Σ^{−2}I_{2}
↓
Y_{1} ⟶ Σ^{−1}I_{1}
↓
Y = Y_{0} ⟶ I_{0}.

By construction, each of the maps in the tower induces zero in π_{∗}^{𝕋} so that π_{∗}^{𝕋}(holim_{←}_{s} Y_{s}) = 0. However we want a stronger statement.

Theorem 3.3.1. (Convergence.) If Y is a bounded below object of 𝕋−Spec/H, then the Adams tower converges in the sense that holim_{←}_{s} Y_{s} ≃ ∗.

As usual, the Whitehead and Hurewicz theorems are the basis for the proof.

Theorem 3.3.2. (Whitehead theorem for 𝕋−Spec/H.) If X and Y are spectra in 𝕋−Spec/H and f : X ⟶ Y induces an isomorphism in π_{∗}^{𝕋} then f is an equivalence.

Proof: The hypothesis is equivalent to saying the cofibre Z of f is π_{∗}^{𝕋}-acyclic, and it is enough to show Z is contractible. Indeed, by the naive Whitehead theorem 2.3.3, it is enough to show that π_{∗}^{H}(Z) = 0. This follows from the Hurewicz theorem 3.3.3 below.

As remarked in its proof, the Whitehead theorem is the statement that a π_{∗}^{𝕋}-acyclic object of 𝕋−Spec/H is in fact contractible. The Hurewicz theorem is a little more quantitative. The extra suspension in the statement is due to that in the Wirthmüller isomorphism. If X is a c-connected object of 𝕋−Spec/H it may be constructed with cells only in dimensions c+1 and above; since π_{∗}^{𝕋}(σ_{H}^{c+1}) is (c+1)-connected it follows that π_{∗}^{𝕋}(X) is (c+1)-connected.

Theorem 3.3.3. (Hurewicz theorem for 𝕋−Spec/H.) If Z is an object of 𝕋−Spec/H and π_{∗}^{𝕋}(Z) is (c + 1)-connected then Z is c-connected. If in addition π_{c+2}^{𝕋}(Z) ≠ 0 then π_{c+1}^{H}(Z) ≠ 0, so the result is sharp.

Proof: We use the cofibre sequence G/H_{+} ⟶ S^{0} ⟶ S^{V (H)}. By the universal Thom isomorphism 2.3.6(a) [S^{V (H)}, Z]_{∗}^{𝕋} ≅ [S^{2}, Z]_{∗}^{𝕋} and the cofibre sequence gives the Gysin sequence

3.3.4.

… ⟵ π_{n−1}^{𝕋}(Z) ⟵^{c_{H}} π_{n+1}^{𝕋}(Z) ⟵ π_{n}^{H}(Z) ⟵ π_{n}^{𝕋}(Z) ⟵^{c_{H}} π_{n+2}^{𝕋}(Z) ⟵ ….

It follows that π_{∗}^{H}(Z) is c-connected as required.

3.4. TORSION ℚ[c_{H}]-MODULES AND GEOMETRIC IMPLICATIONS. 39

Proof of 3.3.1: First note that the Hurewicz theorem 3.3.3 allows us to upgrade the conclusion of 3.2.2 and deduce that if Y is c-connected then Y_{1} is (c + 1)-connected.

Thus the connectivity of the terms Y_{s} tends to infinity, and therefore ∏_{s} Y_{s} ≃ ⋁_{s} Y_{s}. Since the geometric fixed point functor Φ^{K} commutes with wedges, this product is in 𝕋−Spec/H. Finally the cofibre sequence holim_{←}_{s} Y_{s} ⟶ ∏_{s} Y_{s} ⟶ ∏_{s} Y_{s} shows that the spectrum holim_{←}_{s} Y_{s} lies in 𝕋−Spec/H.

We may therefore apply the Whitehead theorem 3.3.2 to deduce convergence from the fact that π_{∗}^{𝕋}(holim_{←}_{s} Y_{s}) = 0.

We have now established the existence and convergence of an Adams resolution of Y by locally finite bounded below wedges of suspensions of the spectra E⟨H⟩, and we obtain a spectral sequence by applying [X, ·]_{∗}^{𝕋}. We find E_{2}^{s,t} = Extℚ[c_{H}]^{s,t}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) by 3.1.3,

and we find that it converges to [X, Y]_{∗}^{𝕋} by the convergence theorem, 3.3.1. Since ℚ[c_{H}] is of global dimension the spectral sequence takes the form of the short exact sequence of 3.1.1. This completes the proof that 3.1.3 implies 3.1.4.

3.4. Torsion ℚ[c_{H}]-modules and geometric implications.

In this section we prove that 3.1.6 implies 3.1.7 in the proof of Theorem 3.1.1. This completes the proof of Theorem 3.1.1.

In fact it simply remains to exploit the fact that realizable modules are torsion (2.4.2), by showing it implies that any topologically realized ℚ[c_{H}]-module can be embedded in a realizable injective, and that all torsion injectives are topologically realizable.

Proposition 3.4.1. If M is a torsion ℚ[c_{H}]-module then there is an embedding M ⟶ ⨁_{i} Σ^{n_{i}}𝕀(H) for suitable indexing set and integers n_{i}. Furthermore, if M is injective the embedding e may be taken to be an isomorphism.

Proof: We consider the filtration

0 = F_{0}M ⊆ F_{1}M ⊆ F_{2}M ⊆ ⋯ ⊆ ⋃_{j} F_{j}M = M

of M, where F_{j}M = {x ∈ M | c_{H}^{j}x = 0}; this exhausts M because M is a torsion module. Now the associated graded module GrM has trivial c_{H}-action, so it is natural to view it as a graded vector space. We shall construct an embedding e : M ⟶ GrM ⊗ 𝕀(H). Since Hom_{ℚ[c_{H}]}(M, N) = lim_{←}_{j} Hom_{ℚ[c_{H}]}(F_{j}M, N) it is sufficient to give e one filtration at a time, using the notation

e_{j} : F_{j}M ⟶ ( ⨁_{0≤i≤j} F_{i}M/F_{i−1}M) ⊗ 𝕀(H)

and starting with the unique homomorphism when j = 0. Now if we suppose that e_{j} has been defined, we form e_{j+1} as follows. The codomain is injective so e_{j} extends to a map ê_{j} on F_{j+1}M with the same codomain, and we use the canonical embedding F_{j+1}M/F_{j}M ⟶ F_{j+1}M/F_{j}M ⊗ 𝕀(H) as the final coordinate. This final coordinate ensures that no element of F_{j+1}M \ F_{j}M lies in the kernel of e.

Now, if M is injective, we show that e is also an epimorphism. Indeed, if e is not surjective we may choose an element y not in the image, with torsion order as small as possible. More precisely we suppose c_{H}^{k}y = 0 and that the image of e includes all elements annihilated by c_{H}^{k−1}; note that k ≥ 2 by construction of e. By hypothesis, we may choose x so that e(x) = c_{H}y, and since M is injective there is an element x̃ so that c_{H} x̃ = x. Now, let ỹ = e(x̃), so that c_{H}(y − ỹ) = 0. By construction of e there is an x′ with e(x′) = y − ỹ; hence e(x′ + ỹ) = y, contradicting the choice of k.

Corollary 3.4.2. Any injective torsion ℚ[c_{H}]-module I is realizable in the sense that there is a 𝕋-spectrum X(I) over H so that π_{∗}^{𝕋}(X(I)) ≅ I.

We may now complete the proof of Theorem 3.1.1. Indeed for any spectrum Y in 𝕋−Spec/H, the ℚ[c_{H}]-module π_{∗}^{𝕋}(Y) is torsion by 2.4.2, and hence by 3.4.1 embeds in π_{∗}^{𝕋}(⋁_{i} Σ^{n_{i}−1}E⟨H⟩) for suitable indexing set and integers n_{i}. By the general injective case 3.1.6, this algebraic embedding is realized by a map k : Y ⟶ I_{0} of spectra. Since ℚ[c_{H}] has global dimension 1, the cofibre of k has injective π_{∗}^{𝕋} and it is thus reasonable to call it I_{1}. Therefore we have a cofibre sequence Y ⟶ I_{0} ⟶ I_{1} realizing the ℚ[c_{H}]-injective resolution of π_{∗}^{𝕋}(Y), and, by 3.1.6, Theorem 3.1.1 holds for calculating [X, I_{0}]_{∗}^{𝕋} and [X, I_{1}]_{∗}^{𝕋}: Theorem 3.1.1 follows for Y.

Of course we could have constructed a long Adams resolution for arbitrary Y without dealing with the bounded below case first, but the proof of the convergence theorem 3.3.1 would have been obscured by algebraic complications.

Since only the zeroth and first terms I_{0} and I_{1} are required, we shall use the letters I and J to denote them. Although the construction is neither canonical nor functorial we shall use the notation

Y ⟶ I(Y) ⟶ J(Y)

for an Adams resolution of an object Y of 𝕋−Spec/H with I(Y) and J(Y) being wedges of suspensions of E⟨H⟩ and with the induced sequence

0 ⟶ π_{∗}^{𝕋}(Y) ⟶ π_{∗}^{𝕋}(I(Y)) ⟶ π_{∗}^{𝕋}(J(Y)) ⟶ 0

an injective resolution of π_{∗}^{𝕋}(Y).

The Adams short exact sequence 3.1.1 and the realization of injectives 3.4.2 are enough to allow us to make algebraic models, both of 𝕋−Spec/H and of 𝕋−Spec/F. In the next section we begin to establish a suitable abstract framework for this categorical reprocessing, but we shall first state the result we are going to prove. This should also provide a particular case to bear in mind in the course of the coming abstraction.

Consider the homotopy category of differential graded ℚ[c_{H}]-modules. Henceforth we often abbreviate ‘differential graded’ to ‘dg’. Next we form the derived category of dg ℚ[c_{H}]-modules: its objects are dg ℚ[c_{H}]-modules, and its morphisms are obtained from the homotopy category of dg ℚ[c_{H}]-modules by inverting homology isomorphisms. More practically, maps in the derived category from M to N are simply [ΓM, N] = [M, ΔN] where ΓM is projective as a ℚ[c_{H}]-module and equipped with a homology isomorphism ΓM ⟶ M, and ΔN is injective as a ℚ[c_{H}]-module and equipped with a homology isomorphism N ⟶ ΔN. In fact we want to restrict attention to torsion modules, and form the derived category of dg torsion ℚ[c_{H}]-modules. Since there are no projective torsion ℚ[c_{H}]- modules we are forced to use the construction of the derived category in terms of injectives; we have seen that a realizable ℚ[c_{H}]-module N admits a realizable dg ℚ[c_{H}]-module ΔN. More precisely, we shall prove the following theorem.

Theorem 3.4.3. There is an equivalence of triangulated categories between 𝕋−Spec/H and the full subcategory D(torsℚ[c_{H}]) of the derived category of dg ℚ[c_{H}]-modules torsion ℚ[c_{H}]-modules:

𝕋−Spec/H ≃ D(torsℚ[c_{H}]).

The proof will be completed in Section 4.3 after we have developed the necessary algebra.

42 3. THE ADAMS SPECTRAL SEQUENCE.

CHAPTER 4 Categorical reprocessing.

An Adams spectral sequence arises from a homology functor on a triangulated category with values in an abelian category. The algebraic archetype of a triangulated category is the derived category of an abelian category. In this chapter we show that the algebraic simplicity of the categories which concern us allows us to deduce the structure of the triangulated categories of spectra from their abelianizations.

We begin in Section 4.1 by recalling the construction of the derived category in an appropriate form. In Section 4.2 we give abstract conditions (one dimensionality being the most important) under which an abelian category determines a triangulated category of which it is the abelianization, and in Section 4.3 we show this applies to the two cases we have encountered so far: torsion ℚ[c_{H}]-modules and F-finite torsion O_{F}-modules. We show in the next chapter that it also applies to a suitable abelianization of the category of all 𝕋-spectra.

4.1. Recollections about derived categories.

For a differential graded algebra R the associated derived category D(R) is obtained from the category of dg R-modules by forming the category of fractions in which homology isomorphisms are inverted. For a category of fractions to be useful one must give a fairly concrete construction: this has the additional merit of proving its existence. Since most references restrict themselves to derived categories of ungraded algebras, and since we need to discuss derived categories of other abelian categories, we shall spend some time outlining well-known constructions. In this section we concentrate on giving a very concrete approach, but the construction is a special case of Quillen’s construction of the homotopy category of a closed model category [21]. We therefore use terminology consistent with Quillen’s, and give details in Appendix B of how to regard our algebraic model categories as closed model categories.

We shall repeatedly use the following formality in constructing relevant derived categories. This is more familiar to topologists in its dual counterpart, where it is used to construct a category in which weakly equivalent spaces are isomorphic. The dual properties are then the cellular approximation theorem, and two theorems of J.H.C.Whitehead.

Suppose given a category ℂ, a collection S of morphisms which we intend to invert, and a collection I of fibrant objects (which have the character of injectives) such that the following conditions hold.

Conditions 4.1.1. (i) For every object X of ℂ there is an element s : X ⟶ ΔX of S with ΔX ∈ I (ii) If s : I ⟶ I′ is an element of S with I, I′ ∈ I then s is an isomorphism, and (iii) For every object I of I and morphism s : X ⟶ Y in S the induced map s^{∗} : ℂ(Y, I) ⟶ ℂ(X, I) is a bijection.

We may then extend Δ to a functor Δ : ℂ ⟶ ℂ by filling in the diagram

X ⟶ ΔX
f ↓  ↓ Δf
Y ⟶ ΔY

using (iii). Next define a category S^{−1}ℂ by giving it the same objects as ℂ and morphisms S^{−1}ℂ(X, Y) = ℂ(ΔX, ΔY), and a functor l : ℂ ⟶ S^{−1}ℂ by using Δ.

Lemma 4.1.2. The morphism l : ℂ ⟶ S^{−1}ℂ constructed above is the localization of the category ℂ away from S.

Now suppose given a graded algebra R with zero differential, and let ℂ be the homotopy category of differential graded R-modules. More generally, we may suppose given a graded abelian category 𝔸, and let ℂ be the category of differential graded objects of 𝔸. For dg 𝔸objects X and Y, we shall denote the dg abelian group of 𝔸-morphisms by Hom(X, Y), and the group of homotopy classes by [X, Y]; it is useful to note that [X, Y] = H_{0}(Hom(X, Y)).

Of course, a short exact sequence of dg 𝔸-objects induces a long exact sequence in homology. It follows that a cofibre sequence of dg 𝔸-objects also induces a long exact sequence in homology, but cofibre sequences have the advantage that they are preserved by application of Hom(·, Y) and Hom(X, ·), and hence they also induce long exact sequences on applying [·, Y] and [X, ·]. We can only expect short exact sequences of dg 𝔸-objects to induce long exact sequences of [·, Y] if Y consists of injective 𝔸-objects and of [X, ·] if X consists of projective 𝔸-objects.

Now let S be the class of homology isomorphisms and I be the collection of differential graded 𝔸-objects which may be formed from injective 𝔸-objects with zero differential by a finite number of cofibre sequences.

Proposition 4.1.3. If 𝔸 is of finite homological dimension, then with these choices of S and I, Conditions 4.1.1 hold and hence the derived category D(𝔸) may be formed from the homotopy category of dg 𝔸-objects by inverting homology isomorphisms.

Proof: The finiteness of homological dimension is only used to verify Condition (i), so we begin without assumption on 𝔸.

Lemma 4.1.4. If J is an injective 𝔸-object with zero differential and E is exact then [E, J] = 0.

Proof: Suppose f : E ⟶ J is a map of dg 𝔸-objects. Since d = 0 on J it follows that f = 0 on im(d) = ker(d). Hence f factors through E/ker(d); by injectivity of J this map may be extended along the inclusion E/ker(d) ⊆ E, providing a map h : E ⟶ J of 𝔸-objects. Since d = 0 on J we find dh + hd = f, so that f is null-homotopic.

Corollary 4.1.5. Conditions 4.1.1 (ii) and (iii) hold for these choices of ℂ, I and S.

Proof: We first show Condition (iii) holds. If f : X ⟶ Y is a homology isomorphism, its cofibre E is exact. Since a cofibre sequence induces a long exact sequence in the first variable it is enough to show [E, I] = 0 for all objects I in I. Since I may be formed from injective 𝔸-objects using finitely many cofibre sequences this follows from Lemma 4.1.4 using exact sequences in the second variable.

Now for Condition (ii) we must find an inverse t to s : I ⟶ I′. By Condition (iii) we have a bijection s^{∗} : [I′, I] ⟶^{≅} [I, I], and similarly with I′ in the second variable. Now choose t so that s^{∗}t = id: this ensures ts = id, and since s^{∗}(st) = s^{∗}(id) we conclude that st = id as well.

Let us now turn to Condition (i) and begin with a useful observation.

Lemma 4.1.6. If X ⟶ Y ⟶ Z is a cofibre sequence and two of the three objects are admit homology isomorphisms to an object of I, then so does the third.

Proof: Since the property is invariant under isomorphism and suspension we may as well suppose that there are homology isomorphisms e : X ⟶ I and f : Y ⟶ J; it is enough by Verdier’s axiom and the 5-lemma to show that we may complete the square up to homotopy

X ⟶ Y
e ↓  ↓ f
I ⋯ > J.

But this is immediate from Condition (iii).

The analogue for short exact sequences now follows.

Corollary 4.1.7. If 0 ⟶ X ⟶^{i} Y ⟶^{q} Z ⟶ 0 is a short exact sequence and two of the three objects admit homology isomorphisms to an object of I, then so does the third.

Proof: The main point is that by Condition (iii), if we have a homology isomorphism A ⟶ B and one of the two objects admits a homology isomorphism to an object of I, so too does the other.

If X is the term not known to admit a homology isomorphism to an object of I, we note that there is a homology isomorphism X ⟶ F (q), where F (q) denotes the mapping fibre of q, and apply 4.1.6. Similarly for Z we use the homology isomorphism C(i) ⟶ Z where C(i) denotes the mapping cone of i.

Finally for Y we use the short exact sequence 0 ⟶ Y ⟶ C(i) ⟶ ΣX ⟶ 0, the homology isomorphism C(i) ⟶ Z and the first case.

Lemma 4.1.8. If 𝔸 is of finite homological dimension then Condition 4.1.1 (i) also holds.

Proof: An arbitrary dg 𝔸-object X has dg sub-object dX of boundaries, and both dX and X/dX have trivial differential. Accordingly, by 4.1.7 and the above remarks, it is sufficient to deal with the case that X has zero differential.

For this we argue by induction on the length of injective resolution. Indeed for an injective object we may use the identity and any 𝔸-object X of positive injective dimension lies in an exact sequence 0 ⟶ X ⟶ I ⟶ Q ⟶ 0, with I injective and Q of lower injective dimension.

This completes the proof of Proposition 4.1.3

Finally it will be convenient to begin the construction of an Adams spectral sequence for objects of D(𝔸). Consider the natural transformation

θ : [X, Y]_{derived} ⟶ Hom_{𝔸}(H_{∗}(X), H_{∗}(Y)).

Lemma 4.1.9. If H_{∗}(Y) is injective then θ is an isomorphism.

Proof: First note that if I is an injective object of 𝔸, regarded as a dg 𝔸-object with zero differential, then

[X, I] = H_{0}(Hom(X, I)) = Hom(H_{∗}(X), I).

Since [X, I] = [X, I]_{derived} when I is injective by construction, this proves the result if Y has zero differential. It also shows that if H_{∗}(Y) is injective, then there is a map Y ⟶ H_{∗}(Y) inducing the identity in homology. Thus [X, Y]_{derived} = [X, H_{∗}(Y)]_{derived}, and the general case follows.

Following the abuse of notation standard in topology we henceforth omit the subscript derived. Thus [X, Y] now denotes maps in the derived category.

4.2. Split linear triangulated categories.

The aim of this section is to show that a sufficiently simple triangulated category is determined by its abelianization. We treat this abstractly for the usual reasons: firstly it becomes apparent exactly what properties are being used, and secondly there are three instances of the phenomenon that we want to treat simultaneously.

Thus we suppose given a triangulated category 𝔻, which might be a derived category of some abelian category or a category of spectra; we shall use X, Y, … to denote objects of 𝔻, and [X, Y] to denote morphisms. We also suppose given a graded abelian category 𝔸, and a functor H_{∗} : 𝔻 ⟶ 𝔸, exact in the sense that triangles are taken to long exact sequences; in the applications H_{∗} is some form of homology. The idea is that H_{∗} is a linearization, so that we require in particular the Inverse Function Theorem holds in the sense that homology isomorphisms are invertible (i.e. that H_{∗} creates isomorphisms).

The example to bear in mind at present is the abelian category 𝔸 = torsℚ[c_{H}]-mod, and the triangulated category 𝔻 = 𝕋−Spec/H, with π_{∗}^{𝕋} providing the linearization.

One can hope to incorporate non-linear phenomena by using derived functors in 𝔸; this is the situation when there is an Adams spectral sequence based on H_{∗}. One could think of this as a formal power series description of 𝔻 with coefficients in 𝔸. However, if two or more derived functors are involved, there will usually be differentials in the Adams spectral sequence, and one cannot hope to recover 𝔻 from 𝔸 alone. Suppose then that 𝔸 has injective dimension 1, so that the Adams spectral sequence takes the form of a short exact sequence

4.2.1.

0 ⟶ Ext_{𝔸}(H_{∗}(ΣX), H_{∗}(Y)) ⟶ [X, Y] ⟶ Hom_{𝔸}(H_{∗}(X), H_{∗}(Y)) ⟶ 0.

This happens in case 𝔻 = 𝕋−Spec/H by Theorem 3.1.1.

The Adams short exact sequence shows that objects X are classified by their images H_{∗}(X) in 𝔸. Indeed, if H_{∗}(X) ≅ H_{∗}(Y) we may lift an algebraic isomorphism to a map f : X ⟶ Y in 𝔻; since it is a homology isomorphism by construction, it is an equivalence in 𝔻. If we are to recover 𝔻 from 𝔸 we should therefore require that H_{∗} is surjective on isomorphism classes of objects. It is sufficient to show enough injectives are in the image.

Lemma 4.2.2. If enough injectives are in the image of H_{∗} then all objects of 𝔸 are in the image of H_{∗}.

By Corollary 3.4.2 this applies when 𝔻 = 𝕋−Spec/H. Proof: First note that if I is a realizable injective then any retract of I is also realizable. Indeed a retract of I is of the form eI for some idempotent e : I ⟶ I. Now if H_{∗}(X(I)) = I then we may lift e uniquely to a map X(e) : X(I) ⟶ X(I) by the Adams short exact sequence, and X(e) is also idempotent. Thus H_{∗}(X(e)X(I)) = eI as required.

Now an arbitrary object M of 𝔸 has an injective resolution 0 ⟶ M ⟶ I ⟶^{f} J ⟶ 0, and we may realize I by X(I), J by X(J) and the map f by a unique map X(f) : X(I) ⟶ X(J) by the Adams short exact sequence. Thus the fibre of X(f) realizes M.

However we still cannot expect to recover 𝔻 from 𝔸 without further hypothesis, because the extension 4.2.1 will not generally be split, In the applications, every object M of 𝔸 has a splitting M = M_{+} ⊕ M_{−} so that Hom(M_{δ}, N_{ϵ}) = 0 and Ext(M_{δ}, N_{ϵ}) = 0 if δ ≠ ϵ, and so that (ΣM)_{ϵ} = Σ(M_{ϵ+1}). For example this applies when 𝔸 = torsℚ[c_{H}]-mod since any module splits as a sum of its even graded and odd graded parts. We shall call such a category 𝔸 a split one dimensional graded abelian category. We say that a triangulated category 𝔻 is split linear if there is a functor H_{∗} : 𝔻 ⟶ 𝔸, creating isomorphisms to a split one dimensional category 𝔸, which is surjective on isomorphism classes of objects and so that there are natural Adams short exact sequences 4.2.1. We shall see that a split one dimensional abelian category 𝔸 is the linearization of a split linear triangulated category 𝔻 unique up to equivalence of categories. Furthermore, amongst categories 𝔻 arising as homotopy categories, this equivalence preserves triangles.

We have motivated the above definitions with the example 𝔻 = 𝕋−Spec/H, 𝔸 = torsℚ[c_{H}]-mod and H_{∗} = π_{∗}^{𝕋}, but there is a general example.

Lemma 4.2.3. If 𝔸 is a split one dimensional abelian category then the derived category D(𝔸) is a split linear triangulated category with linearization 𝔸.

Proof: Homology gives a functor H_{∗} : D(𝔸) ⟶ 𝔸. We see this is surjective on objects by using dg objects with zero differential. The Adams short exact sequence is established in the usual way. First, note that, by 4.1.9, it holds when H_{∗}(Y) is injective, and it only remains to prove that all objects have Adams resolutions by objects of this type. In fact, homology gives a natural map

[X, Y] ⟶ Hom_{𝔸}(H_{∗}(X), H_{∗}(Y)),

where [X, Y] denotes maps in the derived category, which is an isomorphism when H_{∗}(Y) is injective. Now, for an arbitrary object Y in the derived category there is a triangle

Y ⟶ I ⟶ J

so that

0 ⟶ H_{∗}(Y) ⟶ H_{∗}(I) ⟶ H_{∗}(J) ⟶ 0

is an injective resolution of H_{∗}(Y) in 𝔸. To see this, find an embedding e : H_{∗}(Y) ⟶ I with I injective. Regard I as a dg object with zero differential and use the injective case to lift e to a map Y ⟶ I in the derived category. The mapping cone of this map necessarily has injective homology, and hence qualifies as J. This completes the proof that D(𝔸) is a split linear triangulated category with linearization 𝔸.

Proposition 4.2.4. If 𝔸 is a split one dimensional abelian category and 𝔻 is a split linear triangulated category 𝔻 with linearization 𝔸, then 𝔻 is equivalent to D(𝔸). Furthermore, the Adams short exact sequence is split.

Proof: Consider the two split linear triangulated categories, 𝔻 and 𝔼 = D(𝔸). Each is equipped with an exact functor H_{∗} to 𝔸, which is essentially surjective on objects, and which creates isomorphisms. Furthermore we have an exact sequence 4.2.1 for objects of 𝔻, and similarly for 𝔼.

We shall construct a functor p : 𝔻 ⟶ 𝔼 which is an equivalence of categories over 𝔸. First note that the splitting of objects of 𝔸 lifts to splittings of objects of 𝔻 and 𝔼. Indeed if H_{∗}(X) = H_{+} ⊕ H_{−} then, since H_{∗} is essentially surjective, there is an object X_{+} with H_{∗}(X_{+}) ≅ H_{+}. Furthermore, it follows from the Adams short exact sequence 4.2.1 that we may lift the projection H_{∗}(X) ⟶ H_{+} to a map X ⟶ X_{+}; arguing similarly with H_{−} we obtain a map X ⟶ X_{+} ∨ X_{−} which is a homology isomorphism and hence an equivalence. We refer to objects X with X = X_{+} or X = X_{−} as pure parity objects; note that if X and Y are both of pure parity then the Adams short exact sequence either says [X, Y] = Hom_{𝔸}(H_{∗}(X), H_{∗}(Y)) if they are of the same parity, or [X, Y] = Ext_{𝔸}(H_{∗}(ΣX), H_{∗}(Y)) if they are of opposite parity.

Following Adams [1], we may consider d and e invariants of a map f : X ⟶ Y in 𝔻. Any map f has a d invariant d(f) := H_{∗}(f) ∈ Hom_{𝔸}(H_{∗}(X), H_{∗}(Y)), and if d(f) = 0 then the class of the extension

0 ⟶ H_{∗}(Y) ⟶ H_{∗}(C(f)) ⟶ H_{∗}(ΣX) ⟶ 0

gives the e invariant, e(f) ∈ Ext_{𝔸}(H_{∗}(ΣX), H_{∗}(Y)). This much applies to any triangulated category, and in our case the e invariant is the place of a morphism in the Adams short exact sequence 4.2.1.

Now choose such a splitting for each object of 𝔻 and 𝔼, and a preferred representative from each isomorphism class of pure parity objects of 𝔼. Define p on objects by p(X) = p(X_{+})∨p(X_{−}) where p(X_{δ}) is the preferred representative of objects with homology H_{∗}(X_{δ}). On morphisms f : X ⟶ Y between pure parity objects we take p(f) to be the map with the same d or e invariant as f. For arbitrary X and Y we view f as a 2 × 2 matrix of maps between the pure parity summands of X and Y; this gives a splitting of the Adams short exact sequence 4.2.1, so that any map f : X ⟶ Y has a well defined e invariant. We define p(f) by letting p act on each of the four matrix entries. By matrix multiplication, the functoriality of p on all morphisms will follow from that on morphisms between pure parity objects. There are four cases to consider.

Consider maps X ⟶^{f} Y ⟶^{g} Z between pure parity objects. If X, Y and Z have the same parity d(gf) = d(g)d(f) since H_{∗} is a functor, and hence p(gf) = p(g)p(f). Similarly if either f or g is a map between objects of the same parity, then functoriality comes from the naturality of the e invariant in the sense that e(gf) = e(g)d(f) or e(gf) = d(g)e(f) as the case may be. Finally if both f and g are maps between objects with opposite parities then f and g both have Adams filtration 1, so gf has Adams filtration 2 and so gf = 0 because 𝔸 is one dimensional; p(g)p(f) = 0 for analogous reasons. This completes the construction of a functor p : 𝔻 ⟶ 𝔼; a functor in the reverse direction can be constructed in the same way, and their composites are isomorphic to the relevant identity functors. Thus p is an equivalence as required.

It remains to consider the behaviour of p on triangles, and we begin the discussion without further assumptions on the categories. Suppose given a triangle X ⟶^{f} Y ⟶^{g} Z ⟶^{h} ΣX in 𝔻. First note that, because H_{∗} is exact, p commutes with suspensions. The ideal outcome would be to show that p(X) ⟶^{p(f)} p(Y) ⟶^{p(g)} p(Z) ⟶^{p(h)} Σp(X) is a triangle in 𝔼. To do this, we would have to show that a map determines the triangle it lies in, up to automorphism, using only categorical constructions on its d and e invariants.

Lemma 4.2.5. Let f : X ⟶ Y be a morphism in 𝔻. The d and e invariants of f determine (a) its mapping cone Z up to isomorphism and (b) the d invariants of the maps g : Y ⟶ Z and h : Z ⟶ ΣX.

Proof: For (a) we note that it is only necessary to determine the isomorphism class of H_{∗}(Z). Indeed it lies in an extension

0 ⟶ C ⟶ H_{∗}(Z) ⟶ ΣK ⟶ 0,

where C is the cokernel and K the kernel of H_{∗}(f). In fact the extension class is the image of e(f) under the composite

Ext_{𝔸}(H_{∗}(ΣX), H_{∗}(Y)) ⟶ Ext_{𝔸}(H_{∗}(ΣX), C) ⟶ Ext_{𝔸}(ΣK, C).

The construction clearly identifies d(g) and d(h).

To proceed further we wish to complete the diagram

p(X) ⟶^{p(f)} p(Y) ⟶^{g′} Z′  ⟶^{h′} Σp(X) ⟶^{−Σp(f)} Σp(Y)
=↓  =↓ L α⋮ R  =↓  =↓
p(X) ⟶^{p(f)} p(Y) ⟶^{p(g)} p(Z) ⟶^{p(h)} Σp(X) ⟶^{−Σp(f)} Σp(Y)

in which the first row is obtained by embedding p(f) into a triangle in 𝔼. Since p is functorial, p(g)p(f) = p(gf) = 0 and so a map α exists so that the square L commutes; the problem is to choose the map to make the square R commute. If this is possible, then a 5-lemma argument in homology shows that α is an equivalence in 𝔼, and so the second row is isomorphic to the first, and hence is a triangle as required.

The standard method of discussing such problems is that of Toda brackets, but these are not available in an arbitrary triangulated category. Since they are available in the cases of interest to us we shall impose further conditions. Note in particular that D(𝔸) is the homotopy category of the category of dg 𝔸-objects so that it does have Toda brackets.

Theorem 4.2.6. If 𝔸 is a split one-dimensional abelian category, and 𝔻 is a split linear triangulated category of which 𝔸 is the linearisation and which is the homotopy category of a model category then the equivalence

𝔻 ≃ D(𝔸)

of 4.2.4 is an equivalence of triangulated categories.

Proof: We established in 4.2.4 that the derived category 𝔼 of differential graded objects of 𝔸 is a split linear triangulated category with linearization 𝔸, and it was constructed as a homotopy category. We shall show that any other triangulated category 𝔻 as in the statement is equivalent to 𝔼.

Consider the diagram of differential graded objects of 𝔸 and homomorphisms (i.e. not up to homotopy)

A ⟶^{θ} B ⟶^{i} C ⟶^{π} ΣA ⟶^{−Σθ} ΣB
=↓  =↓ L α⋮ R =↓  =↓
A ⟶^{θ} B ⟶^{ϕ} C′ ⟶^{ψ} ΣB ⟶^{−Σθ} ΣB

in which C is the mapping cone of θ, and i and π are the canonical maps.

4.3. THE ALGEBRAICIZATION OF THE CATEGORY OF 𝕋-SPECTRA OVER H. 51

A map α exists so that L strictly commutes if and only if ϕθ ≃ 0, and choices of α correspond exactly to null-homotopies H of ϕθ. The condition that R commutes up to homotopy is then that π ≃ ψα, which is equivalent to the existence of a null-homotopy K of ψϕ so that 1 ≃ ψH − Kθ. In other words, there is a solution α so that both L and R commute if and only if the Toda bracket

⟨ψ, ϕ, θ⟩ ⊆ [ΣA, ΣA]

exists and contains the identity.

Exactly similar arguments apply to the category 𝔻 (see [21, I.3]), so that the result will follow once we show that the map p preserves Toda brackets. This follows from the discussion in [1, Section 5], and specifically from diagram (5.1). In fact we form the analogue of Adams’ diagram (5.1) for calculating the Toda bracket ⟨h, g, f⟩ in the model category whose homotopy category is 𝔻. Since f, g, h are successive maps in a triangle, 1 is an element of the Toda bracket and there are choices of null-homotopies so that the composite HF ≃ 1 in Adams’ notation. Now apply p to the resulting diagram in 𝔻, and conclude that 1 = p(1) ∈ ⟨p(h), p(g), p(f)⟩.

4.3. The algebraicization of the category of 𝕋-spectra over H.

In this section we quickly complete the proof of Theorem 3.4.3, by observing that 𝕋−Spec/H is a split linear triangulated category arising as the homotopy category of a model category. We also prove the analogous theorem for F-spectra, once we have introduced the appropriate abelian category.

Of course we take 𝔻 = 𝕋−Spec/H, 𝔸 = torsℚ[c_{H}]-mod, and H_{∗}(X) = π_{∗}^{𝕋}(X). It is easily verified that torsℚ[c_{H}]-mod is abelian and has injective dimension 1. The split condition arises since ℚ[c_{H}] is concentrated in even degrees, so that for any ℚ[c_{H}]-module M we may take M_{+} to be the even-graded part of M and M_{−} to be the odd graded part. The Adams short exact sequence 3.1.1 shows that 𝔸 is the linearization of 𝔻.

Finally, it follows from the construction of Lewis-May that 𝕋−Spec/H is the homotopy category of a model category. The model category we have in mind is the category of F-spectra, with weak equivalences created by the functor X ⟼ e_{H}π_{∗}^{𝕋}(X). Since π_{∗}^{𝕋}(X) = e_{H}π_{∗}^{𝕋}(X) for objects of 𝕋−Spec/H,we see that homology is well defined. We conclude that Theorem 4.2.6 applies to show there is an equivalence

p : 𝕋−Spec/H ⟶^{≃} D(torsℚ[c_{H}])

of triangulated categories; this completes the proof of Theorem 3.4.3.

It may be instructive to give an example illustrating how maps are represented in D(torsℚ[c_{H}]).

Example 4.3.1. Consider a non-trivial map f : σ_{H}^{1} ⟶ σ_{H}^{0}. First we note that π_{∗}^{𝕋}(σ_{H}^{0}) = Σℚ, and there is an exact sequence

0 ⟶ Σℚ ⟶ Σ𝕀(H) ⟶^{c_{H}} Σ^{3}𝕀(H) ⟶ 0.

Thus p(σ_{H}^{0}) = F (c_{H}), which we may illustrate:

⋮  ⋮
|  ↓
|  ℚ degree 11
↓ ↙ |
ℚ  | degree 10
|  ↓
|  ℚ degree 9
↓ ↙ |
ℚ  | degree 8
|  ↓
|  ℚ degree 7
↓ ↙ |
ℚ  | degree 6
|  ↓
|  ℚ degree 5
↓ ↙ |
ℚ  | degree 4
|  ↓
|  ℚ degree 3
↓ ↙ |
ℚ  | degree 2
↓
ℚ degree 1

Here degree is displayed vertically, the vertical arrows are multiplication by c_{H}, and the differential is displayed diagonally. Now the map f : σ_{H}^{1} ⟶ σ_{H}^{0} has zero d-invariant, and its e-invariant is a non-zero element of ℚ ≅ Ext(Σ^{3}ℚ, Σℚ). Thus p(f) is non-trivial in all even degrees and zero in all odd degrees. The cofibre of f is equivalent to E⟨H⟩^{(2)}, and it is natural to choose an Adams resolution corresponding to the minimal injective resolution 0 ⟶ π_{∗}^{𝕋}(E⟨H⟩^{(2)}) ⟶ Σ𝕀(H) ⟶ Σ^{5}𝕀(H) ⟶ 0. The reader may find it instructive to show that C(p(f)) is quasi-isomorphic to p(E⟨H⟩^{(2)}).

An alternative approach to 𝕋-spectra over H which some might find more natural would be to prove first the analogue for the category 𝕋−Spec/F of all F-spectra, but we have preferred to avoid the relatively unimportant complications of this for as long as possible. Before we state the theorem, we must introduce some further conditions.

If Y is any F-spectrum then Y ≃ EF_{+} ∧ Y so π_{∗}^{𝕋}(Y) is a module over the ring of operations O_{F} := [EF_{+}, EF_{+}]_{∗}^{𝕋}. Using the Decoupling Theorem 2.2.3 we find that O_{F} := ∏_{H} ℚ[c_{H}] in the sense that in degree n we have (O_{F})_{n} = ∏_{H} ℚ[c_{H}]_{n}. Let e_{H} ∈ (O_{F})_{0} denote the idempotent which is projection onto the Hth factor, and let c ∈ (O_{F})_{−2} denote the total cyclotomic class c with Hth coordinate c_{H} (i.e. c_{H} = e_{H}c).

Arbitrary modules over this ring may be quite unpleasant, but by the Decoupling Theorem 2.2.3, the geometrically occurring modules π_{∗}^{𝕋}(Y) for an F-spectrum Y are all rather well behaved. Indeed they all decompose as the sum

π_{∗}^{𝕋}(Y) = ⨁_{H} π_{∗}^{𝕋}(Y (H))

where π_{∗}^{𝕋}(Y (H)) = e_{H}π_{∗}^{𝕋}(Y) is actually a torsion module over ℚ[c_{H}] = e_{H}O_{F}. We therefore consider the following two conditions on an O_{F}-module M.

Condition 4.3.2. M is a torsion O_{F}-module in the sense that for each x ∈ M there is a number N so that c^{N}x = 0.

and

Condition 4.3.3. M is F-finite in the sense that it is the direct sum of its submodules M(H):

M = ⨁_{H} M(H).

We refer to the category of full subcategory of O_{F}-modules satisfying Conditions 4.3.2 and 4.3.3 as torsion F-finite modules, and denote it torsO_{F}^{f}-mod.

Theorem 4.3.4. There is an equivalence

p : 𝕋−Spec/F ⟶^{≃} D(torsO_{F}^{f})

of triangulated categories.

Proof: We take 𝔻 = 𝕋−Spec/F and 𝔸 = torsO_{F}^{f}-mod. We have motivated our definitions by the fact that homotopy gives a functor

π_{∗}^{𝕋} : 𝕋−Spec/F ⟶ torsO_{F}^{f}-mod.

Furthermore it is clear from the Lewis-May construction that 𝕋−Spec/F is the homotopy category of the model category of F-spectra with weak equivalences defined as usual. By the Whitehead Theorem 3.3.2 for 𝕋−Spec/H, weak equivalences of F-spectra may equally well be characterized as π_{∗}^{𝕋}-equivalences.

Since the idempotents e_{H} show that the category torsO_{F}^{f}-mod splits to give an equivalence

torsO_{F}^{f}-mod ≃ ∏_{H} torsℚ[c_{H}]-mod

it follows that torsO_{F}^{f}-mod is a split one dimensional graded abelian category. The Decoupling Theorem 2.2.3 showed that there was a similar splitting of 𝕋−Spec/F, and the map π_{∗}^{𝕋} is compatible with these splittings. Hence we obtain Adams short exact sequences of the appropriate form to show that 𝔸 is the linearization of 𝔻.

54 4. CATEGORICAL REPROCESSING.

CHAPTER 5 Assembly and the standard model.

So far, we have worked entirely locally: one finite subgroup at a time. By the decoupling theorem, that is sufficient to deal with arbitrary F-spectra. However, finite isotropy is related to 𝕋-isotropy, so, to take the 𝕋-fixed part into account, we must relate the parts lying over various finite subgroups. In this chapter, we show how to assemble the finite and infinite isotropy into a single algebraic object giving a complete invariant.

We begin in Sections 5.1 and 5.2 by showing how to deduce Adams spectral sequences for the semifree and general case from the free and F-free cases. This then provides an obvious algebraic candidate for the abelianization, which we introduce formally in Section 5.3. We are then equipped to apply the criteria of Section 4.2, and we show in Section 5.3 that the triangulated category of 𝕋-spectra is equivalent to the derived category of the standard model category. We conclude in Section 5.5 by introducing some explicit notation for particular cases that we need later.

5.1. Assembly.

Now that we understand the categories 𝕋−Spec/𝕋 and 𝕋−Spec/F, it is time to return to the question of how to fit them together. Of course if X is an object of 𝕋−Spec/𝕋, then X ≃ ẼF ∧ X so that if Φ^{𝕋}Y ≃ ∗ (as happens if Y is an object of 𝕋−Spec/F) then [Y, X]_{∗}^{𝕋} = 0. It is therefore enough to understand [X, Y]_{∗}^{𝕋}. Now, as remarked above X ≃ ẼF ∧ X ≃ ẼF ∧ Φ^{𝕋}X, and, since the nonequivariant spectrum Φ^{𝕋}X splits as a wedge of spheres, the essential case is [ẼF, Y]_{∗}^{𝕋}. The obstacle to using our available machinery is that π_{∗}^{𝕋}(ẼF) is not a ℚ[c_{H}]-module.

We begin with the simpler case when Y has only one relevant isotropy group i.e. Y is an object of 𝕋−Spec/H.

Theorem 5.1.1. If X ≃ X ∧ ẼF and Y is in 𝕋−Spec/H there is a short exact sequence

0 ⟶ Ext_{ℚ[c_{H}]}(ℚ[c_{H}, c_{H}^{−1}]⊗π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{ℚ[c_{H}]}(ℚ[c_{H}, c_{H}^{−1}]⊗π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) ⟶ 0.

Proof: As in the construction of the Adams short exact sequence it is enough to construct a natural transformation

ϕ : [X, Y]_{∗}^{𝕋} ⟶ Hom_{ℚ[c_{H}]}(ℚ[c_{H}, c_{H}^{−1}] ⊗ π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y))

and show it is an isomorphism when π_{∗}^{𝕋}(Y) is an injective ℚ[c_{H}]-module. Furthermore, by the splitting in 4.2.4 we may also suppose π_{∗}^{𝕋}(Y) is concentrated in even degrees.

We first explain how to define ϕ. The problem is to get ℚ[c_{H}] acting naturally on some functor of π_{∗}^{𝕋}(X). Observe that for any object Y of 𝕋−Spec/H and any other spectrum Z, ℚ[c_{H}] does act on F (Z, Y), even though it is not usually an F-spectrum.

We shall need a variant on a well known observation.

Lemma 5.1.2. There is an equivalence

E⟨H⟩ ∧ F (E⟨H⟩, X) ≃ E⟨H⟩ ∧ X.

Proof: The cofibre of the natural map from right to left is E⟨H⟩ ∧ F (Ẽ⟨H⟩, X), which is contractible. This follows from the fact that E⟨H⟩ may be formed from cells σ_{H}^{0}, whilst σ_{H}^{0} ∧ F (Ẽ⟨H⟩, X) ≃ F (Dσ_{H}^{0} ∧ Ẽ⟨H⟩, X), and Dσ_{H}^{0} ∧ Ẽ⟨H⟩ is contractible.

Now observe that, since Y ≃ E⟨H⟩ ∧ Y, we have an equivalence Y ∧ DE⟨H⟩ ≃ Y. We may therefore define the natural transformation by taking f : X ⟶ Y, replacing it with f ∧ 1 : X ∧ DE⟨H⟩ ⟶ Y ∧ DE⟨H⟩ ≃ Y and then applying homotopy.

Lemma 5.1.3. There is a natural isomorphism of ℚ[c_{H}]-modules

π_{∗}^{𝕋}(Ẽ⟨H⟩ ∧ DE⟨H⟩ ∧ X) ≅ ℚ[c_{H}, c_{H}^{−1}] ⊗ π_{∗}(Φ^{𝕋}X).

Proof: Since the fibre of Ẽ⟨H⟩ ⟶ ẼF is the wedge of spectra E⟨K⟩ with K ≠ H, and E⟨K⟩ ∧ DE⟨H⟩ ≃ ∗, it follows that π_{∗}^{𝕋}(Ẽ⟨H⟩ ∧ DE⟨H⟩ ∧ X) ≅ π_{∗}^{𝕋}(Φ^{𝕋}(DE⟨H⟩ ∧ X)), and Φ^{𝕋} commutes with smash products it is enough to deal with the case Φ^{𝕋}X = S^{0}. But then π_{∗}^{𝕋}(Ẽ⟨H⟩ ∧ DE⟨H⟩) = π_{∗}^{𝕋}(DE⟨H⟩)[c_{H}^{−1}], by 2.3.8, and we have seen that π_{∗}^{𝕋}(DE⟨H⟩) ≅ ℚ[c_{H}].

It is worth making the general remark that if Z = Z′ ∧ S^{∞V (H)} and Y is a 𝕋-spectrum over H, then c_{H} acts invertibly on F (Z, Y) since c_{H} is the Euler class of V (H) on spectra over H. Since ℚ[c_{H}] is a graded field, the ℚ[c_{H}]-module structure on π_{∗}^{𝕋}(F (Z, Y)) follows from its structure as a vector space. This is relevant to the lemma, but it is also relevant to the proof of the theorem, since we already have the machinery to calculate [X, Y]_{∗}^{𝕋} as a rational vector space when π_{∗}^{𝕋}(Y) is injective. In fact we have seen X ≃ ⋁_{i} Σ^{n_{i}} ẼF, and so [X, Y]^{𝕋} ≅ ∏_{i}[Σ^{n_{i}} ẼF, Y]_{∗}^{𝕋}, it is thus enough to deal with the special case X = ẼF. The cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF thus gives an exact sequence

⋯ ⟵ [S^{0}, Y]_{∗}^{𝕋} ⟵ [ẼF, Y]_{∗}^{𝕋} ⟵ [ΣEF_{+}, Y]_{∗}^{𝕋} ⟵ ⋯,

and we can identify the first term as

[S^{0}, Y]_{∗}^{𝕋} ≅ Hom_{ℚ[c_{H}]}(ℚ[c_{H}], π_{∗}^{𝕋}(Y))

and the third as

[ΣEF_{+}, Y]_{∗}^{𝕋} ≅ Hom_{ℚ[c_{H}]}(ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}], π_{∗}^{𝕋}(Y)).

Since both of these are concentrated in even degrees, the sequence is short exact and the centre term is Hom_{ℚ[c_{H}]}(ℚ[c_{H}, c_{H}^{−1}], π_{∗}^{𝕋}(Y)), at least as a rational vector space. By the remarks above this also respects the ℚ[c_{H}]-module structure.

This proof generalizes readily to F-spectra Y which are nontrivial at only a finite number of subgroups.

5.2. Global assembly.

We now need to discuss the case where Y has nontrivial isotropy at an infinite number of subgroups. As discussed in Section 4.3 above, for an F-spectrum Y we view π_{∗}^{𝕋}(Y) as a module over the ring O_{F} = ∏_{H} ℚ[c_{H}]. Recall that e_{H} ∈ (O_{F})_{0} denotes the idempotent which is projection onto the Hth factor, and c ∈ (O_{F})_{−2} denotes the total cyclotomic class c with Hth coordinate c_{H}.

The element c is particularly useful. Indeed, the representation V (H) only gives rise to an Euler class K-equivariantly for K ⊆ H, but one can pass to limits and obtain a suitable substitute which works for all finite subgroups. As we have seen before, ẼF has a filtration by skeleta ẼF^{(2k−1)} analogous to a putative sphere S^{kV (F)}. Furthermore we have universal Thom isomorphisms for all F-spectra for these generalized spheres, and consequently Euler classes. By passage to limit one sees that c = e(V (F)).

Lemma 5.2.1. (a) For any F-spectrum Y there is an equivalence

Y ∧ S^{V (F)} ≃ Y ∧ S^{2}.

(b) There is an equivalence

S^{V (F)} ∧ S^{kV (F)} ≃ S^{(k+1)V (F)}.

(c) For any F-spectrum Y the composite

Y = Y ∧ S^{0} ⟶ Y ∧ S^{V (F)} ≃ Y ∧ S^{2}

induces multiplication by c.

Proof: All parts are obtained from 2.3.6 and the following discussion by using the splitting of any F-spectrum.

Having emphasized the analogy with spheres we should draw attention to a significant difference: there is no spectrum T so that T ∧ S^{V (F)} ≃ S^{0}. This non-invertibility appears to introduce wrinkles into all approaches.

We continue by analogy with the previous section. The complication is that, since F is infinite, sums and products do not coincide. Once again the module t_{∗}^{F} := π_{∗}^{𝕋}(ẼF ∧ DEF_{+} ∧X) is relevant, but it is not simply O_{F}[c^{−1}]⊗π_{∗}^{𝕋}(X). In fact it is very illuminating to view t_{∗}^{F} as the result of inverting all Euler classes in O_{F}, but we defer this point of view until Section 8.1. The notation t_{∗}^{F} is chosen since it is the coefficient ring of F-Tate homology for S^{0}; this will play no part in what follows, but suggests a number of parallels worth further investigation. Indeed, we need only observe that since O_{F} acts on DEF_{+}, t_{∗}^{F} is a O_{F}-module. To understand the theorem which follows we should be explicit about the O_{F}-module structure.

Lemma 5.2.2. As a graded vector space t_{∗}^{F} = π_{∗}^{𝕋}(ẼF ∧ DEF_{+} ∧ X) is ∏_{H} ℚ in degrees 0, −2, −4, −6, … and ⨁_{H} ℚ in degrees 2, 4, 6, . . .. Furthermore there is a short exact sequence

0 ⟶ O_{F} ⟶ t_{∗}^{F} ⟶^{ϵ} Σ^{2}𝕀 ⟶ 0

of O_{F}-modules, and the map

c : t_{2}^{F} = ⨁_{H} ℚ ⟶ ∏_{H} ℚ = t_{0}^{F}

describing the extension is the natural inclusion of the sum in the product.

Proof: The additive structure of t_{∗}^{F} follows by applying π_{∗}^{𝕋}(• ∧ DEF_{+}) to the cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF; indeed the homotopy groups of EF_{+} ∧ DEF_{+} ≃ EF_{+} follow from 2.3.4 and are in positive degrees, whilst those of DEF_{+} ≃ ∏_{H} DE⟨H⟩ follow from 2.4.1, and are in non-positive degrees. This also gives the short exact sequence.

To determine the effect of c : t_{2}^{F} ⟶ t_{0}^{F} it is enough to consider the square

t_{2}^{F}  ⟶^{c}  t_{0}^{F}
↑  ↑
π_{2}^{𝕋}(F (E⟨H⟩, S^{0}) ∧ ẼF) ⟶^{c_{H}} π_{0}^{𝕋}(F (E⟨H⟩, S^{0}) ∧ ẼF),

where the verticals are induced by the projection EF_{+} ⟶ E⟨H⟩ onto a direct factor. We saw in the previous section that the bottom horizontal was c_{H}.

Although the occurrence of both sums and products might appear unattractive, both seem to play an essential role.

The ring O_{F} is fairly complicated, but we only need to extend the modules M we consider slightly beyond the F-finite torsion modules in the sense of 4.3.2 and 4.3.3 (occurring as π_{∗}^{𝕋}(Y) for an F-spectrum Y) to include modules like t_{∗}^{F}. Note that for an arbitrary module L, Hom_{O_{F}} (L, N(H)) = Hom_{ℚ[c_{H}]}(L(H), N(H)), so that if N(H) is an injective ℚ[c_{H}]- module it is also injective over O_{F}. However since O_{F} is not Noetherian it does not follow that infinite direct sums of modules of this form are injective.

Lemma 5.2.3. If N is F-finite and M is either of the form t_{∗}^{F} ⊗ U for a graded vector space U, or F-finite, then Ext_{O_{F}}^{s} (M, N) = 0 for s ≥ 1 if N(H) is injective for all H or for s ≥ 2 in general.

Proof: Note that if M is F-finite, Hom_{O_{F}} (M, N) = ∏_{H} Hom_{ℚ[c_{H}]}(M(H), N(H)), so that the result holds in this case. The result holds if M = t_{∗}^{F} ⊗ U by 5.2.2, and the fact that O_{F} is projective.

This is sufficient to give us the Adams short exact sequence for arbitrary 𝕋-spectra.

Theorem 5.2.4. If X is F-contractible and Y is in 𝕋−Spec/F there is a short exact sequence

0 ⟶ Ext_{O_{F}} (t_{∗}^{F} ⊗ π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{O_{F}} (t_{∗}^{F} ⊗ π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) ⟶ 0.

Proof: As in the construction of the Adams short exact sequence it is enough to construct a natural transformation

ϕ : [X, Y]_{∗}^{𝕋} ⟶ Hom_{O_{F}} (t_{∗}^{F} ⊗ π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y))

and show it is an isomorphism when π_{∗}^{𝕋}(Y) is an O_{F}-module injective in an appropriate sense.

We define ϕ for arbitrary 𝕋-spectra X. Observe that since Y ≃ EF_{+} ∧ Y and DEF_{+} ∧ EF_{+} ≃ EF_{+}, we have an equivalence Y ∧ DEF_{+} ≃ Y. Now define the natural transformation by taking f : X ⟶ Y, replacing it with f ∧ 1 : X ∧ DEF_{+} ⟶ Y ∧ DEF_{+} ≃ Y and then applying homotopy: explicitly this gives a natural transformation

ϕ : [X, Y]_{∗}^{𝕋} ⟶ Hom_{O_{F}} (π_{∗}^{𝕋}(X ∧ DEF_{+}), π_{∗}^{𝕋}(Y)).

If X ≃ X ∧ ẼF then X ≃ ⋁_{i} Σ^{n_{i}} ẼF, and π_{∗}^{𝕋}(X ∧ DEF_{+}) ≅ t_{∗}^{F} ⊗ π_{∗}^{𝕋}(Φ^{𝕋}X), so that the codomain of ϕ is of the required form. Note also that in this case [X, Y]^{𝕋} ≅ ∏_{i}[Σ^{n_{i}} ẼF, Y]_{∗}^{𝕋}, so that it is enough to deal with the special case X = ẼF; we may also suppose that π_{∗}^{𝕋}(Y) is concentrated in even degrees.

We saw in the proof of 5.2.2 that applying π_{∗}^{𝕋}(•∧DEF_{+}) to the cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF gives a short exact sequence of O_{F}-modules. Hence applying ϕ we obtain a diagram

0 ⟵  [S^{0}, Y]_{∗}^{𝕋}  ⟵  [ẼF, Y]_{∗}^{𝕋}  ⟵  [ΣEF_{+}, Y]_{∗}^{𝕋}  ⟵ 0
≅↓  ↓  ↓≅
0 ⟵ Hom(O_{F}, π_{∗}^{𝕋}(Y)) ⟵ Hom(t_{∗}^{F}, π_{∗}^{𝕋}(Y)) ⟵ Hom(π_{∗}^{𝕋}(ΣEF_{+}), π_{∗}^{𝕋}(Y)) ⟵ 0.

The bottom lefthand zero follows from 5.2.3; the lefthand vertical is tautologically isomorphic, the righthand vertical is isomorphic by 3.1.1. The central vertical is thus an isomorphism by the 5-lemma.

5.3. The standard model category.

The purpose of this section is to describe an abelian category A = A_{F} whose derived category is an algebraic model for the category of 𝕋-spectra, and whose homological algebra provides Adams spectral sequences for both the categories. There is a somewhat simpler abelian category A_{1} whose derived category gives an algebraic model of semifree 𝕋-spectra, and similarly for other collections of proper subgroups.

Definition 5.3.1. The objects of the standard model A are maps N ⟶^{β} t_{∗}^{F} ⊗ V of O_{F}-modules whose kernel and cokernel are F-finite torsion modules. Morphisms are commutative squares

\begin{matrix}M & ⟶^{θ} & N \\ β ↓ & & ↓ γ \\ t_{∗}^{F} ⊗ U & ⟶^{1⊗ϕ} & t_{∗}^{F} ⊗ V \end{matrix} .

We shall refer to N as the nub and V as the vertex of the object N ⟶ t_{∗}^{F} ⊗ V. The map β is called the basing map.

Remark 5.3.2. (a) In due course we will show that F-finite torsion modules are precisely the modules annihilated by inverting the set E of Euler classes. Since t_{∗}^{F} is E local, the condition that β : N ⟶ t_{∗}^{F} ⊗ V has F-finite torsion kernel and cokernel is equivalent to saying that it becomes an isomorphism when E is inverted. Thus the role of β is to give a specific isomorphism E^{−1}N ≅ t_{∗}^{F} ⊗ V. Thus we may view objects of A as given by O_{F} modules together with with a particular vector subspace V of E^{−1}N so the the canonical extension of the inclusion V ⟶ E^{−1}N is an isomorphism. This explains why β is called a basing map.

(b) The standard model category A_{H} for a set H of finite subgroups is obtained by replacing O_{F} by e_{H}O_{F}. The case in which H is finite is significantly simpler, but we shall just make explicit the case H = {1}. Here the objects are ℚ[c]-module maps β : N ⟶ ℚ[c, c^{−1}] ⊗ V which become an isomorphism when c is inverted. It is often helpful to examine any algebraic construction in this case before passing to the general case H = F, and we shall adopt this policy for the more intricate bits of algebra.

Before we study the category A and its derived category, it will be useful to import information into A from categories we understand. For comparison with the model Dℚ of F-contractible spectra, we let ℚ_{∗}-mod denote the category of graded rational vector spaces and define the functor

e : ℚ_{∗}-mod ⟶ A

by

e(V) = (t_{∗}^{F} ⊗ V ⟶^{1} t_{∗}^{F} ⊗ V ) .

Objects of A isomorphic to ones in the image of e will be called torsion free objects. For comparison with the model of F-spectra we define the functor

f : torsO_{F}^{f}-mod ⟶ A

by

f(N) = (N ⟶ 0) .

Objects of A isomorphic to ones in the image of f will be called torsion objects. It is easy to see that both e and f are right adjoints.

Lemma 5.3.3. For any object M ⟶ t_{∗}^{F} ⊗ U of A, any graded vector space V and any F-finite torsion O_{F}-module N we have natural isomorphisms (i)

Hom_{A}((M ⟶ t_{∗}^{F} ⊗ U), e(V)) = Hom_{ℚ}(U, V).

(ii)

Hom_{A}((M ⟶ t_{∗}^{F} ⊗ U), f(N)) = Hom_{O_{F}} (M, N).

Corollary 5.3.4. (i) The functors e and f are full and faithful embeddings. (ii) For any graded vector space W and any F-finite torsion O_{F}-module N

Hom_{A}(f(N), e(W)) = 0.

We can also import injective objects, and they will prove to be sufficient for homological algebra.

Lemma 5.3.5. (i) The object e(V) = (t_{∗}^{F} ⊗V ⟶^{1} t_{∗}^{F} ⊗V) of A is injective for any graded vector space V. (ii) The object f(I) = (I ⟶ 0) of A is injective if I is an F-finite torsion injective.

Proof: This follows from 5.3.3 together with the obvious fact that the functors (M ⟶ t_{∗}^{F} ⊗ V) ⟼ M and (M ⟶ t_{∗}^{F} ⊗ V) ⟼ V are exact.

This is enough for us to prove that A is a split 1-dimensional abelian category.

Theorem 5.3.6. The category A is a split one dimensional graded abelian category in the sense of Section 4.2.

Proof: We must verify that A is abelian, that every object admits an injective resolution of length 1, and that every object X splits as a sum X_{+} ⊕ X_{−} of pure pieces so that Hom(X_{δ}, Y_{ϵ}) = 0 and Ext(X_{δ}, Y_{ϵ}) = 0 if δ ≠ ϵ.

First we make an observation about F-finite torsion O_{F}-modules.

Lemma 5.3.7. If 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an exact sequence of O_{F}-modules then M is F-finite and torsion if and only if M′ and M′′ are both F-finite and torsion.

Now, considering the map in the square displayed in Definition 5.3.1 above, we obtain two short exact sequences:

0 ⟶ K ⟶ M ⟶ M′ ⟶ 0
ξ ↓ β ↓ η ↓
0 ⟶ t_{∗}^{F} ⊗ U ⟶ t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V′ ⟶ 0

and

0 ⟶ M′ ⟶  N  ⟶ C  ⟶ 0
η ↓  γ ↓  σ ↓
0 ⟶ t_{∗}^{F} ⊗ V′ ⟶ t_{∗}^{F} ⊗ W ⟶ t_{∗}^{F} ⊗ X ⟶ 0

By hypothesis, the central verticals have kernel and cokernel F-finite and torsion. It follows from the first diagram that ker(ξ) and cok(η) are F-finite and torsion, and from the second that ker(η) and cok(σ) are F-finite and torsion. Hence η : M′ ⟶ t_{∗}^{F} ⊗ V′ is an object of A. Now we may use the Snake Lemma and closure of F-finite torsion spectra under extensions to deduce that cok(ξ) and ker(σ) are also F-finite and torsion. This completes the proof that A is abelian.

To see that β : M ⟶ t_{∗}^{F} ⊗ V has a resolution of length 1, we recall that 5.3.5 allows us to import injectives from ℚ_{∗}-mod and torsO_{F}^{f}-mod. Now we proceed to construct an injective resolution of length 1. First note that ker(β) is the torsion submodule T M of M, and that there is a resolution 0 ⟶ T M ⟶ I′ ⟶ J′ ⟶ 0 of T M by objects I′ and J′ which are injective as F-finite torsion O_{F}-modules. Furthermore if Q is the image of θ then J′′ = (t_{∗}^{F} ⊗ V)/Q is divisible and an F-finite torsion module, and hence injective in the same sense. In the usual way we form the diagram,

0  0  0
↓  ↓  ↓
0 ⟶ T M ⟶  M  ⟶ Q ⟶ 0
↓  ↓  ↓
0 ⟶ I′ ⟶ I′ ⊕ (t_{∗}^{F} ⊗ V) ⟶ t_{∗}^{F} ⊗ V ⟶ 0
↓  ↓  ↓
0 ⟶ J′ ⟶  J′ ⊕ J′′  ⟶ J′′ ⟶ 0
↓  ↓  ↓
0  0  0

and hence the diagram

0 ⟶ M ⟶ I′ ⊕ (t_{∗}^{F} ⊗ V) ⟶ J′ ⊕ J′′ ⟶ 0
↓  ↓  ↓
0 ⟶ t_{∗}^{F} ⊗ V ⟶  t_{∗}^{F} ⊗ V  ⟶  0  ⟶ 0

This is the required resolution; injectivity of the last term was 5.3.5 (i), and injectivity of the middle term follows since it is an extension of two things known to be injective by 5.3.5.

Finally we just need to observe that any object X = (M ⟶ t_{∗}^{F} ⊗ V) splits as a sum X_{+} ⊕ X_{−} where X_{+} is the even graded part and X_{−} is the odd graded part. It is then clear that Hom(X_{δ}, Y_{ϵ}) = 0 if δ ≠ ϵ, and the fact that Ext(X_{δ}, Y_{ϵ}) = 0 if δ ≠ ϵ follows since the resolution of Y_{ϵ} constructed above is entirely in parity ϵ.

Now that we know the derived category DA exists by 4.1.3 we may observe that the functors e and f induce functors on derived categories.

Corollary 5.3.8. The functors e : ℚ_{∗}-mod ⟶ A and f : torsO_{F}^{f}-mod ⟶ A induce maps of derived categories.

Proof: The functors e and f induce functors on categories of differential graded objects, and, by the universal property of the category of fractions, it is enough to show that the functors e and f take homology isomorphisms to homology isomorphisms. Since e and f are obviously exact functors, this is clear.

5.4. THE ALGEBRAICIZATION OF RATIONAL 𝕋-SPECTRA. 63

5.4. The algebraicization of rational 𝕋-spectra.

We have constructed a split one dimensional abelian category A, and it has a derived category DA by 4.1.3; we shall show in this section that A is the linearization of the category of 𝕋-spectra, and hence that DA is an algebraic model of the category of 𝕋-spectra: this is the main theorem of Part I.

Theorem 5.4.1. (Algebraicization of rational 𝕋-spectra.) There are equivalences of triangulated categories

𝕋−Spec ≃ DA.

Proof: Once again we aim to apply Theorem 4.2.6. It remains to show that 𝕋−Spec has linearization A, and that it arises as the homotopy category of a model category.

It follows from the construction of Lewis-May that 𝕋−Spec is the homotopy category of a model category. It remains to describe a functor 𝕋−Spec ⟶ A, and show it is essentially surjective and a linearization. For this we define

π_{∗}^{A}(X) := π_{∗}^{𝕋}(X ∧ DEF_{+} ⟶ X ∧ ẼF ∧ DEF_{+})

Lemma 5.4.2. The above definition gives a functor

π_{∗}^{A} : 𝕋−Spec ⟶ A.

Proof: The definition is clearly natural, so the only point is to verify that π_{∗}^{A}(X) is an object of A. First we note that π_{∗}^{𝕋}(X ∧ ẼF ∧ DEF_{+}) ≅ t_{∗}^{F} ⊗ π_{∗}^{𝕋}(Φ^{𝕋}X), and we have the natural inclusion π_{∗}(Φ^{𝕋}X) ≅ π_{∗}^{𝕋}(X ∧ẼF) ⟶ π_{∗}^{𝕋}(X ∧ẼF ∧DEF_{+}) ≅ t_{∗}^{F} ⊗π_{∗}^{𝕋}(Φ^{𝕋}X). This shows that the second term has the correct form. Secondly we must show that the kernel K and cokernel C of π_{∗}^{𝕋}(X ∧ DEF_{+}) ⟶ π_{∗}^{𝕋}(X ∧ ẼF ∧ DEF_{+}) are F-finite and torsion. However the fibre of X ∧ DEF_{+} ⟶ X ∧ ẼF ∧ DEF_{+} is X ∧ EF_{+} ∧ DEF_{+} ≃ X ∧ EF_{+}, whose homotopy is F-finite and torsion. Since there is an extension

0 ⟶ Σ^{−1}C ⟶ π_{∗}^{𝕋}(X ∧ EF_{+}) ⟶ K ⟶ 0

the result follows.

For special types of 𝕋-spectra we can go further, and it is useful to relate π_{∗}^{A} to the functors we already understand; this makes it rather easy to calculate π_{∗}^{A}(X) when X is either an F-spectrum or F-contractible.

Lemma 5.4.3. On the subcategories of F-contractible spectra and F-spectra the functor π_{∗}^{A} factorizes through π_{∗}^{𝕋} in the sense that the following two diagrams commute:

𝕋−Spec/𝕋 ⟶^{π_{∗}^{𝕋}} Dℚ  𝕋−Spec/H ⟶^{π_{∗}^{𝕋}} D(torsO_{F}^{f})
↓  ↓ e and  ↓  ↓ f
𝕋−Spec ⟶^{π_{∗}^{A}} DA  𝕋−Spec ⟶^{π_{∗}^{A}}  DA.

Proof: For the first diagram, note that, if Y is an F-contractible spectrum, then Y ∧EF_{+} ≃ ∗, and hence the map Y ∧ DEF_{+} ⟶ Y ∧ ẼF ∧ DEF_{+} is an equivalence. We have seen that π_{∗}^{𝕋}(Y ∧ ẼF ∧ DEF_{+}) = t_{∗}^{F} ⊗ π_{∗}^{𝕋}(Y), so π_{∗}^{A}(Y) = e(π_{∗}^{𝕋}(Y)) as required.

For the second diagram, we use the facts that if X is an F-spectrum then X ∧ ẼF ≃ ∗ and X ∧ DEF_{+} ≃ X, so π_{∗}^{A}(X) = (π_{∗}^{𝕋}(X ∧ DEF_{+}) ⟶ 0) = f(π_{∗}^{𝕋}(X)).

Next we need to observe that the Whitehead Theorem holds; this lemma was a critical point in the recognition of the standard model.

Lemma 5.4.4. (Karlsruhe Lemma). If f : X ⟶ Y is a map of 𝕋-spectra for which π_{∗}^{A}(f) is an isomorphism, then f is an equivalence.

Proof: Using exactness of π_{∗}^{A} it is enough to consider the cofibre Z of f, for which π_{∗}^{A}(Z) = 0, and deduce Z ≃ ∗.

The hypothesis states that (a) π_{∗}^{𝕋}(Z ∧ DEF_{+}) = 0 and (b) π_{∗}^{𝕋}(Z ∧ ẼF ∧ DEF_{+}) = 0. Now we have remarked that quite generally π_{∗}^{𝕋}(X ∧ ẼF ∧ DEF_{+}) = e(π_{∗}(Φ^{𝕋}X)), so we conclude from (b) that Φ^{𝕋}Z ≃ ∗. Thus Z is an F-spectrum, and Z ∧ DEF_{+} ≃ Z, and by 3.3.2 it follows that Z ≃ ∗.

Next we observe that enough injectives are realizable. We have seen in the proof of Theorem 5.3.6 that there are enough injectives amongst those of form f(I) = (I ⟶ 0) for an F-finite torsion injective I and e(V) = (1 : t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V) for a graded vector space V. Objects of the first type occur as f(π_{∗}^{𝕋}(X)) = π_{∗}^{A}(X) for F-spectra X by 3.4.2. Those of the second type occur as e(π_{∗}^{𝕋}(Y)) = π_{∗}^{A}(Y) for F-contractible spectra Y. Once we have established an Adams short exact sequence, essentially surjectivity will follow by 4.2.2.

Theorem 5.4.5. For any rational 𝕋-spectra, there is a natural short exact sequence

0 ⟶ Ext_{A}(π_{∗}^{A}(ΣX), π_{∗}^{A}(Y)) ⟶ [X, Y]_{∗}^{𝕋} ⟶ Hom_{A}(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ⟶ 0,

and it splits unnaturally.

Proof: We are constructing an Adams spectral sequence, so we use the standard operating procedure, and resolve all spectra by injectives for which the theorem holds.

Firstly, we observe that π_{∗}^{A} gives a natural transformation

θ : [X, Y]_{∗}^{𝕋} ⟶ Hom_{A}(π_{∗}^{A}(X), π_{∗}^{A}(Y)).

Next, we note that θ is an isomorphism if Y is an injective F-spectrum or an F-contractible spectrum.

Lemma 5.4.6. If Y ≃ Y ∧ ẼF is F-contractible then θ is an isomorphism.

Proof: First note that, by 5.4.3, π_{∗}^{A}(Y) = e(π_{∗}^{𝕋}(Y)) = e(π_{∗}(Φ^{𝕋}Y)). Now use the commutative diagram

[X, Y ∧ ẼF]_{∗}^{𝕋} ⟶ Hom_{A}(π_{∗}^{A}(X), e(π_{∗}(Φ^{𝕋}Y)))
≅↓  ↓≅
[Φ^{𝕋}X, Φ^{𝕋}Y]_{∗} ⟶^{≅} Hom_{ℚ}(π_{∗}(Φ^{𝕋}X), π_{∗}(Φ^{𝕋}Y)).

The right hand vertical is an isomorphism by obstruction theory, and the left hand vertical is an isomorphism by 5.3.3; the bottom isomorphism is the well known algebraicization of non-equivariant rational spectra.

Lemma 5.4.7. If Y ≃ Y ∧ EF_{+} is an injective F-spectrum then θ is an isomorphism.

Proof: This is a little harder since there are non-zero maps from torsion free objects to torsion objects. Note first that if Y is an F-spectrum then π_{∗}^{A}(Y) = f(π_{∗}^{𝕋}(Y)) by 5.4.3. We now deal separately with various types of spectrum X.

If X is an F-spectrum, π_{∗}^{A}(X) = f(π_{∗}^{𝕋}(X)) by 5.4.3 and the lemma follows from 3.1.1, using commutativity of the diagram

[X, Y ∧ EF_{+}]_{∗}^{𝕋} ⟶ Hom_{A}(f(π_{∗}^{𝕋}(X)), f(π_{∗}^{𝕋}(Y)))
↘  ↓≅
≅  Hom_{O_{F}} (π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)).

The right hand vertical is an isomorphism by 5.3.3.

If X is F-contractible π_{∗}^{A}(X) = e(π_{∗}^{𝕋}(X)) by 5.4.3 and the lemma follows from 5.2.4 by commutativity of the diagram

[X, Y ∧ EF_{+}]_{∗}^{𝕋} ⟶ Hom_{A}(e(π_{∗}^{𝕋}(X)), f(π_{∗}^{𝕋}(Y)))
↘  ↓≅
≅ Hom_{O_{F}} (t_{∗}^{F} ⊗ π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)).

The right hand vertical is an isomorphism by 5.3.3.

The general case follows by applying the 5-lemma to the diagram,

[X ∧ ΣEF_{+}, Y]_{∗}^{𝕋}  ⟶  [X ∧ ẼF, Y]_{∗}^{𝕋}  ⟶  [X, Y]_{∗}^{𝕋}
≅↓  ≅↓  ↓
Hom_{A}(π_{∗}^{A}(X ∧ ΣEF_{+}), π_{∗}^{A}(Y)) ⟶ Hom_{A}(π_{∗}^{A}(X ∧ ẼF), π_{∗}^{A}(Y)) ⟶ Hom_{A}(π_{∗}^{A}(X), π_{∗}^{A}(Y))
⟶  [X ∧ EF_{+}, Y]_{∗}^{𝕋}  ⟶  [X ∧ Σ^{−1}ẼF, Y]_{∗}^{𝕋}
≅↓  ≅↓
⟶ Hom_{A}(π_{∗}^{A}(X ∧ EF_{+}), π_{∗}^{A}(Y)) ⟶ Hom_{A}(π_{∗}^{A}(X ∧ Σ^{−1}ẼF), π_{∗}^{A}(Y)).

Since X ∧ EF_{+} ⟶ X ⟶ X ∧ ẼF is a cofibre sequence the first row is exact, and there is a long exact sequence in π_{∗}^{A}. Since π_{∗}^{A}(Y) is injective, the second row is also exact.

Next we observe that any 𝕋-spectrum Y admits a convergent Adams resolution for π_{∗}^{A} by imported injectives. More precisely, we choose a standard resolution

0 ⟶ π_{∗}^{A}(Y) ⟶ I ⟶ J ⟶ 0

of π_{∗}^{A}(Y) as constructed in 5.3.6 above. Since both I and J are sums of injectives of the form e(V) or f(I), there are spectra I(Y) and J(Y) realizing them. From the injective case proved above for I(Y) and J(Y), we may lift the maps in the resolution to form a sequence

Y ⟶ I(Y) ⟶ J(Y)

which gives the resolution once π_{∗}^{A} is applied. In fact it is a cofibre sequence. Indeed, since the composite is null from the injective case, the map Y ⟶ I(Y) factors through the fibre F of I(Y) ⟶ J(Y), and the map Y ⟶ F is an isomorphism of π_{∗}^{A}; therefore Y is equivalent to the fibre by the Whitehead Theorem 5.4.4.

The theorem follows by applying [X, ·]_{∗}^{𝕋} to Y ⟶ I(Y) ⟶ J(Y).

It is worth recording here the compatibility of the models with the inclusions of categories of spectra.

Corollary 5.4.8. The functors e and f induce full and faithful embeddings

e : Dℚ ⟶ DA and f : D(torsO_{F}^{f}) ⟶ DA

of derived categories. Furthermore, these are compatible with the geometric inclusions in the sense that the diagram

Dℚ  ⟶^{e} DA ⟵^{f} D(torsO_{F}^{f})
≃↓  ≃↓  ≃↓
𝕋−Spec/𝕋 ⟶ 𝕋−Spec ⟵ 𝕋−Spec/F

commutes.

Proof: The fact that e and f induce full and faithful embeddings of derived categories follows from the Adams spectral sequences together with 5.3.3 and the fact that they preserve injectives.

The compatibility with geometry follows from 5.4.3 and the universal properties of categories of fractions.

5.5. Notation.

In this section we provide notation to cover our needs, and make a convenient fibrant approximation explicit: this will be useful in consideration of Mackey functor valued homotopy groups in Section 13.1.

On the one hand we want to refer to an object of A by a single letter like L. We shall write

L = (N_{L} ⟶^{β_{L}} t_{∗}^{F} ⊗ V_{L}) .

We call N_{L} the nub of L and V_{L} the vertex of L. The map β_{L} is called the basing map. By definition K_{L} = ker(r_{L}) and C_{L} = cok(r_{L}) are F-finite torsion modules, and C_{L} is also divisible and hence injective. We also use the notation k_{L} : K_{L} ⟶ N_{L} and c_{L} : t_{∗}^{F} ⊗ V_{L} ⟶ C_{L} for the inclusion and projection. Since all short exact sequences 0 ⟶ Σ^{−1}C_{L} ⟶ T ⟶ K_{L} ⟶ 0 therefore split, it is reasonable to define

T_{L} := Σ^{−1}C_{L} ⊕ K_{L},

and call it the torsion part of L. With this notation, corresponding to the cofibre sequence

X ⟶ X ∧ ẼF ⟶^{q_{X}} X ∧ ΣEF_{+}

we have a cofibre sequence in the derived category

L ⟶ e(V_{L}) ⟶^{q_{L}} f(ΣT_{L}).

To see this, note that there is a map L ⟶ e(V_{L}) which is an isomorphism on vertices, since e is right adjoint to the passage-to-vertex functor. It is clear that the kernel and cokernel are f(K_{L}) and f(C_{L}) respectively, and hence, since C_{L} is injective, the cofibre is f(ΣT_{L}). Explicitly

(q_{L} : e(V_{L}) ⟶ f(ΣT_{L})) = ( \begin{matrix}t_{∗}^{F} ⊗ V_{L} & ⟶^{q′_{L}} & ΣT_{L} \\ ↓ & & ↓ \\ t_{∗}^{F} ⊗ V_{L} & ⟶ & 0 \end{matrix} )

If T_{L} is injective then q′_{L} is simply a homomorphism, but in general it is an object of a derived category of O_{F}-modules that we have not formally introduced. We shall be completely explicit about q′_{L} in 5.5.3 below.

Now, given a 𝕋-spectrum X, we obtain the object L_{X} = π_{∗}^{A}(X), but we would like abbreviations for its terms, so we take

N_{X} = N_{π_{∗}^{A}(X)} = π_{∗}^{𝕋}(X ∧ DEF_{+})

and

V_{X} = V_{π_{∗}^{A}(X)} = π_{∗}(Φ^{𝕋}X).

Similarly, r_{X} : N_{X} ⟶ t_{∗}^{F} ⊗ V_{X} is the map π_{∗}^{𝕋}(X ∧ DEF_{+} ⟶ X ∧ DEF_{+} ∧ ẼF). Finally we let

T_{X} = T_{π_{∗}^{A}(X)};

this has the geometric significance one expects.

Lemma 5.5.1.

T_{X} = π_{∗}^{𝕋}(X ∧ EF_{+}).

Proof: Since EF_{+} ∧ DEF_{+} ≃ EF_{+} we have a cofibre sequence

X ∧ EF_{+} ⟶ X ∧ DEF_{+} ⟶ X ∧ DEF_{+} ∧ ẼF.

Example 5.5.2. Some examples of particular interest arise from the basic cells. Let us write L_{H} := π_{∗}^{A}(σ_{H}^{0}), and similarly for the other associated invariants.

(i) For the group 𝕋 itself we have

L_{𝕋} = (O_{F} ⟶ t_{∗}^{F} )
N_{𝕋} = O_{F}, V_{𝕋} = ℚ and T_{𝕋} = Σ𝕀

(ii) If H is a finite subgroup, and we let ℚ(H) denote the ℚ[c_{H}]-module ℚ, regarded as a O_{F}-module in the natural way, then

L_{H} = (Σℚ(H) ⟶ 0)
N_{H} = Σℚ(H), V_{H} = 0 and T_{H} = Σℚ(H).

In Section 13.1 we use this to calculate the homotopy Mackey functor of an arbitrary object of A.

It is also worth being completely explicit about some of the above constructions. We must begin by talking about injective representations of objects. We use the convention that if M is an arbitrary object then M̂ denotes an injective object with a homology isomorphism M ⟶ M̂. Of course there is usually no canonical choice of M̂. Thus, if T is any F-finite torsion O_{F}-module, with injective resolution 0 ⟶ T ⟶ I ⟶ J ⟶ 0 then we may take T̂ = fibre(I ⟶ J). Of course e(V) is already injective, and { \atop f(T)} may be taken to be f(T̂) by 5.3.5. Thus for an arbitrary object L = (N_{L} ⟶ t_{∗}^{F} ⊗ V_{L}) with zero differential we may take L̂ to be the fibre of q_{L} : e(V_{L}) ⟶ f(ΣT̂_{L}). In what follows we shall use the notation A ⋉ B to denote a dg module which is additively a direct sum of A and B, and which has B as a submodule and A as a quotient. However the sum is twisted by giving the differential a component A ⟶ ΣB. There is thus a short exact sequence

0 ⟶ B ⟶ A ⋉ B ⟶ A ⟶ 0

of dg modules.

Lemma 5.5.3. For an object L of A (with zero differential) an injective equivalent to L in DA is given by

L̂ = ( \begin{matrix}t_{∗}^{F} ⊗ V_{L} & ⋉ & Σ^{−1}C_{L} ⊕ K̂_{L} \\ ↓ & & ↓ \\ t_{∗}^{F} ⊗ V_{L} & ⊕ & 0 \end{matrix} )

where the differential in (t_{∗}^{F} ⊗ V_{L}) ⋉ (Σ^{−1}C_{L} ⊕ K̂_{L}) is −q′_{L} : t_{∗}^{F} ⊗ V_{L} ⟶ C_{L} ⊕ ΣK̂_{L} given explicitly by q′_{L} = {q′_{C}, q′_{K}} where q′_{C} = c_{L} and q′_{K} ∈ Ext(t_{∗}^{F} ⊗ V_{L}, K_{L}) lifts the class of the extension

[N_{L}] = (0 ⟶ K_{L} ⟶ N_{L} ⟶ im(r_{L}) ⟶ 0) .

Furthermore there is a fibre sequence

L̂ ⟶ e(V_{L}) ⟶^{q_{L}} f(ΣT_{L})

where

(q_{L} : e(V_{L}) ⟶ f(ΣT_{L})) = ( \begin{matrix}t_{∗}^{F} ⊗ V_{L} & ⟶^{q′_{L}} & C_{L} ⊕ ΣK̂_{L} \\ ↓ & & ↓ \\ t_{∗}^{F} ⊗ V_{L} & ⟶ & 0 \end{matrix} )

Proof: It is enough to construct a homology isomorphism L ⟶ L̂ since the displayed L̂ is visibly injective, and is the fibre of q_{L}. To simplify the picture in the reader’s mind we now suppose L is even.

We shall construct a diagram

N_{L}  ⟶^{i_{L}} (t_{∗}^{F} ⊗ V_{L}) ⋉ (Σ^{−1}C_{L} ⊕ K̂_{L})
↓  ↓
t_{∗}^{F} ⊗ V_{L} ⟶^{1}  (t_{∗}^{F} ⊗ V_{L}) ⊕ 0

where i_{L} = {i′_{L}, 0, i′′′_{L}} with i′_{L} = β_{L}; the choice of i′′′_{L} needs more careful description. Since the diagram commutes for any choice of i′′′_{L} and the lower arrow is a homology isomorphism, it suffices to choose i′′′_{L} so that i_{L} is a map of dg O_{F}-modules and a homology isomorphism.

For this we display the map i_{L} vertically and find a diagram

0 ⟶ K_{L} ⟶  N_{L}  ⟶  im(r_{L})  ⟶ 0
h_{L} ↓  i_{L} ↓  j_{L} ↓
0 ⟶ K̂_{L} ⟶ (t_{∗}^{F} ⊗ V_{L}) ⋉ (Σ^{−1}C_{L} ⊕ K̂_{L}) ⟶ (t_{∗}^{F} ⊗ V_{L}) ⋉ Σ^{−1}C_{L} ⟶ 0.

We use the natural inclusion K_{L} ⟶ K̂_{L} for h_{L}, and j_{L} is the composite im(β_{L}) ⟶ t_{∗}^{F} ⊗ V_{L} ⟶ (t_{∗}^{F} ⊗ V_{L}) ⋉ Σ^{−1}C_{L}; both of these are obviously dg maps and homology isomorphisms, and the right hand square commutes for any choice of i′′′_{L}. By the 5 lemma, it is enough to choose a map i′′′_{L} so that i_{L} is a chain map and the left hand square commutes.

For this we refer to the diagram

K_{L} ⟶ N_{L} ⟶ im(i_{L}) ⟶ t_{∗}^{F} ⊗ V_{L}
↘ i′′′_{L} ↓  ˜[N_{L}] ↓ ↙
h_{L}  I(K_{L}) ⟶ J(K_{L})  q′_{K}

Here ˜[N_{L}] represents the extension class [N_{L}], and extends to q′_{K} by injectivity of J(K_{L}). Now [N_{L}] is the image of the identity map of K_{L} under the boundary map for the short exact sequence 0 ⟶ K_{L} ⟶ N_{L} ⟶ im(β_{L}) ⟶ 0, and hence there is a map i′′′_{L} in the diagram to make the square commute. Finally, from the Snake Lemma construction of the boundary map, it follows that the left hand composite is the standard inclusion h_{L} : K_{L} ⟶ K̂_{L} as claimed.

Since the square commutes, the map i_{L} = {i′_{L}, 0, i′′′_{L}} commutes with differentials. Since i′′′_{L} extends h_{L} the left hand square in the map of short exact sequences commutes.

70 5. ASSEMBLY AND THE STANDARD MODEL.

CHAPTER 6 The torsion model.

We begin with a section explaining the methods we have actually used for identifying the place of various well known spectra in the standard model: this chapter places the method in a proper framework. The methods are closely tied to standard methods of equivariant topology, as described in the introduction. Whilst their success may be sufficient justification, it is desirable to explain what can and cannot be seen by these methods. To do this we construct the geometrically natural torsion model in Section 6.2, and show algebraically in Section 6.3 that it is equivalent to the standard model. We do not know a direct method for showing the derived category of the torsion model category is equivalent to the category of rational 𝕋-spectra. The method described in Section 6.1 attempts to identify the place of a 𝕋-spectrum in the torsion model. The algebraic drawback of the torsion model is that it is of injective dimension 2, so that the natural Adams spectral sequence associated to this construction does not collapse until E_{3}. This means that not every homomorphism of objects in the torsion model lifts to a geometric map, and hence an object may not be be formal (ie determined by its homology). It is fortunate that for the usefulness of the easy method presented here, that so many spectra of interest are formal.

6.1. Practical calculations.

Given a 𝕋-spectrum, X we may want to identify its place in our classification. In principle this means that we must calculate π_{∗}^{A}(X), but it is often more practical to proceed in several steps. Thus we know that X is the fibre of

q_{X} : X ∧ ẼF ⟶ X ∧ ΣEF_{+},

and hence that the algebraic model π_{∗}^{A}(X) is equivalent to the fibre of the algebraic map

q̂_{X} : π_{∗}^{A}(X ∧ ẼF) ⟶ π_{∗}^{A}(X ∧ ΣEF_{+}),

corresponding to q_{X} in the derived category DA. Furthermore we have seen in 5.4.3 that π_{∗}^{A}(X ∧ ẼF) = e(π_{∗}(Φ^{𝕋}X)) and π_{∗}^{A}(X ∧ ΣEF_{+}) = f(π_{∗}^{𝕋}(X ∧ ΣEF_{+})).

Summary 6.1.1. For any 𝕋-spectrum X the algebraic model π_{∗}^{A}(X) is the fibre of the map

q̂_{X} : e(π_{∗}(Φ^{𝕋}X)) ⟶ f(π_{∗}^{𝕋}(X ∧ ΣEF_{+}))

in the derived category corresponding to q_{X} : X ∧ ẼF ⟶ X ∧ ΣEF_{+}.

To use this method we need to identify Φ^{𝕋}X, calculate π_{∗}^{𝕋}(X ∧ ΣEF_{+}) and identify the map q_{X} : X ∧ ẼF ⟶ X ∧ ΣEF_{+} between them, in algebraic terms. Using 5.4.8, this may be done by identifying q̂_{X} : t_{∗}^{F} ⊗ π_{∗}(Φ^{𝕋}X) ⟶ π_{∗}^{𝕋}(X ∧ ΣEF_{+}) as a map in the derived category. When π_{∗}^{𝕋}(X ∧ ΣEF_{+}) is injective this simply involves finding a homomorphism of O_{F}-modules.

In practice we begin by calculating π_{∗}^{𝕋}(X ∧ ΣEF_{+}) using the spectral sequence of the skeletal filtration of EF_{+}, and some ingenuity to obtain the O_{F}-module structure.

Next, we note that Φ^{𝕋}X is determined by its homotopy groups π_{∗}(Φ^{𝕋}X) = π_{∗}^{𝕋}(X ∧ ẼF). Often the most effective way of calculating this is to use the first step and the long exact homotopy sequence of X ∧ EF_{+} ⟶ X ⟶ X ∧ ẼF. However the best method for calculating Φ^{𝕋}X depends on how X is given. If X is the suspension spectrum of a based space then of course Φ^{𝕋}X is the suspension spectrum of the fixed point space. In the event that X is equipped with Thom isomorphisms X ∧ S^{V (F)} ≃ X ∧ S^{2} (for example if X is an F-spectrum or the equivariant K-theory spectrum), then the model ẼF = S^{∞V (F)} shows π_{∗}(Φ^{𝕋}X) = E^{−1}π_{∗}^{𝕋}(X) where E is the multiplicative set of Euler classes, generated by the Euler classes of V (H) for finite subgroups H. A variation of this is to use the spectral sequence of the filtration

S^{0} ⊆ S^{V (F)} ⊆ S^{2V (F)} ⊆ ⋯ ⊆ ⋃_{k} S^{kV (F)} = S^{∞V (F)} = ẼF.

In calculating q̂_{X} we assume again that it is routine to calculate integer graded π_{∗}^{𝕋}. In any case we will have calculated (q_{X})_{∗} : π_{∗}(Φ^{𝕋}X) ⟶ π_{∗}^{𝕋}(X ∧ ΣEF_{+}) during the previous steps; since q̂_{X}(1⊗x) = (q_{X})_{∗}(x) this determines q̂_{X} on all elements τ ⊗x when τ ∈ t_{∗}^{F} is of negative degree. Many elements x of π_{∗}^{𝕋}(X ∧ ΣE⟨H⟩) are uniquely divisible by c_{H}, which allows one to determine the images of elements like c_{H}^{−1} ⊗ x. However this only accounts for all elements in trivial cases, and it even happen (for example with X = S^{0}) that (q_{X})_{∗} = 0 whilst the map q_{X} (and hence also q̂_{X}) are highly nontrivial. In this case we need to do more work to identify the images of elements c_{H}^{−k} ⊗ x. In fact the definition of c_{H} allows us to see that this is in effect the a part of π_{∗}^{𝕋} twisted by a representation.

Lemma 6.1.2. For any integer k the map

q̂_{X} : ℚ{c_{H}^{−k}} ⊗ π_{∗}(Φ^{𝕋}X) ⟶ Σ^{−2k}π_{∗}^{𝕋}(X ∧ ΣEF_{+})

is the map induced in π_{∗}^{𝕋} by

X ∧ ẼF ≃ X ∧ ẼF ∧ σ^{−kV (H)} ⟶^{q_{X}∧1} X ∧ ΣE⟨H⟩ ∧ σ^{−kV (H)} ≃ X ∧ Σ^{1−2k}E⟨H⟩.

6.2. The torsion model

The reader will have noticed that all motivation and all investigations proceed by the following process. First we understand the F-contractible part (i.e. the vertex), next the F-free part (i.e. the torsion) and finally we identify the comparison map between them. Why then have we not used the abelian category A_{t} of objects (t_{∗}^{F} ⊗ V ⟶ T), and formed a derived category from this? The reason is that A_{t} has injective dimension 2, and this makes the model much less useful. On the one hand, it makes it hard to get a precise hold on morphisms, and on the other, the objects of the abelian model are not adequate to model all 𝕋-spectra.

The main purpose of this section is to introduce the torsion model category A_{t}, and the purpose of the next is to show that DA_{t} is another model for the category of 𝕋-spectra. This makes precise the assertion that the extra information that must be added to an object of A_{t} to specify a 𝕋-spectrum is an extension of the cokernel of the structure map by t_{∗}^{F} ⊗ V; this is exactly what is encoded by the nub in the standard algebraic model. Hence in particular it places our intuition about how to specify a 𝕋-spectrum on a firm and practical basis. Furthermore, it will transpire in Part II that certain functors (the Lewis-May fixed point functor and the quotient functor) are more approachable on the torsion model. Thus the torsion model is also of theoretical value.

Assuming the fact that A_{t} has injective dimension 2, and that the derived category DA_{t} exists, we explain the consequences in more detail, using the natural Adams spectral sequence. Indeed, we have a functor

H_{∗} : DA_{t} ⟶ A_{t},

obtained by taking homology termwise. Since any object of A_{t} may be viewed as an object of DA_{t} with zero differential, this is certainly surjective on objects. However A_{t} is only a quadratic approximation to DA_{t}, so has two linked drawbacks. Firstly, two non-isomorphic objects of DA_{t} may have isomorphic homology (not all objects are ‘formal’ in the sense of Sullivan), and secondly, maps in DA_{t} are calculated by a spectral sequence with a non-trivial d_{2} differential. In fact there is a filtration

0 ⊆ F_{2}(X, Y) ⊆ F_{1}(X, Y) ⊆ F_{0}(X, Y) = [X, Y].

Here (F_{1}/F_{2})(X, Y) = Ext^{1}(H_{∗}(ΣX), H_{∗}(Y)) and there is an exact sequence

0 ⟶ (F_{0}/F_{1})(X, Y) ⟶ Hom(H_{∗}(X), H_{∗}(Y)) ⟶^{d_{2}} Ext^{2}(H_{∗}(ΣX), H_{∗}(Y)) ⟶ F_{2}(X, ΣY) ⟶ 0.

Thus d_{2} gives the obstruction to realizing an algebraic map geometrically, and, even using the splitting by parity there may be extension problems relating (F_{0}/F_{1})(X, Y) and F_{2}(X, Y).

We now turn to the justification of our assertions about A_{t}.

Lemma 6.2.1. (i) For any graded vector space V the object (t_{∗}^{F} ⊗ V ⟶ 0) is injective in A_{t}. (ii) For any F-finite torsion injective I the object

(t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, I) ⟶ I),

whose structure map is evaluation, is injective in A_{t}.

Proof: Part (i) is immediate. For Part (ii) a short calculation verifies that for any F-finite torsion module T

Hom((t_{∗}^{F} ⊗ U ⟶ R), (t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, T) ⟶ T)) = Hom(R, T).

It is plain that the lemma provides enough injectives. Finally, we estimate the injective dimension as follows. For an arbitrary object (t_{∗}^{F} ⊗ V ⟶ T) we have the exact sequence

0 ⟶ 0 ⟶ t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V ⟶ 0
↓  ↓  ↓
0 ⟶ T ⟶ T  ⟶  0  ⟶ 0.

Since (t_{∗}^{F} ⊗ V ⟶ 0) is injective it suffices to show (0 ⟶ T) has dimension ≤ 2. Suppose then that 0 ⟶ T ⟶ I ⟶ J ⟶ 0 is an injective resolution of F-finite torsion modules; we may then form the exact sequence

0 ⟶ 0 ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, I) ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, I) ⟶ 0
↓  ↓  ↓
0 ⟶ T ⟶  I  ⟶  J  ⟶ 0,

and so it suffices to show (t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, I) ⟶ J) has injective dimension ≤ 1. For this we have the exact sequence

0 ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, T) ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, I) ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, J) ⟶ t_{∗}^{F} ⊗ Ext(t_{∗}^{F}, T) ⟶ 0
↓  ↓  ↓  ↓
0 ⟶  0  ⟶  J  ⟶  J  ⟶  0  ⟶ 0.

Finally we note that the bounds can be achieved. For example

Ext^{2}((t_{∗}^{F} ⟶ 0), (0 ⟶ T)) ≅ Ext(t_{∗}^{F}, T),

which need not be zero.

We have seen that we may form the derived category DA_{t} of such a category by inverting homology isomorphisms. The construction of an Adams spectral sequence is then routine. We need only know that enough injectives are realizable and that H_{∗} : [X, I] ⟶ Hom(H_{∗}(X), H_{∗}(I)) is isomorphic when H_{∗}(I) is injective. Now, for any object Y, we construct a sequence

Y ⟶ I_{0} ⟶ I_{1} ⟶ I_{2}

realizing any specified injective resolution

0 ⟶ H_{∗}Y ⟶ H_{∗}I_{0} ⟶ H_{∗}I_{1} ⟶ H_{∗}I_{2} ⟶ 0.

Applying [X, ·] we obtain a spectral sequence of the form described above, which is obviously convergent since it is finite.

We recall that an object Y of DA_{t} is said to be formal it is determined by its homology in the sense that H_{∗}(Y) ≅ H_{∗}(Y′) implies Y ≃ Y′.

Lemma 6.2.2. If H_{∗}(Y) is of injective dimension 1 then Y is formal, and hence in particular Y is formal if the torsion part of H_{∗}(Y) is an injective F-finite torsion O_{F}-module.

6.3. EQUIVALENCE OF DERIVED CATEGORIES OF STANDARD AND TORSION MODELS.75

Proof: The formality of one dimensional objects is clear, since the Adams spectral sequence collapses to let us realise an isomorphism ϕ : H_{∗}(Y′) ⟶ H_{∗}(Y) by a map f : Y′ ⟶ Y. Since f_{∗} = ϕ is a homology isomorphism by construction, it is a weak equivalence.

It remains to observe that any object (t_{∗}^{F} ⊗ V ⟶ I) with I injective is of injective dimension 1. However there is an obvious embedding in the injective (t_{∗}^{F} ⊗ V ⟶ 0) ⊕ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, I) ⟶ I), and the cokernel clearly has zero torsion part.

Remark 6.2.3. Later it will be useful to note that the torsion model category is isomorphic to the category of objects V ⟶ Hom(t_{∗}^{F}, T) by the usual adjunction. The point to be made is that this is also an abelian category, but epimorphisms need not be surjective. In fact a map takes the form

U  ⟶^{f}  V
↓  ↓
Hom(t_{∗}^{F}, S) ⟶^{Hom(1,g)} Hom(t_{∗}^{F}, T),

and its kernel and cokernel are ker(f) ⟶ Hom(1, ker(g)) and cok(f) ⟶ Hom(1, cok(g)).

Note also that in this view the basic injectives are of form (V ⟶ 0) and (Hom(t_{∗}^{F}, T) ⟶^{1} Hom(t_{∗}^{F}, T) for an injective T, which makes them appear more natural.

6.3. Equivalence of derived categories of standard and torsion models.

In this section we establish that the derived category of the torsion model DA_{t} is equivalent to the derived category of thestandard model DA, and hence that it too provides an algebraic model for rational 𝕋-spectra.

Theorem 6.3.1. The torsion model category is equivalent to the derived category DA of A, and hence we have equivalences

DA_{t} ≃ DA ≃ 𝕋−Spec/𝕋

of triangulated categories.

Proof: We want to define functors comparing dgA_{t} and dgA by passage to fibre or cofibre as the case may be. Indeed we may define

fib : dgA_{t} ⟶ dgA

by passage to fibre, in the sense that

fib(t_{∗}^{F} ⊗ V ⟶^{s} ΣT) = (F (s) ⟶ t_{∗}^{F} ⊗ V).

However if we use cofibres and define

cof(N ⟶^{r} t_{∗}^{F} ⊗ V) = (t_{∗}^{F} ⊗ V ⟶ C(r))

the image is not in dgA_{t}. Instead we consider the slight variation dgHA_{t}, consisting of dg objects t_{∗}^{F} ⊗ V ⟶ M with the property that H_{∗}(M) is an F-finite torsion module. Thus we have defined a functor

cof : dgA ⟶ dgHA_{t}.

Once we invert homology isomorphisms the inclusion

i : dgA_{t} ⟶ dgHA_{t}

becomes an equivalence. It is convenient to introduce a similar variation of A: we let dgHA be the category of dg objects N ⟶ t_{∗}^{F} ⊗ V so that the homology is an isomorphism modulo F-finite torsion modules.

Lemma 6.3.2. The derived categories of dgHA_{t} and dgHA exist and the inclusions dgA_{t} ⟶ dgHA_{t} and dgA ⟶ dgHA induce isomorphisms of derived categories which preserve the triangulation.

Proof: In both cases we have the same formal situation. We have a category 𝔸 from which we know how to form the derived category: we first form the category dg𝔸 of dg objects, and then invert the homology isomorphisms. To establish existence we use the class I of objects formed from injective objects of 𝔸 by a finite number of cofiberings.

The category 𝔸 is a subcategory of a larger category 𝔸̂, and we then consider the category dgH𝔸 of dg objects of a 𝔸̂ satisfying the condition that their homology lies in 𝔸. We show that D𝔸 is also the derived category of dgH𝔸 by establishing that Condition 4.1.1 holds with S being the class of homology isomorphisms. No extra work is necessary for Part (ii), and Part (iii) follows by the same proof as given in Section 4.1. It therefore remains only to show that any object M of dgH𝔸 admits a homology isomorphism M ⟶ M̂ with M̂ ∈ I. This is sufficient to establish the existence of the derived category of dgH𝔸, and the fact that the inclusion dg𝔸 ⟶ dgH𝔸 induces an equivalence of derived categories.

The simpler case is when 𝔸 = A; we consider it first, and explain the variations necessary to deal with 𝔸 = A_{t}. First, since A is one dimensional, we may take an injective resolution 0 ⟶ H_{∗}M ⟶ I ⟶ J ⟶ 0 of H_{∗}M. Now view H_{∗}M as dg object with zero differential, and construct a homology isomorphism H_{∗}M ⟶ M̂, where M̂ is the fibre of I ⟶ J, and hence an object of I. From this we obtain the map zM ⟶ H_{∗}M ⟶ M̂ where zM denotes the subobject of cycles. We need to observe that this extends over the inclusion zM ⟶ M to give a homology isomorphism. Now we have a surjective map

[L, M̂] ⟶ Hom(H_{∗}(L), H_{∗}(M)),

and so we may choose a homotopy class M ⟶ M̂ lifting the projection zM ⟶ H_{∗}M; this is represented by a function θ : M ⟶ M̂. The difference between θ and the given function on zM is a map into the injective J, and thus extends to M. Since the differential of M̂ is zero on J this extension necessarily commutes with differentials. Adding the extension to θ we obtain a function in the same homotopy class which agrees exactly with the given function on zM as required.

To deal with 𝔸 = A_{t} we argue slightly less directly. In fact we note that, by the exact sequence (0 ⟶ T) ⟶ (t_{∗}^{F} ⊗V ⟶ T) ⟶ (t_{∗}^{F} ⊗V ⟶ 0) it is enough to deal with objects (t_{∗}^{F} ⊗ V ⟶ 0) and (0 ⟶ T). Those of the first type are already injective, so it is enough to deal with those of the second type. By the above argument applied to the category of F-finite torsion modules we obtain a homology isomorphism T ⟶ T̂ with T̂ the fibre of I ⟶ J where 0 ⟶ H_{∗}T ⟶ I ⟶ J ⟶ 0 is an injective resolution of H_{∗}(T). Thus we have a homology isomorphism (0 ⟶ T) ⟶ (0 ⟶ T̂), and the codomain is an element of dgA_{t} itself, and hence certainly admits a homology isomorphism to an object of I. This completes the argument.

Notice by the long exact sequence associated to a cofibre or fibre sequence the functors fib and cof preserve homology isomorphisms, and hence induce maps of derived categories.

We may now prove the theorem by considering the functors

fib′ : dgHA_{t} ⟶ dgHA and cof′ : dgHA ⟶ dgHA_{t},

which induce functors on derived categories for the same reasons. The standard natural transformations cof′ ∘ fib′ ⟶ 1 and 1 ⟶ fib′ ∘ cof′ induce homology isomorphisms. Hence fib′ and cof′ induce an equivalence between derived categories.

We shall have use for one more result.

Lemma 6.3.3. The functor fib : dgA_{t} ⟶ dgA takes the class I of fibrant objects to fibrant objects.

Proof: It is obvious that fib takes an F-contractible injective to an injective since fib(t_{∗}^{F} ⊗ V ⟶ 0) = (t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V).

On the other hand fib(t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, T) ⟶ T) is the fibre of a map

Σ^{−1}T ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, T)
↓  ↓
0 ⟶ t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, T).

Since fib preserves triangles the result follows.

78 6. THE TORSION MODEL.

Part II

Algebraic counterparts to standard constructions.

80

CHAPTER 7 Introduction to Part II

This chapter begins in Section 7.1 with a summary of the models from Part I, and a broad summary of the contents of Part II. The following three sections provide more substantial introductions to the three main types of example treated in Part II. Finally, in Section 7.5 we formalise a straightforward argument that is needed several times.

7.1. General outline.

In Part I we showed that various categories of rational 𝕋-spectra had algebraic models. Specifically we showed (i) that the category of 𝕋-spectra over H is equivalent to the derived category of torsion ℚ[c_{H}]-modules, (ii) that the category of F-spectra is equivalent to the derived category of F-finite torsion O_{F}-modules and (iii) that the category of semifree spectra is equivalent to the derived category of the category of ℚ[c]-morphisms N ⟶ ℚ[c, c^{−1}] ⊗ V which become isomorphisms when c is inverted and (iv) that the category of all rational 𝕋-spectra is equivalent to the derived category of the one dimensional abelian category A whose objects are morphisms N ⟶ t_{∗}^{F} ⊗ V with F-finite torsion kernel and cokernel.

The main purpose of Part II is to identify the algebraic counterparts of various well known topological functors. Most interesting functors F : 𝕋−Spec ⟶ 𝕋−Spec arise by passage to stable homotopy from a point set level functor f. It is therefore natural to expect that the algebraic counterpart will be the total right derived functor Rf′ : DA ⟶ DA of a functor f′ : A ⟶ A, or its left derived counterpart. (Many may be content to view Rf′ as defined by the formula Rf′X = f′(X̂) where X̂ is a fibrant approximation to X; for further discussion the reader is referred to Appendix B on Quillen closed model category structures). This expectation proves well founded in all cases we have analyzed. There are two caveats to this statement. Firstly, it sometimes seems necessary to replace A by the torsion model A_{t} in order to find an algebraic counterpart at the level of abelian categories. More important is the fact that at present there seems no prospect of showing that the equivalence of topological and algebraic categories arises from a string of adjunctions of Quillen closed model categories. Accordingly there is no prospect of showing that f and f′ correspond, and we must proceed in an indirect way.

Thus the work of this part divides into two. Firstly we have to to do enough algebra to understand the behaviour of the functors f′ which arise, and thus to understand their associated right derived functors. Secondly we have to show how to describe the functor F using only terms which we can model exactly. Of course, if we can identify F X by its homotopy (as happens for the standard model), it is relatively easy to verify the model on objects. There is then the difficulty of indeterminacy in the modelling of morphisms, and we may have to argue rather obliquely to verify this part of the model. However, some interesting functors are hard to describe exactly, even on objects. In this case we may attempt to construct objects X from basic objects X′ using sums and cofibre sequences in such a way that the value F X′ of F on a basic object X′ is modelled by Rf′(M), and so that at each stage we can identify the maps with zero indeterminacy.

Because of Part I, a great deal of the work in Part II is purely algebraic. Of course it is necessary to do a little topology from time to time if we wish to know which topological phenomena we are modelling. On the other hand, there may be readers who feel that after Part I they do not need to talk about actual 𝕋-spectra any more, and that they can work entirely in the algebraic category. Despite the obvious dangers of this attitude, its benefits make it desirable to make it possible to read Part II without intimate acquaintance with Part I. It is thus helpful to recall the model before undertaking a full scale algebraic study.

We therefore begin by recalling that we work almost exclusively over the graded ring O_{F}, which we may view as obtained from the ring (O_{F})_{0} = C(F, ℚ) ≅ ∏_{H} ℚ by adjoining an indeterminate c of degree −2. We let e_{H} denote the idempotent of (O_{F})_{0} corresponding to the projection onto the Hth factor, and we let c_{H} = e_{H}c. We next need the twisting module t_{∗}^{F} which we have approached in Part I through the short exact sequence

0 ⟶ O_{F} ⟶ t_{∗}^{F} ⟶ Σ^{2}𝕀 ⟶ 0

with 𝕀 = ⨁_{H} 𝕀_{H}, and Σ^{2}𝕀_{H} = ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}]. We shall shortly justify a more instructive construction, as follows. First we need to define Euler classes in suitable generality, generalizing the elements c_{1}^{k} ∈ H^{∗}(B𝕋_{+}) for k ≥ 0. For any finite subset ϕ ⊆ F we have an associated idempotent e_{ϕ} ∈ O_{F}, and we have define the Euler class c_{ϕ} = e_{ϕ}c+(1−e_{ϕ}), which is not a homogeneous element of O_{F}. The effect of c_{ϕ} on an O_{F}-module M = e_{ϕ}M ⊕ (1 − e_{ϕ})M is to multiply by c on the first factor and do nothing to the second: thus the result of inverting c_{ϕ} on M is again a graded module. We therefore consider the set

E = {c_{ϕ}^{k}|ϕ ⊆ F finite k ≥ 0}

of Euler classes. We shall see that t_{∗}^{F} = E^{−1}O_{F}. Next, an O_{F}-module M is said to be F-finite if the natural inclusion is an isomorphism ⨁_{H} e_{H}M ≅ M, and it is torsion if M[c^{−1}] = 0. It turns out that M is an F-finite torsion module if and only if E^{−1}M = 0.

We now consider the category A whose objects are morphisms β : N ⟶ t_{∗}^{F} ⊗ V whose kernel and cokernel are Euler torsion. Morphisms are required to be the identity on the tensor factor t_{∗}^{F}. We call N the nub, and V the vertex of the object; the map β is called the basing map. The category A is abelian and one dimensional, and its derived category DA is the standard model for rational 𝕋-spectra (Theorem 5.4.1). Thus there is a short exact sequence for calculating morphisms in either DA or in 𝕋−Spec in terms of Hom and Ext in A, and this sequence is split. This is stated explicitly as Theorem 5.4.5.

Whilst DA is technically very convenient, not all geometrically interesting aspects of 𝕋-spectra are immediately apparent. Thus, even when a construction can be performed in DA, the result may need reinterpretation before it answers interesting questions. There also appear to be cases where DA is not the natural way of modelling 𝕋-spectra. We therefore also need to consider the torsion model category A_{t} whose objects are O_{F}-maps t_{∗}^{F} ⊗ V ⟶ T with T an F-finite torsion module. This is abelian and two dimensional and (Theorem 6.3.1) its derived category is equivalent to DA. This completes the summary of the algebraic models.

The rest of Part 2 is concerned with modelling various geometric constructions. The examples are divided into two groups: those of a general nature and those that change the ambient group of equivariance. It seemed logical to begin with smash products and function spectra, as the most fundamental. Unfortunately they are also the hardest to understand, and we only succeed in identifying them on objects, except for the important special case of products, which we identify precisely. The special case of functional duals is made explicit in Section 15.5. We consider the product spectrum, both because of its own importance, and because it is used as an essential ingredient in our identification of the model of the function spectrum. The topological reader may prefer to skip Chapters 8 and 9 to begin with, and return to the product, smash product and function spectrum after the other examples, because of the substantial amount of algebra they require.

Amongst the functors which change equivariance the trick of identifying a functor by identifying its adjoint is both effective and essential, since it happens several times that only one of a pair of adjoint functors is directly accessible. We begin by considering the forgetful induction and coinduction functors, all of which are quite straightforward. Both types of 𝕋-fixed point functors are built into our models. The most interesting functors are those which relate 𝕋-spectra to 𝕋/K-spectra for finite subgroups K. As would be expected, the geometric K-fixed point functor Φ^{K} is rather easy to describe. However both the inflation functor from 𝕋/K spectra to 𝕋-spectra and its adjoint, the Lewis-May K-fixed point functor are rather subtle, even at the purely algebraic level. It is perhaps only to be expected that quotient functors are only identified on reasonably free objects.

We then move on to the homotopy Mackey functor and Eilenberg-MacLane spectra. In the context of compact Lie groups of positive dimension there are two other variants of Eilenberg-MacLane spectra: co-Eilenberg-MacLane spectra representing ordinary homology, and Brown-Comenetz spectra representing the Mackey duals of homotopy. Since one is used to these coinciding with Eilenberg-MacLane spectra rationally for finite groups of equivariance, it seems worth giving a detailed analysis to show how different they are in the present case.

Although the work of Part II is of interest in its own right, it has further justification in the applications of Part III. In practice the author only undertook the analysis of a functor when forced to by the applications. The resulting clarification and simplification in the special case was most gratifying. However, since there is now a substantial amount of algebra, which may perhaps be of interest in its own right, we have collected it together in chapters of its own. This has the unfortunate effect of separating the algebra from the topology which motivated it. Therefore, to help the reader navigate Part II we will continue this introduction with more complete sketches of the analyses and motivations for the algebraic constructions: we feel justified in talking heuristically at this stage. The structure of the analyses is that we develop the necessary algebra and then explain the the properties of topological constructions which relate it to the algebra. Finally the algebraic and topological functors are shown to correspond using these two parallel structures. The last section of this chapter outlines the general methods we employ for studying functors.

7.2. Modelling of the smash product and the function spectrum.

In Sections 8.3 to 9.3 we identify the algebraic model of the smash product functor and its right adjoint, the function spectrum functor. In essence the smash product is modelled by the left derived tensor product, and the function spectrum by its right adjoint. However there are several important caveats. Firstly, the left derived tensor product does not exist in all the categories modelling spectra: one may instead interpret it as the right derived torsion product. The nomenclature is justified since the construction is the left derived tensor product in a larger category. The next important point is that the right adjoint of the left derived tensor product is not the right derived internal Hom functor, although it is closely related. This fact is a consequence of the caveat about left derived tensor products. Both complications are essential in order to get models of the geometric constructions. Finally, we are only able to describe the models on objects, and not as functors; neither the smash product nor its adjoint preserve pure parity objects in general, so one cannot get a firm grip on morphisms. This means that once we have shown the smash product is modelled by the left derived torsion product it does not follow formally that the function spectrum is modelled by its adjoint. Nonetheless it is true, which seems strong evidence that our models are in fact functorial. To prove functoriality in the present framework, one would need a natural splitting into even and odd pieces. A more satisfactory resolution would be to prove the equivalence of homotopy categories from a string of equivalences at the level of model categories, following the example of Quillen [21, 22]. However this approach appears inaccessible at present.

We also spend some time considering the modelling of arbitrary products of spectra. Section 8.2 identifying the product in the torsion model and its derived category is an instructive introduction to the difficulties faced for the function spectrum. However the identification of the product in the standadard model and its derived category in Section 8.6, is an essential tool in the analysis of the function spectrum. The point is that we can identify the model of the product of spectra as a functor, because the product is right adjoint to the diagonal functor. This gives us control over F (⋁_{i} Y_{i}, Z) ≃ ∏_{i} F (Y_{i}, Z) when F (Y_{i}, Z) is known for all i.

Returning to the smash product and the function spectrum, we shall discuss four different contexts in parallel. The model of free spectra given by torsion ℚ[c]-modules, the model of F-free spectra, given by F-finite torsion modules, the model of semifree spectra, and finally the model of all 𝕋-spectra given by the standard model. Considerations are slightly different in the four cases, but the strategy is common to them all.

For the smash product, the first observation is that it preserves F-spectra and Fcontractible spectra. More precisely X ∧ Y ∧ EF_{+} ≃ (X ∧ EF_{+}) ∧ (Y ∧ EF_{+}) and Φ^{𝕋}(X ∧ Y) ≃ Φ^{𝕋}X ∧ Φ^{𝕋}Y. Since Φ^{𝕋}(X ∧ Y) = Φ^{𝕋}(X) ∧ Φ^{𝕋}(Y), it is immediate that

π_{∗}(Φ^{𝕋}(X ∧ Y)) = π_{∗}(Φ^{𝕋}X) ⊗ π_{∗}(Φ^{𝕋}Y),

and we shall show that π_{∗}^{𝕋}(EF_{+} ∧ X ∧ Y) is the ‘left derived tensor product’ of π_{∗}^{𝕋}(EF_{+} ∧ X) and π_{∗}^{𝕋}(EF_{+} ∧ Y). This might suggest the use of the torsion model, but note that ΣEF_{+} ∧ ΣEF_{+} ≃ Σ^{2}EF_{+}; the extra suspension makes the gluing map inaccessible. We therefore use the standard model. On the other hand, since t_{∗}^{F} = E^{−1}O_{F} it is clear that the definition

(M ⟶ t_{∗}^{F} ⊗ U) ⊗ (N ⟶ t_{∗}^{F} ⊗ V) = (M ⊗ N ⟶ (t_{∗}^{F} ⊗ U) ⊗ (t_{∗}^{F} ⊗ V) = t_{∗}^{F} ⊗ U ⊗ V)

preserves objects of A. There are enough flat objects in A, and the left derived torsion product ⊛ can be calculated using these. The caveats mentioned above suggest alternative ways of doing this which are unavoidable in some of the contexts.

The function spectrum F (X, Y) is easily described if Y is F-contractible, as F (Φ^{𝕋}X, Φ^{𝕋}Y)∧ ẼF. Similarly if X is F-contractible, so is the function spectrum, and F (X, Y) is described by its homotopy groups. However in the general case, even if X and Y are both F-spectra, F (X, Y) is not: in fact the right adjoint of the smash product with an F-spectrum X (regarded as a functor from F-spectra to F-spectra) is Y ⟼ F (X, Y)∧EF_{+}. This expression should prepare for the algebraic surprise in modelling function spectra.

The function spectrum is recognized as the right adjoint of the smash product, and hence the natural expectation (which proves to be correct on objects) is that the model should be given by the right adjoint to the total left derived functor ⊛. The basic warning is that this right adjoint is not the total right derived functor of the internal Hom functor.

To illustrate the issues in the simplest context, we continue the discussion by concentrating on free spectra, modelled by torsion ℚ[c]-modules. To begin with we must identify the internal Hom functor on the category of torsion modules. This functor is characterized by the adjunction

Hom(R ⊗ S, T) = Hom(R, IntHom(S, T)),

and hence the internal Hom functor between torsion modules S and T is given by

IntHom(S, T) = Γ_{c} Hom(S, T).

In fact the right adjoint of ⊛ is R′Hom := RΓ_{c} RHom and not RIntHom. This bewildered the author for some time. First we give an example where RIntHom(S, T) does not model the function spectrum whilst RΓ_{c} RHom(S, T) does. This shows in particular that the two algebraic functors are different, and concentrates our attention on the correct one, which we shall show gives a model in general.

Example 7.2.1. Let X = Y = E𝕋_{+}, so that we have models S = T = Σ𝕀. We observe that RIntHom(S, T) = Γ_{c}ℚ[c] = 0. On the other hand Hom(S, T) = ℚ[c] is not injective, so that to calculate RΓ_{c} RHom(S, T) = RΓ_{c} Hom(S, T) we must take an injective approximation fibre(ℚ[c, c^{−1}] ⟶ Σ^{2}𝕀) to ℚ[c] before applying Γ_{f}. We thus obtain RΓ_{c} RHom(S, T) = Σ𝕀, in agreement with F (E𝕋_{+}, E𝕋_{+}) ∧ E𝕋_{+} ≃ E𝕋_{+}.

The explanation of this is that the internal Hom functor is only recognized as such by virtue of the intermediate category of all modules. In fact we consider the pairs

torsℚ[c] − mod ⇄^{i} ℚ[c] − mod ⇄^{⊗S} ℚ[c] − mod

of adjoint functors (with left adjoints displayed at the top). The composite of the two lower functors is IntHom(S, ·) = Γ_{c} Hom(S, ·) with its domain extended to all ℚ[c]-modules. In fact the right adjoint of ⊛S : D(ℚ[c]) ⟶ D(ℚ[c]) is R′Hom(S, ·) := RΓ_{c} RHom(S, ·). And if S and T model the free spectra X and Y, then R′Hom(S, T) models F (X, Y) ∧ E𝕋_{+}.

We follow our standard operating procedure, by spending Chapter 8 developing the algebra necessary to understand the model. Section 9.1 provides topological input to the final comparison, and finally, in Section 9.2 we prove smash products are modelled by left derived tensor products, and in Section 9.3 we prove that function spectra are modelled by its right adjoint.

7.3. Modelling functors changing equivariance.

Now that we can model categories of G-spectra when G is any subquotient of the circle 𝕋, we may attempt to understand the functors between these categories in algebraic terms.

Of course the categories of H-spectra are very simple when H is a finite subgroup, so functors involving these are rather straightforward to understand: thus we begin in Section 10.1 by modelling the induction and coinduction functors from H-spectra to 𝕋spectra. There is still a potential difficulty in understanding morphisms, but we recognize the algebraic functors as being left and right adjoint to the forgetful functor, which is easy to identify. It is clear in this case that the vertex plays no real part so it is unimportant whether we work with the standard or the torsion model.

More important are the fixed point functors. Of course there are two types of fixed point functors: the geometric fixed point functors and the Lewis-May fixed point functors. Whichever we use, if we take 𝕋-fixed points the result is a non-equivariant spectrum, and hence determined simply by its ordinary homotopy groups. These are easily deduced from the model. We therefore concentrate on K-fixed points regarded as a functor from 𝕋-spectra to 𝕋/K-spectra.

The geometric K-fixed point functor is designed to pick out the isotropy parts corresponding to subgroups of 𝕋 containing K: it is nearly obvious that this means the vertex is unchanged and that the effect on EF_{+}∧X ≃ ⋁_{H} E⟨H⟩∧X is to throw away the summands indexed by the finite subgroups H not containing K. There is an obvious way of doing this algebraically for O_{F}-modules: one simply applies the idempotent e ∈ C(F, ℚ) with support consisting of subgroups H containing K, and identifies this set of subgroups of 𝕋 with the subgroups of 𝕋/K in the obvious way. We prove in Section 10.2 that this not only gives the right functor for F-spectra, but that simply applying e to the standard model category models the geometric fixed point functor.

The Lewis-May K-fixed point functor Ψ^{K} is well known to be much more complicated, and we spend Chapters 11 and 12 in studying it. It is well known that at one extreme, the geometric K-fixed point functor can be obtained from the Lewis-May fixed point functor by Φ^{K}(X) = Ψ^{K}(X ∧ E[⊉ K]_{+}). At the other extreme, the Lewis-May fixed point functor on a free spectrum is the quotient. One might hope that an understanding of these two extreme pieces of behaviour would lead to an identification of the algebraic model of the functor. Because of its intuitive appeal we present this approach in Section 12.3, but it appears only to give the answer on objects, and not as a functor.

Instead, we find it better to recognize the Lewis-May fixed point functor as the right adjoint of the inflation functor. Terminology is not quite consistent in the literature, so we must explain that if U is a complete 𝕋-universe we mean the functor which takes a 𝕋/K-spectrum X indexed on U^{K}, regards it as a 𝕋-spectrum indexed on U^{K} by pullback along the quotient map q : 𝕋 ⟶ 𝕋/K, and then builds in representations by the relative stabilization functor corresponding to the inclusion j : U^{K} ⟶ U. The resulting spectrum is often written j_{∗}q^{∗}X, q^{#}X or simply j_{∗}X. However, since we want to concentrate on the important features we write simply X ⟼ inf_{𝕋/K}^{𝕋}X for this functor. The behaviour of the inflation functor is very simple on suspension spectra in the sense that inf_{𝕋/K}^{𝕋}X = X for a space X, but its general behaviour is rather complicated, and this is reflected algebraically. Actually, its behaviour is simple on F-contractible spectra, in the sense that it is essentially the identity. It is also simple on F-spectra, in that it simply increases the multiplicity of the various factors. We may view an F-spectrum Y ≃ ⋁_{H} Y (H) as classified by a function H ⟼ N(H), where N(H) = π_{∗}^{𝕋}(Y (H)) is a ℚ[c] module. Now, if X is an almost free 𝕋/K spectrum classified by the function \overline{H} ⟼ M(\overline{H}), then inf_{𝕋/K}^{𝕋}X is classified by the function H ⟼ M(\overline{H}), where \overline{H} is the image of H in 𝕋/K. Thus the summand X(\overline{H}), which occurs once in X, is replaced in inf_{𝕋/K}^{𝕋}X by one copy for each subgroup H of 𝕋 whose image in 𝕋/H is \overline{H}. The difficulty comes in splicing these two pieces together, and this leads us on an algebraic digression to the relatively well behaved category of Hausdorff O_{F}-modules. This enables us to identify an algebraic inflation functor on the torsion model category, and then to show that it models the topological inflation functor. It then follows that the right adjoint of the geometric inflation functor models the Lewis-May fixed point functor. Following standard operating procedure, we begin in Chapter 11 by studying the algebraic inflation and its adjoints. In Section 12.1 we establish the essential features of topological inflation and the Lewis-May fixed point functor. Finally we show the algebraic and topological inflation correspond in Section 12.2; it follows that the left adjoint of the algebraic inflation models the Lewis-May fixed point functor. Finally we spend Section 12.4 considering the functor on the standard model: of the functors we have considered this is the only one for which we do not have a good description in the standard model.

The topological quotient functor is only approachable on K-free spectra, but in that case it is left adjoint to inflation. It follows that the left adjoint to the algebraic inflation functor models passage to the quotient. Since we are working rationally one might hope to extend the domain of good behaviour of the quotient functor to all F-spectra. Although we do not do this, supporting evidence comes from the fact that the algebraic inflation functor does have a left adjoint on the category of all F-finite torsion O_{F}-modules: we discuss these matters in Section 12.5.

7.4. Modelling Eilenberg-MacLane spectra and related objects.

A cohomology theory k_{G}^{∗}(·) satisfying the dimension axiom is characterised by the Mackey functor M : G/H ⟼ k_{G}^{0}(G/H_{+}) describing its values on homogeneous spaces. It is represented by the Eilenberg-MacLane spectrum HM. A homology theory l_{∗}^{G}(·) satisfying the dimension axiom is characterised by the coMackey functor N : G/H ⟼ l_{0}^{G}(G/H_{+}) describing its values on homogeneous spaces. It is represented by the co-Eilenberg-MacLane spectrum JN. Finally, given an injective Mackey functor I we can form the Brown-Comenetz cohomology theory by taking hI_{𝕋}^{n}(X) = Hom(\underline{π}_{n}^{𝕋}(X), I), and this is represented by a spectrum hI. Since an arbitrary Mackey functor M admits an injective resolution of length 1, we can also define the spectrum hM. For finite groups the orbit category is self-dual, so that there is a natural way of identifying Mackey functors and co-Mackey functors. Thus a Mackey functor M gives rise to three representing spectra, HM, JM and hM. It is a deeply ingrained fact that the these three spectra coincide. However, it relies on the facts that G/H is a 0-dimensional manifold, and that H^{∗}(BH; ℚ) is concentrated in degree 0. Both these facts fail for positive dimensional compact Lie groups. We shall identify the Eilenberg-MacLane, co-Eilenberg-MacLane and Brown-Comenetz spectra for the circle group, and it is apparent that they are completely different in general. However it is interesting to observe that all three classes of spectra are formal in the torsion model: in fact their torsion parts are injective.

We begin with ordinary cohomology, and the usual Eilenberg-MacLane spectrum. An Eilenberg-MacLane spectrum is recognized by the fact that its homotopy groups are only nonzero in a single degree. It is characterized by the homotopy groups in that degree, regarded as a Mackey functor. We therefore begin in Section 13.1 by making explicit how to recover the Mackey functor homotopy groups from the standard model. By way of illustration we deduce a functorial construction of the model of an Eilenberg-MacLane spectrum from a Mackey functor. In Section 13.2 we take the opposite approach and begin with ordinary cohomology and deduce a model, and provide decompositions of Eilenberg-MacLane spectra which are interesting from a topological point of view. We follow this by Section 13.3 which gives an analogous approach to the analysis of co-Mackey functors. It does not seem worthwhile to make the algebraic approach to co-Eilenberg-MacLane spectra explicit.

We find Brown-Comenetz spectra to be particularly interesting, and devote Section 13.4 to them. One reason for interest is that by definition these spectra are well suited to the construction of Adams spectral sequences, and the author’s first approach to the algebraic modelling of 𝕋-spectra therefore used them. However the extreme complexity of the Brown-Comenetz spectra seems to make the resulting Adams spectral sequence impractical except in certain special cases. On the other hand the spectra do form a natural class of unbounded spectra, and they therefore provide an interesting test case for the present methods.

7.5. Functors between split triangulated categories.

We conclude the introduction to Part II by explaining our general method for modelling functors.

We repeatedly want to construct algebraic counterparts of topological functors between categories of spectra. In other words we have a functor F : Spec_{1} ⟶ Spec_{2} between two categories of spectra, and we want to find the algebraic counterpart F′ : DA_{1} ⟶ DA_{2}, in the sense that we complete the square

Spec_{1} ⟶ Spec_{2}
p ↓≃  ≃↓ p
DA_{1} ⋯ > DA_{2}

so that it commutes up to natural isomorphism. Typically, it is reasonably easy to guess a candidate algebraic functor F′ : DA_{1} ⟶ DA_{2}, and in fact F′ is usually the total right derived functor of some functor f′ : A_{1} ⟶ A_{2}. The difficulty arises because the equivalences p were only defined indirectly and by making certain choices. Thus it is only easy to identify the homotopy of objects, and the d-invariant of morphisms. There are two grades of conclusion that we may hope to achieve. We may only be able to prove that F′ is correct on objects, in the sense that if X is modelled by M then F X is modelled by F′M. However, in especially favourable cases, we will be able to prove that it is correct on morphisms in the sense that the above square commutes up to natural isomorphism.

Since the relevant functors F and F′ preserve cofibre sequences, the idea is to show that F and F′ agree on certain basic objects X′, and that arbitrary objects X can be formed from these basic objects using only constructions in which every step can be modelled without indeterminacy. It is often sufficient to use mapping cones of maps f : X′_{1} ⟶ X′_{2} in which F (f) is determined by its d-invariant. If this can be done for objects we shall say that F is object-accessible, and if it can also be done for morphisms we shall we shall simply say that F is accessible.

Our only comment about object accessibility is that, if we can use a one dimensional model category A_{2}, it suffices to calculate the homology of the image object, since then all objects are formal in the sense that they are determined by their homotopy. In the torsion model object-accessibility is not always clear, and, even in the standard model, considerable work is involved in showing that the smash product and function spectrum functors are both object accessible.

It is often possible to work with one dimensional models, and it is therefore valuable to record the minimal data that needs to be checked to ensure the diagram commutes up to natural isomorphism. In practice we compare the two composites F_{1} = pF and F_{2} = F′p from the top left to the bottom right.

Theorem 7.5.1. Given a split linear triangulated category Spec and a one dimensional split abelian category A, two functors F_{1}, F_{2} : Spec ⟶ DA are naturally isomorphic provided the following conditions are satisfied.

(1) Both functors preserve (a) triangles, (b) injectives, and (c) pure parity objects

(2) The functors agree up to isomorphism (a) on enough injective objects and (b) on maps into injective objects.

We remark that in the motivating case it is enough to check Condition 1 for the functors F and F′ since the functors p are equivalences of triangulated categories, and used to define the notions of parity and injectivity.

Proof: First note that for any object Y we have F_{1}(Y) ≃ F_{2}(Y). Indeed we may choose an injective resolution Y ⟶ I ⟶^{d} J, and thus a triangle F_{i}Y ⟶ F_{i}I ⟶^{F_{i}d} F_{i}J. Since F_{i}J is injective by (1)(b), it follows that the morphism F_{i}d : F_{i}I ⟶ F_{i}J is determined by its d-invariant. Once the objects are identified using (2)(a) the two maps F_{1}d and F_{2}d are homotopic by (2)(b). Hence F_{1}Y ≃ F_{2}Y as required.

By property (1)(c) it is enough to check that the two maps [X, Y] ⟶ [F X, F Y] agree when X and Y are of even parity. In this case the Adams short exact sequence for [X, Y] is split by parity. Now suppose we have an Adams resolution Y ⟶ I ⟶ J with image F Y ⟶ F I ⟶ F J. Suppose first that f : X ⟶ Y is of even degree, and consider the composite X ⟶ Y ⟶ I. Since F I is injective by (1)(b), the two images of the composite F X ⟶ F Y ⟶ F I are equal. However the indeterminacy in the map F X ⟶ F Y with such a composite is the image of the group [F X, Σ^{−1}F J], which is zero since F X and F J are of even parity and F J is injective. Since there is no indeterminacy F_{1}f = F_{2}f.

Next, if f : X ⟶ Y is of odd degree, it factors as X ⟶ Σ^{−1}J ⟶ Y, and the image of this is the composite F X ⟶ Σ^{−1}F J ⟶ F Y. The first map is a map into an injective, and so F_{1} and F_{2} agree; the second map is part of the triangle F Y ⟶ F I ⟶ F J.

CHAPTER 8 Basic algebra for models and their derived categories.

This chapter contains the most substantial algebra of the entire work. However the first section is elementary and will interest all readers. It gives the long-awaited discussion of Euler classes in the global setting. In particular, it gives the elementary proof that F-finite torsion modules are exactly the Euler-torsion modules. The rest of the chapter is only necessary in our analysis of the product, smash product and function spectrum, and some readers will want to skip to Chapter 10.

In Section 8.2 we warm up with a treatment of products in the torsion model and its derived category, before giving an abstract treatment of the tensor-Hom adjunction in Section 8.3, illustrated by the rather straightforward case of categories of torsion modules. Before applying this to deal with the derived category of the standard model, we need to understand the tensor product and internal Hom on the standard model itself: we study these constructions in Section 8.4, and it involves a little work, principally in the analysis of the appropriate ‘torsion functor’. After that, it is not hard to verify in Section 8.5 that the standard model fits into the framework layed out in Section 8.3. Finally, in Section 8.6 we complete the analysis of products in the standard model and its derived category.

8.1. Euler classes and F-finite torsion O_{F}-modules.

In this section we explain the occurrence of F-finite torsion O_{F}-modules, by showing that they are precisely the modules annihilated when Euler classes are inverted. We also summarise some elementary but useful algebra, and introduce associated notation.

For brevity, let ℂ denote the category of all O_{F}-modules, and let ℂ_{f} denote the full subcategory of F-finite modules. Furthermore we let ℂ_{t} denote the category of modules M for which each element is c-power torsion, and we let ℂ_{ft} denote the F-finite torsion modules. We thus have inclusions of full subcategories in the arrangement

\begin{matrix}ℂ_{ft} & ⟶ & ℂ_{f} \\ ↓ & & ↓ \\ ℂ_{t} & ⟶ & ℂ \end{matrix}

In fact these inclusions are all left adjoint to useful functors. For an arbitrary module M we let Γ_{c}M denote the submodule of c-power torsion elements, and we define ϕM = ⨁_{H} e_{H}M. The following statements are readily verified.

Lemma 8.1.1. (i) ϕΓ_{c}M = Γ_{c}ϕM (ii) If L is F-finite then

Hom(L, M) = Hom(L, ϕM)

(iii) If T is a torsion module then

Hom(T, M) = Hom(T, Γ_{c}M).

We conclude that ϕ provides a right adjoint to both verticals, and that Γ_{c} provides a right adjoint to both horizontals. By the first part, there is only one composite to consider, and it is worth considering it in more detail.

We recall that there is an Euler class associated to any representation V of 𝕋 with V^{𝕋} = 0. This is based on the map e(V) : S^{0} ⟶ S^{V}, and exists when we have Thom isomorphisms. We let χ(V) denote the associated Euler class. Provided the requisite Thom isomorphisms exist, the fact that e(V ⊕ W) = 1 ∧ e(V) e(W) shows they are multiplicative in the sense that χ(V ⊕ W) = χ(V)χ(W). Evidently, if V^{𝕋} ≠ 0 then χ(V) = 0, since e(V) is null.

Let us turn to the particular case of F-spectra. So far we have restricted attention to the effect on 𝕋-spectra over H when V = V (H), but now we need to take all isotropy groups into account. Thus we consider the map e(V)∧1 : S^{0} ∧EF_{+} ⟶ S^{V} ∧EF_{+}, and note that, if we split EF_{+} into its summands E⟨H⟩, we obtain c_{H}^{v(H)} : E⟨H⟩ ⟶ Σ^{2v(H)}E⟨H⟩ where v(H) = dim_{ℂ}(V^{H}). For any complex representation V with V^{𝕋} = 0 the Euler class χ(V) is multiplication by c_{H}^{v(H)} on the H’th summand. If V^{𝕋} ≠ 0 then χ(V) = 0. Note also that V^{H} = 0 for almost all H, so that any O_{F}-module M splits as M(H_{1}) ⊕ ⋯ ⊕ M(H_{n}) ⊕ M′ where V^{H} = 0 unless H = H_{1}, …, H_{n−1} or H_{n}, and thus the Euler class acts on any O_{F}-module. To avoid undue dependence on representations we will use an alternative notation.

Definition 8.1.2. If ϕ ⊆ F we let e_{ϕ} be the idempotent with support ϕ, and we let c_{ϕ} = e_{ϕ}c + (1 − e_{ϕ}). Thus, if V is one dimensional with kernel H, we have χ(V) = c_{ϕ} where ϕ = {K | K ⊆ H}. More generally, if v : F ⟶ ℤ_{≥0} is any function, we may let c^{v} ∈ O_{F} be the non-homogeneous element which is c_{H}^{v(H)} over H. For any representation V we may associate the function v(H) = dim_{ℂ}(V^{H}), and then we have χ(V) = c^{v}.

It is also natural to use the corresponding notation in topology, so that, for any dimension function v, we define S^{v} by the cofibre sequence

⋁_{H} E⟨H⟩^{(2v(H))} ⟶ S^{0} ⟶ S^{v}.

Evidently, we can also make analogous definitions in algebra and topology when v also takes negative integer values, provided it is only negative on finitely many subgroups.

Finally, we let E denote the multiplicative set of all Euler classes:

E = {c_{ϕ}^{k} | ϕ ⊆ F finite k ≥ 0},

with associated torsion functor

Γ_{f}M = {x ∈ M | ex = 0 for some nonzero Euler class e }.

8.1. EULER CLASSES AND F-FINITE TORSION O_{F}-MODULES. 93

Lemma 8.1.3. We have equalities

Γ_{f}M = Γ_{c}ϕM = ϕΓ_{c}M

Proof: It is sufficient to note that no element of M with infinitely many coordinates nonzero can be annihilated by an Euler class.

We may now formalise the connection between the discussion in the introduction and the work in Part I by observing that

E^{−1}O_{F} = t_{∗}^{F}.

This lets us regard objects (N ⟶ t_{∗}^{F} ⊗ V) of the standard model category A as O_{F}- modules N together with a basis of E^{−1}N (i.e. a subspace V of E^{−1}N giving an isomorphism E^{−1}N ≅ t_{∗}^{F} ⊗ V).

Lemma 8.1.4. The functor Γ_{f} is left exact, and only has one right derived functor; this is given by R^{1}Γ_{f}M = E^{−1}M/M.

Proof: First note that Euler-torsion modules may be resolved by Euler-torsion injectives, so that R^{i}Γ_{f}T = 0 for i > 0, if T is Euler-torsion. Thus the short exact sequence 0 ⟶ Γ_{f}M ⟶ M ⟶ M/Γ_{f}M ⟶ 0 shows that for i ≥ 1 we have R^{i}Γ_{f}M ≅ R^{i}Γ_{f}M/Γ_{f}M. Now, by definition

0 ⟶ Γ_{f}M ⟶ M ⟶ E^{−1}M

is exact, and this gives the short exact sequence

0 ⟶ M/Γ_{f}M ⟶ E^{−1}M ⟶ E^{−1}M/M ⟶ 0

Since E^{−1}M/M is an F-finite torsion module, which proves the lemma, once we observe that R^{i}Γ_{f}E^{−1}M = 0 for all i. This is clear when i = 0, and follows in general since an Euler-local module admits an injective resolution by Euler-local modules.

The reason for care is that the inclusions ℂ_{ft} ⟶ ℂ_{t} and ℂ_{f} ⟶ ℂ do not preserve products or injectives. From the above lemma we see that the product of F-finite modules M_{α} in ℂ_{f} is ϕ ∏_{α} M_{α}, and similarly, for torsion modules M_{α} the product is Γ_{c} ∏_{α} M_{α}.

Example 8.1.5. Let I be an injective F-finite torsion module, zero below degree −2k. As usual, injectivity may be tested by extending maps θ : J ⟶ I defined on an ideal J over the whole ring O_{F}. Now an ideal is specified by the sequence of vector spaces J_{0}, J_{−2}, …, and if we identify (O_{F})_{2n} with (O_{F})_{0} by using multiplication by c^{n} the sequence of subspaces is increasing. We first extend θ over the ideal corresponding to the constant sequence of subspaces J_{−2k}, J_{−2k}, J_{−2k}, J_{−2k}, … using c-divisibility of I. We are thus reduced to the problem of showing that I_{0} is an injective module over C(F, ℚ).

However this need not be the case. For example the function θ on the ideal (e_{1}, e_{2}, …) defined by θ(e_{i}) = (0, …, 0, 1, 0 …) takes values in ⨁_{H} ℚ_{H}, but does not extend over ℚ^{F} because there is no common bound to the length of the vectors θ(v) for v in the ideal.

The main positive result is that because injectivity of F-finite modules is detected by divisibility, the inclusion ℂ_{ft} ⟶ ℂ_{f} does preserve injectives.

Lemma 8.1.6. If M is an F-finite module, injective in the category ℂ_{ft} then it is also injective in the category ℂ_{f}.

8.2. Products in the torsion model.

It is well known that, although it is easy to show that products exist in categories of spectra, it can be hard to understand anything about them that the categorical properties of products do not guarantee. However the category of 𝕋-spectra is simple enough that we can give a complete analysis, and the author has found the process of understanding extremely instructive. In any case, the analysis of products is important for two other reasons: firstly the case of the standard model allows us to give an analysis of function spectra (on objects), and secondly it allows one to understand inverse limits over a sequence of spectra. The algebraic and categorical issues which are important here occur again in a more complicated form in our discussion of the tensor and Hom functors. This section may therefore be helpful preparation for Section 8.3.

Suppose then that A is an indexing set, and that we wish to understand the product ∏_{α∈A} Y_{α} of 𝕋-spectra Y_{α} in terms of the algebraic models of the various spectra Y_{α}. The reason products are amenable to analysis is that the the functor ∏^{ℂ} : ℂ^{A} ⟶ ℂ is right adjoint to the very well behaved diagonal functor Δ : ℂ ⟶ ℂ^{A}. Since we know that the model of the topological diagonal is the algebraic diagonal, we know that their right adjoints correspond, and hence we need only identify the product in the algebraic context.

Now suppose that ℂ = dg𝔸 for one of our finite dimensional abelian categories 𝔸 (this includes the standard and torsion model categories as well as the categories of torsion k[c] and O_{F}-modules). We shall see that there is a product ∏^{𝔸} : 𝔸^{A} ⟶ 𝔸, which therefore induces a functor ∏^{dg𝔸} : dg𝔸^{A} ⟶ dg𝔸, which is also the product functor. Since this functor preserves the homotopy relation, it follows that we can define a functor ∏^{D𝔸} : DA^{A} ⟶ DA by applying ∏^{dg𝔸} to a fibrant approximation of an object (ie ∏^{D𝔸} = R ∏^{dg𝔸}). Furthermore, since ∏^{dg𝔸} is a right adjoint, it preserves injectives and pullbacks, and hence it is still right adjoint to the diagonal functor. In other words if M_{α} is a model of Y_{α} then a model of ∏_{α} Y_{α} is

∏ _{α}^{D𝔸}(M_{α}) = ∏ _{α}^{dg𝔸}M̂_{α}.

The remaining problem is to find the homology of the model. Although this description makes the construction appear straightforward, the reader may find some surprises along the way. In this section we treat only the F-finite torsion modules and the torsion model: we shall apply the same method to the standard model in Section 8.6 when we have established the necessary algebraic foundations.

We begin with the category F-finite torsion modules, leaving the easier category of torsion k[c]-modules to the reader. This is an ingredient in the discussion of the torsion model, and also illustrates the points which arise. In fact, one simply notes that if Y_{α} is a F-spectrum for each α then the product in the category of F-spectra is EF_{+} ∧ ∏_{α} Y_{α}: indeed, if T is a F-spectrum,

[T, ∏_{α} Y_{α}]^{𝕋} = [T, EF_{+} ∧ ∏_{α} Y_{α}]^{𝕋}.

Part (i) of the following proposition is immediate once we note that the inclusion i : torsO_{F}^{f}-mod ⟶ O_{F}-mod has right adjoint Γ_{f} : O_{F}-mod ⟶ torsO_{F}^{f}-mod given by taking the sub-module of elements annihilated by some Euler class.

Proposition 8.2.1. If 𝔸 is the category of F-finite torsion O_{F}-modules, then

(i) ∏ _{α}^{𝔸}M_{α} = Γ_{f} ∏_{α} M_{α} and (ii) ∏ _{α}^{D𝔸}M_{α} = RΓ_{f} ∏_{α} M_{α}.

Proof: It remains only to show that right derived functors respect composition. For this we need only check that the relevant functors preserve injectives. This is a little delicate, because ∏ (as a functor on families of F-finite torsion modules) is not a right adjoint.

However, we saw in Section 8.1 that

Γ_{f} ∏ = Γϕ ∏ .

Now ϕ ∏ is the product in the category of F-finite modules, and thus a right adjoint. It is exact because it is the composite of exact functors. Thus we have R(Γ_{f} ∏) = RΓ∘R(ϕ ∏) = RΓ ∘ (ϕ ∏), and the result follows.

Since objects of dg𝔸 are formal in this case, it is obviously useful to record the homology of the product.

Corollary 8.2.2. The homology of the product in the derived category is calculated by a split short exact sequence

0 ⟶ R^{1}Γ_{f} ∏_{α} H_{∗}(Σ^{−1}M_{α}) ⟶ H_{∗}(∏ _{α}^{D𝔸}M_{α}) ⟶ Γ_{f} ∏_{α} H_{∗}(M_{α}) ⟶ 0.

Thus, for example, if M_{α} is a dg object in even degrees, then the product has model Γ_{f} ∏_{α} M_{α} ⊕ Σ^{−1}R^{1}Γ_{f} ∏_{α} M_{α}.

Next consider the torsion model 𝔸 = A_{t}. We may reflect that we already know the torsion part of the model for the product ∏_{α} Y_{α}, and we almost know the vertex. Indeed

∏_{α}(Y_{α} ∧ EF_{+}) ⟶ ∏_{α} Y_{α}

is an F-equivalence, and therefore the torsion part of the product is given by the product of the torsion parts in the category of F-finite torsion modules. The vertex can almost be calculated from the exact sequence

⋯ ⟶ π_{∗}^{𝕋}(EF_{+} ∧ ∏_{α} Y_{α}) ⟶ π_{∗}^{𝕋}(∏_{α} Y_{α}) ⟶ π_{∗}^{𝕋}(ẼF ∧ ∏_{α} Y_{α}) ⟶ · · · .

However, the categorical procedure makes this a redundant prop to our confidence.

Proposition 8.2.3. The product of objects M_{α} = (t_{∗}^{F} ⊗V_{α} ⟶ T_{α}) in the torsion model category is ∏_{α} M_{α} = (t_{∗}^{F} ⊗ V_{Π} ⟶^{s_{Π}} T_{Π}) where (i)T_{Π} = Γ_{f} ∏_{α} T_{α}, (ii) The graded vector space V_{Π} and structure map are described by the pullback square

V_{Π}  ⟶  ∏_{α} V_{α}
↓  ↓
Hom(t_{∗}^{F}, Γ_{f} ∏_{α} T_{α}) ⟶ Hom(t_{∗}^{F}, ∏_{α} T_{α})

where the lower horizontal is induced by the inclusion Γ_{f} ∏_{α} T_{α} ⟶ ∏_{α} T_{α}. The behaviour on morphisms is as implied by the above categorical description.

Proof: It is convenient to replace the objects (t_{∗}^{F} ⊗ V_{α} ⟶ T_{α}) by the corresponding objects (V_{α} ⟶ Hom(t_{∗}^{F}, T_{α})). It is then immediate that, in the category of objects V ⟶ Hom(t_{∗}^{F}, M) with M an arbitrary O_{F}-module, the product is (∏_{α} V_{α} ⟶ Hom(t_{∗}^{F}, ∏_{α} T_{α})). The fact that V_{Π} is as claimed follows immediately from the universal property of the pullback.

As remarked above, this automatically gives a model for the product of 𝕋-spectra. However we would like to know the homology of the product and, more exactly, its vertex and torsion part. We shall let V Π and T Π denote the functors giving the vertex and torsion part of the product. For the torsion part T Π_{α}M_{α} = ∏^{D𝔸}T_{α}, and so we already have an answer, provided by Corollary 8.2.2 above.

Now we turn to the vertex. We note that, as in Section 8.1, it may be useful to factorise Γ_{f} = Γϕ. We would then be concerned with pullback along the inclusions i and j as displayed in the following diagram:

i^{∗}j^{∗} ∏_{α} V_{α}  ⟶  j^{∗} ∏_{α} V_{α}  ⟶  ∏_{α} V_{α}
↓  ↓  ↓
Hom(t_{∗}^{F}, Γϕ ∏_{α} T_{α}) ⟶^{i} Hom(ϕ ∏_{α} T_{α}) ⟶^{j} Hom(∏_{α} T_{α}).

The advantage is that if, for example, all torsion objects were supported on the same finite set of subgroups, then ϕ ∏_{α} M_{α} = ∏_{α} M_{α}. It is sometimes convenient to let k : Γ_{f}M ⟶ M denote the composite ji.

Again we work with the adjoint form of torsion model objects. We learn from the torsion case, and begin with the composite ϕ ∏. In the following display we use the maps Γϕ ∏_{α} T_{α} ⟶^{i} ϕ ∏_{α} T_{α} ⟶^{j} ∏_{α} T_{α}. We display the functor as a composite:

(( \begin{matrix}V_{α} \\ ↓ \\ Hom(t_{∗}^{F}, T_{α}) \end{matrix} )_{α}) ϕ ∏ ; ⟼ ( \begin{matrix}j^{∗} ∏_{α} V_{α} \\ ↓ \\ Hom(t_{∗}^{F}, ϕ(∏_{α} T_{α})) \end{matrix} ) ⟼^{Γ_{f}} ( \begin{matrix}i^{∗}j^{∗} ∏_{α} V_{α} \\ ↓ \\ Hom(t_{∗}^{F}, Γϕ(∏_{α} T_{α})) \end{matrix} ) ⟶ i^{∗}j^{∗} ∏_{α} V_{α}.

To understand derived functors, we must specify the categories concerned. We consider objects V ⟶ Hom(t_{∗}^{F}, M); the category of these with M arbitrary will be denoted Â_{t}, the category with M F-finite will be denoted Ã_{t}. We view these inclusions as giving functors

Â_{t} ⟶^{j^{∗}} Ã_{t} ⟶^{i^{∗}} A_{t}

We begin by noting that the functors i^{∗} and j^{∗} are both left exact. It is obvious that they preserve monomorphisms. Next, notice that given V ⟶^{λ} Hom(t_{∗}^{F}, M) we have v ∈ i^{∗}V if and only if λ(v)(1) ∈ ϕM. This is because elements of t_{∗}^{F} of positive degree have finite support. Similarly v ∈ j^{∗}V if and only if λ(v)(1) ∈ ΓM. The half exactness then follows since if M ⊆ N then ϕM = M ∩ ϕN and similarly ΓM = M ∩ ΓN. Since both ϕ and Γ are right adjoints, they both preserve injectives, and so R(i^{∗}j^{∗}) = Ri^{∗}Rj^{∗}. Also, both i^{∗} and j^{∗} have the property that they are the identity on torsion free injectives.

It is now rather straightforward to calculate the right derived functors. If t_{∗}^{F} ⊗ V ⟶ M is the torsion object and 0 ⟶ M ⟶ I ⟶ J ⟶ 0 is an injective resolution we obtain the following two exact sequences for the vertex of the right derived functors of j^{∗}.

0 ⟶ Hom(t_{∗}^{F}, M) ⟶ A ⟶ Hom(t_{∗}^{F}, ΓM) ⟶ Ext(t_{∗}^{F}, M) ⟶ V Π^{2} ⟶ 0

and

0 ⟶ V Π^{0} ⟶ V ⊕ Hom(t_{∗}^{F}, ΓI) ⟶ A ⟶ V Π^{1} ⟶ 0

There are precisely analogous exact sequences for either the right derived functors of i^{∗} and of i^{∗}j^{∗}.

There is also a product in the standard model, and for use in our approach to function spectra this will be more important. As in the cases we have treated here we find ∏^{𝔸} = Γ̂ ∏𝔸̂, where Γ̂ : 𝔸̂ ⟶ 𝔸 is right adjoint to the inclusion and ∏^{D𝔸} = RΓ̂ R ∏𝔸̂. The work is involved in making these descriptions more explicit. However the right adjoint to the inclusion functor i : 𝔸 ⟶ 𝔸̂ is much harder to describe in this case, so we defer the treatment until Section 8.6.

8.3. The tensor hom adjunction.

Once again we are led to consider an abstract categorical situation because it arises in several algebraic ways. Thus we let 𝔸 denote an abelian sub-category of an abelian category 𝔸̂. We shall treat four examples, all of which arise as algebraic models.

Example 8.3.1. 𝔸 is the category of torsion k[c]-modules, for a field k and 𝔸̂ is the category of all k[c]-modules.

Example 8.3.2. 𝔸 is the category A of k[c]-morphisms (N ⟶^{s} t ⊗ V) which become an isomorphism when c is inverted, where t = k[c, c^{−1}] and 𝔸̂ is the category of all maps (N ⟶ t ⊗ V).

Example 8.3.3. 𝔸 is the category of F-finite torsion O_{F}-modules, 𝔸̂ is the category of all F-finite O_{F}-modules.

Example 8.3.4. 𝔸 is the category A of objects (N ⟶^{s} t_{∗}^{F} ⊗ V) which become an isomorphism when E is inverted, 𝔸̂ is the category of all such maps.

We shall illustrate the general discussion with the two single object examples, and return to the examples with objects of the form (N ⟶ t ⊗ V) in the next section, since they are much more substantial.

We begin by summarising the framework in which we operate. We shall show in due course that the above examples all fit in this framework. Firstly, suppose that the inclusion i : 𝔸 ⟶ 𝔸̂ is exact, and has a right adjoint Γ̂ : 𝔸̂ ⟶ 𝔸. This assumption holds in all the examples above for general reasons, but we shall identify the adjoints explicitly below. Next, we assume that for any object N of 𝔸̂ there is a tensor product functor ⊗N : 𝔸̂ ⟶ 𝔸̂, and that, if N is in 𝔸, this functor restricts to a functor on 𝔸. Finally, we suppose that the tensor product functor is associative, and has a right adjoint Hom(N, ·) : 𝔸̂ ⟶ 𝔸̂; we use bold face to emphasise that Hom(N, P) is is an object of 𝔸̂, and not just the abelian group of homomorphisms. Of course, it then follows that if N is an object of 𝔸 the functor ⊗N : 𝔸 ⟶ 𝔸 has right adjoint IntHom(N, ·) = Γ̂ Hom(N, ·). To summarise, we have the diagram of adjoint pairs of functors

𝔸 ⇄^{i} 𝔸̂ ⇄^{⊗N} 𝔸̂,

where the left adjoints are displayed at the top.

Example 8.3.1 continued: We see that the assumptions are satisfied. The right adjoint to inclusion is the c-power torsion functor: for an arbitrary k[c]-module N̂ we have Γ̂N = Γ_{c}N. Here Hom is the usual Hom functor, and the internal Hom functor on the category of torsion k[c]-modules is

IntHom(N, P) = Γ_{c} Hom(N, P).

The next example is very little different. Example 8.3.3 continued: We see that the assumptions are satisfied. The right adjoint to inclusion is the E-torsion functor: for an arbitrary F-finite module N̂ we have Γ̂N = Γ_{f}N. Here Hom is the F-finite part of the usual Hom functor:

Hom(N, P) = ϕHom(N, P),

and the internal Hom functor on the category of torsion F-finite O_{F}-modules is

IntHom(N, P) = Γ_{f} Hom(N, P).

The point of factorizing the internal Hom functor is twofold. Firstly, as the examples illustrate, the functor Hom is much closer to the usual group of homomorphisms, and thus easier to identify. Much more important is the second reason: we wish to pass to derived categories and retain the adjointness of functors.

To proceed further, we shall need the following condition to be satisfied: we see below that this covers our examples.

Condition 8.3.5. (i) The category 𝔸 has enough injectives and has injective dimension 1. (ii) The category 𝔸̂ has enough injectives and enough flat objects, and has both injective and flat dimension 1. (iii) Either 𝔸 has enough flat objects, or I ⊗ M = 0 for all injectives I in 𝔸. (iv) For any object C of 𝔸 R^{1}Γ̂(iC) = 0, and hence RΓ̂(iX) = X for any object X of D𝔸.

We begin by stating the result and then discuss what is involved in making sense of it.

Proposition 8.3.6. Suppose given adjoint pairs of functors in the above framework, and suppose in addition that Condition 8.3.5 is satisfied. For objects M, N and P of 𝔸 we have the natural isomorphism

[M ⊗^{L} N, P] = [M, RΓ̂ RHom(N, P)].

The two points to notice here concern the way the composite functors have been treated. The first is that it is not immediately clear how to interpret M ⊗^{L} N as an object of 𝔸, since 𝔸 may not have enough projective or flat objects. The second point is that on the right hand side

RΓ̂ RHom(N, P) ≠ RIntHom(N, P)

in general. This can happen because the functor Hom(N, ·) does not preserve injectives, as we saw in Example 7.2.1. Proof: We now outline the proof of the proposition. It turns out that both pairs l : 𝔻 ⇄ 𝔼 : r of adjoint functors do indeed pass to adjoint functors Ll : D𝔻 ⇄ D𝔼 : Rr. For the pair i, e this is because i is exact, and therefore the functor on dg categories preserves both cofibrations and acyclic cofibrations. For the tensor-Hom pair the reason is different. First note that RHom exists since 𝔸̂ has enough injectives. If 𝔸̂ has enough projectives we know that ⊗^{L} exists, but it also suffices that 𝔸̂ has enough flat objects. Now consider the functor of three variables

Hom(M ⊗ N, P) = Hom(M, Hom(N, P))

to graded abelian groups. It follows from associativity of the tensor product that, if M is flat M⊗ preserves flat objects, and so the total right derived functor on the left can be calculated as a composite. Similarly, if P is injective Hom(·, P) takes flat objects to injectives so that the functor on the right can also be calculated as a composite. This gives the equality

RHom(M ⊗^{L} N, P) = RHom(M, RHom(N, P))

and the required equality comes by taking 0th homology. This gives

D𝔸̂(M ⊗^{L} N, P) = D𝔸(M, RΓ̂ RHom(N, P)).

Finally if M, N and P are in 𝔸 then

D𝔸(M ⊗^{L} N, P) = D𝔸̂(M ⊗^{L} N, P).

Note first that the meaning of M ⊗^{L} N on the left is unclear if 𝔸 does not have enough flat objects. However, the tensor product functor is right exact and has only one left derived functor, the torsion product functor. In the cases that 𝔸 does not have enough flat objects we have the equality

M ⊗^{L} N = ΣRTor(M, N)

in D𝔸̂ provided M or N is in 𝔸, by Condition 8.3.5 (iii). Now the category 𝔸 does have enough injectives, so the right hand side can be interpreted in D𝔸. Furthermore in the relevant cases, 𝔸-injective objects are also 𝔸̂-injective, so that ΣRTor(M, N) has the same meaning in 𝔸̂ and in 𝔸.

It therefore suffices to show that D𝔸̂(L, P) = D𝔸(L, P) when L and P are in 𝔸. For this it suffices to note that it 0 ⟶ P ⟶ Î ⟶ Ĵ ⟶ 0 is an injective resolution of P in 𝔸̂ then 0 ⟶ P ⟶ Γ̂Î ⟶ Γ̂Ĵ ⟶ 0 is also an injective resolution. Since Γ̂ is a right adjoint the terms are injective, and by Condition 8.3.5 (iv) the final map is surjective.

We end this section by completing the justification for the two single space examples. Example 8.3.1 continued. (The modifications for Example 8.3.3 are minimal.)

The short exact sequence

0 ⟶ k[c] ⟶ k[c, c^{−1}] ⟶ Σ^{2}𝕀 ⟶ 0

immediately gives the following.

Lemma 8.3.7. For any k[c]-module M there is an exact sequence

0 ⟶ Tor(M, Σ^{2}𝕀) ⟶ M ⟶ M[c^{−1}] ⟶ M ⊗ Σ^{2}𝕀 ⟶ 0,

and hence, for any torsion module T, we have T ⊗ Σ^{2}𝕀 = 0 and Tor(T, Σ^{2}𝕀) = T.

Note that this means in particular that we may calculate Tor_{∗}(S, T) for torsion modules S and T using an injective resolution of S. Indeed if 0 ⟶ S ⟶ I ⟶ J ⟶ 0 is an injective resoloution using sums of suspensions of 𝕀 we obtain the exact sequence

0 ⟶ Tor(S, T) ⟶ Tor(I, T) ⟶ Tor(J, T) ⟶ S ⊗ T ⟶ 0.

Our problem is that there are not enough projectives amongst torsion modules, so that it is not clear that the left derived torsion product exists. However the category of k[c]-modules is one dimensional, and thus the torsion product functor is left exact, so we may use the right derived torsion product instead.

Lemma 8.3.8. On the category of k[c]-modules, the left derived torsion product is equivalent to the suspension of the right derived torsion product on torsion modules. More precisely, if M or N is a torsion module

M ⊗^{L} N = ΣRTor(M, N).

Proof: It is enough to establish that k[c] ⊗^{L} N = ΣRTor(k[c], N). Consider the exact sequence

0 ⟶ k[c] ⟶ k[c, c^{−1}] ⟶ Σ^{2}𝕀 ⟶ 0.

Since the first two terms are flat, it gives the calculation

Tor(Σ^{2}𝕀, N) = ker(N ⟶ N[c^{−1}]).

On the other hand, k[c, c^{−1}] is both flat and injective, so that RTor(k[c, c^{−1}], N) = 0. Thus RTor(k[c], N) = Σ^{−1}RTor(Σ^{2}𝕀, N). Moreover, since 𝕀 is injective RTor(Σ^{2}𝕀, N) = Tor(Σ^{2}𝕀, N).

We write M ⊛ N for this functor, and refer to it as the left derived tensor product. If M and N are torsion modules it should be interpreted as the suspension of the right derived torsion product: by the lemma this is not ambiguous.

Lemma 8.3.9. The right adjoint of ⊛B : D(torsk[c]) ⟶ D(torsk[c]) is R′Hom(B, ·) := RΓ_{c} RHom(B, ·).

Proof: The adjoint pair i : k[c] − mod ⇆ torsk[c] − mod : Γ_{c} passes to derived categories since i is exact. The adjoint pair ⊗B : k[c] − mod ⇆ k[c] − mod : Hom(B, ·) passes to derived categories by use of projectives. This shows the required adjunction, or more properly, that

D(k[c])(A ⊛ B, C) = D(torsk[c])(A, RΓ_{c} RHom(B, C)).

The remaining point is to show that

D(torsk[c])(A ⊛ B, C) = D(k[c])(A ⊛ B, C)

when A, B and C are torsion. The point is that we may calculate A ⊛ B = RTor(A, B) on the right using torsion injectives.

8.4. Hom, tensor and torsion functors in standard models.

In this section we treat Examples 8.3.2 and 8.3.4, showing that they satisfy the basic assumptions of the previous section. In the following section we shall deal with passage to derived categories.

We have already considered the objects e(V) = (t ⊗ V ⟶ t ⊗ V) and f(T) = (T ⟶ 0) which are left adjoints to the vertex and nub functors. We now need to consider objects good as domains. Inspired by topology, we consider spheres analogous to the the representation spheres S^{nV} with V the natural representation:

S^{nλ} := (Σ^{2n}O ⟶^{c^{−n}} t).

Of course we also permit the notation S^{m+nλ} = Σ^{m}S^{nλ}. Because t is ℚ in each even dimension we must emphasize that degree 0 is special, and t is an abbreviation of t ⊗ ℚ.

Thus Σ^{m}S^{nλ} = (Σ^{m+2n}O ⟶^{c^{−n}} t ⊗ Σ^{m}ℚ).

We begin by investigating the functor S^{nλ} corepresents, so let C = (P ⟶^{γ} t ⊗ W) be some object of 𝔸̂.

Lemma 8.4.1.

Hom(S^{nλ}, C) = γ^{−1}(c^{−n} ⊗ Σ^{−2n}V).

Proof: A map

Σ^{2n}O ⟶^{θ} P
c^{−n} ↓  ↓ γ
t ⟶^{1⊗ϕ} t ⊗ W

is obviously determined by θ(Σ^{2n}1). However it is subject to the constraint that γ(θ(Σ^{2n}1)) = (1 ⊗ ϕ)(c^{−n} ⊗ 1) = c^{−n} ⊗ ϕ(1).

Thus we can begin to reconstruct C from its subspaces P (c^{−n}) = γ^{−1}(c^{−n} ⊗ V); this will motivate the construction of of the functor Γ̂ : 𝔸̂ ⟶ 𝔸 below. In fact it is convenient to introduce a dummy variable χ of degree 0, so we may write pχ^{n} for an element p ∈ P if we consider it as an element of P (c^{n}). The action of c is then given by c·(pχ^{−n}) = (cp)χ^{−(n−1)}, and this map is co-represented by the map c : S^{(n−1)λ} ⟶ S^{nλ}

Σ^{2n+2}O ⟶^{c} Σ^{2n}O
c^{−(n+1)} ↓  ↓ c^{−n}
t  ⟶^{1} t

The only elements of the nub P which lie in several distinct P (c^{n}) are those lying in the kernel of γ. Of course, if C is in 𝔸, the kernel of γ is c-power torsion, and can be detected by mapping out of objects (T ⟶ 0), and the vertex of C can also be calculated as follows.

Lemma 8.4.2. For an arbitrary object C of 𝔸, we may recover t⊗W as lim_{→}(⨁_{n} P (c^{n}) ⟶^{c}

Σ^{2} ⨁_{n} P (c^{n}) ⟶^{c} Σ^{4} ⨁_{n} P (c^{n}) ⟶^{c} ⋯). Furthermore, the vertex W and the map W = 1 ⊗ W ⟶ t ⊗ W may be recovered as W = im(P (c^{0}) ⟶ lim_{→}(Σ^{2n}P (c^{n}))).

The advantage of the category 𝔸̂ is that it has certain extra objects. It is obvious that the object Ê = (0 ⟶ t) co-represents the vertex functor. To co-represent the nub one would want to consider the dual t′ = Hom(t, k) and an object of the form (O ⟶ t⊗t′) with 1 ⟼ Σc^{n} ⊗ c_{n}; however the sum is infinite and so this putative object must be replaced by the spheres S^{nλ} of which it would be a limit.

We may now describe the functors that concern us. The right adjoint Γ̂ to the inclusion of categories i : 𝔸 ⟶ 𝔸̂ can be described using direct limits as follows. For any object C in 𝔸̂ we may consider the category of objects x : T ⟶ C over C with T in 𝔸. Restricting to the sub-category of objects with cardinality less than card(t)card(C) so as to obtain a category with small skeleton, we may take the direct limit Γ̂(C) = lim_{→} T. The following description is more helpful.

Lemma 8.4.3. If C = (P ⟶^{γ} t ⊗ W) then Γ̂(C) = (P′ ⟶^{γ′} t ⊗ W′) may be described as follows. The nub of Γ̂C is P′ = ⨁_{n∈ℤ} P (c^{n})/K where K is the submodule generated by differences p(χ^{m}−χ^{n}) with p being c-power torsion. The vertex of Γ̂C is W′ = im(P (c^{0}) ⟶ lim_{→}_{n}(Σ^{2n}P (c^{n})). The natural map t ⊗ W′ ⟶ lim_{→}_{n}(Σ^{2n}P (c^{n})) is an isomorphism and the structure map is the natural map.

Proof: We may see that this definition is forced since for A in 𝔸 we have Hom(A, C) = Hom(A, Γ̂C). Taking A = S^{nλ} we see that (P′)(c^{n}) = P (c^{n}). We then find t ⊗ W ≅

P′[c^{−1}] = lim_{→} Σ^{2n}P′(c^{n}).

We write pχ^{n} for an element of P′ represented by p ∈ P with γ(p) ∈ c^{n}⊗W. The elements of W′ are thus represented by elements of form, pχ^{0} and the map t⊗W′ ⟶ lim_{→}_{n}(Σ^{2n}P (c^{n})) is given by c^{n} ⊗ pχ^{0} ⟼ c^{n}pχ^{n}. This is obviously surjective; to see it is injective it suffices to restrict to c^{0} ⊗ W′ ⊆ t ⊗ W′, and this is clearly injective.

If C is in 𝔸 then Γ̂C = C, and we may take the unit of the adjunction to be the identity. The counit iΓ̂C ⟶ C of the adjunction is given on nubs by the inclusions P (c^{n}) ⟶ P, and this extends uniquely over t ⊗ W′ since t ⊗ W is c-local. The extension takes 1 ⊗ W′ into 1 ⊗ W by construction. Note that Γ̂iΓ̂ = Γ̂ and iΓ̂i = i; the triangular identities are easily checked.

Example 8.4.4. Although we have chosen the notation Γ̂ to suggest the analogy with the torsion functor, the functor can behave in quite different ways. Thus it annihilates the object Ê co-representing the vertex Γ̂(0 ⟶ t) = 0. On the other hand,

Γ̂(M ⟶ 0) = (M[χ, χ^{−1}]/(Γ_{c}M · (χ^{m} − χ^{n})|m, n ∈ ℤ) ⟶ t ⊗ \overline{M}) ,

where \overline{M} = M/Γ_{c}M.

Note also that the functor Γ̂ does admit a description analagous to the description lim_{→}_{n} Hom(A/I^{n}, ·) of the I-power torsion functor of an ideal I in a commutative ring A. Unfortunately it is not given by an internal Hom, but rather by taking the direct limit in nub and torsion parts. More precisely, we consider the inverse system and ⋯ ⟶ f(O/c^{3}) ⟶ f(O/c^{2}) ⟶ f(O/c), and adjoin to the diagram the maps S^{kλ} ⟶ Σ^{2k}f(O/c^{l}) for all k and l. Note that the spheres S^{kλ} have no incoming arrows. Thus if we apply Hom(·, C) and take direct limits we obtain the nub of Γ̂C. For the vertex we have the inverse system ⋯ ⟶ S^{−2λ} ⟶ S^{−λ} ⟶ S^{0}, which admits a comparison map to the nub system; applying Hom(·, C) and taking direct limits we obtain the vertex part of Γ̂C.

For the tensor Hom adjunction we need objects A = (M ⟶ t ⊗ U), B = (N ⟶ t ⊗ V) and C = (P ⟶ t ⊗ W). The tensor product is the obvious one: A ⊗ B = (M ⊗ N ⟶ (t ⊗ U) ⊗ (t ⊗ V) = t ⊗ (U ⊗ V)). This clearly preserves 𝔸.

Next, we describe the internal Hom functor in 𝔸̂. In fact we define an object Hom(B, C) = (Q ⟶ t ⊗ H), and then show Hom(B, ·) gives the right adjoint to B⊗. Of course we identify H and Q by assuming the defining adjunction

Hom(A ⊗ B, C) = Hom(A, Hom(B, C)).

Taking A = Ê = (0 ⟶ t) we see H = Hom(V, W). Taking A = S^{nλ} we see that Q(c^{−n}) = Σ^{2n}Hom(S^{nλ} ⊗ B, C). Furthermore, the action of c on Q is corepresented by c : S^{(n−1)λ} ⟶ S^{nλ}. The structure map is given by the map Q(c^{−n}) = Σ^{2n}Hom(S^{nλ} ⊗ B, C) ⟶ c^{n} ⊗ Hom(V, W) taking a homomorphism S^{nλ} ⊗ B ⟶ C to its vertex part. We could summarise all this by saying that Hom(B, C) is obtained by applying Hom(·⊗B, C) to the diagram

⋯ ⟶ S^{−λ} ⟶ S^{0} ⟶ S^{λ} ⟶ ⋯
↖ ↑ ↗
Ê

We may therefore describe the unit η : A ⟶ Hom(B, A⊗B) and counit ϵ : Hom(B, C)⊗ B ⟶ C. On vertices we simply use the familiar maps U ⟶ Hom(V, U ⊗ V) and evaluation Hom(V, W) ⊗ V ⟶ W. On nubs we may consider the part over c^{−n} where we have Hom(S^{nλ}, A) ⟶ Hom(S^{nλ} ⊗ B, A ⊗ B) obtained by tensoring with B. For the counit we consider the part arising from i + j = n, giving Hom(S^{iλ} ⊗ B, C) ⊗ Hom(S^{jλ}, B) ⟶ Hom(S^{nλ}, C), and since S^{iλ} ⊗S^{jλ} ≅ S^{nλ} we may use composition. It is readily verified that the nub and vertex components are compatible.

Lemma 8.4.5. The unit η : A ⟶ Hom(B, A ⊗ B) and counit

ϵ : Hom(B, C) ⊗ B ⟶ C

just described, show that Hom(B, ·) is right adjoint to B⊗.

Proof: The triangular identities on vertices are the familiar ones, and on nubs they are easily checked.

This completes the account of the tensor product, Hom and Γ̂ functors in Example 8.3.2. In the next section we will discuss passage to derived categories.

We now turn to the full standard model, and a certain amount of extra work is required. As in the simpler model, we begin with objects which are good as domains. Inspired by topology we should perhaps expect these objects to be analogous to spheres, and we have chosen the notation to emphasize this analogy.

Recall from Section 8.1 that if V^{𝕋} = 0, there is an associated Euler class χ(V). Algebraically the analogue of the representation V is the dimension function v : F ⟶ ℤ_{≥0}, given by v(H) = dim_{ℂ}(V^{H}), which is only positive on the finitely many subgroups fixing a non-zero vector in V.

Now let v : F ⟶ ℤ be an arbitrary function. Consider elements of t_{∗}^{F} in each degree as functions on F, and take the sub-O_{F}-module O_{F}(v) consisting in degree 2n of functions on F which are only non-zero at H if n ≤ v(H). For any such a dimension function we consider the object

S^{v} = (O_{F}(v) ⟶ t_{∗}^{F})

of 𝔸̂, where the map is inclusion. Note that, if v has finite support, the map becomes an isomorphism when E is inverted, and hence S^{v} is an object of 𝔸. We extend the notation to include suspensions in the obvious way: S^{n+v} = Σ^{n}S^{v}. Note that if λ is the dimension function of the natural representation (restricted to the trivial subgroup) this notation is consistent with the notation used for the semifree case. Note also that if c^{v} is c_{H}^{v(H)} over H and we view O_{F} as a submodule of t_{∗}^{F}, then O_{F}(v) = c^{−v}O_{F}.

Lemma 8.4.6. For any representation V with V^{𝕋} = 0, and dimension function v(H) = dim_{ℂ}(V^{H}) the algebraic object S^{v} models the topological stable sphere S^{V}.

In general, it is quite hard to describe the functor S^{v} corepresents, but the idea is that Hom(S^{v}, C) is specified by elements of the nub of C which are c^{v}-divisions of elements of γ^{−1}(c^{0} ⊗ W). For example S^{0} co-represents γ^{−1}(c^{0} ⊗ W). The other essential case is when v is constant at n > 0 on a finite set ϕ of subgroups and zero on its complement, so that c^{v} = c_{ϕ}^{n} = e_{ϕ}c^{n} + (1 − e_{ϕ}). In this case

Hom(S^{v}, C) = γ^{−1}(c_{ϕ}^{−n} ⊗ Σ^{−2n}W) × γ^{−1}(c_{F\ϕ}^{0} ⊗ W)).

This suffices to find P (τ) = γ^{−1}(τ ⊗ W) for any homogeneous τ ∈ t_{∗}^{F}. Indeed if τ is a monomial of positive degree it is c_{ϕ}^{n} for some n, and hence P (c_{ϕ}^{n}) = e_{ϕ}Σ^{2n}Hom(S^{v}, C). On the other hand if τ is of negative degree P (τ) = τP (c^{0}) = τHom(S^{0}, C).

8.5. Hom, tensor and torsion functors on derived categories.

We now take the functors on 𝔸 and 𝔸̂ and show that they pass to derived categories where they continue to give adjoint pairs. Condition 8.3.5 explains what is necessary.

For the i-Γ̂ adjunction this is immediate, since i is an exact functor. Thus it preserves cofibrations, and acyclic cofibrations so that i = Li. However for the tensor-Hom adjunction a little more comment is necessary. In one respect the situation is simpler than in the single object examples.

Lemma 8.5.1. The category 𝔸 has enough flat objects, and it is of flat dimension 1.

Proof: It is obvious that S^{nλ} is flat for all n. By the description of Hom(S^{nλ}, B), it is easy to construct a map from a sum of spheres to B which is surjective on nubs. It is then necessarily surjective on vertices since inverting c is exact. This shows there are enough flat objects.

Next, if Σ is a sum of spheres, any subobject R is flat. Indeed, the nub of R is a subobject of a free k[c]-module and hence free, and the vertex is necessarily flat.

Obviously spheres remain flat in 𝔸̂. Furthermore the object Ê co-representing the vertex is flat, so there are also enough flat objects in 𝔸̂, which is also of flat dimension 1.

Lemma 8.5.2. For B ∈ 𝔸 the left derived tensor product functor ⊗B : D𝔸 ⟶ D𝔸 exists, and may be calculated by flat resolutions. The analogous statement is true for the category 𝔸̂.

Proof: In the absence of projective objects, we must say something to establish the existence of the left derived tensor product. We have seen that any epimorphism F ⟶ A with F flat gives a flat resolution 0 ⟶ R ⟶ F ⟶ A ⟶ 0, and if F′ ⟶ A is a second, we may compare the two using the sum F ⊕ F′ ⟶ A. This shows that cofibre(R ⊗ B ⟶ F ⊗ B) gives a well defined functor on the derived category. The rest of the proof that this is the total left derived functor proceeds as with cofibrant approximations: the essential point is that tensor product with a flat object preserves weak equivalences.

We continue the tradition of writing ⊛ for this functor. It follows directly as in the general discussion that the tensor-Hom adjunction passes to derived categories. There is one last point.

Lemma 8.5.3. For objects A, B and C of 𝔸 we have

D𝔸(A ⊛ B, C) = D𝔸̂(A ⊛ B, C)

Proof: In the semifree case this is immediate since, on the one hand, the notation A ⊛ B is unambiguous, and, on the other hand, i does preserve injectives.

Now in the general case, Example 8.3.4, we begin by remarking that S^{v} is flat when v takes finitely many values. Indeed the module O_{F}(v) is projective as a O_{F}-module, since it is a finite sum of suspensions of e_{ϕ}O_{F} where ϕ ⊆ F is a set on which v is constant.

Corollary 8.5.4. There are enough flat objects in the standard model A.

Lemma 8.5.5. The standard model A has flat dimension 1.

Proof: Since the category has injective dimension 1 it suffices to show that the standard injectives have flat dimension 1. The injectives e(V), are themselves flat. Observe next that f(T) is flat if T is flat as a O_{F}-module, and so a flat resolution of T gives one of f(T).

However the last point needs a little more care than in previous cases.

Lemma 8.5.6. For objects A, B and C of the standard model A,

D𝔸(A ⊛ B, C) = D𝔸̂(A ⊛ B, C)

Proof: Firstly the object A ⊛ B is again unambiguous and lies in D𝔸, since there are enough flat objects in A, and they remain flat in Â_{t}. However injectives in A are not necessarily injective in Â_{t}: it suffices to prove the following lemma instead.

Lemma 8.5.7. If C is an object of A then R^{1}Γ̂(C) = 0, and hence for any object X of DA, we have RΓ̂ iX = X.

Proof: For an object C of A, take an injective resolution 0 ⟶ C ⟶ Î ⟶ Ĵ ⟶ 0 in Â_{t}. We obtain an exact sequence 0 ⟶ C ⟶ Γ̂Î ⟶ Γ̂Ĵ, and we claim it is an injective resolution in A. The injectivity of Γ̂Î and Γ̂Ĵ is immediate since Γ̂ is a right adjoint. It remains to see that the last map is surjective. To see this, take an injective resolution of C in A. This will involve objects e(V) and f(I). Those of the first type remain injective in Â_{t}, but those of the second have ⨁_{H} I(H) replaced by ∏_{H} I(H). However this is sufficient to give an injective resolution in Â_{t}, and applying Γ̂ we recover the original resolution in A.

This completes the proof that the left derived tensor product has a right adjoint.

We conclude by continuing Example 8.4.4.

Example 8.5.8. We evaluate RΓ̂f(M) for an arbitrary O_{F}-module M. Indeed, we consider the fibre sequence M ⟶ E^{−1}M ⟶ E^{−1}M/M, and apply RΓ̂f(·). Since E^{−1}M is torsion free and E-local we have Γ̂f(E^{−1}M) = e(E^{−1}M). Furthermore, an E-local module admits an injective resolution by E-local injectives, so RΓ̂ = Γ̂ on the category of objects f(N) with N E-local. Finally, since E^{−1}M/M is E-torsion it is in A and we find RΓ̂f(E^{−1}M/M) = Γ̂f(E^{−1}M/M) = f(E^{−1}M/M) by 8.5.7. Thus we have a fibre sequence RΓ̂f(M) ⟶ e(E^{−1}M) ⟶ f(E^{−1}M/M).

8.6. Products in the standard model.

The main purpose of this short section is to record the product in the standard model and its derived category. It completes the work of Section 8.2, which calculated the products for other standard examples and the torsion model. As in that case, the product in DA automatically models the product of 𝕋-spectra, since both are adjoint to their respective diagonal functors which obviously correspond.

The present section is almost a special example of the Hom functor, since of course products of a single object can be constructed as ∏_{i} C = Hom(⋁_{i} S^{0}, C). Nonetheless the product functor will have special significance because it is the only case of the function spectrum for which we have a model as a functor. This will be essential in identifying the model of the function spectrum on objects.

The method of the present section is the same as that of Section 8.2, and exploited also for Hom in Section 8.3. By definition the product is the right adjoint to the diagonal functor. As in the case of the Hom functor it is convenient to identify the product functor ∏𝔸̂ in 𝔸̂ and then note that Γ̂ ∏𝔸̂ is the product in 𝔸. However, this time there is no problem whatever in identifying the maps in the derived category, since the diagonal functor is exact. Thus the product on D𝔸 is RΓ̂ R ∏𝔸̂. To avoid confusion we shall only use the undecorated symbol ∏ for a functor which is the product on underlying sets.

Proposition 8.6.1. Example 8.3.1 revisited. The product of objects M_{i} in the derived category of torsion k[c]-modules is

RΓ_{c} ∏_{i} M_{i}.

The modifications for the second single-object model are straightforward, and were made in 8.2.1.

The two-object models are a little more complicated. Let us begin with the case of k[c]-modules, with 𝔸̂ being the category of k[c]-morphisms N ⟶ t ⊗ V. As before we swiftly calculate the product in the category 𝔸̂. Thus if C_{i} = (P_{i} ⟶ t ⊗ W_{i}), and if we assume ∏𝔸̂ ; i C_{i} = (Q ⟶ t ⊗ H) then we apply Hom(Ê, ·) to deduce H = ∏_{i} W_{i}. Applying Hom(S^{nλ}, ·) we deduce Q(c^{−n}) = ∏_{i} P_{i}(c^{−n}). We give a more direct construction. We let δ : t ⊗ ∏_{i} W_{i} ⟶ ∏_{i}(t ⊗ W_{i}) denote the diagonal, and we let δ^{∗} be the pullback functor:

·  ⟶  ∏_{i} C_{i}
↓  ↓
t ⊗ ∏_{i} W_{i} ⟶^{δ} ∏_{i}(t ⊗ W_{i})

Lemma 8.6.2. The product on the category 𝔸̂ is ∏𝔸̂ = δ^{∗} ∏.

Proof: Immediate from the universal property of pullbacks.

Proposition 8.6.3. The product on the semifree category D𝔸 is R(Γ̂δ^{∗}) ∏ = RΓ̂Rδ^{∗} ∏.

Proof: Since the product of sets is exact, the only point at issue is that δ^{∗} preserves injectives. This is immediate since the injectives are either of the form (N ⟶ N), which δ^{∗} takes to a map of the same form, or of form (I ⟶ 0), on which δ^{∗} is the identity.

All is just the same for the full standard model.

Lemma 8.6.4. The product on the category 𝔸̂ is ∏𝔸̂ = δ^{∗}ϕ ∏.

Proposition 8.6.5. The product on the derived category of the standard model category DA is R(Γ̂δ^{∗}) ∏ = RΓ̂Rδ^{∗}ϕ ∏.

CHAPTER 9 Products, smash products and function 𝕋-spectra.

This chapter begins with a section fundamental to the structure of the derived category of the standard model category: it gives factorizations for all maps between basic injectives. This is then applied in Section 9.2 to show that the smash product is modelled by the left derived tensor product, and in Section 9.3 to show the function spectrum is modelled by the right adjoint of the left derived tensor product.

9.1. Maps between injective spectra.

We have shown that every 𝕋-spectrum Y admits an Adams resolution

Y ⟶ I(Y) ⟶^{δ(Y)} J(Y),

and furthermore I(Y) = ⋁_{α} I_{α} and J(Y) = ⋁_{β} J_{β} where the spectra I_{α} and J_{β} are suspensions of the standard spectra E⟨H⟩ for various H ∈ F and ẼF. Furthermore, no F-contractible summands are required in J(Y). Thus the map δ(Y) is described by its components δ_{α} : I_{α} ⟶ J(Y) = ⋁_{β} J_{β}. Each such component is thus a map E⟨H⟩ ⟶ J or ẼF ⟶ J for an F-free injective J. It is the purpose of the present section to show that such maps can be described as composites of inclusions of direct summands and a few maps of very special form.

Note first that [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} = ℚ[c_{H}], so that any graded self-map of E⟨H⟩ is a rational mulitple of c^{n} : E⟨H⟩ ⟶ Σ^{2n}E⟨H⟩ for some n ≥ 0. On the other hand [ẼF, ΣEF_{+}]_{∗}^{𝕋} = t_{∗}^{F}; in degree 0 and negative degrees, the maps factor through the quotient map ẼF ⟶ ΣEF_{+}, and are thus described by a map ΣEF_{+} ⟶ Σ^{2n+1}EF_{+} for some n ≥ 0. However in positive degrees each element has a finite support. We thus choose a subgroup H containing all the supporting subgroups, and the element factors through the composite ẼF ≃ ẼF ∧S^{−nV (H)} ⟶ ΣEF_{+}∧S^{−nV (H)}. Here ΣEF_{+}∧S^{−nV (H)} is a wedge of suspensions Σ^{−2vK}E⟨K⟩, where v(K) = n if K ⊆ H and v(K) = 0 otherwise.

Lemma 9.1.1. If J is a wedge of suspensions of spectra E⟨H⟩ for various H then

(i) Any map E⟨H⟩ ⟶ J factors as

E⟨H⟩ ⟶ Σ^{2n}E⟨H⟩ ⟶ J

110 9. PRODUCTS, SMASH PRODUCTS AND FUNCTION 𝕋-SPECTRA.

for some n, where the first map is a scalar multiple of c^{n} and the second is the inclusion of a direct summand. (ii) Any map ẼF ⟶ J factors as

ẼF ⟶ ΣEF_{+} ∧ S^{−nV (H)} ≃ ⋁_{K} Σ^{−2v(K)+1}E⟨K⟩ ⟶ ⋁_{K} Σ^{−2v′(K)+1}E⟨K⟩ ⟶ J

where the first map is as described above, the second has components c^{v(K)−v′(K)} : E⟨K⟩ ⟶ Σ^{2v(K)−2v′(K)}E⟨H⟩ and the third is the inclusion of a direct summand.

Proof: (i) First note that the map is described by its effect in homotopy. Next note that the image in homotopy is a quotient of 𝕀(H) = π_{∗}^{𝕋}(E⟨H⟩), and hence is of the form Σ^{2n}𝕀(H). This is injective, and so the inclusion of the image is split. Since all objects concerned are injective we may realise the splitting geometrically.

(ii) The map is described by a homomorphism θ : t_{∗}^{F} ⟶ π_{∗}^{𝕋}(J). Of course the image of 1 is torsion, and hence c^{n}θ(1) = 0 for some n. Hence θ factors through one of the maps ẼF ⟶ Σ^{−2n+1}EF_{+} described above; this is clear algebraically, and thanks to injectivity may be realized geometrically. The resulting map Σ^{−2n+1}EF_{+} ⟶ J is described by its components, each of which can be described as in Part (i).

9.2. Models of smash products.

In this section we show that the left derived tensor product as discussed in Sections 8.4 and 8.5 is indeed the model of the smash product, in the category of free, semifree, almost free or arbitrary 𝕋-spectra, as appropriate. Unfortunately, we are only able to describe the model on objects; neither the smash product nor its adjoint preserve pure parity objects in general, so one cannot get a firm grip on morphisms. To resolve this difficulty in the present framework, one would need a natural splitting into even and odd pieces. A more satisfactory resolution would be to prove the equivalence of homotopy categories from a string of equivalences at the level of model categories, following the example of Quillen [22].

The first observation is that the smash product preserves F-spectra and F-contractible spectra. More precisely X∧Y ∧EF_{+} ≃ (X∧EF_{+})∧(Y ∧EF_{+}) and Φ^{𝕋}(X∧Y) ≃ Φ^{𝕋}X∧Φ^{𝕋}Y. Since Φ^{𝕋}(X ∧ Y) = Φ^{𝕋}(X) ∧ Φ^{𝕋}(Y), it is immediate that

π_{∗}(Φ^{𝕋}(X ∧ Y)) = π_{∗}(Φ^{𝕋}X) ⊗ π_{∗}(Φ^{𝕋}Y),

and we shall show that π_{∗}^{𝕋}(EF_{+} ∧X ∧Y) is the left derived tensor product of π_{∗}^{𝕋}(EF_{+} ∧X) and π_{∗}^{𝕋}(EF_{+} ∧Y). This might suggest the use of the torsion model, but we warn that there is a suspension which makes the gluing map inaccessible.

The connection with topology arises from the fact that Σ𝕀 ⊛ M = M for any F-finite torsion module M, where ⊛ denotes the left derived torsion product functor. Indeed, we have E⟨H⟩ ∧ Y = Y, and hence

π_{∗}^{𝕋}(E⟨H⟩ ∧ Y) = ΣTor(π_{∗}^{𝕋}(E⟨H⟩), π_{∗}^{𝕋}(Y)) = π_{∗}^{𝕋}(E⟨H⟩) ⊛ π_{∗}^{𝕋}(Y).

We may now prove the special case of the theorem in which both spectra are F-spectra.

Proposition 9.2.1. For F-spectra X and Y, their smash product is modelled by the dg O_{F}-module π_{∗}^{𝕋}(X) ⊛ π_{∗}^{𝕋}(Y). In particular there is a short exact sequence

0 ⟶ π_{∗}^{𝕋}(X) ⊗ π_{∗}^{𝕋}(Y) ⟶ π_{∗}^{𝕋}(X ∧ Y) ⟶ ΣTor(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) ⟶ 0.

Proof: We have observed that the case when Y is injective is immediate. For the general case use an Adams resolution Y ⟶ I(Y) ⟶ J(Y), realising π_{∗}^{𝕋}(Y) ⟶ I ⟶^{d} J in homotopy, and obtain a cofibre sequence X ∧ Y ⟶ X ∧ I(Y) ⟶ X ∧ J(Y). The model of X ∧ Y is thus the fibre of (S ⊛ I ⟶ S ⊛ J) = Σ(Tor(S, I) ⟶ Tor(S, J)). Providing we identify the map as 1 ⊛ d = ΣTor(1, d) the proof is complete. However both I(Y) and J(Y) are wedges of suspensions of E⟨H⟩, so it suffices to show for an arbitrary map f : Σ^{n}E⟨H⟩ ⟶ E⟨H⟩ that the map 1 ∧ f : Σ^{n}X ∧ E⟨H⟩ ⟶ X ∧ E⟨H⟩ induces 1 ⊛ p(f). But we have classified all such maps f: they are multiplications by c_{H}^{m} for some m. Identifying X ∧ E⟨H⟩ with X, the map 1 ∧ f is identified with c_{H}^{m} by definition of the ℚ[c_{H}]-module structure.

Theorem 9.2.2. For arbitrary rational 𝕋-spectra X and Y, their smash product is modelled by π_{∗}^{A}(X) ⊛ π_{∗}^{A}(Y). In particular there is a short exact sequence

0 ⟶ π_{∗}^{A}(X) ⊗ π_{∗}^{A}(Y) ⟶ π_{∗}^{A}(X ∧ Y) ⟶ ΣTor(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ⟶ 0.

Proof: We have observed that the case when Y is injective is immediate. For the general case use an Adams resolution Y ⟶ I(Y) ⟶ J(Y), realising π_{∗}^{A}(Y) ⟶ I ⟶^{d} J in homotopy, and obtain a cofibre sequence X ∧ Y ⟶ X ∧ I(Y) ⟶ X ∧ J(Y). The model of X ∧ Y is thus the fibre of (π_{∗}^{A}(X) ⊛ I ⟶ π_{∗}^{A}(X) ⊛ J). It remains to identify the map as 1 ⊛ d.

However I(Y) is a wedge of suspensions of spectra E⟨H⟩, so it suffices to deal with one factor δ : E⟨H⟩ ⟶ J(Y) where J(Y) is a F-free injective. In Section 9.1 we identified such maps as composites of certain standard maps and retractions. The smash product and its model respect retracts, so it suffices to check that the standard maps are accurately modelled. These are of two types. The first type are the maps c^{n} : E⟨H⟩ ⟶ Σ^{2n}E⟨H⟩, which is correctly modelled since the model is built with O_{F}-modules. The second type are the maps g : ẼF ⟶ ΣEF_{+}∧S^{−nV (H)}, which is the cofibre of the map f : S^{−nV (H)} ⟶ ẼF which is the identity on geometric fixed points. Of course the codomain of f is injective, so f is determined by its induced map. To describe the model let v be the function with v(K) = n when K ⊆ H, and v(K) = 0 otherwise, and let O_{F}(−v) be the submodule of t_{∗}^{F}, which in degree m consists of the product of the factors with m ≤ −v(K). Thus g is modelled by the map depicted thus

t_{∗}^{F} ⟶ t_{∗}^{F}/O_{F}(−v)
↓  ↓
t_{∗}^{F} ⟶  0.

In other words, this is the cokernel of the inclusion ϕ : S^{−v} ⟶ e(ℚ), in the notation of Section 8.3. The modelling process preserves cofibre sequences, so it suffices to show that 1 ∧ ϕ : π_{∗}^{A}(X) ⊛ S^{−v} ⟶ π_{∗}^{A}(X) ⊛ e(ℚ) models 1 ∧ f : X ∧ S^{−nV (H)} ⟶ X ∧ ẼF. It suffices to deal with the case when X is even. However in this case X ∧ S^{−nV (H)} and X ∧ ẼF are also both even, so the map 1 ∧ f is determined by its induced map. Furthermore, since X ∧ ẼF is F-contractible, this is determined by its effect on vertices, which is the identity. This is faithfully modelled by 1 ⊛ ϕ.

9.3. Models of function spectra.

The preceding sections we have identified the candidate for an algebraic model of function spectra, and established various formal properties. In this section we establish that the function spectrum is modelled by the right adjoint R′Hom to the left derived tensor product in all four of our examples.

Theorem 9.3.1. If Y and Z are rational 𝕋-spectra modelled by B and C in the standard model, then F (Y, Z) is modelled by RΓ̂ RHom(B, C).

The analogous results hold for free, almost free and semi-free spectra. We shall give the proof in the free case by way of motivating the proof of 9.3.1. The other cases are similar.

Proposition 9.3.2. If Y and Z are free spectra modelled by torsion ℚ[c]-modules S and T then the internal function spectrum F (Y, Z) ∧ E𝕋_{+} is modelled by the dg torsion module RΓ_{c} RHom(S, T).

Proof: We note that this result could be proved from the general case using a little algebra, since the theorems on function spectra and smash products assert that F (Y, Z) ∧ E𝕋_{+} is modelled by RΓ̂ RHom(f(S), f(T)) ⊛ Σ𝕀. Nonetheless it seems worth giving a more direct proof, by way of preparation for the case of arbitrary spectra. Special cases of the proposition admit simpler proofs, but the rather indirect approach does appear necessary. The above discussion showed that we ought to concentrate on the case when ⊗S preserved cofibrations. Whilst this only happens in degenerate cases, the torsion product functor preserves cofibrations when S is injective.

This suggests that it is natural to identify F (Y, Z) ∧ E𝕋_{+} as the cofibre of F (I(Y), Z) ∧ E𝕋_{+} ⟵ F (J(Y), Z) ∧ E𝕋_{+} as might otherwise seem rather unnatural: we refer to this as the Worms method. In our case we know that both I(Y) and I(Z) are wedges of suspensions of E𝕋_{+}, and so F (I(Y), Z) ∧ E𝕋_{+} and F (J(Y), Z) ∧ E𝕋_{+} are internal products of suspensions of spectra F (E𝕋_{+}, Z)∧E𝕋_{+} ≃ Z ∧E𝕋_{+}. Since we have modelled products, it suffices to identify the map between the products. However the map I(Y) ⟶ J(Y) is specified by its components, and any map E𝕋_{+} ⟶ J(Z) factors as E𝕋_{+} ⟶ Σ^{2n}E𝕋_{+} ⟶ J(Z) for some n where the first map is multiplication by c^{n} and the second is the inclusion of a direct summand. The remaining verifications will be treated in more detail in the proof of the general case.

However this gives very little indication of the general answer. The vertices are especially hard to describe since they involve both torsion and torsion free parts. For example if X and Y are both F-spectra we may calculate [X, Y]_{∗}^{𝕋} using the Adams short exact sequence, and F (X, Y) ∧ ẼF has homotopy groups E^{−1}[X, Y]_{∗}^{𝕋}. Roughly speaking this suggests a contribution E^{−1}RHom(S, T) from the torsion parts of a general spectrum. It is not immediately clear how this should be mixed with the vertex contribution Hom(V, W).

Remark 9.3.3. We have proved that if Y and Z are F-spectra modelled by F-finite torsion modules, the vertex of F (Y, Z) is modelled by E^{−1}RHom(S, T). We claim that it is also modelled by RHom(t_{∗}^{F}, ΣRΓ_{f}RHom(S, T)). This is a manifestation of Warwick duality.

To be precise, if E is a multiplicatively closed set in a commutative ring A we define the chain complex K(E) by the fibre sequence

K(E) ⟶ A ⟶ E^{−1}A.

Denoting projective approximation by the letter P, Warwick duality is the statement that there is a quasi-isomorphism

Hom(P E^{−1}A, ΣK(E) ⊗ X) ≃ Hom(P K(E), X) ⊗ E^{−1}A.

The derived analogue of this is

RHom(t_{∗}^{F}, ΣRΓ_{f}X) ≃ E^{−1}RHom(Σ𝕀, X).

The required equivalence thus follows provided X = Hom(S, T) is complete in the sense that the natural map

X = RHom(O_{F}, X) ⟶ RHom(Σ𝕀, X)

is an equivalence. However for X = Hom(S, T) we may use the tensor Hom adjunction to establish completeness:

RHom(O_{F}, RHom(S, T)) ⟶ RHom(Σ𝕀, RHom(S, T)) ⟶^{≃} RHom(Σ𝕀⊛S, T) = RHom(S, T).

The obvious conjecture is that the function spectrum corresponds to the internal Hom functor before passage to homotopy, and hence that in the homotopy category the function spectrum is modelled by something closely related to RHom. The case of F-spectra shows that the model will not be RHom itself, but should be recognized as the right adjoint of the total left derived tensor product. Of course, this would follow immediately if the identification of the smash product with the total left derived functor of tensor product was functorial. Nonetheless, we have identified the right adjoint R′Hom of the total left derived functor of tensor product. It is a homotopy functor that preserves cofibre sequences, and it suffices to show that the function spectrum is object-accessible and is modelled by this functor on generators. Proof of Theorem 9.3.1: One might expect the proof to use the cofibration F (Y, Z) ⟶ F (Y, I(Z)) ⟶ F (Y, J(Z)), where Z ⟶ I(Z) ⟶ J(Z) is an Adams resolution. However it is not clear how to eliminate indeterminacy in the description of the map F (Y, I(Z)) ⟶ F (Y, J(Z)). Instead, we therefore use our analysis of products to apply the Worms method as follows.

In the usual way, we are going to prove that the functor F (Y, Z) is object-accessible, by constructing it from particular cases F (Y′, Z′), by taking mapping cones of maps which can be identified with no indeterminacy at any stage. Of course the algebraic model, R′Hom is a functor on the homotopy category and preserves cofibres, so it suffices to show that it has the correct value on the building blocks F (X′, Y′). We begin by outlining the argument. Step 1: For an arbitrary Z we give a description of F (ẼF, Z) and F (E⟨H⟩, Z) for all H. This is reasonably straightforward. The description is not claimed to be natural in Z; however, it is automatically natural in the first variable, since all maps between spectra of the form ẼF or E⟨H⟩ are essentially given by multiplication by powers of c or Euler classes as we saw in Section 9.1. Step 2: Consider an Adams resolution Y ⟶ I(Y) ⟶ J(Y) of Y. Thus both I(Y) = ⋁_{α} I_{α} and J(Y) = ⋁_{β} J_{β} are wedges of suspensions of ẼF and E⟨H⟩ for various H. Furthermore J(Y) does not involve any copies of ẼF. Step 3: Using the description of products we can describe the models of F (I(Y), Z) = ∏_{α} F (I_{α}, Z) and F (J(Y), Z) = ∏_{β} F (J_{β}, Z). We therefore have a cofibre sequence

F (Y, Z) ⟵ ∏_{α} F (I_{α}, Z) ⟵^{δ^{∗}} ∏_{β} F (J_{β}, Z).

Step 4: Describe the map δ^{∗} between the products. For this, note that the map δ : I(Y) ⟶ J(Y) is categorically described by its components δ_{α} : I_{α} ⟶ J(Y), so the map of function spectra is also described by its components F (δ_{α}, Z). It therefore suffices to describe these. Step 5: Since I_{α} is either a suspension of ẼF or of E⟨H⟩ for some H, we have shown in Section 9.1 that any map into a spectrum of the form J(Y) can be described as the composite of a very explicit map followed by the inclusion of a direct summand. The description in Step 1 is natural for these maps, and the product description is categorical.

The point of this analysis is that it can be carried out in parallel in the algebraic and topological categories, and the primitive pieces and constructions necessarily correspond.

We proceed with Step 1.

Lemma 9.3.4. The algebraic model of F (ẼF, Z) is correctly described by the theorem. Consequently, if the model of Z is C = (P ⟶ t_{∗}^{F} ⊗W) with torsion part T then the model of F (ẼF, Z) is torsion free and the vertex is the homology of the fibre of a certain map

W ⟶ RHom(t_{∗}^{F}, T),

whose composite with the projection to Hom(t_{∗}^{F}, T) is adjoint to the homomorphism t_{∗}^{F} ⊗ W ⟶ T.

Proof: We take the model e(ℚ) of ẼF, and establish the lemma in turn for Z being F-contractible, F-free and arbitrary.

If Z is F-contractible F (ẼF, Z) ≃ Z. On the other hand C = e(W), which is already injective, so that RHom(e(ℚ), C) = Hom(e(ℚ), C) = e(W). Since this too is injective RΓ̂e(W) = Γ̂e(W), and e(W) is already in the standard model, so Γ̂ does not alter it.

If Z is an F-spectrum in even degrees then F (ẼF, Z) is F-contractible, and has homotopy Hom(t_{∗}^{F}, T) in even degrees and Ext(Σt_{∗}^{F}, T) in odd degrees. On the other hand C = f(T), and we may choose an injective resolution f(T) ⟶ f(I) ⟶ f(J). Thus

RHom(e(ℚ), f(T)) = fibre (Hom(e(ℚ), f(I)) ⟶ Hom(e(ℚ), f(J))) .

Since t_{∗}^{F} is flat, the two terms are both injective, so that

RΓ̂RHom(e(ℚ), f(T)) = fibre(Γ̂Hom(e(ℚ), f(I)) ⟶ Γ̂Hom(e(ℚ), f(J))).

Of course Hom(e(ℚ), f(I)) has zero vertex, and Hom(t_{∗}^{F}, I) is torsion free, so that

Γ̂Hom(e(ℚ), f(I)) = e(E^{−1}Hom(t_{∗}^{F}, I)).

However, Euler classes are already invertible, so the functor E^{−1} may be omitted. The same comments apply to J. Hence RΓ̂RHom(e(ℚ), f(T)) is Hom(t_{∗}^{F}, I) in even degrees, and Ext(t_{∗}^{F}, I) in odd degrees, as required.

For the general case we must consider the structure map q_{Z} : Z ∧ ẼF ⟶ Z ∧ ΣEF_{+}. Now note that the induced map F (ẼF, q) : F (ẼF, Z ∧ ẼF) ⟶ F (ẼF, Z ∧ ΣEF_{+}) is determined by the diagam

F (ẼF, Z ∧ ẼF) ⟶^{F (ẼF,q)} F (ẼF, Z ∧ ΣEF_{+})
≃↓  ↓
F (S^{0}, Z ∧ ẼF) ⟶^{q_{X}} F (S^{0}, Z ∧ ΣEF_{+})

since all maps from an F-contractible spectrum to F (EF_{+}, T) are null. This argument applies equally well in the algebraic category, where it describes the map induced in the model.

Lemma 9.3.5. The algebraic model of F (E⟨H⟩, Z) is correctly described by the theorem.

Remark 9.3.6. For calculational purposes we care more about knowing the the torsion part and vertex of F (E⟨H⟩, Z), both of which are easy. Indeed, the torsion part is RΓ_{c} RHom(Σ𝕀(H), T) where T is the torsion part of Z, and the vertex is E^{−1}RHom(Σ𝕀(H), T). On the other hand, this does not serve the present purpose.

Proof: Note first that both the algebraic and geometric model are trivial when Z is Fcontractible. Indeed F (E⟨H⟩, Z ∧ ẼF) ≃ ∗, and on the other hand e(W) is injective, so that RHom(Σ𝕀(H), e(W)) = Hom(Σ𝕀(H), e(W)) = 0. We may thus assume Z is an F-spectrum, and C = f(T). Similarly, we may assume that Z is a 𝕋-spectrum over H. For notational simplicity we assume H = 1 and drop the subscripts. Now if Z is semifree, the function spectrum F (E⟨H⟩, Z) is semifree, and similarly in algebra. We may therefore conduct the rest of the proof with semifree objects: it is clear that the algebraic and topological functors including semifree categories in the full categories correspond.

Now take an injective resolution 0 ⟶ T ⟶ I ⟶ J ⟶ 0. We thus find

RΓ̂RHom(f(𝕀), f(T)) = fibre (RΓ̂ Hom(f(𝕀), f(I)) ⟶ RΓ̂ Hom(f(𝕀), f(J))) .

Now Hom(f(𝕀), f(I)) has zero vertex and is thus f(H(I)) where H(I) = Hom(f(𝕀), I). Since H(I) is torsion free, we can easily construct an injective resolution 0 ⟶ H(I) ⟶ H(I)[c^{−1}] ⟶ K(I) ⟶ 0, and hence

RΓ̂f(H(I)) = fibre (Γ̂f(H(I)[c^{−1}]) ⟶ Γ̂f(K(I))) .

However Γ̂f(H(I)[c^{−1}]) = e(H(I)[c^{−1}]), and Γ̂f(K(I)) = K(I); in particular we have calculated RΓ̂RHom(f(𝕀), I), and it is even; similarly RΓ̂RHom(f(𝕀), J) is even, and the map between them is classified by its effect in homotopy. Since the relevant maps are composites of retractions and collapse maps c^{n} : E⟨H⟩ ⟶ Σ^{2n}E⟨H⟩, it is immediate that they are modelled by the corresponding algebraic maps.

This completes Step 1. Steps 2, 3 and 4 require no further comment. We proceed to Step 5.

Lemma 9.3.7. The identifications in 9.3.4 and 9.3.5 are natural for the maps in 9.1.1.

Proof: The identifications are certainly natural for retractions. This leaves the question of naturality for the maps c^{n} : E⟨H⟩ ⟶ Σ^{2n}E⟨H⟩ and ẼF ⟶ ΣEF_{+} ∧S^{−nV (H)}. Naturality for those of the first type is clear since the modelling is all done in the category of O_{F}- modules.

For those of the second type we may compose with projection onto the Kth factor, and it is sufficient to prove naturality for these. There are two types, depending on whether K is contained in H or not. If K is not contained in H, the map is equal to the composite ẼF ⟶ ΣEF_{+} ⟶ ΣE⟨K⟩, and thus a special case of the second type.

We may therefore suppose that f : ẼF ⟶ Σ^{−2n+1}E⟨H⟩ is given, and that we want to understand the map f^{∗} : F (Σ^{−2n+1}E⟨H⟩, Z) ⟶ F (ẼF, Z). Applying F (f, ·) to the cofibre sequence Z ∧ EF_{+} ⟶ Z ⟶ Z ∧ ẼF, we obtain the diagram

F (Σ^{−2n+1}E⟨H⟩, Z ∧ EF_{+}) ⟶^{≃} F (Σ^{−2n+1}E⟨H⟩, Z) ⟶ F (Σ^{−2n+1}E⟨H⟩, Z ∧ ẼF)
↓  ↓  ↓
F (ẼF, Z ∧ EF_{+})  ⟶  F (ẼF, Z)  ⟶  F (ẼF, Z ∧ ẼF),

in which the three verticals are induced by f. Since F (E⟨H⟩, Z ∧ ẼF) ≃ ∗, the induced map is determined by the composite α : F (Σ^{−2n+1}E⟨H⟩, Z ∧ EF_{+}) ⟶ F (ẼF, Z ∧ ẼF). Next, the codomain of α is F-contractible, and so α is determined by its effect in homotopy. Splitting Z ∧ EF_{+} into even and odd parts, we see that the Adams short exact sequence shows that the effect of f in homotopy determines the effect of α in homotopy. Since the model is based on the effect of f in homotopy, this completes the proof.

CHAPTER 10 Induction, coinduction and geometric fixed points.

This chapter treats the simplest change of groups functors. In Section 10.1 we give algebraic models of the forgetful functor from 𝕋-spectra to H-spectra, and its adjoints, the induction and coinduction functors. In Section 10.2 we give the algebraic model of the geometric K-fixed point functor.

10.1. Forgetful, induction and coinduction functors.

It is the purpose of this section to describe the algebraic counterparts of the forgetful functor U : 𝕋−Spec ⟶ H−Spec, the induction functor ind : H−Spec ⟶ 𝕋−Spec, X ⟼ 𝕋⋉_{H}X, and the coinduction functor coind : H−Spec ⟶ 𝕋−Spec, X ⟼ F_{H}[𝕋, X). The induction and coinduction functors are left and right adjoint to the forgetful functor respectively, so that if X is a 𝕋-spectrum and Y an H-spectrum

[𝕋 ⋉_{H} Y, X]^{𝕋} = [Y, UX]^{H} and [UX, Y]^{H} = [X, F_{H}[𝕋, Y)]^{𝕋}.

We must begin by describing the category H−Spec. First, we recall that an arbitrary rational H-spectrum splits as a product of Eilenberg-MacLane spectra; thus H−Spec is equivalent to the derived category of the 0-dimensional abelian category of rational H-Mackey functors.

For any finite group H, the rational Mackey functors are easy to describe, and are sums of the functors arising from representations of the Weyl groups N_{H}(K)/K for K ⊆ H. In our case H is abelian, so we need only explain that a module V for the quotient H/K gives rise to a Mackey functor R_{K}^{H}(V) defined by

R_{K}^{H}(V)(L) = \{ \begin{matrix}V^{L} & if & L ⊇ K \\ 0 & if & L ⊉ K \end{matrix}

The restriction maps are given by inclusions of fixed point sets, and the transfer maps are given by coset sums. We let 𝔐_{H} denote the category of all rational Mackey functors, and 𝔐_{H}^{triv} denote the full subcategory of Mackey functors with trivial Weyl group action (i.e. sums of functors of form R_{K}^{H}(ℚ). Thus

H−Spec ≃ D(𝔐_{H}).

Because the category of rational Mackey functors is of global dimension 0, and the free functors are realizable, all spectra are generalized Eilenberg-MacLane spectra and it is easy to see that the equivalence is given by X ⟼ \underline{π}_{∗}^{H}(X). Therefore the algebraic counterpart of the forgetful functor 𝕋−Spec ⟶ H−Spec is the functor DA ⟶ D(𝔐_{H}) given by the the same condition. However, it is perhaps clearer to compose with the equivalence

D𝔐_{H} ⟶^{≃} ∏_{K} ℚH/K − mod

whose Kth component is M ⟼ e_{K}M(K).

Summary 10.1.1. The forgetful map 𝕋−Spec ⟶ H−Spec corresponds to the functor DA ⟶ ∏_{K} D(ℚH/K − mod) with Kth component M ⟼ [L_{K}, M], where L_{K} = (Σℚ(H) ⟶ 0) is the algebraic counterpart to the basic cell σ_{K}^{0}.

We have two methods open to us to identify the algebraic induction and coinduction functors. We could calculate the behaviour of the geometric functors, but instead we shall simply guess the answers and prove they have the requisite adjointness properties. Actually, we ‘guess’ the functors by a calculation on geometric objects. Because H-spectra split as a wedge of Eilenberg-MacLane spectra, and the induction functor commutes with wedges, the induction will be determined by its values on Eilenberg-MacLane spectra. Finally, it is useful to use the fact that F_{H}[𝕋, X) ≃ Σ^{−1}𝕋 ⋉_{H} X. Now we simply observe that any induced spectrum 𝕋 ⋉_{H} X is an F-spectrum, and is thus determined by its homotopy groups. π_{∗}^{𝕋}(𝕋 ⋉_{H} Y) = π_{∗}^{𝕋}(ΣF_{H}[𝕋, Y)) = π_{∗}^{H}(ΣY).

Lemma 10.1.2. If the H-Mackey functor M corresponds to V = (V_{K}) with V_{K} a ℚH/Kmodule (ie M = ⨁_{K} R_{K}^{H}V_{K}), then

M(H) = ⨁_{K}(V_{K})^{H}.

Definition 10.1.3. If V_{K} is a graded ℚH/K-module for each K ⊆ H, then we define induction and coinduction on V = (V_{K}) by

ind(V) = Σ(⨁_{K}(V_{K})^{H} ⟶ 0)

and

coind(V) = (⨁_{K}(V_{K})^{H} ⟶ 0),

where the Chern class c acts as zero in both cases.

Evidently these are both exact functors ∏_{K} ℚH/K − mod ⟶ A, and hence induce functors on the derived category.

Proposition 10.1.4. The induction and coinduction functors are left and right adjoint to the forgetful functor A ⟶ ∏_{K} ℚH/K −mod, and the same holds at the level of derived categories.

Before sketching the proof we record the desired corollary.

Corollary 10.1.5. The algebraic and topological induction and coinduction functors correspond in the sense that the diagrams

H−Spec  ⟶^{𝕋⋉_{H}} 𝕋−Spec  H−Spec  ⟶^{F_{H}[𝕋,·)} 𝕋−Spec
≃↓  ↓≃ and  ≃↓  ↓≃
∏_{K} ℚH/K − mod ⟶^{ind} DA  DA ∏_{K} ℚH/K − mod ⟶^{coind}  DA

are commutative up to natural isomorphism.

Proof: As usual we must record the units and counits, and we leave the verification of the triangular identities to the reader. We must first recall the natural exact sequence

0 ⟶ e_{K}M/ce_{K}M ⟶ [L_{K}, M] ⟶ Σ^{−1}ann(c, e_{K}M) ⟶ 0.

To describe the counit indUM ⟶ M suppose M = (N ⟶ t_{∗}^{F} ⊗ W). It is enough to describe the map of nubs since the domain is c torsion and hence maps to zero in t_{∗}^{F} ⊗ W, so this is necessarily consistent on vertices. The counit has Kth component the evident map

Σ([L_{K}, M]) ⟶ Σ(Σ^{−1}ann(c, e_{K}N)) ⟶ e_{K}N.

The unit V ⟶ UindV has as its Kth component the composite

V_{K} ⟶ (V_{K})^{H} ⟶ [L_{K}, indV] ≅ (V_{K})^{H} ⊕ Σ(V_{K})^{H}

where the first map is given by coset averages, and the second uses the fact that the short exact sequence for [L_{K}, indV] splits naturally in V.

Next, to describe the unit M ⟶ coindUM it is obviously enough to discuss nubs, since the codomain has zero vertex. Its Kth component is

e_{K}N ⟶ e_{K}N/ce_{K}N ⟶ [L_{K}, M].

The counit UcoindV ⟶ V has Kth component the composite

[L_{K}, coind(V)] ⟶ (V_{K})^{H} ⟶ V_{K}.

10.2. Geometric fixed points.

We give an analysis of the geometric H-fixed point functor, regarded as a functor from 𝕋-spectra to 𝕋/K-spectra for a finite subgroup K. The functor Φ^{𝕋} is corresponds to taking the vertex, and is an integral part of our analysis. It therefore requires no further comment.

To minimize confusion in the coming discussion we let \overline{𝕋} = 𝕋/K and use bars to indicate reference to the ambient group \overline{𝕋}; when necessary, we denote the quotient map by q : 𝕋 ⟶ \overline{𝕋}. For example \overline{F} is the family of finite subgroups of \overline{𝕋}, and \overline{H} is the image in \overline{𝕋} of the subgroup H of 𝕋. Note that the \overline{𝕋}-space E\overline{F}_{+} may be regarded as a 𝕋-space, and as such it is EF_{+}. Nonetheless, we shall use use the notation which best indicates which group is acting.

In one sense, the functor Φ^{K} is obvious from our construction. Indeed, we analyze X using the cofibre sequence

X ⟶ X ∧ ẼF ⟶ X ∧ ΣEF_{+},

and if we apply Φ^{K} we obtain

Φ^{K}X ⟶ Φ^{K}(X ∧ ẼF) ⟶ Φ^{K}(X ∧ ΣEF_{+}).

Now we use the fact that X ∧ ẼF = (Φ^{𝕋}X) ∧ ẼF so that Φ^{K}(X ∧ ẼF) = (Φ^{𝕋}X) ∧ Ẽ\overline{F}. On the other hand, we have conducted our analysis using the stable splitting

EF_{+} ≃ ⋁_{H} E⟨H⟩

where E⟨H⟩ is the space which is the cofibre of the universal map E[⊂ H]_{+} ⟶ E[⊆ H]_{+}. Since Φ^{K} is the extension of the K-fixed point space functor the basic fact is the following.

Lemma 10.2.1. For any finite subgroups H and K of 𝕋, we have the equivalence

(E⟨H⟩)^{K} = \{ \begin{matrix}E⟨\overline{H}⟩ & if & H ⊇ K \\ ∗ & if & H ⊉ K \end{matrix}

of based \overline{𝕋}-spaces.

Proof: The characterization of universal spaces by their fixed point spaces gives (EH)^{K} = E\overline{H} for any family H of subgroups of 𝕋, where \overline{H} = \{\overline{H} | H ∈ H\}.

Restated in the stable language, this states that, whenever K ⊆ H,

Φ^{K}E⟨H⟩ ≃ E⟨\overline{H}⟩.

The natural guess is now that the invariants for Φ^{K}(X ∧ EF_{+}) are obtained from the sequence of spectra X(H) = X ∧ E⟨H⟩ and their characteristic ℚ[c_{H}]-modules π_{∗}^{𝕋}(X(H)) by ignoring those terms with H ⊉ K. This turns out to be correct. To make sense of it, we need to observe the there is a natural identification of the rings of operations ℚ[c_{H}] and ℚ[c\overline{H}].

The following observation is repeatedly useful.

Lemma 10.2.2. If K ⊆ H and Y is any 𝕋-spectrum over H then Φ^{K} gives an isomorphism

Φ^{K} : [X, Y]_{∗}^{𝕋} ⟶^{≅} [Φ^{K}X, Φ^{K}Y]\overline{𝕋} ; ∗.

Proof: Since

[Φ^{K}X, Φ^{K}Y]\overline{𝕋} ; ∗ = [X, Ẽ[⊉ K] ∧ Y]_{∗}^{𝕋},

and we have the cofibre sequence

E[⊉ K]_{+} ⟶ S^{0} ⟶ Ẽ[⊉ K],

it is enough to show E[⊉ K]_{+} ∧ Y ≃ ∗. However Y ≃ Y ∧ E⟨H⟩, and, by considering fixed points, E[⊉ K]_{+} ∧ E⟨H⟩ is a contractible space.

10.2. GEOMETRIC FIXED POINTS. 121

Using the fact that Φ^{K}E⟨H⟩ ≃ E⟨\overline{H}⟩, we obtain an identification of rings of operations. Corollary 10.2.3. Whenever K ⊆ H, Φ^{K} induces an isomorphism

ℚ[c_{H}] = [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} ⟶^{≅} [E⟨\overline{H}⟩, E⟨\overline{H}⟩]\overline{𝕋} ; ∗ = ℚ[c\overline{H}].

Furthermore, we can now confirm intuition.

Corollary 10.2.4. If K ⊆ H then Φ^{K} induces an isomorphism

π\overline{𝕋} ; ∗((Φ^{K}X)(\overline{H})) = π_{∗}^{𝕋}(X(H))

of ℚ[c_{H}]-modules. Proof: We calculate

(Φ^{K}X)(\overline{H}) = (Φ^{K}X) ∧ E⟨\overline{H}⟩ = Φ^{K}(X ∧ E⟨H⟩) ≃ Φ^{K}(X(H)).

This shows how to construct the algebraic analogue. We consider the cofibre sequence

M ⟶ e(V_{M}) ⟶^{q̂_{M}} f(ΣT_{M})

and apply Φ^{K} to obtain

Φ^{K}M ⟶ Φ^{K}e(V_{M}) ⟶^{Φ^{K} q̂_{M}} Φ^{K}f(ΣT_{M}).

Thus, it is enough to identify Φ^{K} on e(V_{M}) and f(T_{M}). Since X(𝕋) = Φ^{𝕋}X ∧ ẼF and Φ^{𝕋}X = Φ\overline{𝕋}Φ^{K}X, we take Φ^{K}e(V_{M}) = \overline{e}(V_{M}); in other words, vertices are identical V_{Φ^{K}M} = V_{M}. On the other hand, we take Φ^{K}f(T_{M}) = \overline{f}(\overline{T_{M}}), where T_{Φ^{K}M} = \overline{T_{M}} is obtained by deleting the factors not containing K. More succinctly, we let e denote the idempotent of O_{F} with support the set of subgroups containing K. Now view \overline{O}\overline{F} as eO_{F} and take \overline{T_{M}} = eT_{M}. Also, once we have observed that t\overline{F} ; ∗ ≅ et_{∗}^{F} we can say q̂_{Φ^{K}M} = \overline{q̂_{M}} is eq̂_{M}

It remains to identify this as a functor on arbitrary objects.

Definition 10.2.5. Let e ∈ O_{F} be the idempotent whose support is the set of finite subgroups containing K. Using the identifications \overline{O}\overline{F} ≅ eO_{F} and t\overline{F} ; ∗ ≅ et_{∗}^{F}, we define the functor

Φ^{K} : A ⟶ \overline{A}

by

Φ^{K}(N_{M} ⟶^{r_{M}} t_{∗}^{F} ⊗ V_{M}) := (eN_{M} ⟶^{er_{M}} et_{∗}^{F} ⊗ V_{M}),

or more briefly Φ^{K}M := eM .

We may now set about showing the algebraic functor has the desired properties.

Theorem 10.2.6. The algebraic functor Φ^{K} induces a functor

Φ^{K} : DA ⟶ D\overline{A}

so that the diagram

𝕋−Spec ⟶^{Φ^{K}} \overline{𝕋}−Spec
≃↓  ↓≃
DA ⟶^{Φ^{K}} D\overline{A}

commutes up to natural isomorphism.

Before we prove this, we should verify that the identifications

\overline{O}\overline{F} ≅ eO_{F} and t\overline{F} ; ∗ ≅ et_{∗}^{F}

correspond to suitable geometric statements. However, some care is necessary at this point: several naive expectations are false.

First, note that we may view a \overline{𝕋}-space Y as a 𝕋-space, and hence we may form the dual D_{𝕋}Y of Y as a 𝕋-spectrum; we may also form the dual \overline{𝕋}-spectrum D\overline{𝕋}Y and view it as a 𝕋-spectrum by building in representations. (The inflation functor functor building in representations is often written j_{∗} where j : U^{K} ⟶ U is the inclusion of universes, but in this section we shall follow the convention for suspension spectra and omit the notation j_{∗}, which will always be made clear from the context. When notation is required for emphasis we write infD\overline{𝕋}Y.) Furthermore, by regarding the \overline{𝕋}-map D\overline{𝕋}Y ∧ Y ⟶ S^{0} as a 𝕋-map, we obtain a comparison map

μ : D\overline{𝕋}Y ⟶ D_{𝕋}Y

of 𝕋-spectra. We warn that this is definitely not an equivalence for Y = E\overline{F}_{+}; the easiest way to see this is to observe that the 𝕋-equivariant homotopy groups of Ẽ[⊉ K] ∧ D\overline{𝕋}E\overline{F}_{+} are zero in positive degrees, whilst those of Ẽ[⊉ K] ∧ D_{𝕋}E\overline{F}_{+} are not. This means that

Φ^{K}DEF_{+} ≄ DE\overline{F}_{+}.

However, the following positive result is what we need.

Lemma 10.2.7. The map

μ : D\overline{𝕋}E\overline{F}_{+} ⟶ D_{𝕋}EF_{+}

is an F-equivalence.

Proof: Apply the forgetful functor to the construction of μ; since EF_{+} is H-equivariantly S^{0} for all finite subgroups H, the lemma follows from the space level construction of the Spanier-Whitehead dual of a finite \overline{𝕋}-complex.

We may now find the geometric basis for the algebraic definition of Φ^{K} on nubs.

Lemma 10.2.8. There is a natural equivalence eDEF_{+} ≃ D\overline{𝕋}(E\overline{F}_{+}) ∧ Ẽ[⊉ K], of 𝕋spectra and hence

X ∧ eD_{𝕋}EF_{+} ≃ Φ^{K}X ∧ D\overline{𝕋}E\overline{F}_{+} ∧ Ẽ[⊉ K],

and

eπ_{∗}^{𝕋}(X ∧ D_{𝕋}EF_{+}) ≅ π\overline{𝕋} ; ∗(Φ^{K}X ∧ D\overline{𝕋}E\overline{F}_{+}).

Proof: First observe that EF_{+} = eEF_{+} ∨ (1 − e)EF_{+}, and that (1 − e)EF_{+} ≃ E[⊉ K]_{+} whilst eEF_{+} ≃ EF_{+} ∧ Ẽ[⊉ K]. Thus E[⊉ K]_{+} ∧ eD_{𝕋}EF_{+} ≃ (1 − e)EF_{+} ∧ eD_{𝕋}EF_{+} ≃ ∗, so that eD_{𝕋}EF_{+} ≃ Ẽ[⊉ K] ∧ eD_{𝕋}EF_{+}.

We may therefore consider the natural map ν which is the composite

D\overline{𝕋}E\overline{F}_{+} ∧ Ẽ[⊉ K] ⟶^{μ∧1} D_{𝕋}EF_{+} ∧ Ẽ[⊉ K] ⟶ eD_{𝕋}EF_{+} ∧ Ẽ[⊉ K] ≃ eD_{𝕋}EF_{+}.

Since μ is an F-equivalence, it follows from the definition of e that ν is an F-equivalence.

For the rest, we want to show that the map Ẽ[⊉ K] ∧ D\overline{𝕋}E\overline{F}_{+} ⟶ eD_{𝕋}EF_{+} induces a bijection of [S^{0}, ·]_{∗}^{𝕋}. Now diagrams

S^{0}
↙  ↘
Ẽ[⊉ K] ∧ D\overline{𝕋}E\overline{F}_{+}  ⟶  eD_{𝕋}EF_{+}

correspond under the adjunction to diagrams

eEF_{+}
↙  ↘
Ẽ[⊉ K] ∧ D\overline{𝕋}E\overline{F}_{+} ∧ eEF_{+}  ⟶  S^{0}.

Finally,

Ẽ[⊉ K] ∧ D\overline{𝕋}E\overline{F}_{+} ∧ eEF_{+} ≃ Ẽ[⊉ K] ∧ eEF_{+} ≃ eEF_{+},

and the horizontal is induced by the collapse map EF_{+} ⟶ S^{0}. The cofibre of the horizontal thus becomes (1 − e)EF_{+} when smashed with EF_{+}, and hence the horizontal induces a bijection in [eEF_{+}, ·]_{∗}^{𝕋} as required.

Proof of 10.2.6: For the first statement, we need only observe that the above definition gives an exact functor on A; it then induces a functor on the category of dg A-objects, and preserves homology isomorphisms. The existence of the functor then follows from the universal property of a category of fractions.

To see the algebraic Φ^{K} is compatible with the topological one we apply the functor comparison theorem 7.5.1. Note first that Condition 1 holds. Indeed, it is obvious that both functors preserve cofibre sequences. It is also obvious that the algebraic functor preserves both parity and injectives; the analogous fact for the topological functor follows from the agreement of the functors on objects used to motivate the definition. This gives (2)(a), and finally we must check (2)(b), that if X and Y are each either F-contractible or injective F-spectra, then for any map f : X ⟶ Y we have Φ^{K}(π_{∗}^{A}(f)) = π\overline{A} ; ∗(Φ^{K}f).

If Y is F-contractible, we use the diagram

[X, Y]_{∗}^{𝕋}  ⟶^{≅}  Hom(π_{∗}(Φ^{𝕋}X), π_{∗}(Φ^{𝕋}Y))
Φ^{K} ↓≅  ↓=
[Φ^{K}X, Φ^{K}Y]\overline{𝕋} ; ∗ ⟶^{≅} Hom(π_{∗}(Φ\overline{𝕋}Φ^{K}X), π_{∗}(Φ\overline{𝕋}Φ^{K}Y)),

which commutes since Φ\overline{𝕋}Φ^{K} = Φ^{𝕋}. If Y is F-free and injective we use the diagram

[X, Y]_{∗}^{𝕋}  ⟶^{≅}  Hom(π_{∗}^{𝕋}(X ∧ DEF_{+}), π_{∗}^{𝕋}(Y))
Φ^{K} ↓  ↓ Φ^{K}
[Φ^{K}X, Φ^{K}Y]\overline{𝕋} ; ∗ ⟶^{≅} Hom(π_{∗}^{𝕋}(Φ^{K}X ∧ DE\overline{F}_{+}), π\overline{𝕋} ; ∗(Φ^{K}Y)).

We must explain why the diagram commutes. First note that, since Y is an F-spectrum e, is defined as a self-map of Y and eY ≃ Y ∧ Ẽ[⊉ K] so that [Φ^{K}X, Φ^{K}Y]\overline{𝕋} ; ∗ = [X, eY]_{∗}^{𝕋}. The diagram would therefore commute if the right hand vertical was replaced by Hom(π_{∗}^{𝕋}(X ∧ DEF_{+}), π_{∗}^{𝕋}(Y)) ⟶ Hom(π_{∗}^{𝕋}(X ∧ DEF_{+}), eπ_{∗}^{𝕋}(Y)). It therefore suffices to prove that multiplication by e on nubs corresponds to the geometric construction, which was 10.2.8 above.

It seems worth recording one consequence of the above discussion.

Lemma 10.2.9. If Y ⟶ I(Y) ⟶ J(Y) is an Adams resolution, then it remains so after taking fixed points.

CHAPTER 11 Algebraic inflation and deflation.

This chapter develops the algebra necessary for modelling the Lewis-May fixed point functor. Section 11.1 introduces the class of Hausdorff O_{F}-modules, which is large enough to cover the modules that concern us, and small enough to allow a suitable definition of inflation. In Section 11.2 we show the inflation and deflation functors have the requisite adjointness properties on the category of Hausdorff O_{F}-modules, and in Section 11.2 this is extended to the torsion model category.

11.1. Algebraic inflation and Hausdorff O_{F}-modules.

Given a quotient homomorphism G ⟶ G/N one obtains various change of group results, and we generically refer to functors from G/N-equivariant structure to G-equivariant structure as inflation, and to functors in the reverse direction as deflation. This section is devoted to a particular algebraic case of this which will be important in the analysis of Lewis-May fixed points. The reader principally interested in the topological applications may prefer to look at the motivation in Section 12.3 before reading further.

We suppose given a finite subgroup K of 𝕋 and follow the conventions of Section 10.2 on notation by using bars to indicate the ambient group is the quotient \overline{𝕋} = 𝕋/K. First, note that the quotient map q : 𝕋 ⟶ 𝕋/K = \overline{𝕋} induces a map of subgroups, and in particular a surjective map q_{∗} : F ⟶ \overline{F}. Since any subgroup mapping to a fixed subgroup \overline{H} of \overline{𝕋} lies inside q^{−1}(\overline{H}), it is clear that the fibres of q_{∗} are finite.

Accordingly, since O_{F} = C(F, ℚ[c]) the map q_{∗} induces a map

q^{∗} : \overline{O}\overline{F} ⟶ O_{F},

and since the fibres of q_{∗} are finite this extends to

q^{∗} : t\overline{F} ; ∗ ⟶ t_{∗}^{F}.

Explicitly, q^{∗} is the diagonal map on each factor in the sense that q^{∗}(e\overline{H}) = Σ_{\overline{L}=\overline{H}}e_{L}. Pullback along q^{∗} induces a deflation functor

def : O_{F} − mod ⟶ \overline{O}\overline{F} − mod.

If we think of an F-finite module M as corresponding to a sequence of modules e_{L}M, then the effect of deflation is to collect together all the summands e_{L}M with \overline{L} = \overline{H} and make this the summand e\overline{H}defM. After this section we shall often omit notation for deflation, since the underlying set is unchanged.

We also want to obtain an inflation functor

inf : \overline{O}\overline{F} − mod ⟶ O_{F} − mod,

in the reverse direction. On F-finite modules we would want to define inf(\overline{N}) to be the Ffinite module with e_{H}inf(\overline{N}) = e\overline{H} \overline{N}. Thus the inflation functor simply gives each summand e\overline{H} \overline{N} the multiplicity of the fibre of q_{∗}^{−1}(\overline{H}):

inf(\overline{N}) = ⨁_{H} e\overline{H} \overline{N},

so the word inflation is also suggestive of the construction. However, the construction is not given by pullback along a ring homomorphism, so it is not obvious how to extend it to arbitrary O_{F}-modules. For our purposes it is sufficient to define it on all modules occurring as nubs.

For any O_{F}-module M, we may form the module M̂ = ∏_{H} e_{H}M. We have chosen the notation since this can be viewed as a completion, and it is therefore natural to call a module M for which the natural map

M ⟶ M̂

is injective, a Hausdorff module. The convenient property of a Hausdorff module M is that maps into M are determined by their component maps into e_{H}M. An example of a non-Hausdorff module is the quotient ∏_{H} ℚ_{H}/ ⨁_{H} ℚ_{H}, where O_{F} acts on ℚ_{H} via the idempotent e_{H}. We shall define the inflation functor on all Hausdorff modules. Note that, by construction, the completion functor M ⟼ M̂ is exact.

Lemma 11.1.1. (i) All F-finite O_{F}-modules are Hausdorff. (ii) The ring O_{F} is Hausdorff as a module. (iii) Any submodule of a Hausdorff module is Hausdorff. (iv) The class of Hausdorff modules is closed under extensions, in the sense that if 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an exact sequence of O_{F}-modules with M′ and M′′ Hausdorff, then M is also Hausdorff.

(v) The class of Hausdorff modules is closed under arbitrary direct sums or direct products.

Proof: Parts (i) and (ii) are clear. Parts (iii) and (iv) follow from the Snake Lemma by exactness of completion. In Part (v), closure under direct products is obvious. For direct sums the inclusion

λ : ⨁_{i} M_{i} ⟶ ∏_{i} ∏_{H} e_{H}M_{i}

is injective since it is the composite of the map

⨁_{i} M_{i} ⟶ ⨁_{i} ∏_{H} e_{H}M_{i},

injective by hypothesis, and the inclusion of the sum in the product. However, λ also factors through completion, which is therefore also injective.

11.1. ALGEBRAIC INFLATION AND HAUSDORFF O_{F}-MODULES. 127

Corollary 11.1.2. The modules N and t_{∗}^{F} ⊗ V occurring in any object N ⟶ t_{∗}^{F} ⊗ V of the standard abelian category A are both Hausdorff.

Proof: Since N is an extension of an F-finite module by a submodule of t_{∗}^{F} ⊗ V it is sufficient, by Parts (iii) and (iv) of the lemma, to observe that t_{∗}^{F} ⊗ V is Hausdorff. The fact that t_{∗}^{F} is Hausdorff follows from Part (iv), since t_{∗}^{F} is the extension of O_{F} over the F-finite module Σ^{2}𝕀, and the fact that t_{∗}^{F} ⊗ V is Hausdorff now follows from Part (v).

We are now equipped to give a sufficiently general definition of inflation.

Definition 11.1.3. If \overline{N} is a Hausdorff \overline{O}\overline{F}-module with completion ˆ \overline{N} we proceed as follows. On the complete module ˆ \overline{N} we define inflation by inf ˆ\overline{N} = ∏_{H} e\overline{H} \overline{N} in the obvious way. Now define inf\overline{N} to be the set of sequences (n_{H}) ∈ inf ˆ\overline{N} so that for all sections s : \overline{F} ⟶ F of q_{∗} the sequence (n_{s(\overline{H})}) is an element of \overline{N}.

Note that each section s defines a ring homomorphism s^{∗} : O_{F} ⟶ \overline{O}\overline{F}, and hence we may view \overline{N} and ˆ \overline{N} as modules over O_{F}. Furthermore s induces a projection inf ˆ\overline{N} ⟶ ˆ \overline{N}, and this is a map of O_{F}-modules when ˆ \overline{N} is a module by pullback along s^{∗}. Taking the product over all sections we see that the definition states that there is an exact sequence

0 ⟶ inf\overline{N} ⟶ inf ˆ\overline{N} ⟶ ∏_{s} ˆ \overline{N}/\overline{N},

or equivalently that inf\overline{N} is a pullback

inf\overline{N} = inf ˆ\overline{N} ×_{∏_{s} ˆ \overline{N}} ∏_{s} \overline{N}.

This establishes that inf\overline{N} is indeed a O_{F}-module.

Lemma 11.1.4. The above definition extends to a left exact functor

inf : \overline{O}\overline{F}^{Haus} − mod ⟶ O_{F}^{Haus} − mod.

Proof: The observation just made ensures that inf is a functor on Hausdorff modules. Evidently inf\overline{N} ⟶ inf ˆ\overline{N} is completion, so that inf\overline{N} is Hausdorff.

The functor is certainly exact on complete modules. Given a short exact sequence 0 ⟶ \overline{N}′ ⟶ \overline{N} ⟶ \overline{N}′′ ⟶ 0 of Hausdorff modules, the sequence of quotients 0 ⟶ ˆ\overline{N}′/\overline{N}′ ⟶ ˆ \overline{N}/\overline{N} ⟶ ˆ \overline{N}′′/\overline{N}′′ ⟶ 0 is also exact by the Snake Lemma. Left exactness also follows from the Snake Lemma, using the exact sequence formulation of the definition.

11.2. Algebraic inflation and deflation of O_{F}-modules.

We now establish that inflation, as defined in the previous section, has the properties we require: it behaves well on model categories, and has good adjointness properties.

Lemma 11.2.1. (i) For any graded vector space V we have inf(t\overline{F} ; ∗ ⊗ V) = t_{∗}^{F} ⊗ V. (ii) For any \overline{F}-finite module \overline{N} we have inf(\overline{N}) = ⨁_{H} e\overline{H} \overline{N}.

Note in particular that this means inflation gives an exact functor on the torsion model category A_{t}. More surprising is that it is also exact on modules occurring in the standard model (i.e. on modules isomorphic modulo Euler-torsion to those of the form t_{∗}^{F} ⊗ V).

Lemma 11.2.2. Inflation is exact on modules occurring as nubs, and hence induces an exact functor \overline{A} ⟶ A.

Proof: Before beginning, we warn that the category of Hausdorff modules is not abelian. However the category of modules occurring as nubs is abelian, and furthermore, F-finite injective modules and modules of the form t_{∗}^{F} ⊗ V are injective in this category. We may therefore use right derived functors in the course of the proof.

If we have an object \overline{β} : \overline{N} ⟶ t\overline{F} ; ∗ ⊗V of \overline{A}, there are short exact sequences 0 ⟶ \overline{K} ⟶ \overline{N} ⟶ \overline{I} ⟶ 0 and 0 ⟶ \overline{I} ⟶ t\overline{F} ; ∗ ⊗ V ⟶ \overline{C} ⟶ 0 as usual, where the \overline{K} and \overline{C} are F-finite torsion modules and \overline{I} is the image of \overline{β}. Since inflation is exact on the category of F-finite modules we find R^{1}inf\overline{K} = 0, and so it suffices, by the first short exact sequence, to show that R^{1}inf\overline{I} = 0. From the second exact sequence and exactness on modules t\overline{F} ; ∗ ⊗V we see that it is enough to check surjectivity of inf t\overline{F} ; ∗ ⊗ V ⟶ inf\overline{C}. Since inf\overline{C} is F-finite, this is true provided each idempotent summand e_{H}inf t\overline{F} ; ∗ ⊗ V ⟶ inf\overline{C} is surjective. But this map is e\overline{H}t\overline{F} ; ∗ ⊗ V ⟶ e\overline{H} \overline{C}, which is surjective by hypothesis.

The following adjunction is the algebraic core of the section.

Proposition 11.2.3. The functors inf and def are both left and right adjoint to each other when restricted to the subcategories of Hausdorff modules. Thus if M is a Hausdorff O_{F}-module and \overline{N} is a Hausdorff \overline{O}\overline{F}-module we have isomorphisms

Hom(defM, \overline{N}) = Hom(M, inf\overline{N}) and Hom(inf\overline{N}, M) = Hom(\overline{N}, defM).

Proof: We shall give the unit and counit of each adjunction and leave the reader to verify the triangular identities. We begin by considering the F-finite case, for which we may restrict attention to a single subgroup \overline{H} of \overline{𝕋}, and to the finite set of subgroups L with \overline{L} = \overline{H}.

Letting R = ℚ[c], and supposing there are k subgroups L with \overline{L} = \overline{H}, we are thus concerned with R-modules \overline{N} and R^{k}-modules M and the diagonal homomorphism R ⟶ R^{k}. It is probably clearest to think of M as a k-tuple (M_{1}, M_{2}, …, M_{k}) of R-modules. The inflation map is thus pullback along the diagonal ring homomorphism q^{∗} : R ⟶ R^{k} with

11.2. ALGEBRAIC INFLATION AND DEFLATION OF O_{F}-MODULES. 129

associated map def(M_{1}, …, M_{k}) = M_{1}⊕⋯⊕M_{k}. On the other hand inf(\overline{N}) = (\overline{N}, …, \overline{N}). The unit η : M ⟶ inf ∘ defM of the first adjunction is the map

(M_{1}, …, M_{k}) ⟶ (M_{1} ⊕ ⋯ ⊕ M_{k}, …, M_{1} ⊕ ⋯ ⊕ M_{k})

given in the ith term M_{i} ⟶ M_{1} ⊕ ⋯ ⊕ M_{k} by the canonical injection. The counit ϵ : def ∘ inf\overline{N} ⟶ \overline{N} of the first adjunction is the folding map

\overline{N} ⊕ … ⊕ \overline{N} ⟶ \overline{N}.

The unit η : \overline{N} ⟶ def ∘ inf\overline{N} of the second adjunction is the diagonal map

\overline{N} ⟶ \overline{N} ⊕ ⋯ ⊕ \overline{N}.

The counit ϵ : inf ∘ defM ⟶ M of the second adjunction is the map

(M_{1} ⊕ ⋯ ⊕ M_{k}, …, M_{1} ⊕ ⋯ ⊕ M_{k}) ⟶ (M_{1}, …, M_{k})

given in the ith coordinate by the projection M_{1} ⊕ ⋯ ⊕ M_{k} ⟶ M_{i}.

The above descriptions of units and counits all extend in an obvious way to complete modules, by taking products over the various subgroups \overline{H}. This makes it obvious that the triangular identities continue to hold in this context. It remains to consider general Hausdorff modules, and we simply need to verify that if an element lies in the relevant uncompleted module then so does its image. We use here the fact that completion commutes with inflation by definition, and with deflation since the fibres of q_{∗} are finite.

To use the above work it is convenient to enumerate the subgroups of \overline{𝕋} and arrange them in a row, say \overline{H}^{1}, \overline{H}^{2}, . . .. Below each subgroup \overline{H}^{i} we enumerate the subgroups L with \overline{L} = \overline{H}, say as L_{1}^{i}, L_{2}^{i}, …, L_{k(i)}^{i}. Thus we have an infinite number of columns with a finite and bounded number of rows in each column. Now, for a \overline{O}\overline{F}-module \overline{N}, we may consider its idempotent summands \overline{N}^{1}, \overline{N}^{2}, … arranged in a row, and for an O_{F}-module M we consider its idempotent summands M_{j}^{i} as arranged in an array patterned after the subgroups. The above discussion was essentially the case of a single column. Now for a Hausdorff module \overline{N} the module inf\overline{N} has components (inf\overline{N})_{j}^{i} = \overline{N}^{i} independent of j. A section s is simply a function selecting a row s(i) for each column i, and the condition that an element (x_{j}^{i}) ∈ inf ˆ\overline{N} lie in inf\overline{N} is that for each section s the element (x_{s(i)}^{i}) ∈ ˆ \overline{N} actually lies in \overline{N}.

We may now verify that the units and counits restrict to give maps between uncompleted modules. For the unit of the first we suppose (x_{j}^{i}) ∈ M; then η(x)_{j}^{i} = x_{j}^{i} ∈ M_{1}^{i} ⊕⋯⊕M_{k(i)}^{i}. But for any s we find (η(x)_{s(i)}^{i}) = (x_{s(1)}^{1}, x_{s(2)}^{2}, …) does lie in defM since it is e_{s}x, where e_{s} ∈ O_{F} is the idempotent with support the image of s.

For the counit of the first adjunction we must be a little more careful. We choose y ∈ def ∘ inf\overline{N}, which has ith component y^{i} = (y_{j}^{i}) ∈ \overline{N}^{i} ⊕ ⋯ ⊕ \overline{N}^{i}. Then we have ϵ(y)^{i} = y_{1}^{i} + ⋯ + y_{k(i)}^{i}. Suppose that the maximum number of rows in a column is k; we shall express ϵ(y) as a sum of k elements known to be in \overline{N}. Indeed, we divide the columns into sets C^{1}, C^{2}, …, C^{k} where C^{j} is the set of columns with exactly j rows. Now choose sections s_{1}^{j}, … s_{j}^{j} so that they exhaust the j rows in each column from C^{j}, and let

y(j) = e_{C^{j}}((y_{s_{1}^{j}(i)}^{i}) + (y_{s_{2}^{j}(i)}^{i}) + ⋯ + (y_{s_{j}^{j}(i)}^{i}))

The idempotent e_{C^{j}} lies in \overline{O}\overline{F}, and the j terms lie in \overline{N} since y lies in def ∘ inf\overline{N}. Now ϵ(y) = y(1) + ⋯ + y(k).

It is trivial to verify that the unit of the second adjunction preserves the uncompleted submodule, since if y ∈ \overline{N} then η(y)^{i} = (y^{i}, …, y^{i}) ∈ \overline{N} ⊕ ⋯ ⊕ \overline{N}. However the counit is again quite tricky. Suppose that x ∈ inf ∘ defM. Then ϵ(x)_{j}^{i} = π_{j}(x_{j}^{i}) where x_{j}^{i} ∈ (inf ∘ defM)_{j}^{i} = M_{1}^{i} ⊕ ⋯ ⊕ M_{k(i)}^{i} and π_{j} is projection onto the jth factor. In effect, for each subgroup \overline{H}^{i} we have a k(i) × k(i) square of entries. The hypothesis is that however we select a row from each such square we obtain an element (x_{s(i)}^{i}) ∈ defM, and we need to know that if we select the diagonal elements of each square we obtain an element of M. However if we let e_{1}, e_{2}, …, e_{k} be the idempotents defined by requiring e_{i} is supported on the ith element along each diagonal and if we let j denote the section which selects the jth row whenever it exists and the first row otherwise, then

ϵ(x) = e_{1}(x_{1}^{i}) + ⋯ + e_{k}(x_{k}^{i}),

and for each fixed j, (x_{j}^{i}) ∈ defM by hypothesis.

11.3. Inflation and its right adjoint on the torsion model category.

The use we make of the results of the previous section is in constructing functors to correspond to the inflation functor \overline{𝕋} − Spec ⟶ 𝕋 − Spec building in representations, and its right adjoint. This right adjoint is the Lewis-May fixed point functor [18, II.4.4]. It is difficult to identify the Lewis-May fixed point functor directly, since it does not satisfy all the hypotheses of Theorem 7.5.1; instead we shall identify the algebraic inflation functor, which does satisfy the relevant conditions, and then identify the Lewis-May fixed point functor as its adjoint. The second curious feature is that it seems extremely difficult to define the Lewis-May fixed point functor using the standard abelian models A and \overline{A}. Instead we define it using the torsion models A_{t} and \overline{A}_{t}; this means that our control over its behaviour on morphisms in DA is somewhat indirect. Because we are forced to approach the Lewis-May fixed point functor through the torsion model, we do the same for inflation.

The definition of the inflation functor

inf : \overline{A}_{t} ⟶ A_{t}

is the obvious termwise extension of the functor on Hausdorff modules:

inf(t_{∗}^{F} ⊗ V ⟶ T) = (t_{∗}^{F} ⊗ V = inf(t\overline{F} ; ∗ ⊗ V) ⟶ infT).

We have seen in 11.2.2 this is an exact functor.

We now define its right adjoint,

Ψ^{K} : A_{t} ⟶ \overline{A}_{t},

11.3. INFLATION AND ITS RIGHT ADJOINT ON THE TORSION MODEL CATEGORY. 131

which we call the algebraic Lewis-May fixed point functor.

Suppose that M = (t_{∗}^{F} ⊗ V ⟶ T) we take

Ψ^{K}M = (t\overline{F} ; ∗ ⊗ V ⟶^{q^{∗}⊗1} t_{∗}^{F} ⊗ V ⟶ T)

where t_{∗}^{F} and T are regarded as \overline{O}\overline{F}-modules by pullback along q^{∗}. This definition extends in an obvious way to morphisms, and is visibly exact. The reason for naming this after the Lewis-May fixed point functor will appear in Section 12.1 below, although the following proposition may seem sufficient explanation in view of [18, II.4.4].

Proposition 11.3.1. The algebraic Lewis-May fixed point functor

Ψ^{K} : A_{t} ⟶ \overline{A}_{t}

is right adjoint to the inflation functor

inf : \overline{A}_{t} ⟶ A.

Thus we have natural isomorphisms

Hom(inf\overline{N}, M) = Hom(\overline{N}, Ψ^{K}M)

for any object \overline{N} of \overline{A}_{t} and any object M of A_{t}.

Proof: As usual it suffices to construct the unit

η′ : \overline{N} ⟶ Ψ^{K}inf\overline{N}

and the counit

ϵ′ : infΨ^{K}M ⟶ M

and verify the triangular identities. Before we begin, notice that the map q^{∗} : t\overline{F} ; ∗ ⟶ t_{∗}^{F} = def ∘ inft\overline{F} ; ∗ is the unit η of the second adjunction of 11.2.3.

For the unit, suppose \overline{N} = (t\overline{F} ; ∗ ⊗ V ⟶^{\overline{s}} \overline{T}) and define η′ by the diagram

t\overline{F} ; ∗ ⊗ V ⟶^{1} t\overline{F} ; ∗ ⊗ V
|  ↓ q^{∗} ⊗ 1
\overline{s} | t_{∗}^{F} ⊗ V
↓  ↓ inf(\overline{s})
\overline{T}  ⟶^{η} def ∘ inf(\overline{T})

where the lower horizontal is the unit of the adjunction of 11.2.3. The commutativity of the square states η ∘\overline{s} = inf(\overline{s})∘(q^{∗} ⊗1), which follows from the triangular identity of 11.2.3 together with naturality of η. For the counit suppose M = (t_{∗}^{F} ⊗ V ⟶^{s} T) and define ϵ′ by the diagram

t_{∗}^{F} ⊗ V  ⟶^{1} t_{∗}^{F} ⊗ V
inf(q^{∗}) ⊗ 1  ↓  |
inf ∘ def(t_{∗}^{F}) ⊗ V  | s
↓  ↓
inf ∘ def(T) ⟶^{ϵ} T

where the lower horizontal is the counit of the adjunction of 11.2.3. The commutativity of the square states s = inf(s) ∘ (inf(q^{∗}) ⊗ 1), which follows from the triangular identity of 11.2.3 together with the naturality of ϵ. The triangular identities are immediate from those of 11.2.3.

We remark that we cannot use a similar construction for a left adjoint of inf because of the role of q^{∗}. We would need to know that the unit t_{∗}^{F} ⟶ inf ∘ def t_{∗}^{F} of the first adjunction of 11.2.3 was the inflation of some map t\overline{F} ; ∗ ⟶ def t_{∗}^{F}, which is not the case. In fact, inf does not have a left adjoint, since it does not preserve arbitrary products. We can substantiate this when we have studied products a little more, but the claim is the algebraic counterpart of the fact that the natural map

inf(∏_{n} Σ^{−n}E\overline{𝕋}_{+}) ⟶ ∏_{n}(inf Σ^{−n}E\overline{𝕋}_{+})

is not an equivalence, where n runs through the natural numbers. It will be easier to verify this later. The convenient fact is that π_{∗}^{𝕋}(Y) = π\overline{𝕋} ; ∗(Y ∧ Ψ^{K}S^{0}) and, since the Lewis-May fixed point spectrum Ψ^{K}S^{0} is a wedge of S^{0} and terms including E\overline{𝕋}_{+}, we need to consider the comparison map

E\overline{𝕋}_{+} ∧ ∏_{n} Σ^{−n}E\overline{𝕋}_{+} ⟶ ∏_{n} E\overline{𝕋}_{+} ∧ Σ^{−n}E\overline{𝕋}_{+}.

This is not an equivalence since its fibre is

Ẽ\overline{𝕋} ∧ ∏_{n} Σ^{−n}E\overline{𝕋}_{+}.

The homotopy groups of this fibre are obtained from ∏_{n} Σ^{1−n}\overline{𝕀} by inverting c_{1}, and are thus nonzero.

However we shall see later that inflation does have a left adjoint on suitable subcategories.

We end this section with a discussion of the induced functors on derived categories. The point is that both inf : \overline{A}_{t} ⟶ A_{t} and Ψ^{K} : A_{t} ⟶ \overline{A}_{t} are exact functors, and hence the functors on dg categories that they induce preserve homology isomorphisms. They therefore induce functors inf : D\overline{A}_{t} ⟶ DA_{t} and def : DA_{t} ⟶ D\overline{A}_{t} on derived categories in the usual way. It is reasonable to use the notation inf and def, as we have done, because of the exactness of the functors on abelian categories. However, for certain purposes we also want to view these as functors on derived categories of the standard model.

Now the Lewis-May fixed point functor is not defined in an obvious way on A, and so it is not misleading or ambiguous to continue to use the same notation for the functor DA ⟶ D\overline{A}. But notice that if M is an object of A, then Ψ^{K}M is calculated by taking an injective approximation, moving it into dgHA_{t}, moving into dgA_{t}, applying inf and moving back to dgA_{t}. We shall make the end result of this process more explicit on objects in Section 12.3.

However inflation is also defined as a functor \overline{A} ⟶ A by applying inf termwise, and it is still exact by 11.2.2. It therefore induces a functor D\overline{A} ⟶ DA on derived categories. It will be valuable to know that this corresponds with the original functor under the equivalence between standard and torsion model categories; this is obvious by the fibre construction of the equivalences fib : dgA_{t} ⟶^{≃} dgA and fib : dg\overline{A}_{t} ⟶^{≃} dg\overline{A}.

Lemma 11.3.2. The diagram

D\overline{A} ⟶^{inf} DA
≃↓  ↓≃
D\overline{A}_{t} ⟶^{inf} DA_{t}

commutes up to natural isomorphism.

134 11. ALGEBRAIC INFLATION AND DEFLATION.

CHAPTER 12 Inflation, Lewis-May fixed points and quotients.

We begin in Section 12.1 by recording the formal properties of the inflation and Lewis-May fixed point functors in topology, and deducing the facts we need later in the chapter. There are then two natural routes through the chapter: to simply obtain a model for the Lewis-May fixed point functor, continue with Section 12.2. There we apply the Functor Comparison Theorem 7.5.1 to deduce that the algebraic inflation functor models the topological one, and conclude that their right adjoints, the Lewis-May fixed point functors, also correspond. However this route is rather indirect, and some readers will prefer a more concrete approach. We therefore give a more direct analysis of the Lewis-May fixed point functor in Section 12.3, by considering the F-free and F-contractible pieces separately. In the same spirit, Section 12.4 makes the functor explicit in the standard model: this is very much based at the derived category level, and does not approach morphisms at all. However, it does illustrate the behaviour of the functor, and it gives a splitting theorem.

The chapter ends with the very short Section 12.5, drawing attention to the fact that we have also modelled the quotient on sufficiently free spectra, since it is left adjoint to inflation.

12.1. The topological inflation and Lewis-May fixed point functors.

We continue to use the notation introduced in Section 10.2, with K being a finite subgroup of 𝕋, and q : 𝕋 ⟶ \overline{𝕋} the quotient map. We have explained in Section 7.3 that the inflation functor inf = inf𝕋 ; \overline{𝕋} : \overline{𝕋}−Spec ⟶ 𝕋−Spec regarding a \overline{𝕋}-spectrum as 𝕋-spectrum is the composite of the pull-back along the quotient q : 𝕋 ⟶ \overline{𝕋} and the functor building in representations which are not K-fixed.

The purpose of this section is to provide the various facts we shall need about the topological inflation and Lewis-May fixed point functors. This will give us enough information to apply the Functor Comparison Theorem 7.5.1 to the algebraic and geometric inflation functors. Although we are first concerned with inflation, there are various points where it seems convenient to use properties of Lewis-May fixed points, so we discuss the two functors together. Because of its important place in the discussion, we include notation for the inflation functor throughout this section.

The fundamental fact [18, II.4.4] is that for any \overline{𝕋}-spectrum Y, there is an adjunction

[inf Y, X]_{∗}^{𝕋} = [Y, Ψ^{K}X]\overline{𝕋} ; ∗.

The crudest implication, which we shall often use, is that, since S^{0} is an inflation, π_{∗}^{𝕋}(X) = π\overline{𝕋} ; ∗(Ψ^{K}X). We shall also need to understand the fixed point functor on an inflated spectrum. Thus if we take X = inf Y, the 𝕋-equivariant identity map on the left corresponds to the unit η : Y ⟶ Ψ^{K}inf Y of the adjunction. Accordingly for any 𝕋-spectrum Y we may consider the composite

Ψ^{K}(X) ∧ Y ⟶ Ψ^{K}(X) ∧ Ψ^{K}(inf Y) ⟶ Ψ^{K}(X ∧ inf Y).

The following curious lemma is the key to the analysis.

Lemma 12.1.1. If Y is a \overline{𝕋}-spectrum then the natural map

Ψ^{K}(X) ∧ Y ⟶ Ψ^{K}(X ∧ inf Y)

is an equivalence of \overline{𝕋}-spectra.

Proof: We have a natural map which commutes with direct limits and cofibre sequences in the variable Y, so it is enough to verify it is an equivalence when Y is a cell, \overline{𝕋}/\overline{H}_{+} for some subgroup \overline{H} ⊆ \overline{𝕋}. However, the lemma for any finite \overline{𝕋}-complex Y is clear by playing with adjunctions, since the \overline{𝕋}-dual of Y regarded as a 𝕋-spectrum is the 𝕋-dual.

The particular case X = S^{0} is often useful:

Corollary 12.1.2. For any \overline{𝕋}-spectrum Y we have a natural equivalence

Ψ^{K}(S^{0}) ∧ Y ⟶^{≃} Ψ^{K}inf Y,

where Ψ^{K}S^{0} denotes the Lewis-May fixed point spectrum.

This becomes most useful in conjunction with tom Dieck’s calculation of Ψ^{K}S^{0}, which we recall in Example 12.4.6 below.

Now we turn to the behaviour of inflation, and begin by considering how basic cells behave. Recall that there are only finitely many subgroups L with \overline{L} = \overline{H}.

Lemma 12.1.3. If we inflate the basic \overline{𝕋}-cell σ0 ; \overline{H} to a 𝕋-spectrum we find an equivalence

infσ0 ; \overline{H} ≃ ⋁_{\overline{L}=\overline{H}} σ_{L}^{0}

of 𝕋-spectra.

Proof: Let H = q^{−1}(\overline{H}) and consider the quotient map q : H ⟶ \overline{H}. This induces a map q^{∗} : A(\overline{H}) ⟶ A(H), and for any x ∈ A(\overline{H}), whenever L ⊆ H, we have ϕ_{L}(q^{∗}(x)) = ϕ\overline{L}(x). Thus in particular

q^{∗}(e\overline{H}) = Σ_{\overline{L}=\overline{H}}e_{L}.

The lemma follows.

12.1. THE TOPOLOGICAL INFLATION AND LEWIS-MAY FIXED POINT FUNCTORS. 137

Since the the \overline{𝕋}-space E\overline{F}_{+} is EF_{+} when regarded as a 𝕋-space (i.e. inf(E\overline{F}_{+}) ≃ EF_{+}), we immediately deduce the required consequence.

Corollary 12.1.4. If E⟨\overline{H}⟩ is regarded as a 𝕋-spectrum we have an equivalence

E⟨\overline{H}⟩ ≃ ⋁_{\overline{L}=\overline{H}} E⟨L⟩

of 𝕋-spectra.

It follows from this and the calculation of the self maps of E⟨H⟩, that the inflation map

ℚ[c\overline{H}] = [E⟨\overline{H}⟩, E⟨\overline{H}⟩]\overline{𝕋} ; ∗ ⟶ [E⟨\overline{H}⟩, E⟨\overline{H}⟩]_{∗}^{𝕋} = ∏_{\overline{L}=\overline{H}} ℚ[c_{L}]

is the diagonal inclusion c\overline{H} ⟼ (c_{L})_{L}.

Corollary 12.1.5. The inflation functor

\overline{O}\overline{F} = [E\overline{F}_{+}, E\overline{F}_{+}]\overline{𝕋} ; ∗ ⟶ [EF_{+}, EF_{+}]_{∗}^{𝕋} = O_{F}

induces the ring homomorphism q^{∗}.

Since inf is exact on F-finite modules we deduce the following.

Corollary 12.1.6. For an \overline{F}-spectrum X

π_{∗}^{𝕋}(infX) = inf π\overline{𝕋} ; ∗(X).

Proof: We apply inflation to an Adams resolution of X. Now use the fact that 12.1.4 also identifies inflations of maps between F-injectives with their algebraic counterparts. We will give a detailed proof of a generalization of this immediately below.

We would like to make the analogous statement for arbitrary spectra. The remaining obstacle is the identification of maps between injectives. The key to understanding this is a calculation.

First note that we have a projection map infE⟨\overline{H}⟩ ⟶ E⟨H⟩, with adjoint E⟨\overline{H}⟩ ⟶ Ψ^{K}E⟨H⟩. Similarly the injection map E⟨H⟩ ⟶ infE⟨\overline{H}⟩ can be used to form

E⟨H⟩ ∧ infD\overline{𝕋}E⟨\overline{H}⟩ ⟶ infE⟨\overline{H}⟩ ∧ infD\overline{𝕋}E⟨\overline{H}⟩ ≃ inf(E⟨\overline{H}⟩ ∧ D\overline{𝕋}E⟨\overline{H}⟩) ⟶ S^{0}

with adjoint infD\overline{𝕋}E⟨\overline{H}⟩ ⟶ D_{𝕋}E⟨H⟩; the adjoint of this is a map D\overline{𝕋}E⟨\overline{H}⟩ ⟶ Ψ^{K}(D_{𝕋}E⟨H⟩).

Lemma 12.1.7. The natural maps described above give equivalences of \overline{𝕋}-spectra

E⟨\overline{H}⟩ ⟶^{≃} Ψ^{K}E⟨H⟩ and DE⟨\overline{H}⟩ ⟶^{≃} Ψ^{K}(DE⟨H⟩)

Proof: Both facts follows from 10.2.2. The second begins with the calculation

[A, Ψ^{K}(D_{𝕋}E⟨H⟩)]\overline{𝕋} = [infA, D_{𝕋}E⟨H⟩]^{𝕋} = [infT ∧ E⟨H⟩, S^{0}]^{𝕋} = [infT ∧ E⟨H⟩, E⟨H⟩]^{𝕋},

and continues by saying this is the same as

[T, D\overline{𝕋}E⟨\overline{H}⟩]\overline{𝕋} = [T ∧ E⟨\overline{H}⟩, S^{0}]\overline{𝕋} = [T ∧ E⟨\overline{H}⟩, E⟨\overline{H}⟩]\overline{𝕋}.

Under passage to geometric fixed points this is an isomorphism by 10.2.2. Since E⟨H⟩ is a space over K, geometric fixed points coincides with Lewis-May fixed points. One must now check that this map coincides with that induced by the map described above. The first equivalence is slightly easier.

Corollary 12.1.8. Under the identification of ℚ[c_{H}] and ℚ[c\overline{H}], for any map f : X ⟶ Y of \overline{𝕋}-spectra with Y an \overline{F}-spectrum the map

infX ∧ D_{𝕋}E⟨H⟩ ⟶ infY ∧ D_{𝕋}E⟨H⟩ ≃ infY ∧ E⟨H⟩

induces the same map in π_{∗}^{𝕋} as

X ∧ D\overline{𝕋}E⟨\overline{H}⟩ ⟶ Y ∧ D\overline{𝕋}E⟨\overline{H}⟩ ≃ Y ∧ E⟨\overline{H}⟩

in π\overline{𝕋} ; ∗.

Proof: We use the fact that π_{∗}^{𝕋}(Z) = π\overline{𝕋} ; ∗(Ψ^{K}Z) and Ψ^{K}(inf Y ∧ Z) ≃ Y ∧ Ψ^{K}Z. We now obtain a diagram

X ∧ Ψ^{K}(D_{𝕋}E⟨H⟩) ⟶ Y ∧ Ψ^{K}(E⟨H⟩) ∧ Ψ^{K}(D_{𝕋}E⟨H⟩) ⟶ Y ∧ Ψ^{K}(E⟨H⟩)
↑  ↑  ↑
X ∧ D\overline{𝕋}E⟨\overline{H}⟩ ⟶  Y ∧ E⟨\overline{H}⟩ ∧ D\overline{𝕋}E⟨\overline{H}⟩  ⟶ Y ∧ E⟨\overline{H}⟩

in which the first and last verticals are equivalences by 12.1.7 above. The commutativity of the left square is naturality of the equivalence Ψ^{K}(inf Y ∧ Z) ≃ Y ∧ Ψ^{K}Z. For the right hand square, we first explain that the upper horizontal is the composite of

Ψ^{K}E⟨H⟩ ∧ Ψ^{K}(D_{𝕋}E⟨H⟩) ⟶ Ψ^{K}(E⟨H⟩ ∧ D_{𝕋}E⟨H⟩)

and the K fixed points of evaluation to S^{0}, which lifts uniquely to E⟨H⟩. Commutativity now follows, since the square with Y omitted is the adjoint of the diagram

inf Ψ^{K}(E⟨H⟩) ∧ Ψ^{K}(D_{𝕋}E⟨H⟩) ⟶ E⟨H⟩ ∧ D_{𝕋}E⟨H⟩ ⟶ E⟨H⟩
↑  ↑
infE⟨\overline{H}⟩ ∧ D_{𝕋}E⟨H⟩  −  −  ⟶ infE⟨\overline{H}⟩,

whose commutativity is clear from the definitions.

12.2. CORRESPONDENCE OF ALGEBRAIC AND GEOMETRIC INFLATION MAPS. 139

Proposition 12.1.9. For an arbitrary \overline{𝕋}-spectrum X,

π_{∗}^{𝕋}(infX ∧ D_{𝕋}EF_{+}) = infπ\overline{𝕋} ; ∗(X ∧ D\overline{𝕋}E\overline{F}_{+}).

Hence also we have

π_{∗}^{A}(infX) = infπ\overline{A} ; ∗(X).

Proof: We have already seen this is true if X is injective. The result follows once we know that topological inflation corresponds to algebraic inflation for maps between two injective spectra X. This is clear if both copies of X are F-spectra, so the remaining case is covered by considering a map

f : Ẽ\overline{F} ⟶ Σ^{n}E\overline{F}_{+}.

Since we are considering maps into a Hausdorff module it suffices to consider its idempotent parts. In other words we need to show that the map

ẼF ∧ D_{𝕋}E⟨H⟩ ≃ Ẽ\overline{F} ∧ e_{H}D_{𝕋}EF_{+} ⟶ Σ^{n}E\overline{F}_{+} ∧ e_{H}D_{𝕋}EF_{+} ≃ Σ^{n}E⟨H⟩

induces the same as

Ẽ\overline{F} ∧ D\overline{𝕋}E⟨\overline{H}⟩ ≃ Ẽ\overline{F} ∧ e\overline{H}D\overline{𝕋}E\overline{F}_{+} ⟶ Σ^{n}E\overline{F}_{+} ∧ e\overline{H}D\overline{𝕋}E\overline{F}_{+} ≃ Σ^{n}E⟨\overline{H}⟩.

This is a special case of 12.1.8.

We could at this point give a direct approach to the Lewis-May fixed point functor on objects. To emphasize the logical structure of the argument, we defer this to Section 12.3. However, the reader is recommended to look at this by way of motivation for the definition of the algebraic Lewis-May fixed point functor.

12.2. Correspondence of Algebraic and geometric inflation maps.

Theorem 12.2.1. The algebraic inflation functor induces a functor

inf : DA_{t} ⟶ D\overline{A}_{t}

so that the diagram

\overline{𝕋}−Spec ⟶^{inf} 𝕋−Spec
≃↓  ↓≃
D\overline{A}_{t}  ⟶^{inf} DA_{t}

commutes.

Proof: The existence of the functor on derived categories is immediate from the fact the algebraic functor \overline{A}_{t} ⟶ A_{t} of Section 11.2 is exact.

We want to apply the Functor Comparison Theorem 7.5.1 to the two functors \overline{𝕋}−Spec ⟶ DA. We are therefore implicitly using the equivalence DA_{t} ⟶^{≃} DA to identify DA_{t} and DA.

We work through Condition 1, starting with the topological functor \overline{𝕋}−Spec ⟶ 𝕋−Spec ⟶ DA. It certainly preserves triangles. Enough topological injectives are obtained from wedges of suspensions of Ẽ\overline{F} and the various spectra E⟨\overline{H}⟩. The inflation of Ẽ\overline{F} is ẼF and, by 12.1.4, the inflation of E⟨\overline{H}⟩ is ⋁_{L} E⟨L⟩ where the wedge is over the subgroups L with \overline{L} = \overline{H}. For the algebraic functor the analogous facts were built into the definition. To see that objects of pure parity are preserved is immediate from exactness of inflation as manifested in 11.3.2.

The fact that the topological functor preserves objects of pure parity is immediate from 12.1.9.

We now turn to Condition 2 of 7.5.1. The definition of the algebraic inflation functor was designed to agree with the topological one on enough injectives, as one sees from 12.1.9. Finally it remains to prove that they agree for maps into injectives, and we recall that 11.2.2 showed the algebraic inflation functor on the standard model was also termwise application of inflation. Suppose that X and Y are \overline{𝕋}-spectra. First consider the case when Y is \overline{F}-contractible, we use the diagram

[X, Y]_{∗}^{𝕋} ⟶^{≅} Hom(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ≅ Hom(π_{∗}(Φ^{𝕋}X), π_{∗}(Φ^{𝕋}Y))
inf ↑≅ ↑ inf ↑=
[X, Y]\overline{𝕋} ; ∗ ⟶^{≅} Hom(π\overline{A} ; ∗(X), π\overline{A} ; ∗(Y)) ≅ Hom(π_{∗}(Φ\overline{𝕋}X), π_{∗}(Φ^{𝕋}Y)).

The diagram commutes since the horizontals factor through passage to total geometric fixed points. Both algebraic and topological inflation functors are the identity on vertices.

Next we suppose that Y is an injective \overline{F}-spectrum, and use the diagram

[X, Y]_{∗}^{𝕋} ⟶^{≅} Hom(π_{∗}^{A}(X), π_{∗}^{A}(Y)) ≅ Hom(π_{∗}^{𝕋}(X ∧ DEF_{+}), π_{∗}^{𝕋}(Y))
inf ↑ ↑ inf ↑
[X, Y]\overline{𝕋} ; ∗ ⟶^{≅} Hom(π\overline{A} ; ∗(X), π\overline{A} ; ∗(Y)) ≅ Hom(π\overline{𝕋} ; ∗(X ∧ DE\overline{F}_{+}), π\overline{𝕋} ; ∗(Y)).

We must explain why the diagram commutes. In other words we must show that if π\overline{A} ; ∗(X) = (\overline{N} ⟶ t\overline{F} ; ∗ ⊗V) and the map f : X ⟶ Y corresponds to θ : \overline{N} ⟶ π_{∗}^{𝕋}(Y) so that

θ = π\overline{𝕋} ; ∗(X∧DE\overline{F}_{+} ⟶ Y ∧DE\overline{F}_{+} ≃ Y) then inf(θ) is π_{∗}^{𝕋}(X∧DEF_{+} ⟶ Y ∧DEF_{+} ≃ Y).

A map into a Hausdorff O_{F}-module (such as π_{∗}^{𝕋}(Y)) is determined by its idempotent parts. It is thus sufficient to show that π_{∗}^{𝕋}(e_{H}(X ∧ DEF_{+}) ⟶ e_{H}Y) is the same as

π\overline{𝕋} ; ∗(e\overline{H}(X ∧ DE\overline{F}_{+}) ⟶ e\overline{H}Y), but since e_{H}X ∧ DEF_{+} ≃ X ∧ e_{H}DEF_{+} ≃ X ∧ DE⟨H⟩

and similarly e\overline{H}X ∧ DE\overline{F}_{+} ≃ X ∧ DE⟨\overline{H}⟩ this follows from 12.1.8.

It is now formal to deduce the correspondence between the algebraic and topological Lewis-May fixed point functors.

Theorem 12.2.2. The algebraic Lewis-May K-fixed point functor Ψ^{K} induces a functor

Ψ^{K} : DA_{t} ⟶ D\overline{A}_{t}

so that the diagram

𝕋−Spec ⟶^{Ψ^{K}} \overline{𝕋}−Spec
≃↓  ↓≃
DA_{t}  ⟶^{Ψ^{K}} D\overline{A}_{t}

commutes.

Proof: The existence of the functor on derived categories is immediate from the fact the Ψ^{K} is an exact functor A_{t} ⟶ \overline{A}_{t}.

We have seen that the algebraic Lewis-May fixed point functor is right adjoint to inflation at the level of abelian categories. The unit and counit of the adjunction induce maps of derived categories, and the triangular identities continue to hold. Since the topological Lewis-May fixed point functor is right adjoint to inflation by [18, II.4.4], and since we have seen in 12.2.1 that the left adjoints correspond, the right adjoints also correspond.

12.3. A direct approach to the Lewis-May fixed point functor.

In the previous section we completed the proof that the algebraic Lewis-May K-fixed point functor corresponds to the geometric one, and for logical purposes all further discussion could take place in the algebraic models. However, we think it is important to make the link between algebraic and geometric forms of the functor more direct, and that is the purpose of this section. This can be viewed as motivation for the definition of the algebraic Lewis-May fixed point functor, and can be used to give an alternative proof of the theorem for objects. However it seems hard to get a sufficiently tight grasp on morphisms to give a complete proof of the theorem by this route.

The construction of the model is reasonably obvious if the question is viewed correctly. The point is that our decomposition X(F) ≃ ⋁_{H} X(H) follows the decomposition of a Mackey functor as an hSB-module. Thus

π_{∗}^{H}(X) = ⨁_{L⊆H} π_{∗}^{H}(X(L)).

The point is that when H ⊇ K we may calculate π\overline{H} ; ∗(Ψ^{K}X) = π_{∗}^{H}(X), so that whatever the subgroup \overline{H}, this group contains the summands for X(L) whenever L ⊆ K. The end result is that the when we decompose (Ψ^{K}X)(F) as a wedge of spectra (Ψ^{K}X)(\overline{H}), the term (Ψ^{K}X)(\overline{1}) contains information equivalent to ⋁_{L⊆K} X(L), and in general the term (Ψ^{K}X)(\overline{H}) contains information equivalent to X(L) for all L with \overline{L} = \overline{H}.

Once again we need to give a context in which to make this meaningful. The basic idea is that there is a natural way of identifying ℚ[c_{H}] and ℚ[c\overline{H}]. Indeed, we gave one identification in 10.2.3 in terms of geometric fixed points; for our present purpose we must show the Φ^{K} fixed point map ℚ[c_{H}] ⟶ ℚ[c\overline{H}] coincides with passage to Lewis-May fixed points.

Lemma 12.3.1. If H is a finite subgroup containing K, then passage to Lewis-May Kfixed points defines the isomorphism

ℚ[c_{H}] = [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} ⟶^{≅} [E⟨\overline{H}⟩, E⟨\overline{H}⟩]\overline{𝕋} ; ∗ = ℚ[c\overline{H}]

also described as passage to geometric fixed points above.

Proof: Since E⟨H⟩ is concentrated over K in the sense that E⟨H⟩ ≃ E⟨H⟩ ∧ Ẽ[⊉ K], Lewis-May fixed points coincide with Φ^{K} fixed points.

Note that by 12.1.5 the inflation map

ℚ[c\overline{H}] = [E⟨\overline{H}⟩, E⟨\overline{H}⟩]\overline{𝕋} ; ∗ ⟶ [E⟨\overline{H}⟩, E⟨\overline{H}⟩]_{∗}^{𝕋} = ∏_{\overline{L}=\overline{H}} ℚ[c_{L}]

is the diagonal inclusion c\overline{H} ⟼ (c_{L})_{L}. This gives sense to the following consequence.

Corollary 12.3.2. If H is a finite subgroup of 𝕋 containing K then

π\overline{𝕋} ; ∗((Ψ^{K}X)(\overline{H})) ≅ ⨁_{\overline{L}=\overline{H}} π_{∗}^{𝕋}(X(L))

The curious Lemma 12.1.1 also lets us deduce the F-contractible part.

Corollary 12.3.3. For any 𝕋-spectrum X we have an equivalence

Φ\overline{𝕋}(Ψ^{K}X) ≃ Φ^{𝕋}X.

In particular

π\overline{𝕋} ; ∗(Ψ^{K}(X) ∧ Ẽ\overline{F}) ≅ π_{∗}^{𝕋}(X ∧ ẼF).

Proof: Let Y = Ẽ\overline{F} in 12.1.1, to find that Ψ^{K}(X) ∧ Ẽ\overline{F} ≃ Ψ^{K}(X ∧ Ẽ\overline{F}). Since Ẽ\overline{F} regarded as a 𝕋-space is ẼF, the result follows.

The result may seem peculiar to begin with, so we outline an alternative heuristic argument. We note that π\overline{𝕋} ; ∗(Ψ^{K}(X) ∧ Ẽ\overline{F}) is the localization of π\overline{𝕋} ; ∗(Ψ^{K}X) = π_{∗}^{𝕋}(X) so as to invert Euler classes of all representations \overline{V} of \overline{𝕋} with \overline{V} \overline{𝕋}. On the other hand π_{∗}^{𝕋}(X ∧ ẼF) is the localization so as to invert all representations V of 𝕋 with V^{𝕋} = 0. However the Euler class of V divides the Euler class of V^{⊗n} and if K is of order n then V^{⊗n} is a representation of \overline{𝕋}. The two localizations are therefore equal.

We are now in a position to give the algebraic analogue of the Lewis-May fixed point functor. The basic ingredient is the ring homomorphism

q^{∗} : \overline{O}\overline{F} = [E\overline{F}_{+}, E\overline{F}_{+}]\overline{𝕋} ; ∗ ⟶ [EF_{+}, EF_{+}]_{∗}^{𝕋} = O_{F}

defined by the inflation map. We have already seen that it is given explicitly by the diagonal maps

ℚ[c\overline{H}] ⟶ ∏_{\overline{L}=\overline{H}} ℚ[c_{L}].

Because the vertex and torsion are treated so differently, it proves difficult to define the Lewis-May fixed points in the standard model. More precisely, it is not hard to identify the homotopy type of the fixed point object, even at the level of the abelian category; however it seems impossible to do this sufficiently naturally to obtain a functor. The remedy is to retreat to the differential graded category, and once we have done that, it is much simpler to work with the torsion model. We have given a functor fib : dgA_{t} ⟶ dgA which induces an equivalence of derived categories so it is easy to translate the construction into the standard category once we have the torsion model form of an object or morphism. However we recall that the inverse functor takes values in the category dgHA_{t}, so the easiest way to get back into A_{t} is to take homology: this loses control of morphisms, and exactly corresponds to the difficulties described above.

We now complete the alternative approach to Lewis-May fixed points by checking directly that the algebraic construction is correct on objects.

Lemma 12.3.4. The algebraic Lewis-May fixed point functor is compatible with the geometric one on objects in the sense that

Ψ^{K}(p(X)) ≃ \overline{p}(Ψ^{K}X)

for any 𝕋-spectrum X.

Proof: Of course we motivated the definition by arranging this was true if X is either F-free or F-contractible.

However, because the definitions on torsion and vertices was so different, the core of the matter is that the structure maps are correct. For this we consider the diagram

Ψ^{K}(X) ∧ DE\overline{F}_{+} ∧ Ẽ\overline{F} ⟶ Ψ^{K}(X) ∧ DE\overline{F}_{+} ∧ ΣE\overline{F}_{+} ⟵^{≃} Ψ^{K}(X) ∧ ΣE\overline{F}_{+}
≃↓  ≃↓
Ψ^{K}(X ∧ DE\overline{F}_{+}) ∧ Ẽ\overline{F} ⟶ Ψ^{K}(X ∧ DE\overline{F}_{+}) ∧ ΣE\overline{F}_{+}
↓ ↓ ↓
Ψ^{K}(X ∧ DEF_{+}) ∧ Ẽ\overline{F} ⟶ Ψ^{K}(X ∧ DEF_{+}) ∧ ΣE\overline{F}_{+}
≃↓  ≃↓
Ψ^{K}(X ∧ DEF_{+} ∧ ẼF) ⟶ Ψ^{K}(X ∧ DEF_{+} ∧ ΣEF_{+}) ⟵^{≃} Ψ^{K}(X ∧ ΣEF_{+})

in which the equivalences come from 12.1.1. Since the top horizontal is q and the bottom horizontal is \overline{q}, and the right hand vertical is an equivalence, it remains to remark that the central vertical induces inf in homotopy.

This in turn follows from the fact that DE\overline{F}_{+} ⟶ Ψ^{K}(DEF_{+}) induces inf : \overline{O}\overline{F} ⟶ O_{F}, as is clear from the fact that it is two adjunctions from the 𝕋-map EF_{+} ≃ EF_{+}∧DE\overline{F}_{+} ⟶ S^{0}.

12.4. Lewis-May fixed points on objects in the standard model.

It is useful to identify the behaviour of the fixed point functor on objects, using the fact that objects of DA correspond to objects of A in the standard model. It does not seem possible to extend this construction to a functor on the abelian category A itself, because of the choices involved in the definition: it is entirely based on a homotopy level analysis.

We suppose given an object M = (N ⟶^{β} t_{∗}^{F} ⊗ V) of the standard model A. Letting I denote the image of β we consider the modules associated to M by the exact sequences

0 ⟶ K ⟶ N ⟶ I ⟶ 0 and 0 ⟶ I ⟶ t_{∗}^{F} ⊗ V ⟶ C ⟶ 0;

the torsion part is then given by T = K ⊕ Σ^{−1}C. The definition is based on the fact that the torsion part of the Lewis-May fixed point object Ψ^{K}M is simply T regarded as an \overline{O}\overline{F}-module by pullback along q^{∗}, and its vertex is the same as that for M.

We shall make a definition of an object \overline{Ψ}^{K}M of \overline{A}, which will be of the homotopy type of Ψ^{K}M; the notation for the associated modules will be systematically indicated by bars. Thus \overline{Ψ}^{K}M has torsion module \overline{T} = q^{∗}T, and vertex \overline{V} = V. From now on we regard all O_{F}-modules as \overline{O}\overline{F}-modules, and omit the notation q^{∗}. We therefore need to define an \overline{O}\overline{F}-homomorphism \overline{s} : t\overline{F} ; ∗ ⊗ V ⟶ ΣT; it is natural to use the composite

t\overline{F} ; ∗ ⊗ V ⟶^{inf⊗1} t_{∗}^{F} ⊗ V ⟶^{β} ΣT.

This gives a map of short exact sequences

\begin{aligned}0 & ⟶ & t\overline{F} ; ∗ ⊗ V & ⟶ & t_{∗}^{F} ⊗ V & ⟶ & t_{∗}^{F}/t\overline{F} ; ∗ ⊗ V & ⟶ & 0 \\ & & ↓ & & c ↓ & & ↓ \\ 0 & ⟶ & C & ⟶ & C & ⟶ & 0 & ⟶ & 0 \end{aligned} .

In particular, we note that although \overline{T} = T, the contributions from the kernel and cokernel of the basing map are quite different: \overline{C} is bigger than C and \overline{K} is correspondingly smaller than K.

Now consider the map inf ⊗ 1 : t_{∗}^{F} ⊗ V ⟶ t\overline{F} ; ∗ ⊗ V; it is injective and we view it as an inclusion, so that we may take \overline{I} = (t\overline{F} ; ∗ ⊗ V) ∩ I. Thus we have an exact sequence

0 ⟶ \overline{I} ⟶ I ⟶ (t_{∗}^{F}/t\overline{F} ; ∗) ⊗ V ⟶ \overline{D} ⟶ 0.

Now let N′ = β^{−1}(\overline{I}), so that we have an exact sequence

0 ⟶ K ⟶ N′ ⟶ \overline{I} ⟶ 0.

Definition 12.4.1. We define the crude Lewis-May fixed point object of an object M =

(N ⟶^{β} t_{∗}^{F} ⊗ V) of A by taking \overline{Ψ}^{K}M = (\overline{N} ⟶^{\overline{β}} t\overline{F} ; ∗ ⊗ V), where \overline{N} = Σ^{−1}\overline{D} ⊕ N′ and \overline{β} is

the composite \overline{N} = Σ^{−1}\overline{D} ⊕ N′ ⟶ N′ ⟶ \overline{I} ⟶ t\overline{F} ; ∗ ⊗ V.

Remark 12.4.2. With the above definition we have a splitting

\overline{Ψ}^{K}M ≅ f(Σ^{−1}\overline{D}) ⊕ Λ^{K}M,

with \overline{D} injective. Note in particular that, even if M is even, \overline{Ψ}^{K}M will not usually be even. The fact that the splitting is not canonical is the reason this construction is not sufficient to construct the algebraic Lewis-May fixed points as a functor.

We should begin by observing that \overline{Ψ}^{K}M is actually an object of \overline{A}.

Lemma 12.4.3. The kernel and cokernel of the map \overline{β} just defined are both \overline{F}-finite and torsion, and hence \overline{Ψ}^{K}M is an object of \overline{A}.

Proof: By construction we see that, since \overline{C} is injective, C = \overline{C} ⊕ \overline{D} and also \overline{K} = K ⊕ Σ^{−1}\overline{D}. This establishes that \overline{K} and \overline{C} are \overline{F}-finite and torsion.

It now makes sense to claim that the \overline{Ψ}^{K} construction describes Lewis-May fixed points.

Proposition 12.4.4. The \overline{Ψ}^{K} construction gives the Lewis-May fixed point functor on objects in the sense that if we identify objects of DA up to isomorphism with those of A, then

Ψ^{K}M ≃ \overline{Ψ}^{K}M.

Proof: The object \overline{N} ⟶^{\overline{β}} t\overline{F} ; ∗ ⊗ V is determined by the image of \overline{β}, and its extension class in Ext(im(\overline{β}), ker(\overline{β})) = Ext(\overline{I}, K ⊕ Σ^{−1}\overline{D}). The image of \overline{β} is the kernel of the composite t\overline{F} ; ∗ ⊗ V ⟶ t_{∗}^{F} ⊗ V ⟶ ΣT by construction, as required. Since \overline{D} is injective, the extension class of \overline{N} is the pullback of that of N in Ext(I, K) by construction.

Combining 12.2.2, 12.4.2 and 12.4.4 we obtain a topological splitting result.

Corollary 12.4.5. There is an unnatural splitting

Ψ^{K}X ≃ Σ^{−1}Δ^{K}X ∨ Λ^{K}X

with Δ^{K}X an injective \overline{F}-spectrum.

Example 12.4.6. Consider the algebraic 0-sphere, L_{𝕋} = (O_{F} ⟶ t_{∗}^{F}). We find C = Σ^{2}𝕀, K = 0 and I = O_{F}. Thus \overline{I} = \overline{O}\overline{F}, \overline{C} = Σ^{2}\overline{𝕀}, and \overline{K} = ΣI/\overline{I}. More explicitly, I/\overline{I} is a sum of injectives 𝕀(\overline{H}), with multiplicity one less than the number of subgroups L with \overline{L} = \overline{H}. For example, if K is of order 2 there is a single summand 𝕀(\overline{H}) whenever \overline{H} is of odd order.

This should be compared with the equivalence of [18, V.11.1], which gives

Ψ^{K}S^{0} ≃ ⋁_{Λ⊆K} E[∩K/Λ = 1]_{+}/(K/Λ),

where the quotient of the universal K/Λ-free 𝕋/Λ-space, E[∩K/Λ = 1]_{+} by K/Λ is viewed as a \overline{𝕋}-space using the isomorphism \overline{𝕋} = 𝕋/K ≅ (𝕋/Λ)/(K/Λ). Note first that the summand with Λ = K is S^{0}, and that all other terms are \overline{F}-spaces.

It seems worth making the consistency with the algebraic description explicit. Let K^{∗} = K/Λ, and, more generally, use asterisks to denote reference to 𝕋/Λ. We have observed that if Λ = K the summand is S^{0}, so now suppose Λ is a proper subgroup of K. Since K^{∗} is finite, the quotient of a K^{∗}-free universal space is again rationally a universal space. Indeed

E[∩K^{∗} = 1] = ⋁_{L^{∗}∩K^{∗}=1} E⟨L^{∗}⟩,

and when L^{∗} ∩ K^{∗} = 1, E⟨L^{∗}⟩/K^{∗} = E⟨\overline{L}⟩. Thus

E[∩K^{∗} = 1]_{+}/K^{∗} ≃ ⋁_{L^{∗}∩K^{∗}=1} E⟨\overline{L}⟩.

Thus, a subgroup L gives rise to a summand E⟨\overline{L}⟩ for each subgroup Λ ⊂ K with (|L^{∗}|, |K^{∗}|) = 1; this happens exactly once for each subgroup not containing K. The multiplicity of the summand E⟨\overline{L}⟩ is thus |q_{∗}^{−1}(\overline{L})| − 1.

It seems worth recording the following.

Lemma 12.4.7. If Y ⟶ I(Y) ⟶ J(Y) is an Adams resolution, then it remains so after taking Lewis-May fixed points.

Proof: We have seen that passage to Lewis-May fixed points preserves F-contractible objects and F-free objects. By the classification of torsion F-finite injectives we see that the fixed points of a F-free injective is also injective.

Finally we must observe that

0 ⟶ π\overline{A} ; ∗(Ψ^{K}Y) ⟶ π\overline{A} ; ∗(Ψ^{K}I(Y)) ⟶ π\overline{A} ; ∗(Ψ^{K}J(Y)) ⟶ 0

is exact. This is a diagram chase using the fact that the first map is an isomorphism of vertices, and that, because the torsion sequence is exact, the sequences of kernels and of cokernels are also both exact.

Finally we remark that it is not hard to prove directly that d invariants of the algebraic and topological Lewis-May fixed points agree. This falls short of a complete direct analysis because Lewis-May fixed points do not preserve pure parity objects.

12.5. Quotient functors.

It is notorious that the quotient functor is only well behaved on sufficiently free spectra. This phenomenon has already shown itself in the algebra. Topologically, it seems impossible even to calculate even the homotopy groups of quotient of arbitrary spectra.

In fact the properties of the deflation functor show that def : D(torsO_{F}^{f}) ⟶ D(tors\overline{O}\overline{F}^{f}) is left adjoint to inf : D(tors\overline{O}\overline{F}^{f}) ⟶ D(torsO_{F}^{f}). Hence, whenever the topological quotient is left adjoint to the topological inflation functor, the algebraic deflation models the effect on homotopy groups of the topological quotient. Applying [18, I.3.8 and II.2.8] we immediately deduce the required result.

Proposition 12.5.1. The algebraic K-deflation functor corresponds to the topological quotient by K on the category of K-free spectra.

It seems reasonable to refer to the topological functor /K defined by the diagram

𝕋−Spec/F ⟶^{/K} 𝕋−Spec/\overline{F}
≃↓  ↓≃
D(torsO_{F}^{f}) ⟶^{def} D(tors\overline{O}\overline{F}^{f})

as the quotient by K. We have seen that it agrees with the quotient by K on K-free spectra, and it extends this functor on rational spectra to arbitrary F-spectra. It would be interesting to know if there is a topological definition of this functor.

148 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.

CHAPTER 13 Homotopy Mackey functors and related constructions.

This chapter investigates a number of constructions associated with a Mackey functor, and the reader is advised to glance at Appendix A before reading further.

Section 13.1 makes explicit the homotopy Mackey functor applied to objects of the standard model, and deduces the models of Eilenberg-MacLane spectra as a consequence. The following section studies Eilenberg-MacLane spectra from a topological point of view, concentrating on cell structures and useful cofibre sequences involving them. Section 13.3 does the same for coMackey functors, the representing objects of ordinary homology. Finally, in Section 13.4, we study the Brown-Comenetz spectra associated to a Mackey functor; these are usually unbounded, and illuminate the rational Segal conjecture.

13.1. The homotopy Mackey functor on A.

We have the information to calculate the homotopy Mackey functor of any object of DA. We shall make use of the notation described in Section 5.5, and in particular we recall from Example 5.5.2 that the algebraic basic cells are L_{H} = (Σℚ(H) ⟶ 0) and L_{𝕋} = (O_{F} ⟶ t_{∗}^{F}). We use the obvious injective form of the algebraic fixed cell L̂_{𝕋} = (t_{∗}^{F} ⋉ Σ𝕀 ⟶ t_{∗}^{F}) where the the semidirect product t_{∗}^{F} ⋉ Σ𝕀 is additively the direct sum, but with differential given by the augmentation t_{∗}^{F} ⟶ Σ^{2}𝕀. We proceed in steps, using the notation π_{∗}^{𝕋}(M) = [L_{𝕋}, M] and eπ_{∗}^{H}(M) = [L_{H}, M] for an object M of DA. The symbol abbreviates the more descriptive e_{H}π_{∗}^{H}, the point being that L_{H} is the algebraic counterpart of the basic cell σ_{H}^{0} rather than G/H_{+}. Since M ≃ H_{∗}(M) in DA we may work with objects with zero differential.

Lemma 13.1.1. If M is an object with zero differential and s_{M} is the composite

V_{M} ⟶ t_{∗}^{F} ⊗ V_{M} ⟶^{d(q_{M})} ΣT_{M},

then there is a natural isomorphism

π_{∗}^{𝕋}(M) ≅ Σ^{−1}cok(s_{M}) ⊕ ker(s_{M}).

Proof: This is straightforward from the triangle M ⟶ e(V_{M}) ⟶ f(ΣT_{M}). Indeed [L_{𝕋}, e(V_{M})] = [V_{𝕋}, V_{M}] = Hom(ℚ, V_{M}) = V_{M}, and because N_{𝕋} = O_{F} is projective we have [L_{𝕋}, f(ΣT_{M})] = [N_{𝕋}, ΣT_{M}] = Hom(O_{F}, ΣT_{M}) = ΣT_{M}. The map is easily identified.

The splitting is constructed as follows. If v ∈ ker(s_{M}) then a map L_{𝕋} ⟶ M is explicitly represented by the homomorphism L_{𝕋} ⟶ M̂ displayed in the diagram

O_{F} ⟶^{{1⊗v,0}} (t_{∗}^{F} ⊗ V_{M}) ⋉ T̂_{M}
↓  ↓
t_{∗}^{F}  ⟶^{1⊗v}  t_{∗}^{F} ⊗ V_{M}.

The homotopy groups π_{∗}^{H}(M) are a little easier, since L_{H} is torsion.

Lemma 13.1.2. If M is an object with zero differential and there is a short exact sequence

0 ⟶ T_{M}(H)/c_{H} ⟶ eπ_{∗}^{H}(M) ⟶ Σ^{−1}ann(c_{H}, T_{M}(H)) ⟶ 0,

where T_{M}(H) = e_{H}T_{M} is regarded as a module over ℚ[c_{H}] ≅ e_{H}O_{F} in the obvious way. The sequence splits unnaturally.

Proof: This follows from the Adams short exact sequence by calculating Hom(ℚ(H), T_{M}) = ann(c_{H}, T_{M}(H)) and Ext(ℚ(H), T_{M}) = Σ^{2}T_{M}(H)/c_{H}.

Before continuing, let us see what this implies about objects with homotopy groups only nonzero in a single degree: the Eilenberg-MacLane objects.

Example 13.1.3. If M is an Eilenberg-MacLane object with zero differential and homotopy groups π_{∗}^{𝕋}(M) = W (𝕋) and eπ_{∗}^{H}(M) = W (H), concentrated in degree zero then

V_{M} = W (𝕋) ⊕ ⨁_{H} Σ^{2}𝕀(H) ⊗ W (H)

and

ΣT_{M} = ⨁_{H} Σ^{2}𝕀(H) ⊗ W (H).

The map s_{M} : V_{M} ⟶ ΣT_{M} is surjective.

Proof: Suppose M is an Eilenberg-MacLane object with non-zero homotopy in degree zero. Since any non-zero object of with zero differential has non-zero homotopy we may suppose M is even or odd, and the following discussion leads to the conclusion M = 0 if M is odd. We therefore suppose M is even.

Now e(W) is an Eilenberg-MacLane object as in the statement if W is concentrated in degree 0. If M is not torsion free, then T_{M} ≠ 0. Next, we see that T_{M} is injective and in odd degrees. Indeed, T_{M} decomposes into even and odd parts; since T_{M} is torsion, if the even part is non-zero it gives a non-zero homotopy group in odd degree arising from Σ^{−1}ann(c_{H}, T_{M}(H)) for some H, so we may suppose T_{M} is odd. For the same reason, if the odd part is non-zero it definitely has homotopy groups in even degrees, and hence the odd contribution must be zero, and T_{M}(H)/c_{H} = 0 for all H; it is therefore divisible and hence injective.

Hence T_{M}(H) is entirely entirely odd and T_{M}(H) = Σ𝕀(H) ⊗ W (H) where W (H) is concentrated in degree zero. Now consider s_{M} : V_{M} ⟶ ΣT_{M} = Σ^{2}𝕀(H) ⊗ W (H). The cokernel gives homotopy in odd degrees, so that s_{M} is onto. Since π_{∗}^{𝕋}(M) is concentrated in degree 0, s_{M} is an isomorphism, except perhaps in degree 0.

13.1. THE HOMOTOPY MACKEY FUNCTOR ON A. 151

To determine the entire Mackey functor we need to identify the restriction map π_{∗}^{𝕋}(M) ⟶ eπ_{∗}^{H}(M) induced by the generator p_{H} : L_{H} ⟶ L_{𝕋} of [L_{H}, L_{𝕋}] ≅ ℚ. We must begin by being explicit that p_{H} is represented by the homomorphism

(p̂_{H} : L_{H} ⟶ L̂_{𝕋}) = ( \begin{matrix}Σℚ(H) & ⟶^{\{0,ι_{H}\}} & t_{∗}^{F} ⋉ Σ𝕀 \\ ↓ & & ↓ \\ 0 & ⟶ & t_{∗}^{F} \end{matrix} )

where ι_{H} : ℚ(H) ⟶ 𝕀 is the inclusion of the degree 0 part of e_{H}𝕀. This is readily verified from the definitions.

Proposition 13.1.4. Restriction map π_{∗}^{𝕋}(M) ⟶ eπ_{∗}^{H}(M) is the direct sum of the natural quotient map

res′ : Σ^{−1}cok(s_{M}) ⟶ T_{M}(H)/c_{H}

given by res′(\overline{t}) = \overline{\overline{t}} and the map

res′′ : ker(s_{M}) ⟶ Σ^{−1}ann(c_{H}, T_{M}(H))

given by res′′(v) = β_{M}(c_{H}^{−1} ⊗ v).

Proof: The proof is by unravelling definitions. In fact we suppose given a map θ : L_{𝕋} ⟶ M, which we have seen can actually be represented by a homomorphism of the same name. Using hats to denote fibrant approximation as usual, we then find a homomorphism θ̂ so that the diagram

L_{𝕋} ⟶^{θ} M
↓  ↓
L_{H} ⟶^{p̂_{H}} L̂_{𝕋} ⟶^{θ̂} M̂

commutes. Then res(θ) is determined by θ̂p_{H}(1) = θ̂(0, ι_{H}(1)) = θ̂(d(c_{H}^{−1}, 0)) = dθ̂(c_{H}^{−1}, 0). Now θ̂ takes the form

t_{∗}^{F} ⋉ Σ𝕀 ⟶^{θ̂′} (t_{∗}^{F} ⊗ V_{M}) ⋉ T̂_{M}
↓  ↓
t_{∗}^{F}  ⟶^{θ′′}  t_{∗}^{F} ⊗ V_{M}.

Next observe that if the map of nubs is θ′ = {θ′_{V} , θ′_{T} } : O_{F} ⟶ (t_{∗}^{F} ⊗ V_{M}) ⋉ T̂_{M} then the extension of θ′ over t_{∗}^{F} determines θ̂′ since the differential t_{∗}^{F} ⟶ Σ𝕀 is surjective, and the map into the t_{∗}^{F} ⊗ V_{M} factor must be θ′′. It remains to specify an extension θ̃: t_{∗}^{F} ⟶ T̂_{M} of θ′_{T} and verify that the result commutes with differentials. Fortunately we need only do this when θ is of two particularly simple forms.

For res′ we suppose θ corresponds to \overline{t} ∈ cok(s_{M} : V_{M} ⟶ ΣT_{M}) for some element t ∈ T_{M} (which may be of even or odd degree). In this case θ factors through f(T_{M}) ⟶ M. Now suppose T̂_{M} = fibre(I ⟶ J), and view t as an element of I. We may then extend the map θ : O_{F} ⟶ I to θ̃: t_{∗}^{F} ⊗ I, and any such θ̃ gives rise to a map L̂_{𝕋} ⟶ f(T̂_{M}) and hence to θ̂ as the composite L̂_{𝕋} ⟶ f(T̂_{M}) ⟶ M̂. Therefore res′(\overline{t}) is represented by dθ̃(c_{H}^{−1}) ∈ J, which is an element dy where y ∈ I satisfies c_{H}y = t; the result follows by considering the resolution resulting from the resolution beginning 0 ⟶ T_{M}(H) ⟶ T_{M}(H)/c_{H} ⊗ 𝕀(H).

For res′′ we use the explicit splitting given in 13.1.1 above. Suppose that θ corresponds to v ∈ V_{M}, so that θ′_{V} = 1⊗v, θ′_{T} = 0 and θ′′ = 1⊗v. We naturally choose θ̃ = 0, and it is immediate that this defines a map θ̂′ commuting with differentials. Thus θ̂′(c_{H}^{−1}) = (c_{H}^{−1}⊗v, 0); the result follows.

This allows us to complete the discussion of Eilenberg-MacLane objects.

Example 13.1.5. We continue with the notation of Example 13.1.3. First observe that since s_{M} is surjective res′ = 0. Thus for v ∈ π_{0}^{𝕋}(M) = W (𝕋) we have res(v) = β_{M}(c_{H}^{−1} ⊗v). The map q′_{M} : t_{∗}^{F} ⊗ V_{M} ⟶ ΣT_{M} may thus be chosen to have components

q′_{𝕋} : t_{∗}^{F} ⊗ W (𝕋) ⟶ ⨁_{H} Σ^{2}𝕀(H) ⊗ W (H)

and

q′_{H} : t_{∗}^{F} ⊗ Σ^{2}𝕀(H) ⊗ W (H) ⟶ ⨁_{H} Σ^{2}𝕀(H) ⊗ W (H)

described as follows. First q′_{𝕋} is the composite of projection t_{∗}^{F} ⊗ W (𝕋) ⟶ Σ^{2}𝕀 ⊗ W (𝕋) ≅ ⨁_{H} Σ^{2}𝕀(H)⊗W (𝕋) and the map which is restriction on each factor. Next q′_{H} is the composite of projection t_{∗}^{F} ⊗Σ^{2}𝕀(H)⊗W (H) ⟶ e_{H}t_{∗}^{F} ⊗Σ^{2}𝕀(H)⊗W (H) ≅ ℚ[c_{H}, c_{H}^{−1}]⊗Σ^{2}𝕀(H)⊗ W (H) and the map c_{H}^{k}⊗c_{H}^{l}⊗w ⟼ c_{H}^{k+l}⊗w, where c_{H}^{k+l} is interpreted as zero if k+l ≥ 0.

13.2. Eilenberg-MacLane spectra.

In the previous section we worked entirely in the algebraic model to identify objects with homotopy only in one degree. In the present section we study Eilenberg-MacLane spectra as is natural from a topological point of view. The identification of the model of a topological Eilenberg-MacLane spectrum gives an alternative to the algebraic deduction, but the point of view is sufficiently different for this second method to be worth including.

We shall use notation and terminology for Mackey functors established in Appendix A, which the reader should refer to as necessary. In particular we recall that a Mackey functor M corresponds to a collection of vector spaces V (H), one for each finite subgroup H and a vector space V (𝕋) with restriction maps V (𝕋) ⟶ V (H).

The easy example is the Eilenberg-MacLane spectra associated to the Mackey functor L[U] concentrated over 𝕋, for which we obviously have H(L[U]) ≃ ẼF ∧ S^{0}[U]. This deals with all F̃-injectives. On the other hand, the basic F-injective is the functor R_{K} concentrated over 𝕋 and K. An arbitrary F-injective is a product of injectives which are sums of the basic injective R_{K}, so the same is true of the corresponding Eilenberg-MacLane spectra. Therefore the following examples describe the structure of Eilenberg-MacLane spectra for all projective and injective spectra.

Example 13.2.1. (i) For the indecomposable projective Mackey functor P_{𝕋} (the ‘constant’ functor) we have a cofibre sequence

ΣẼF ∧ S^{0}[ℚF] ⟶ S^{0} ⟶ HP_{𝕋}.

(ii) For any finite subgroup K with corresponding indecomposable projective Mackey functor P_{K} concentrated over K we have a cofibre sequence

ΣẼF ⟶ E⟨K⟩ ⟶ HP_{K}.

(iii) For any finite subgroup K with corresponding injective Mackey functor R_{K} we have cofibre sequences

ΣHR_{K} ⟶ σ_{K}^{0} ⟶ HP_{K}

and

HP_{K} ⟶ HR_{K} ⟶ ẼF.

Proof: The case of HP_{𝕋} is straightforward: we kill homotopy groups in positive degrees to obtain a map S^{0} ⟶ HP_{𝕋}, and observe its fibre is F-contractible.

In fact the same idea works for HP_{K}: kill homotopy groups in positive degrees to obtain a map E⟨K⟩ ⟶ HP_{K}. The first cofibre sequence for HR_{K} follows since σ_{K}^{0} is a two stage Postnikov tower. The second follows from the defining extension 0 ⟶ P_{K} ⟶ R_{K} ⟶ L[ℚ] ⟶ 0 for R_{K}, as in A.17.

One may obtain a cell picture for an arbitrary Eilenberg-MacLane spectrum HM by taking a minimal projective resolution 0 ⟶ P_{1} ⟶ P_{0} ⟶ M ⟶ 0 and using the associated cofibre sequence HP_{1} ⟶ HP_{0} ⟶ HM.

To complete the analysis of HM in our framework we need to analyze HM ∧ EF_{+} and HM ∧ ẼF. We let M(F) = lim_{→}_{H} M(H) = ⊕_{H}V (H).

Lemma 13.2.2. For any Mackey functor M there is an equivalence

HM ∧ EF_{+} ≃ ⋁_{H} E⟨H⟩ ∧ S^{0}[V (H)],

and hence HM ∧EF_{+} corresponds to the O_{F}-module M(F)⊗_{(O_{F})_{0}}Σ𝕀 = ⨁_{H} V (H)⊗Σ𝕀(H).

Proof: By construction, (HM ∧EF_{+})(K) = HM ∧E⟨K⟩. Recall that E⟨K⟩ = σ_{K}^{0} ∪ σ_{K}^{0} ∧ e^{2} ∪ σ_{K}^{0} ∧e^{4} ∪…; now consider the spectral sequence obtained by applying [S^{0}, HM ∧•]_{∗}^{𝕋} to the skeletal filtration. Because of the suspension in the Wirthmüller isomorphism we find [S^{0}, HM ∧ σ_{K}^{0}]_{∗}^{𝕋} = ΣV (K), and thus the spectral sequence collapses to show that the homotopy groups are V (K) in each positive odd dimension.

It remains to show that c_{K} gives an isomorphism π_{2n+1}^{𝕋}(HM ∧ E⟨K⟩) ⟶ π_{2n−1}^{𝕋}(HM ∧ E⟨K⟩) for n ≥ 1. For this we use the long exact sequence of the cofibering

HM ∧ E⟨K⟩ ∧ σ_{K}^{0} ⟶ HM ∧ E⟨K⟩ ∧ S^{0} ⟶^{e} HM ∧ E⟨K⟩ ∧ S^{V (K)}

in which e induces multiplication by c_{K}. The result follows from the fact that HM ∧ E⟨K⟩ is K-equivariantly an Eilenberg-MacLane spectrum, so that π_{∗}^{𝕋}(HM ∧ E⟨K⟩ ∧ σ_{K}^{0}) is concentrated in degree 1.

The homotopy type of the rational spectrum Φ^{𝕋}X is determined by its homotopy groups π_{∗}(Φ^{𝕋}X) ≅ π_{∗}^{𝕋}(ẼF ∧ X). With due caution about duality we may calculate Φ^{𝕋}HM for any Mackey functor M with associated hSB-module V.

Lemma 13.2.3.

π_{k}(Φ^{𝕋}HM) = \{ \begin{matrix}M(𝕋) & if k = 0 \\ M(F) & if k is even and positive \\ 0 & otherwise \end{matrix}

Proof: This follows immediately from the cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF and 13.2.2.

Note in particular that Φ^{𝕋}HM is only an Eilenberg-MacLane spectrum if M is an F̃- injective. By contrast, the idempotent description of the geometric fixed point functor given in Theorem 10.2.6 shows that Φ^{K}HM is an Eilenberg-MacLane spectrum for all finite K.

Corollary 13.2.4. The map HM ∧ ẼF ⟶ HM ∧ ΣEF_{+} is classified by the O_{F}- morphism

q : π_{∗}(Φ^{𝕋}HM) ⊗ t_{∗}^{F} ⟶ M(F) ⊗_{(O_{F})_{0}} Σ^{2}𝕀

described as follows, in which ϵ : t_{∗}^{F} ⟶ Σ^{2}𝕀 denotes the quotient. If n ≥ 1 and x ∈ π_{2n}(Φ^{𝕋}HM) = M(F) then q(x ⊗ y) = x ⊗ ϵ(y); if x ∈ π_{0}(Φ^{𝕋}HM) = M(𝕋), then c(x ⊗ y) = res_{F}^{𝕋}(x) ⊗ ϵ(y).

Note that res_{F}^{𝕋} in the statement is a map M(𝕋) ⟶ ∏_{H} V (H), whose codomain is the product, but that ϵ(y) is only nonzero in finitely many coordinates.

Proof: Suppose first that x is of positive degree. From the proof of 13.2.3, we know that q(x ⊗ 1) = x and the result follows since non-zero elements of 𝕀(H) are uniquely divisible.

For x of degree zero it is enough to show that q(x ⊗ c_{H}^{−1}) = res_{H}^{𝕋}(x). For this we examine the map

HM ∧ ẼF ∧ σ^{−V (H)} ⟶ HM ∧ ΣE⟨H⟩ ∧ σ^{−V (H)}

in [S^{0}, ·]^{𝕋}. Transposing σ^{−V (H)} across to its dual in the domain, and replacing the spaces in the codomain by suitable skeleta we must examine

[σ^{V (H)}, HM ∧ Ẽ⟨H⟩^{(2)}]^{𝕋} ⟶ [σ^{V (H)}, HM ∧ σ_{H}^{1}]^{𝕋}.

Now recall that Ẽ⟨H⟩^{(2)} = σ^{V (H)}, and note that if x ∈ M(𝕋), it is represented in the first group by x ∧ 1 : S^{0} ∧ σ^{V (H)} ⟶ HM ∧ σ^{V (H)}. Its image in the second is obtained by composing with the map HM ∧ σ^{V (H)} ⟶ HM ∧ σ_{H}^{1}. This corresponds to res_{H}^{𝕋}(x) ∈ M(σ_{H}^{0}) = e_{H}M(H) as required.

13.3. coMackey functors and spectra representing ordinary homology.

A coMackey functor is the algebraic object providing coefficients for ordinary homology. It is therefore a covariant additive functor hSO ⟶ Ab. By exactly the same argument as for Mackey functors, we see that the category of coMackey functors is equivalent to the category of hSB^{op}-modules. An hSB^{op}-module W is thus described by vector spaces W (𝕋) and W (H) together with induction maps W (H) ⟶ W (𝕋) for each finite subgroup H.

The treatment of Mackey functors in Appendix A is the algebraic model for the theory of co-Mackey functors, so the reader should refer to it as necessary. In view of the similarity to the structure of Mackey functors, we omit the proofs of the following facts.

One may make explicit the condition that an hSB^{op}-module W is injective or projective, and give canonical resolutions.

Lemma 13.3.1. (i) A hSB^{op}-module W is projective if and only if the map ⨁_{H} W (H) ⟶ W (𝕋) is a monomorphism. (ii)A hSB-module W is injective if and only if the maps W (H) ⟶ W (𝕋) are all epimorphisms.

Note in particular that if W (H) = 0 for all H then W is projective, and if W (𝕋) = 0 then W is injective. The principal projectives are (a) Q_{𝕋} defined by Q_{𝕋}(H) = 0 for all H and Q_{𝕋}(𝕋) = ℚ and (b) Q_{H} defined by Q_{H}(H) = ℚ, Q_{H}(𝕋) = ℚ (with induction being the identity), and Q_{H}(K) = 0 if H ≠ K. It is not hard to see how to construct canonical projective and injective resolutions of length 1.

The existence and uniqueness of 𝕋-spectra JN representing homology with coefficients in N is guaranteed by Brown representability. For any coMackey functor N we let N|_{F} denote the quotient zero at 𝕋; this has zero structure maps involving 𝕋 and can therefore be regarded as a Mackey functor. In fact JN is a two stage Postnikov tower with

\underline{π}_{k}^{𝕋}(JN) = \{ \begin{matrix}N|_{F} & if k = −1 \\ L[N(𝕋)] & if k = 0 \\ 0 & otherwise \end{matrix} ,

where L[N(𝕋)] is the Mackey functor concentrated at 𝕋 where it takes the value N(𝕋). There is thus a cofibre sequence

ẼF ∧ S^{0}[N(𝕋)] ⟶ JN ⟶ Σ^{−1}H(N|_{F}).

Dually, the cofibre sequence

ẼF ∧ S^{0}[M(𝕋)] ⟵ HM ⟵ H(M|_{F})

can be regarded as a decomposition of HM in terms of spectra JN.

Finally we place JN in the torsion model.

Lemma 13.3.2. For any coMackey functor N there is an equivalence

JN ∧ EF_{+} ≃ ⋁_{H} Σ^{−1}E⟨H⟩ ∧ S^{0}[W (H)],

and hence JN ∧EF_{+} corresponds to the injective O_{F}-module N(F)⊗_{(O_{F})_{0}} 𝕀 = ⨁_{H} W (H)⊗ 𝕀(H).

Proof: The proof follows that of 13.2.2, but is slightly simpler.

The behaviour of the homological Eilenberg-MacLane spectra JN is rather different from their more familiar cohomological counterparts, and this is manifested in the fixed points. Suppose N is a coMackey functor with corresponding hSB^{op}-module W, and let N(F) = lim_{→}_{H} N(H) = ⨁_{H} W (H). Let us define D(N) and E(N) by the exact sequence

0 ⟶ D(N) ⟶ N(F) ⟶ N(𝕋) ⟶ E(N) ⟶ 0.

Thus E(N) is measures the failure of an F-induction theorem for N, and D(N) measures the precision of such a theorem.

Lemma 13.3.3.

π_{k}(Φ^{𝕋}JN) = \{ \begin{matrix}E(N) & if k = 0 \\ D(N) & if k = 1 \\ N(F) & if k is at least 3 and odd \\ 0 & otherwise \end{matrix}

Proof: This follows immediately from the cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF and 13.3.2.

Note in particular that Φ^{𝕋}JN is only an Eilenberg-MacLane spectrum if N is a F̃- projective.

Because N(F) ⊗_{(O_{F})_{0}} 𝕀 is uniquely divisible on all non-zero elements the assembly map is easily identified in this case.

Corollary 13.3.4. The map JN ∧ ẼF ⟶ JN ∧ ΣEF_{+} is classified by the O_{F}- morphism

q : π_{∗}(Φ^{𝕋}JN) ⊗ t_{∗}^{F} ⟶ N(F) ⊗_{(O_{F})_{0}} Σ𝕀

defined by q(x ⊗ y) = q(x) ⊗ ϵ(y), where ϵ : t_{∗}^{F} ⟶ Σ^{2}𝕀 is the quotient map.

13.4. Brown-Comenetz spectra.

Another interesting class of spectra are the Brown-Comenetz spectra. If I is any injective Mackey functor we may define the functor hI_{𝕋}^{∗}(•) by

hI_{𝕋}^{n}(X) = Hom(\underline{π}_{n}^{𝕋}(X), I).

Since I is injective this is an exact functor of X and hence it is represented by a spectrum hI. The reader may find the homotopy groups of hI more complicated than expected, and in particular hI is very rarely bounded below. For further information on Mackey functors, and associated notation the reader is referred to Appendix A. In particular, A.15 shows that any injective is a sum of those of two types, and it is enough to deal with F-injectives and F̃-injectives.

Lemma 13.4.1. (i) If I is an F-injective Mackey functor

\underline{π}_{k}^{𝕋}(hI) = \{ \begin{matrix}I & if k = 0 \\ I|_{F} & if k = −1 \\ 0 & otherwise, \end{matrix}

where I|_{F} is the largest subfunctor of I which is zero at 𝕋. (ii) If I = L[U] is an F̃-injective Mackey functor

\underline{π}_{∗}^{𝕋}(hI) = \{ \begin{matrix}L[U] & if k = 0 \\ CF[U] & if k = −1 \\ L[U^{F}] & if k is odd and ≤ −3 \\ 0 & otherwise. \end{matrix}

Here U^{F} is the vector space of U-valued functions on F, and CF[U] is the F-injective defined by CF[U](𝕋) = U^{F} and CF[U](H) = U with restriction maps being the relevant projections.

Proof: First recall the homotopy functors of cells from A.16. Since \underline{π}_{0}^{𝕋}(S^{0}) = P_{𝕋}, and \underline{π}_{0}^{𝕋}(σ_{H}^{0}) = P_{H}, it follows from the Yoneda lemma that \underline{hI}_{𝕋}^{0} = I whichever sort of injective I is.

On the other hand, in positive degrees \underline{π}_{∗}^{𝕋}(S^{0}) is L[ℚF] in each odd degree and 0 in each even degree. It is easy to verify that Hom(L[U], M) = Hom(U, C(M)) where the core is defined by C(M) = ker(M(𝕋) ⟶ ∏_{H} M(H)). Observe that C(L[U]) = U and C(I) = 0 if I is a F-injective.

For each finite subgroup we have \underline{π}_{1}^{𝕋}(σ_{H}^{0}) = R_{H}, and an extension

0 ⟶ P_{H} ⟶ R_{H} ⟶ L[ℚ] ⟶ 0.

Since Hom(P_{H}, L[U]) = 0 and Hom(L[ℚ], I) = 0 if I is F-injective, this is enough to calculate the homotopy groups. In the case that I is F-injective this also determines the functors. Finally we need to understand I = L[U] and in particular I^{∗} := \underline{π}_{−1}^{𝕋}(hI), or more precisely the restriction maps I^{∗}(𝕋) = Hom(ℚF, U) ⟶ U = I^{∗}(H).

Unravelling definitions we see that the restriction map is induced by applying Hom( , L[U]) to the map π_{∗} : R_{H} = \underline{π}_{1}^{𝕋}(σ_{H}^{0}) ⟶ \underline{π}_{1}^{𝕋}(S^{0}) = L[ℚF] induced by the projection σ_{H}^{0} ⟶ S^{0}. Since Hom(V, L[U]) = Hom(V (𝕋), U), it is only the part of π_{∗} at 𝕋 that is relevant, which is dealt with in the following lemma.

Lemma 13.4.2. The map π_{∗} : ℚ = π_{1}^{𝕋}(σ_{H}^{0}) ⟶ π_{1}^{𝕋}(S^{0}) = ℚF is the inclusion of the Hth factor.

Proof: The tom Dieck splitting isomorphism states that a certain natural map from a direct sum to π_{1}^{𝕋}(X) is an isomorphism, for any space X. The Hth factor is π_{1}(E(𝕋/H_{+}) ∧_{𝕋/H} ΣX^{H}) ≅ π_{0}(X^{H}), and the map Φ^{H}(σ_{H}^{0}) ⟶ (G/H_{+})^{H} ⟶ (S^{0})^{H} induces a bijection of π_{0}.

Note in particular that \underline{π}_{0}^{𝕋}(hJ) = J, so that if M is an arbitrary Mackey functor one may define a spectrum hM by taking a resolution 0 ⟶ M ⟶ J_{0} ⟶ J_{1} ⟶ 0 of Mackey functors and realizing it as a cofibre sequence hM ⟶ hJ_{0} ⟶ hJ_{1}. It is easy to see there is a short exact sequence

0 ⟶ Ext(\underline{π}_{n}^{𝕋}(ΣX), M) ⟶ hM_{𝕋}^{n}(X) ⟶ Hom(\underline{π}_{n}^{𝕋}(X), M) ⟶ 0.

We remark that it is extremely easy to construct an Adams resolution by spectra hJ, and its convergence is immediate from the ordinary Whitehead theorem. Its E_{2} term is also reasonably computable. However the spectral sequence does not seem useful except perhaps for maps from bounded below spectra to finite Postnikov towers.

Lemma 13.4.3. (i) For any F-injective Mackey functor I with associated hSB-module V there is an equivalence

hI ∧ EF_{+} ≃ ⋁_{H} σ_{H}^{−1} ∧ S^{0}[V (H)].

We might reasonably write σ_{F}^{−1}[I] for this spectrum. (ii) For any F̃-injective Mackey functor I = L[U] there is an equivalence

hI ∧ EF_{+} ≃ Σ^{−1}EF_{+}[U].

Proof: Note that

[S^{0}, hI ∧ σ_{H}^{0}]_{k}^{𝕋} = [σ_{H}^{−1}, hI]_{k}^{𝕋} = \underline{π}_{k−1}^{𝕋}(hI)(σ_{H}^{0}),

and refer to 13.4.1 for the values this takes.

For Part (ii) we argue as in the proof of 13.2.2. If I is an F̃-injective the displayed group is only nonzero for k = 0. The homotopy groups follow as before, and the action of c_{H} follows from the fact that hI ∧ E⟨H⟩ is H-equivariantly an Eilenberg-MacLane spectrum.

For Part (i) the shape is slightly different. As above we see that π_{∗}^{𝕋}(hI ∧ σ_{H}^{0}) = V (H) ⊕ ΣV (H). Now the attaching maps σ_{H}^{2n+1} ⟶ σ_{H}^{2n} in E⟨H⟩ are non-trivial in π_{∗}^{𝕋}, and hence so are their duals. It follows from the definition of hI_{𝕋}^{∗}(•) that the map

[S^{2n+1}, hI ∧ σ_{H}^{2n+1}]^{𝕋} ⟶ [S^{2n+1}, hI ∧ σ_{H}^{2n}]^{𝕋}
∥  ∥
Hom(\underline{π}_{1}^{𝕋}(σ_{H}^{0}), I) ⟶ Hom(\underline{π}_{0}^{𝕋}(σ_{H}^{0}), I)
∥  ∥
V (H)  ⟶  V (H)

is an isomorphism. Hence, by induction on n, the limit π_{∗}^{𝕋}(hI ∧ E⟨H⟩) is concentrated in degree 0, where it is V (H).

Lemma 13.4.4. (i) For any F-injective Mackey functor I with associated hSB-module V we have

π_{k}(Φ^{𝕋}hI) = \{ \begin{matrix}∏_{H} V (H)/ ⊕_{H} V (H) & if k = 0 \\ 0 & otherwise \end{matrix}

(ii) For any F̃-injective Mackey functor I = L[U] we have

π_{k}(Φ^{𝕋}hI) = \{ \begin{matrix}UF & if k is odd and ≥ 3 \\ \overline{UF} & if k = 1 \\ U^{F} & if k is odd and ≤ −1 \\ 0 & otherwise \end{matrix}

where \overline{UF} is the kernel of the natural map UF ⟶ U.

Proof: In both cases we consider the cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF, and use 13.4.3 and 13.4.1 to give the groups in the long exact sequence in π_{∗}^{𝕋}. The connecting maps are identified using the definition of hI as it applies to identify the map hI_{𝕋}^{0}(σ_{H}^{−1}) ⟶ hI_{𝕋}^{0}(S^{0}) as in 13.4.3 (i).

Corollary 13.4.5. If I is an F-injective, then in the notation of 13.4.3

hI ≃ σ_{F}^{−1}[I] ∨ ẼF ∧ S^{0}[∏_{H} V (H)/ ⨁_{H} V (H)].

In particular, if V (H) is only nonzero for finitely many H, then hI ≃ σ_{F}^{−1}[I] is an Fspectrum.

Proof: Since t_{∗}^{F} and π_{∗}^{𝕋}(σ_{F}^{−1}[I]) are in even degrees, the assembly map is detected as an element f ∈ Hom(t_{∗}^{F} ⊕ ∏_{H} V (H)/ ⨁_{H} V (H), ⨁_{H} V (H)). But if f ≠ 0 with f(x) nonzero in the Hth component then f(e_{H}x) ≠ 0; but e_{H}x = c_{H} x̃ for some x̃ with degree 2, and by dimension f(x̃) = 0.

Corollary 13.4.6. If I = L[U] the map hI ∧ ẼF ⟶ hI ∧ ΣEF_{+} is classified by the O_{F}-morphism

q : π_{∗}(Φ^{𝕋}hI) ⊗ t_{∗}^{F} ⟶ U ⊗ Σ𝕀

defined by the fact that q(x ⊗ c_{H}^{k}) is x(H) if it lies in a nonzero group; this applies whether x is in UF, \overline{UF} or U^{F}.

Proof: This follows from the above discussion unless k is negative. Furthermore, since U ⊗ Σ𝕀 is uniquely divisible in positive degrees, the result follows unless x is of negative degree.

Let us therefore suppose x ∈ U^{F} comes from π_{−2k+1}(Φ^{𝕋}hI), with k ≥ 1. It suffices to show that q(x ⊗ c_{H}^{−k}) is x(H). For this we must examine the map

hI ∧ ẼF ∧ σ^{−kV (H)} ⟶ hI ∧ Σ^{1}E⟨H⟩ ∧ σ^{−kV (H)}

in π_{−2k+1}^{𝕋} = [S^{−2k+1}, ·]^{𝕋}. Transposing σ^{−kV (H)} across into its dual in the domain, and using the fact that Ẽ⟨H⟩ ⟶ E⟨H⟩ is the direct limit of maps σ^{lV (H)} ⟶ ΣE⟨H⟩^{(2l−2)} with dual σ^{−lV (H)} ⟵ Σ^{−2l}E⟨H⟩^{(2l−2)} we must examine the direct system

[σ^{kV (H)} ∧ σ^{−lV (H)}, Σ^{2k−1}hI]^{𝕋} ⟶ [σ^{kV (H)} ∧ Σ^{−2l}E⟨H⟩^{(2l−2)}, Σ^{2k−1}hI]^{𝕋}.

Of course [X, Σ^{2k−1}hI] = Hom(π_{2k−1}^{𝕋}(X), U) because I = L[U], so we only need to understand

σ^{(k−l)V (H)} ≃ σ^{kV (H)} ∧ σ^{−lV (H)} ⟵ σ^{kV (H)} ∧ Σ^{−2l}E⟨H⟩^{(2l−2)} ≃ Σ^{2k−2l}E⟨H⟩^{(2l−2)}

in π_{2k−1}. It is more convenient to consider the previous term in the cofibre sequence Σ^{−1} ∧ σ^{kV (H)}, and observe this has zero homotopy in positive odd degrees. The required map is surjective for each l and hence also in the limit.

We may describe hI for an F̃-injective I in more familiar terms.

Lemma 13.4.7. If I = L[U] there is a cofibre sequence

ẼF ∧ S^{0}[U] ⟶ hI ⟶ Σ^{−1}F (EF_{+}, S^{0}[U]).

Proof: By 13.4.3 we have a map

hI ∧ EF_{+} = Σ^{−1}EF_{+} ∧ S^{0}[U] ⟶ Σ^{−1}S^{0}[U],

and hence, by adjunction a map hI ⟶ Σ^{−1}F (EF_{+}, S^{0}[U]). By construction this is an F-equivalence, so that the fibre is an F-contractible spectrum. The result follows provided we check that the map hI ⟶ Σ^{−1}F (EF_{+}, S^{0}[U]) is an isomoprhism in negative dimensional homotopy. This is easily verified by adjunction, since diagrams S^{k} ⟶ hI ⟶ Σ^{−1}F (EF_{+}, S^{0}[U]) correspond to diagrams S^{k} ∧ EF_{+} ⟶ hI ∧ EF_{+} ⟶ Σ^{−1}S^{0}[U].

Part III Applications.

162

CHAPTER 14 Introduction to Part III.

14.1. General Outline

The material in Part III is a collection of applications of the general theory developed in Parts I and II. Accordingly, the chapters are completely independent of each other. Since most of the work has already been incorporated in the general framework, the sections are rather short, and uncluttered by technicalities.

We begin with Chapter 15, which consists of five independent sections answering the main questions motivating this study. Perhaps the most obvious problem of all, in the light of Haeberly’s example, is to understand the behaviour of the Atiyah-Hirzebruch spectral sequence. We show that it collapses at E_{2} for all F-spaces if and only if π_{∗}^{𝕋}(K ∧ EF_{+}) is injective. In general, the differentials encode the Adams short exact sequence, but there are differentials of arbitrary length. For arbitrary spaces, the main message is that the Atiyah-Hirzebruch spectral sequence is not a natural tool. Alternatively, we can always use a cellular decomposition of the domain to try to understand maps. In our case, the graded orbit category can be made quite explicit, and we can therefore do homological algebra over the category of additive functors on this category. We may then construct a spectral sequence, whose E_{2} term is calculable and given by homological algebra over the graded orbit category. It is also obviously convergent, but it does not seem a practical tool in general, because it is usually a half-plane spectral sequence. The moral is that we should not decompose spectra by Eilenberg-MacLane spectra or by cells, but rather by injectives in the standard model.

Another basic construction of spectra is by taking the suspension spectrum of a space. Equivariantly it is known that suspension spectra have certain special properties, such as tom Dieck splitting, which are not enjoyed by all spectra. In particular, the inclusion of the H-fixed points can be regarded as a map of 𝕋-spaces, and this has implications for the model. However, since our model has no record of purely unstable information, we cannot hope for a precise characterization of suspension spectra.

The special case of K-theory is interesting because it has Bott periodicity, and hence Euler classes of its own. The representing spectrum also turns out to be formal in the torsion model, so that the K-theory of any spectrum depends only on its homology in the torsion model. On the other hand, our analysis is really only the beginning of a study melding the formalisms of the present work with the geometric information of K-theory: in particular it would be interesting to understand the Chern character in more detail and to relate our results to the work of Brylinski and collaborators [3].

After our analysis of function spectra, the rational Segal conjecture is little more than an elementary example: in the torsion model DE𝕋_{+} is formal, and represented by the natural map

t_{∗}^{F} ⊗ t_{∗}^{F} ⟶ t_{∗}^{F} ⟶ t_{∗}^{F}/O_{F} = Σ^{2}𝕀 ⟶ ℚ[c, c^{−1}]/ℚ[c].

We also present a more naive approach by way of comparison.

In Chapter 16 we consider the well known 𝕋-equivariant cohomology theory given by cyclic cohomology. This is very simple rationally; more generally rational Tate spectra are also rather simple, but we may make certain intriguing algebraic connections. Finally we are able to identify the integral Tate spectrum t(Kℤ) of integral complex K-theory Kℤ. This is of interest because t(Kℤ) is known to be H-equivariantly rational for all finite subgroups H. In fact, writing Kℤ for integral complex K-theory for emphasis, we identify the spectra t(Kℤ) ∧ ẼF and t(Kℤ) ∧ ΣEF_{+} together with the map of which t(Kℤ) is the fibre. The first is obtained from K-theory with suitable coefficients by inflating and smashing with ẼF, and the second is rational, and we can identify it as an injective Eulertorsion O_{F}-module.

The final Chapter 17 is more substantial. We turn to examples gaining their importance from algebraic K-theory. Bökstedt, Hsiang and Madsen define the topological cyclic cohomology of a ring or a space [2]. It is obtained by performing various constructions on the topologicial Hochschild homology spectrum and can be used as a close approximation to the completed algebraic K-theory of suitable rings. Hesselholt and Madsen [17] identify the structure required of a 𝕋-spectrum X for one to be able to construct T C(X). Spectra X with this structure are called cyclotomic spectra. The motivation for the notion of a cyclotomic spectrum comes from the free loop space ΛX = map(𝕋, X) on a 𝕋-fixed space X. This has the property that if we take K-fixed points we obtain the 𝕋/K-space map(𝕋/K, X), and if we identify the circle 𝕋 with the circle 𝕋/K by the |K|th root isomorphism we recover ΛX. For spectra, one also needs to worry about the indexing universe, but a cyclotomic spectrum is basically one whose geometric fixed point spectrum Φ^{K}X, regarded as a 𝕋-spectrum, is the original 𝕋-spectrum X. After the suspension spectrum of a free loop space, the principal example comes from the topological Hochschild homology of T HH(F) of a functor F with smash products. Given such a cyclotomic spectrum X, one may construct the topological cyclic spectrum T C(X) of Bökstedt-Hsiang-Madsen [2], which is a non-equivariant spectrum. An intermediate construction of some interest is the 𝕋-spectrum T R(X). Although these constructions are principally of interest profinitely, it is instructive to identify the cyclotomic spectra in our model and follow the constructions through. We identify the rational cyclotomic spectra in the torsion model: they are the spectra X so that the function [N] : F ⟶ torsionℚ[c] − modules modelling EF_{+} ∧ X is constant, and so that the structure map t_{∗}^{F} ⊗ V ⟶ ΣN commutes with any translation of the finite subgroups. It therefore factors through t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F}/O_{F} ⊗ V and the map O_{F}/O_{F} ⊗ V ⟶ ΣN is a direct sum of copies of ℚ[c, c^{−1}]/ℚ[c] ⊗ V ⟶ Σ[N](1). Furthermore, we may recover Goodwillie’s theorem [7] that for any cyclotomic spectrum X we have T C(X) = X^{h𝕋}: topological cyclic cohomology coincides with cyclic cohomology in the rational setting.

14.2. Prospects and problems.

The main theoretical problem is to show that the equivalence of the category of 𝕋-spectra and the derived category of the standard model can be obtained from a chain of equivalences arising from adjoint pairs of functors on underlying Quillen model categories. This would inevitably be linked with a better understanding of the meaning of the standard model.

One of the most interesting prospective applications is that of understanding rational 𝕋-equivariant elliptic cohomology. Constructions have recently been given by Grojnowski [8] and by Ginzburg-Kapranov-Vasserot [6], and the cohomology of any 𝕋-space is a sheaf over an elliptic curve. One can ask if these theories are represented. This would involve considering sheaves of 𝕋-spectra over an elliptic curve, and it would seem a sensible first step to consider sheaves of objects of A.

Both in this case and that of K-theory, there is the task of relating the general model to the geometry of the cohomology theory: in practice this will involve concentration on the Chern character, and comparison with the work of Brylinski [3]. There are a number of other classes of spectra which we do not understand as well as we should like, such as suspension spectra, free loop spaces, THH, ......

In the present work we have concentrated entirely on the circle group 𝕋. Although it is unlikely to be possible to give so complete a picture as we have done for 𝕋-spectra, we hope to consider other small groups in due course. The continuous quaternion and dihedral groups are prime candidates, both by virtue of their simplicity and the prospects for applications. However, consideration of the case of Mackey functors [11] shows that it is necessary to replace the underlying algebra of O_{F}-modules by that of sheaves over spaces of subgroups, since the topology on the space of subgroups can no longer be ignored. For groups of rank greater than 1, the injective dimension of the relevant algebraic categories will be greater than 1. There is therefore no prospect of obtaining splittings for formal reasons, and models must be based on a more complete geometric understanding than we have used here.

166 14. INTRODUCTION TO PART III.

CHAPTER 15 Classical miscellany.

The sections in this chapter are independent of each other; each answers a natural question about rational 𝕋-spectra.

In Section 15.1 we give a complete analysis of the behaviour of the Atiyah-Hirzebruch spectral sequence for F-spectra, generalising the study in Section 1.4. Section 15.2 sets up a calculable spectral sequence for calculating maps between 𝕋-spectra from a cellular decomposition, based on the graded orbit category. Section 15.3 shows how the existence of tom Dieck splitting makes the models of suspension spectra very special. Section 15.4 finally returns to complex 𝕋-equivariant K-theory, and identifies its place in the torsion model, showing that it is formal. Finally, Section 15.5 identifies the functional dual DE𝕋_{+}, giving the rational analogue of the geometric equivariant Segal conjecture.

15.1. The collapse of the Atiyah-Hirzebruch spectral sequence.

The purpose of this section is to analyse the Atiyah-Hirzebruch spectral sequence for F-spectra.

We suppose given an arbitrary 𝕋-equivariant cohomology theory K_{𝕋}^{∗}(·), and consider the Atiyah-Hirzebruch spectral sequence

E_{2}^{s,t} = H_{𝕋}^{s}(X; \underline{K}_{𝕋}^{t}) ⟹ K_{𝕋}^{∗}(X).

This may be constructed either by using the skeletal filtration of X or, in complete generality, by using the Postnikov filtration of K. It is conditionally convergent if X is bounded below.

First, let us suppose that K = E𝕋_{+}^{(2n)}; we observe that, if X is free, the only relevant part of the Mackey functor \underline{K}_{𝕋}^{∗} is the nonequivariant homotopy of K. Thus the spectral sequence is concentrated on the lines q = 1 and q = 2n + 2. There are therefore no differentials except

d_{2n+2} : H^{p}(X/𝕋) = H_{𝕋}^{p}(X; \underline{K}_{𝕋}^{2n+2}) ⟶ H_{𝕋}^{p+2n+2}(X; \underline{K}_{𝕋}^{1}) = H^{p+2n+2}(X/𝕋).

If we take the special case X = E𝕋_{+}^{(2m)}, we see that the differential must be given by multiplication by c^{n+1} in order to give the correct answer, as calculated by the Adams short exact sequence. In particular, the differential is non-zero if and only if m > n.

Theorem 15.1.1. If X is an F-spectrum then the Atiyah-Hirzebruch spectral collapses at E_{2} if π_{∗}^{𝕋}(K) is injective over O_{F}. Conversely, if the Atiyah-Hirzebruch Spectral sequence collapses at E_{2} for all F-spectra X, then π_{∗}^{𝕋}(K) is injective.

More precisely, any differential d_{2i+1} is zero, and the nonzero differentials d_{2n+2} are all explained by the above example, in a sense to be made precise in the proof.

Proof: The first observation is that if X is an F-spectrum then K_{𝕋}^{∗}(X) = [X, K]_{𝕋}^{∗} = [X, K ∧ EF_{+}]_{𝕋}^{∗}. Since the spectral sequence is natural, we may suppose that both X and K ∧ EF_{+} have homotopy in even degrees. We argue that if π_{∗}^{𝕋}(K) is injective then the E_{2} term is entirely in even total degrees, and hence the spectral sequence collapses.

First note that for any F-spectrum T,

K_{𝕋}^{∗}(T) = Hom(π_{∗}^{𝕋}(T), π_{∗}^{𝕋}(K ∧ EF_{+})).

Thus, taking T = σ_{H}^{0}, we see that, for any finite subgroup H, the H-equivariant basic homotopy groups are purely in odd degrees, since π_{∗}^{𝕋}(σ_{H}^{0}) = Σℚ is odd. Thus the part of the graded Mackey functor \underline{K}_{𝕋}^{∗} over F is entirely in odd degrees. On the other hand, since X is an F-spectrum, [X, HM]_{𝕋}^{∗} = [X, HM ∧ EF_{+}]_{𝕋}^{∗}, and HM ∧ EF_{+} is a wedge of copies of E⟨H⟩, with one factor for each basis element of V (H) = e_{H}M(H). Thus, if we let 𝕀 ⊗ M = ⨁_{H} 𝕀(H) ⊗ V (H) we have

H_{𝕋}^{∗}(X; M) = Hom(π_{∗}^{𝕋}(X), Σ𝕀 ⊗ M),

which is entirely in odd degrees. Thus E_{2}^{∗,∗} = H_{𝕋}^{∗}(X; \underline{K}_{𝕋}^{∗}) is in even total degrees as claimed.

Now suppose that the Atiyah-Hirzebruch spectral sequence does not collapse, and that x ∈ E_{r}^{p,q} supports a non-zero differential d_{r}(x) = y ≠ 0. We shall prove that r = 2n + 2 for some n, and that the differential is explained by naturality and the differentials described above.

We may pick a representative x′ ∈ E_{1}^{p,q} = [X^{(p)}/X^{(p−1)}, K]_{𝕋}^{p+q} for x. This shows that x′ is supported on a map σ_{H}^{p} ⟶ X/X^{(p−1)} ⟶ Σ^{p+q}K. Replacing X by X/X^{(p−1)} and suspending, we may assume that X is (−1)-connected and p = 0. We may therefore replace K by its connective cover K_{0}^{∞} without changing the fate of x in the spectral sequence. Now, letting K_{m}^{n} denote the Postnikov section of K with non-zero homotopy groups in degree i with m ≤ i ≤ n, we consider the Postnikov tower of K:

K_{r−1}^{r−1} ⟶ K_{0}^{r−1} ⟶ ΣK_{r}^{r}
↓
K_{r−2}^{r−2} ⟶ K_{0}^{r−2} ⟶ ΣK_{r−1}^{r−1}
↓
K_{2}^{2} ⟶ K_{0}^{2} ⟶ ΣK_{3}^{3}
↓
K_{1}^{1} ⟶ K_{0}^{1} ⟶ ΣK_{2}^{2}
↓
X − −^{x} ⟶ K_{0}^{0} ⟶ ΣK_{1}^{1}

By hypothesis, x : X ⟶ K_{0}^{0} lifts to x^{(r)} : X ⟶ K_{0}^{r−2} so that the composite x^{(r)} : X ⟶ K_{0}^{r−2} ⟶ ΣK_{r−1}^{r−1} is essential and represents y. Since X is an F-spectrum, the behaviour is unaltered if the diagram is smashed with EF_{+}. Since HM ∧ EF_{+} is injective, all maps at the E_{2}-term are detected by their d-invariant, and it is thus appropriate to examine the effect of taking homotopy of the above diagram smashed with EF_{+}. The basic observation is that,

π_{∗}^{𝕋}(HM ∧ EF_{+}) = Σ𝕀 ⊗ M,

which is in odd degrees. Furthermore, x is detected by π_{1}^{𝕋}, and we need only look at the odd graded part. This immediately shows that all odd differentials are zero, so that r = 2n + 2 for some n.

Next, we note that the maps

π_{∗}^{𝕋}(K_{0}^{2s+1} ∧ EF_{+}) ⟶ π_{∗}^{𝕋}(K_{0}^{2s} ∧ EF_{+})

are injective in odd degrees, whilst the maps

π_{∗}^{𝕋}(K_{0}^{2s+2} ∧ EF_{+}) ⟶ π_{∗}^{𝕋}(K_{0}^{2s+1} ∧ EF_{+})

are surjective in odd degrees. Furthermore, the image of the composite consists of elements divisible by c. We thus find the diagram

π_{∗}^{𝕋}(K_{0}^{2n} ∧ EF_{+})  ⟶ π_{∗}^{𝕋}(ΣK_{2n+1}^{2n+1} ∧ EF_{+}) = Σ^{2n+3}𝕀 ⊗ \underline{K}_{𝕋}^{2n+1}
↗  ↓
π_{∗}^{𝕋}(X) ⟶  π_{∗}^{𝕋}(K_{0}^{0} ∧ EF_{+}) = Σ𝕀 ⊗ \underline{K}_{𝕋}^{0}  ,

and we know that some element z̃ of π_{2n+3}^{𝕋}(X) maps to z ∈ π_{2n+3}^{𝕋}(K_{0}^{2n} ∧ EF_{+}), and c^{n+1}z detects x, whilst the image of z in π_{2n+3}^{𝕋}(ΣK_{2n+1}^{2n+1} ∧ EF_{+}) detects y. We therefore find a map

π_{∗}^{𝕋}(E⟨H⟩^{(2n+2)}) = Σ𝕀_{0}^{2n+2} ⟶ π_{∗}^{𝕋}(X)

with z̃ as the image of the top class, and x̃ as the image of the bottom class. By the Adams short exact sequence, this is realised by a map E⟨H⟩^{(2n+2)} ⟶ X.

15.2. Orbit category resolutions.

Integrally one expects the cellular decomposition to be unhelpful in global calculations because one does not know the stable homotopy groups of spheres. Rationally, everything is much simpler. For finite groups, cells are Eilenberg-MacLane spectra, and hence the cellular decomposition is simply another way of viewing the complete splitting [14]. In the present 𝕋-equivariant context, cells are not all Eilenberg-MacLane spectra, but one may understand the entire graded category of natural or basic cells. We shall concentrate on the graded category hSB_{∗} of basic cells (i.e. the full subcategory of the graded stable category with the basic cells as objects).

One thus views the entire homotopy functor \underline{π}_{∗}^{𝕋}(X) of X as a module over the graded category hSB_{∗}. By the Yoneda lemma, the case when X is a cell plays the role of a free object, and a resolution is form of cellular approximation. We understand maps of degree 0 from the discussion of Mackey functors presented in Appendix A. Referring to 2.1.4, we see that composition in hSB_{∗} is usually zero for dimensional reasons. In fact, there are no maps of degree 2 between any pair of objects, and the only case with maps of degree more than 1 is [σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{∗}^{𝕋} = ℚ⊕Σ(ℚF[c^{−1}]). For maps of degree 1 we have [σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{1}^{𝕋} = ℚF, [σ_{𝕋}^{0}, σ_{H}^{0}]_{1}^{𝕋} = ℚ, [σ_{H}^{0}, σ_{𝕋}^{0}]_{1}^{𝕋} = 0 and [σ_{H}^{0}, σ_{H}^{0}]_{1}^{𝕋} = ℚ; for maps of degree 0 we have [σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{0}^{𝕋} = ℚ, [σ_{𝕋}^{0}, σ_{H}^{0}]_{0}^{𝕋} = 0, [σ_{H}^{0}, σ_{𝕋}^{0}]_{0}^{𝕋} = ℚ and [σ_{H}^{0}, σ_{H}^{0}]_{0}^{𝕋} = ℚ. It therefore remains to deal with the composite of a degree 1 morphism (which must be of form x_{𝕋}^{𝕋} : σ_{𝕋}^{1} ⟶ σ_{𝕋}^{0}, x_{H}^{H} : σ_{H}^{1} ⟶ σ_{H}^{0} or a multiple of the transfer tr_{H}^{𝕋} : σ_{𝕋}^{1} ⟶ σ_{H}^{0}) and a degree 0 morphism (which is either a multiple of the relevant identity or a multiple of the projection pr_{𝕋}^{H} : σ_{H}^{0} ⟶ σ_{𝕋}^{0}). We now deal with these remaining cases.

Lemma 15.2.1. The composites of the x’s and y’s are as follows.

(i) x_{𝕋}^{𝕋}pr_{𝕋}^{H} = 0 and pr_{𝕋}^{H}x_{H}^{H} = 0, (ii) tr_{H}^{𝕋}pr_{𝕋}^{H} ≠ 0 and pr_{𝕋}^{H}tr_{H}^{𝕋} corresponds to the inclusion of the Hth factor in ℚF = [σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{1}^{𝕋}.

Proof: Part (i) is clear since the composites lie in the zero group. The first fact in Part (ii) follows from the explicit geometric construction of the transfer. The second fact in Part (ii) follows by construction of the isomorphism in tom Dieck splitting [18, V.11].

It is thus natural to take τ_{H} = tr_{H}^{𝕋}pr_{𝕋}^{H} as the basic generator of [σ_{H}^{0}, σ_{H}^{0}]_{1}^{𝕋} and δ_{H} = pr_{𝕋}^{H}tr_{H}^{𝕋} as a standard basis element of [σ_{𝕋}^{0}, σ_{𝕋}^{0}]_{1}^{𝕋}.

Now, suppose given any 𝕋-spectrum X, we consider [·, X]_{∗}^{𝕋} as a contravariant functor on the graded orbit category. As such, we may form a projective resolution, and we may realise it. In fact we may construct a map P_{0} ⟶ X, which is surjective in graded equivariant homotopy for all subgroups of 𝕋, and in which P_{0} is a wedge of cells. Now let X_{1} be the cofibre of this, and iterate to form the diagram

X = X_{0} ⟶ X_{1} ⟶ X_{2} ⟶ ⋯
↑  ↑  ↑
P_{0}  P_{1}  P_{2}

By construction, all the maps X_{s} ⟶ X_{s+1} are zero in H-equivariant homotopy for all subgroups H, and so holim_{→}_{s} X_{s} is contractible by the Whitehead theorem.

It is convenient to form the dual diagram with X^{p−1} = fibre(X ⟶ X_{p}), so that X^{−1} = ∗ and X ≃ holim_{→} X^{p}:

∗ = X^{−1} ⟶ X^{0} ⟶ X^{1} ⟶ ⋯
↑  ↑  ↑
Σ^{−1}P_{0}  Σ^{−1}P_{1}  Σ^{−1}P_{2}

Replacing the maps by inclusions, we view this as a filtration of X with subquotients X^{p}/X^{p−1} = P_{p}. We may now construct a spectral sequence by applying [·, Y]_{𝕋}^{∗} to the diagram. It has E_{1}^{p,q} = [P_{p}, Y]_{𝕋}^{p+q}, and D_{1}^{p,q} = [X^{p}, Y]_{𝕋}^{p+q}. The spectral sequence lies in the right half-plane, and the differentials are cohomological, so that d_{r} : E_{r}^{p,q} ⟶ E_{r}^{p+r,q−r+1}, and it is evidently conditionally convergent. Finally, we see by construction that the sequence

⋯ ⟶ \underline{π}_{∗}^{𝕋}(Σ^{−2}P_{2}) ⟶ \underline{π}_{∗}^{𝕋}(Σ^{−1}P_{1}) ⟶ \underline{π}_{∗}^{𝕋}(P_{0}) ⟶ \underline{π}_{∗}^{𝕋}(X) ⟶ 0

is exact. Hence we may identify the E_{2} term as an Ext group, and the spectral sequence takes the form

E_{2}^{p,q} = ExthSB_{∗}^{p,q}(\underline{π}_{∗}^{𝕋}(X), \underline{π}_{∗}^{𝕋}(Y)) ⟹ [X, Y]_{𝕋}^{p+q}.

The fact that the category of Mackey functors is one dimensional gives a vanishing line if X is bounded below. Indeed, since cells are also (−1)-connected we may ensure P_{0} is (−1)- connected, and hence that X_{1} is 0-connected. This formality proves that if the resolution is dimensionally minimal, X_{p} is (p − 1)-connected. However, if we use basic cells and the fact that the category of Mackey functors is of projective dimension 1, we may ensure that X_{2} is 2-connected. By iteration we see that X_{2s} and P_{2s} are (3s − 1)-connected; similarly X_{2s+1} and P_{2s+1} are 3s-connected. Thus the map X^{2s−1} ⟶ X is (3s − 2)-connected, and the map X^{2s} ⟶ X is (3s − 1)-connected. Unfortunately this only seems to be useful if the homotopy of the spectrum Y is bounded above. Thus if Y_{H}^{q} = 0 for q ≤ −1 and all H, then the nonzero entries are in the first quadrant and lie on or above the line q = (p − 1)/2.

15.3. Suspension spectra.

In this section we suppose given a based 𝕋-space Z, and we identify the place of its suspension spectrum in our classification. We follow our usual convention of omitting notation for the suspension spectrum functor, and using the notation Ψ^{𝕋} for Lewis-May fixed point functor.

Our basic tool is tom Dieck splitting, which states that the Lewis-May fixed points of the suspension spectrum of Z is

Ψ^{𝕋}Z = Z ∨ ⋁_{K} E𝕋/K_{+} ∧_{𝕋/K} ΣZ^{K};

furthermore this is natural, and applies to stable retracts of spaces. The crucial simplification for spaces is that there is a map Z^{𝕋} ⟶ Z of 𝕋-spectra, and hence a diagram

ẼF ∧ Z^{𝕋} ⟶ ΣEF_{+} ∧ Z^{𝕋}
≃↓  ↓
ẼF ∧ Z ⟶ ΣEF_{+} ∧ Z.

The structure map ẼF ∧ Z ⟶ ΣEF_{+} ∧ Z thus factors through the corresponding map for Z^{𝕋}, which we understand completely, since Z^{𝕋} is rationally a wedge of spheres.

On the other hand, by naturality of tom Dieck splitting, we find

π_{∗}^{𝕋}(Z ∧ E⟨H⟩) = π_{∗}(E𝕋/H_{+} ∧_{𝕋/H} ΣZ^{H}) = H_{∗}(E𝕋/H_{+} ∧_{𝕋/H} ΣZ^{H}),

which we may certainly regard as computable.

Summary 15.3.1. The algebraic model of the suspension spectrum of a space Z is

t_{∗}^{F}⊗H_{∗}(Z^{𝕋}) ⟶ Σ^{2}𝕀⊗H_{∗}(Z^{𝕋}) = Σ^{2} ⨁_{H} H_{∗}(E𝕋/H_{+}∧_{𝕋/H}Z^{𝕋}) ⟶ Σ^{2} ⨁_{H} H_{∗}(E𝕋/H_{+}∧_{𝕋/H}Z^{H}).

The first map in the diplayed composite necessarily has zero e-invariant, and is simply induced by the quotient t_{∗}^{F} ⟶ Σ^{2}𝕀. However the second map is induced by the inclusion X^{𝕋} ⟶ X, and may have non-zero d and e invariant. Again we may be satisfied that the d-invariant is given by a homology calculation. For the e invariant, since the above discussion applies to retracts of spaces, we may assume that X^{𝕋} and EF_{+} ∧ X are of pure parity, and then identify the e invariant with the Borel homology of the cofibres X^{H}/X^{𝕋} as in [1].

There remains the question of whether this characterizes suspension spectra. More precisely, we cannot distinguish 𝕋-fixed suspensions so that we are asking if every model of the above sort is the model of a spectrum Σ^{n}Σ^{∞}X for some 𝕋-space X and some integer n. We do not have available the option of identifying the suspension spectrum functor, since there is no algebraic model of rational 𝕋-spaces. One might hope to realize finitely generated examples by explicit construction, but one would expect a certain amount of suspension to be necessary to achieve stability in each case. To obtain a global realization one would need a bound on these suspensions; since the model contains no data relevant to the achievment of stability, there is no ready way to do this.

15.4. K-theory revisited.

Let us consider the structure of the 𝕋-spectrum K representing rational equivariant K-theory. We know that K_{∗}^{H} = R(H)[β, β^{−1}] for all subgroups H, where R(H) is the rationalization of the complex representation ring and β ∈ K_{−2} is the Bott element. Now R = R(𝕋) = ℚ[z, z^{−1}], and R(H) = ℚ[z]/(z^{n} − 1) when H is of order n. The restriction maps are implicit in the notation here, and the induction maps to 𝕋 are zero (holomorphic induction maps are not part of the structure). Now, by Bott periodicity we have K-theory Euler classes c(H) of degree −2 for each finite subgroup H, obtained by applying the Thom isomorphism to the image of e(V (H)) in K-theory. We may apply Bott periodicity to obtain the usual K-theory Euler class λ(H) ∈ R(𝕋). In other words if H is of order n we have

λ(H) = 1 − z^{n} and c(H) = βλ(H).

Furthermore

λ(H) = ∏_{d|n} Φ_{d},

where Φ_{d} is the dth cyclotomic polynomial. Let S be the multiplicative set generated by the Euler classes 1 − z^{n}, and T be generated by the cyclotomic polynomials Φ_{d}; in practice the geometry localizes so as to invert S, which is algebraically the same as inverting T, and the latter is easier to understand. Let F = S^{−1}R = T^{−1}R, so that

π_{∗}^{𝕋}(K ∧ ẼF) ≅ F [β, β^{−1}] and π_{∗}^{𝕋}(K ∧ ΣEF_{+}) ≅ (F/R)[β, β^{−1}].

Since the cyclotomic polynomials are coprime, an element of F/R can be written uniquely as a sum of terms f_{d}(z)/Φ_{d}(z)^{n} for n ≥ 1 where f_{d}(z) ∈ R is not divisible by Φ_{d}(z). Hence

F/R = ⨁_{n} R[1/Φ_{n}]/R.

We should relate this to our geometric decompositions.

Lemma 15.4.1. If K(H) denotes the part of K in 𝕋−Spec/H as usual then

π_{∗}^{𝕋}(K(H)) = (R[1/Φ_{n}]/R)[β, β^{−1}]

and c_{H} acts as multiplication by βΦ_{n} times an automorphism.

Remark 15.4.2. Our map c_{H} is defined up to a non-zero rational number, whilst the K-theory Euler class is defined absolutely. The multiple is therefore well defined up to a non-zero scalar, but its exact value is not relevant to us at present. Crabb considers studies this value in greater detail [5].

Proof: The only part requiring proof is that the action of c_{H} is as claimed. We shall show that c_{H} acts as c(V (H)) times an automorphism on K ∧ E⟨H⟩. To do this, we let V = V (H), and unravel definitions.

We have a K-theory Thom isomorphism t : K ∧ S^{V} ⟶^{≃} K ∧ S^{|V |}, and the F-spectrum Thom isomorphism τ : S^{V} ∧ E⟨H⟩ ⟶^{≃} S^{|V |} ∧ E⟨H⟩; we need to know they are compatible in the sense that the composites

K ∧ S^{0} ∧ E⟨H⟩ ⟶ K ∧ S^{V} ∧ E⟨H⟩ ⟶ K ∧ S^{|V |} ∧ E⟨H⟩

are equal. Now, we observe that both are maps of K-module spectra, and hence it is sufficient to show both induce the same map S^{0} ∧ E⟨H⟩ ⟶ K ∧ S^{|V |} ∧ E⟨H⟩. Maps of this form are classified by K_{𝕋}^{|V |}(E⟨H⟩) ≅ lim_{←}_{n} K^{|V |}(E⟨H⟩^{(2n)}). Indeed the maps are classified by what they induce in homotopy:

[E⟨H⟩ ∧ K, S^{|V |} ∧ E⟨H⟩ ∧ K]_{0}^{K,𝕋} ⟶^{≅} Hom_{R}(R/Φ_{n}^{∞}, R/Φ_{n}^{∞}).

We know the K-theory Thom isomorphism gives multiplication by Φ_{n}. We may express the action of c_{H} in its Φ_{n}-adic expansion as multiplication by x_{0} + x_{1}Φ_{n} + x_{2}Φ_{n}^{2} + ⋯, where x_{i} ∈ R. It suffices to prove that the F-spectrum Euler class is multiplication by λΦ_{n} mod Φ_{n}^{2} for a non-zero scalar λ, i.e. that x_{0} = 0 and x_{1} = λ.

First, we know that the map σ_{H}^{0} ⟶ σ_{H}^{0} ∧ K induces the permutation module map ℚ = e_{H}A(H) ⟶ e_{H}R(H) = R/Φ_{n}. Now, we need to understand something of the map E⟨H⟩^{(2n)} ⟶ K ∧ E⟨H⟩^{(2n)}, and we can infer enough by considering the diagram

0  ℚ  ℚ
∥  ∥  ∥
π_{2k+1}^{𝕋}(E⟨H⟩^{(2k−2)}) ⟶ π_{2k+1}^{𝕋}(E⟨H⟩^{(2k)}) ⟶^{≅} π_{2k+1}^{𝕋}(σ_{H}^{2k})
↓ ↓ ↓
π_{2k+1}^{𝕋}(E⟨H⟩^{(2k−2)} ∧ K) ⟶ π_{2k+1}^{𝕋}(E⟨H⟩^{(2k)} ∧ K) ⟶ π_{2k+1}^{𝕋}(σ_{H}^{2k} ∧ K)
∥  ∥  ∥
R/Φ_{n}^{k}  R/Φ_{n}^{k+1}  R/Φ_{n}.

This shows that the image g_{k} of the generator of π_{2k+1}^{𝕋}(E⟨H⟩^{(2k)}) in π_{2k+1}^{𝕋}(E⟨H⟩^{(2k)}) = R/Φ_{n}^{k+1} is λ_{k} modulo Φ_{n}^{k} for a non-zero rational number λ_{k}.

Now consider the diagram

ℚ  ℚ
∥  ∥
π_{2k+1}^{𝕋}(E⟨H⟩^{(2k)}) ⟶^{≅} π_{2k+1}^{𝕋}(Σ^{kV} E⟨H⟩^{(0)})
≅↓  ↓≅
π_{2k+1}^{𝕋}(E⟨H⟩) ⟶^{c_{H}^{k}} π_{2k+1}^{𝕋}(Σ^{kV} E⟨H⟩)
↓  ↓
π_{2k+1}^{𝕋}(E⟨H⟩ ∧ K) ⟶ π_{2k+1}^{𝕋}(Σ^{kV} E⟨H⟩ ∧ K)
∥  ∥
R/Φ_{n}^{∞}  R/Φ_{n}^{∞}.

We see that c_{H}^{k} takes the image of g_{k} ∈ R/Φ_{n}^{k+1} to the image of 1 ∈ R/Φ_{n}. Now g_{k} is mapped to λ_{k}/Φ_{n}^{k+1} modulo elements annihilated by Φ_{n}^{k}. We conclude from the case k = 1 that x_{1} = λ_{1} as required. The general case shows that λ_{k} = λ_{1}^{k}.

Corollary 15.4.3. The ℚ[c_{H}]-module π_{∗}^{𝕋}(K(H)) is injective. Indeed, if L_{n}^{i} is the space of Laurent polynomials in z with poles of order at most i at a primitive nth root of unity, then multiplication by Φ_{n} gives an isomorphism L_{n}^{i+1}/L_{n}^{i} ⟶ L_{n}^{i}/L_{n}^{i−1}. Thus π_{∗}^{𝕋}(K(H)) ≅ I(H) ⊗ (L_{n}^{1}/L_{n}^{0})[β, β^{−1}], and hence

K(H) ≃ E⟨H⟩ ∧ S^{0}[ (L_{n}^{1}/L_{n}^{0})[β, β^{−1}] ].

Accordingly K is characterized by t_{∗}^{F} ⊗ π_{∗}^{𝕋}(K ∧ ẼF), π_{∗}^{𝕋}(K ∧ ΣEF_{+}) and the homomorphism between them. To make sense of the following statement, note that (F/R)[β, β^{−1}] is a module over O_{F}, since it is F-finite; more explicitly, if x ∈ R[1/Φ_{n}]/R and H is of order n, then c_{H}x = Φ_{n}xβ as mentioned above. Of course, multiplication by Φ_{n}^{−1} is not defined on F/R, but it makes sense for F.

Theorem 15.4.4. The 𝕋-spectrum K is the unique 𝕋-spectrum for which t_{∗}^{F} ⊗ π_{∗}^{𝕋}(K ∧ ẼF) ⟶ π_{∗}^{𝕋}(K ∧ ΣEF_{+}) is the map

q̂_{K} : t_{∗}^{F} ⊗ F [β, β^{−1}] ⟶ (F/R)[β, β^{−1}]

described as follows. For x ∈ O_{F} of degree −2k we have q̂_{K}(x ⊗ fβ^{l}) = x\overline{f}β^{l+k}, and q̂_{K}(c_{H}^{−k} ⊗ fβ^{l}) = \overline{Φ_{n}^{−k}f}β^{l−k}.

Proof: The value of q̂_{K}(x ⊗ fβ^{l}) is immediate from our method of calculating π_{∗}^{𝕋}(K ∧ ΣEF_{+}).

For q̂_{K}(c_{H}^{−k} ⊗ fβ^{l}) we apply 6.1.2, using the compatibility statement in 15.4.1 to relate it to our present naming of elements. Consider the diagram

K  ⟵^{1∧e}  K ∧ S^{−kV (H)}  ⟶^{≃}  K ∧ S^{−2k}  ⟶^{≃}  K
↓  ↓  ↓  ↓
K ∧ ẼF ⟵^{≃} K ∧ ẼF ∧ S^{−kV (H)}  ⟶^{≃} K ∧ ẼF ∧ S^{−2k}  ⟶^{≃} K ∧ ẼF
↓  r ↓  ↓  ↓
K ∧ ΣEF_{+} ⟵ K ∧ ΣEF_{+} ∧ S^{−kV (H)} ⟶^{≃} K ∧ ΣEF_{+} ∧ S^{−2k} ⟶^{≃} K ∧ ΣEF_{+}.

15.5. THE GEOMETRIC EQUIVARIANT RATIONAL SEGAL CONJECTURE FOR 𝕋. 175

The first horizontal is induced by e : S^{−kV (H)} ⟶ S^{0}, the second is the K-theory Thom class, and the third is multiplication by the integer Bott class. By 6.1.2 the relevant map is induced by r.

Applying π_{∗}^{𝕋} we see by definition that the composite from right to left in the first row is multiplication by λ(kV (H)), and hence this is also true in the second row. Since the bottom right hand vertical induces projection, we identify the second vertical r in the lower ladder as stated.

It is tempting to rewrite the description of q̂_{K} above as q̂_{K}(x ⊗ fβ^{l}) = \overline{xf}β^{l+k} for all x ∈ t_{∗}^{F}, but this makes no sense, since F is not an O_{F}-module. This suggests we should perform some algebraic completion to F. This can be achieved geometrically by replacing K with F (EF_{+}, K). This is a reasonable thing to do since the fibre of the completion map is the F-contractible spectrum F (ẼF, K), which is determined by its homotopy groups.

15.5. The geometric equivariant rational Segal conjecture for 𝕋.

In this section we aim to analyse the functional dual DE𝕋_{+} = F (E𝕋_{+}, S^{0}); the title is something of a misnomer since neither Segal nor anyone else has made a conjecture about DE𝕋_{+}. It completes the description given in [9, 12] of the integral functional dual. It would be possible to give an entirely algebraic treatment since we have a model for function spectra, but we shall first present a more direct treatment so as to avoid relying on the more complicated bits of algebra.

Since E𝕋_{+} = E⟨1⟩, we should really discuss the more general question of identifying DE⟨H⟩. Indeed, we should also consider DEF_{+} ≃ D(⋁_{H} E⟨H⟩) ≃ ∏_{H} DE⟨H⟩. Since the initial stages of the analysis are easier to understand for EF_{+}, we begin with that.

We already know EF_{+} ∧DEF_{+} ≃ EF_{+}, and that π_{∗}^{𝕋}(DEF_{+}) = t_{∗}^{F}; accordingly we have the cofibre sequence

DEF_{+} ⟶ ẼF ∧ S^{0}[t_{∗}^{F}] ⟶^{a} ΣEF_{+},

where a induces projection onto the positive dimensional part. In the algebraic model DEF_{+} is thus described by the element â ∈ Hom_{O_{F}} (t_{∗}^{F} ⊗t_{∗}^{F}, Σ^{2}𝕀) given by smashing a with DEF_{+} and looking in π_{∗}^{𝕋}. For definiteness we emphasize that the second copy of t_{∗}^{F} is the new one.

Lemma 15.5.1. The map â : t_{∗}^{F} ⊗ t_{∗}^{F} ⟶ Σ^{2}𝕀 is given by â(x ⊗ y) = a_{∗}(xy).

Note that â realizes Tate duality between negative and positive parts of t_{∗}^{F}. Proof: Given an O_{F}-map θ : t_{∗}^{F} ⟶ M, so that θ(1) = m the value of θ on O_{F} ⊆ t_{∗}^{F} follows, and if the components of m are uniquely divisible by the relevant c_{H} then θ is determined. The result follows provided x is of positive degree. The real content of the lemma is that the formula is valid also when a_{∗}(x) = 0.

To begin with we note how a nontrivial map ẼF ⟶ Σ^{2k+1}EF_{+} is detected. As motivation, we observe that the obvious example is the quotient of ẼF = S^{∞V (F)} by S^{kV (F)}. If it were possible to simply desuspend by smashing with a putative spectrum S^{(−k−1)V (F)}, then the map would be detected in homotopy. Since the spectrum S^{V (F)} is not invertible we must be slightly less direct by concentrating on a single subgroup H at a time, and using σ^{kV (H)} instead of S^{kV (F)}. This part of the analysis is given in the proof of 15.5.2 below.

The following identifies DE⟨H⟩, and the special case H = 1 gives DE𝕋_{+}.

Proposition 15.5.2. There is a cofibre sequence

DE⟨H⟩ ⟶ ẼF ∧ S^{0}[ ℚ[c_{H}, c_{H}^{−1}] ] ⟶^{b} ΣE⟨H⟩

and the extension is determined by the fact that

b̂ ∈ Hom_{O_{F}} (t_{∗}^{F} ⊗ ℚ[c_{H}, c_{H}^{−1}], Σ^{2}𝕀(H)) = Hom_{ℚ[c_{H}]}(ℚ[c_{H}, c_{H}^{−1}] ⊗ ℚ[c_{H}, c_{H}^{−1}], Σ^{2}𝕀(H))

is given by b̂(x ⊗ y) = b_{∗}(xy), which represents a perfect duality of ℚ[c_{H}, c_{H}^{−1}]

Proof: We smash the standard cofibre sequence EF_{+} ⟶ S^{0} ⟶ ẼF with DE⟨H⟩; the terms are identified with those in the statement by the following lemma.

Lemma 15.5.3. (i)There is an equivalence

EF_{+} ∧ DE⟨H⟩ ≃ E⟨H⟩.

(ii)There is an isomorphism

π_{∗}^{𝕋}(Φ^{𝕋}DE⟨H⟩) ≅ ℚ[c_{H}, c_{H}^{−1}].

Remark: Part (i) of 15.5.3 with H = 1 corrects statements in the rational analysis of [9]. More precisely, the space EF_{+} should be replaced by EG_{+} in 1.6, Theorem B, 4.8, and on the right hand side of 4.5 and page 359 line −3. The correction is discussed in more detail in [12]. Proof of 15.5.3: (i) The equivalence EF_{+} ∧DE⟨H⟩ ≃ E⟨H⟩ follows since EF_{+} ∧DE⟨H⟩ is a retract of EF_{+} ∧ DEF_{+} ≃ EF_{+}. Indeed EF_{+} ≃ ⋁_{H} E⟨H⟩, and the idempotent for all subgroups K ≠ H annihilates DE⟨H⟩. Hence EF_{+} ∧ DE⟨H⟩ is a retract of E⟨H⟩, and by homotopy groups it is an equivalence. (ii) The identification of Φ^{𝕋}DE⟨H⟩ is immediate from 2.4.1.

It remains to show that b̂(x⊗y) = b_{∗}(xy); this follows when x is of positive degree exactly as in 15.5.1. Now suppose x = c_{H}^{−k} for k ≥ 0; the verification that b̂(c_{H}^{−k} ⊗ y) = b_{∗}(c_{H}^{−k}y) in this case will also complete the proof of 15.5.1.

We take the cofibre sequence in the statement and smash it with σ^{nV (H)}. Since σ^{nV (H)} is formed from S^{0} and various basic cells with isotropy H, we have σ^{nV (H)} ∧ ẼF ≃ ẼF; by the Thom isomorphism E⟨H⟩ ∧ σ^{nV (H)} ≃ E⟨H⟩ ∧ S^{2n}, and also

σ^{nV (H)} ∧ DE⟨H⟩ ≃ D(E⟨H⟩ ∧ σ^{−nV (H)}) ≃ D(E⟨H⟩ ∧ S^{−2n}).

Thus the cofibre sequence becomes

S^{2n} ∧ DE⟨H⟩ ⟶ ẼF ∧ S^{0}[ ℚ[c_{H}, c_{H}^{−1}] ] ⟶^{1∧b} Σ^{2n+1}E⟨H⟩.

Because the final term is 2n-connected and the first has π_{∗}^{𝕋} only in degrees ≤ 2n it follows that π_{∗}^{𝕋}(1 ∧ b) is surjective for all n. Taking n ≤ −k − 1 establishes the required formula for b̂.

Now let us turn to the general question of what can be said about DX = F (X, S^{0}) if we already understand X in the standard model. Thus we suppose that X has model B = (N ⟶ t_{∗}^{F} ⊗ V), and we let Q ⟶ t_{∗}^{F} ⊗ H denote the torsion model of DX.

First we recall that S^{0} has model (O_{F} ⟶ t_{∗}^{F}), this has torsion part Σ𝕀, and the natural fibrant model is as the fibre of the map e(ℚ) ⟶ f(Σ^{2}𝕀). We have seen in Section 9.3 that if S is the torsion part of X, the torsion part of DX is

RΓ_{f} Hom(S, Σ𝕀).

The vertex is described by the fibre sequence

H ⟶ V^{∗} ⟶ E^{−1}Hom(t_{∗}^{F} ⊗ V, Σ^{2}𝕀).

However the real aim here is to give a description of the complete model. We show that the obvious cofibre sequence DX ⟶ DX ∧ ẼF ⟶ DX ∧ ΣEF_{+} is also natural from the algebraic point of view.

Lemma 15.5.4. If B = (N ⟶ t_{∗}^{F} ⊗ V), then the functional dual of B is described by

RΓ̂ RHom(B, S^{0}) = fibre (e(V^{∗}) ⟶ RΓ̂ f(Hom(N, Σ^{2}𝕀))) .

Proof: We use the injective resolution S^{0} ⟶ e(ℚ) ⟶ f(Σ^{2}𝕀), and deduce that

RHom(B, S^{0}) = fibre (Hom(B, e(ℚ)) ⟶ Hom(B, Σ^{2}f(𝕀))) .

Now Hom(B, e(ℚ)) is the injective e(V^{∗}), whilst Hom(B, Σ^{2}f(𝕀)) = f(Hom(N, Σ^{2}𝕀)). Applying Γ̂ we obtain the result.

We can say a little more about the term RΓ̂ f(Hom(N, Σ^{2}𝕀)) in the description For an arbitrary O_{F}-module M we proved in Example 8.5.8 that there is a fibre sequence RΓ̂f(M) ⟶ e(E^{−1}M) ⟶ f(E^{−1}M/M).

178 15. CLASSICAL MISCELLANY.

CHAPTER 16 Cyclic and Tate cohomology.

This is a short chapter, but fits naturally between its neighbours. The first section identifies rational cyclic cohomology, the second gives an algebraic description of the Tate construction on rational spectra, and the third gives a description of the Tate spectrum of integral complex equivariant K-theory.

16.1. Cyclic cohomology.

In this section we consider periodic cyclic cohomology. We begin by observing that the rationalisation of the integral cyclic cohomology is the cyclic cohomology of the rationals, so that the two possible interpretations coincide.

It was proved in [14] that the representing spectrum for cyclic cohomology with coefficients in an abelian group A is the Tate spectrum t(HA) = F (E𝕋_{+}, HA) ∧ Ẽ𝕋. The following lemma is special to bounded cohomology theories.

Lemma 16.1.1. For any Mackey functor A the rationalization of t(HA) is t(H(A ⊗ ℚ)).

Proof: The essential point is that F (E𝕋_{+}, HA) = holim_{←}_{n} F (E𝕋_{+}^{(n)}, HA), and that the maps induced in [X, ·]^{𝕋} by those of the inverse system are ultimately isomorphisms for each finite X. This inverse limit therefore commutes with direct limit under degree zero selfmaps of HA.

Thus

t(HA) ∧ S^{0}ℚ = holim_{→}_{m}( F (E𝕋_{+}, HA) ∧ Ẽ𝕋, m! )
≃ F (E𝕋_{+}, holim_{→}_{m}(HA, m!)) ∧ Ẽ𝕋
≃ F (E𝕋_{+}, HA ⊗ ℚ) ∧ Ẽ𝕋 = t(H(A ⊗ ℚ)).

Henceforth we suppose A is rational.

Lemma 16.1.2. Provided A is rational, t(HA) is F-contractible. We therefore have an equivalence

t(HA) ≃ ẼF ∧ S^{0}[π_{∗}^{𝕋}(t(HA))].

Proof: Indeed t(HA)|_{H} = t(HA|_{H}), and the rational Tate cohomology of any finite group is 0.

For any abelian group A we have π_{∗}^{𝕋}(t(HA)) = ĤC_{∗} ⊗ A = A[c, c^{−1}].

16.2. Rational Tate spectra.

In this section we discuss the Tate construction of [14], which generalizes the periodic cyclic cohomology discussed in the previous section. Recall that the Tate construction on a 𝕋-spectrum is defined by t(X) = F (E𝕋_{+}, X) ∧ Ẽ𝕋. This simplifies considerably in the rational case, and it seems worth giving a complete description of the Tate construction in the category of rational 𝕋-spectra.

We begin with the warning that if X is integral, the map t(X) ⟶ t(X ∧ S^{0}ℚ) need not be a rational equivalence, so that Lemma 16.1.1 above is special to suitably bounded theories like HA. An example is given by complex K-theory, since t(Kℤ)|_{H} is non-trivial and rational for all non-trivial finite subgroups H [10, 14, 15]; the following lemma shows this is false for t(Kℚ).

We revert to our global assumption that all spectra are rational.

Lemma 16.2.1. The natural map

t(X) = F (E𝕋_{+}, X) ∧ Ẽ𝕋 ⟶ F (E𝕋_{+}, X) ∧ ẼF

is an equivariant equivalence. Thus t(X) is an F-contractible spectrum determined by its homotopy groups.

Proof: We give two proofs. Firstly, the Tate construction commutes with restriction: t(X)|_{H} = t(X|_{H}). The lemma follows from the fact that the Tate construction is trivial on rational spectra for finite groups. One way of seeing this is to use the fact that if H is finite and e ∈ A(H) is the idempotent with support 1 then EH_{+} = eS^{0} and ẼH = (1 − e)S^{0}.

For the second proof, we compare the cofibre sequence E𝕋_{+} ⟶ S^{0} ⟶ Ẽ𝕋 with EF_{+} ⟶ S^{0} ⟶ ẼF. We see that the lemma is equivalent to showing that the natural map f : F (E𝕋_{+}, X) ∧ E𝕋_{+} ⟶ F (E𝕋_{+}, X) ∧ EF_{+} is an equivalence. However, the cofibre of f is a wedge of terms F (E𝕋_{+}, X) ∧ E⟨H⟩ with H ≠ 1; this is contractible, as one sees from the fact that F (E𝕋_{+}, X) ∧ σ_{H}^{0} ≃ ∗ by using cofibre sequences and passing to direct limits.

Proposition 16.2.2. If X is a rational 𝕋-spectrum with associated module M = π_{∗}^{𝕋}(X∧ E𝕋_{+}) over ℚ[c_{1}], then t(X) is the F-contractible spectrum with homotopy groups

Ĥ_{(c_{1})}^{0}(M) ⊕ ΣĤ_{(c_{1})}^{−1}(M)

where Ĥ_{(c_{1})}^{∗} denotes local Tate cohomology in the sense of [10].

16.3. THE INTEGRAL 𝕋-EQUIVARIANT TATE SPECTRUM FOR COMPLEX K-THEORY. 181

Remark 16.2.3. There are two methods for calculating the local Tate cohomology; since (c_{1}) is principal both are very simple. The second description simplifies further because M is torsion. In fact, the local Tate cohomology is only non-zero in codegrees 0 and −1, and for these cases we have

Ĥ_{(c_{1})}^{−i}(M) = (L_{i}^{(c_{1})}M)[1/c_{1}] = lim_{←}^{i}(M, c_{1}),

where L_{∗}^{(c_{1})} denotes the left derived functors of completion at (c_{1}). Note that this is only likely to be equal to (c_{1})-adic completion when M is finitely generated. However, since M is torsion, when it is finitely generated it is already complete; the Tate cohomology therefore vanishes, as we know it must for geometric reasons.

Proof: Observe F (E𝕋_{+}, X) ≃ F (E𝕋_{+}, X ∧ E𝕋_{+}), so that if M = π_{∗}^{𝕋}(X ∧ E𝕋_{+}), there is an exact sequence

0 ⟶ Ext(Σ^{2}𝕀, M) ⟶ [E𝕋_{+}, X]_{∗}^{𝕋} ⟶ Hom(Σ𝕀, M) ⟶ 0.

This is precisely parallel to the algebraic situation. We may split X into even and odd parts, and thus the exact sequence splits. Therefore F (E𝕋_{+}, X) is modelled by the complex Hom(P K(c_{1}), M) where P K(c_{1}) is a complex of projectives approximating the stable Koszul complex ℚ[c_{1}] ⟶ ℚ[c_{1}, c_{1}^{−1}]; the homology of this complex calculates the left derived functors of c_{1}-completion [13]. In particular, when X is even, [E𝕋_{+}, X]_{∗}^{𝕋} is L_{0}^{(c_{1})}M in even degrees and ΣL_{1}^{(c_{1})}M in odd degrees.

Now conclude that there is a split exact sequence

0 ⟶ Ext(Σ^{2}𝕀, M)[1/c] ⟶ t(X)_{∗}^{𝕋} ⟶ Hom(Σ𝕀, M)[1/c] ⟶ 0.

Therefore, if T T_{(c_{1})}(M) is the complex of the second avatar in the notation of [10], t(X) is modelled by the corresponding torsion free model, e(T T_{(c_{1})}(M)). Thus, if X is even, t(X)_{∗}^{𝕋} is Ĥ_{(c_{1})}^{0}(M) in even degrees and ΣĤ_{(c_{1})}^{−1}(M) in odd degrees.

16.3. The integral 𝕋-equivariant Tate spectrum for complex K-theory.

In this section we apply the general theory to identify the Tate spectrum of complex equivariant K-theory Kℤ integrally. However, we note that t(Kℤ) is not rational, and its rationalization is not t(Kℚ), so this is not an application of the previous section.

Before we state the theorem, recall that the representation ring R(𝕋) = ℤ[z, z^{−1}], and that the Euler class of the representation z^{n} is 1−z^{n}. In particular, we let χ = 1−z and find R(𝕋)_{(χ)}^{∧} = ℤ[[χ]]; indeed, z = 1 − χ is invertible in ℤ[χ]/(χ^{n}), so that ℤ[χ] ⟶ ℤ[χ, z^{−1}] = ℤ[z, z^{−1}] induces an isomorphism of (χ)-completions. We write ℤ((χ)) for the localization ℤ[[χ]][χ^{−1}], and S for the mulitiplicative set generated by the Euler classes. Note that if n ≥ 2 the Euler class 1 − z^{n} is a multiple of χ. However, although the multiplier is a unit in ℚ((χ)), it is not a unit in ℤ((χ)) ⊗ ℚ.

Theorem 16.3.1. The Tate spectrum t(Kℤ) is F-equivalent to a rational spectrum, and is thus determined by the homotopy type of Φ^{𝕋}Kℤ and its rational type. There is an equivalence of Kℤ-module spectra

Φ^{𝕋}Kℤ ≃ KS^{−1}ℤ((χ)).

The rational spectrum t(Kℤ) ∧ S^{0}ℚ is classified in the torsion model by

t_{∗}^{F} ⊗ S^{−1}ℤ((χ))[β, β^{−1}] ⟶ S^{−1}ℤ((χ))/ℤ((χ))[β, β^{−1}] = ⨁_{H≠1} ℤ((χ))/Φ_{|H|}^{∞}[β, β^{−1}],

where Φ_{n} is the nth cyclotomic polynomial, and the structure map is described as in 15.4.4.

Proof: First note that t(Kℤ) = F (E𝕋_{+}, Kℤ)∧Ẽ𝕋, so that we may calculate its coefficient ring from the Atiyah-Segal completion theorem. First, equivariant K-theory has coefficients R(𝕋)[β, β^{−1}], with R(𝕋) = ℤ[z, z^{−1}], and the K-theory Euler class of z^{n} is 1 − z^{n}. By the Atiyah-Segal completion theorem, π_{∗}^{𝕋}(F (E𝕋_{+}, Kℤ)) = R(𝕋)_{(χ)}^{∧}[β, β^{−1}].

Consider the cofibre sequence

t(Kℤ) ⟶ t(Kℤ) ∧ ẼF ⟶ t(Kℤ) ∧ ΣEF_{+}.

We shall identify t(Kℤ) ∧ ẼF, t(Kℤ) ∧ ΣEF_{+}, and the map between them in turn. Firstly, since χ is an Euler class,

π_{∗}^{𝕋}(t(Kℤ) ∧ ẼF) = S^{−1}ℤ((χ))[β, β^{−1}],

where S is the multiplicative set generated by 1 − z^{n} for n ≥ 1. Now S^{−1}ℤ((χ)) is flat over ℤ, and hence the coefficients of Φ^{𝕋}t(Kℤ) are the same as those of K-theory with coefficients in S^{−1}ℤ((χ)).

Lemma 16.3.2. There is an equivalence

Φ^{𝕋}Kℤ ≃ KS^{−1}ℤ((χ))

of non-equivariant Kℤ-module spectra.

Proof: First note that Φ^{𝕋}t(Kℤ) is a module over Kℤ. Now let MS^{−1}ℤ((χ)) be a nonequivariant Moore spectrum, and construct a map f : MS^{−1}ℤ((χ)) ⟶ t(Kℤ) inducing an isomorphism in π_{0}^{𝕋}. Now form the composite

KS^{−1}ℤ((χ)) = K ∧ MS^{−1}ℤ((χ)) ⟶ K ∧ Φ^{𝕋}t(Kℤ) ⟶ Φ^{𝕋}t(Kℤ)

in which the first map is obtained from f by applying K ∧ Φ^{𝕋}(·) to f, and the second uses the module structure. By construction this induces an isomorphism in homotopy, and is therefore an equivalence.

Next, we claim that t(Kℤ) ∧ EF_{+} is rational. This is immediate from the fact that, t(Kℤ)|_{H} is rational for all finite subgroups H [10, 14, 15]. Indeed, we know that t(Kℤ) ∧ 𝕋/H_{+} is induced from t(Kℤ)|_{H}. Rationally EF_{+} ≃ ⋁_{H} E⟨H⟩, so t(Kℤ) ∧ EF_{+} has a corresponding splitting. The summand for E⟨1⟩ = E𝕋_{+} is trivial, so we may choose H ≠ 1 and consider t(Kℤ) ∧ ΣE⟨H⟩. From the identification of c_{H}, we find it has homotopy groups ℤ((χ))/Φ_{n}^{∞} in each even degree, and in particular it is injective.

Finally, the map t(Kℤ)ẼF ⟶ t(Kℤ) ∧ ΣEF_{+} factors through the rationalization t(Kℤ) ∧ ẼF ⟶ t(Kℤ) ∧ ẼF ∧ S^{0}ℚ, and the resulting map t(Kℤ) ∧ ẼF ∧ S^{0}ℚ ⟶ t(Kℤ) ∧ ΣEF_{+} is classified by its d-invariant since the codomain is injective. The map from t_{∗}^{F} ⊗ S^{−1}ℤ((χ))[β, β^{−1}] is the analogue of that in Theorem 15.4.4.

184 16. CYCLIC AND TATE COHOMOLOGY.

CHAPTER 17 Cyclotomic spectra and topological cyclic cohomology.

In this chapter, we study various 𝕋-spectra arising from algebraic K-theory. Various constructions are used to define suitable targets for trace maps from algebraic K-theory, and the most sophisticated takes Bökstedt’s Topological Hochschild homology of a ring, and forms the associated topological cyclic spectrum in the sense of Bökstedt-Hsiang-Madsen [2]. Madsen has recently given a very helpful general survey [20].

The topological cyclic construction can be applied to any 𝕋-spectrum with appropriate extra structure, and we begin in Section 17.1 by identifying the extra structure involved in specifying such a ‘cyclotomic’ spectrum. In the following section, we illustrate this by considering the basic examples: free loop spaces on a 𝕋-fixed space, and topological Hochschild homology of a functor with smash products. Finally, in Section 17.3 we analyse the topological cyclic construction on rational cyclotomic spectra.

17.1. Cyclotomic spectra.

We must begin by recalling the definition of a cyclotomic spectrum. The basic idea is that it is a spectrum X with the property analogous to that of the free loop space ΛZ, on a 𝕋-fixed based space Z, namely that for any finite subgroup K the fixed point set (ΛZ)^{K} is equivalent to the original space ΛZ. The analogue should be that any fixed point spectrum Φ^{K}X is equivalent to X again. Of course Φ^{K}X is really a 𝕋/K-spectrum, so we must begin by explaining exactly how we interpret it as a 𝕋-spectrum indexed on the original universe. In addition, we want to avoid redundant structure, so we simply require that the resulting equivalences are transitive.

We wish to consider the group 𝕋 and all its quotients \overline{𝕋} = 𝕋/K by finite subgroups K. We want transitive systems of structure, so we first let ρ = ρ_{K} : 𝕋 ⟶^{≅} \overline{𝕋} be the isomorphism given by taking the |K|th root. If we index our 𝕋-spectra on a complete universe U, we index our \overline{𝕋}-spectra on the complete universe U^{K}. However we want these universes to be comparable, so we say that a complete 𝕋-universe U is cyclotomic if it is provided with isomorphisms U ⟶^{≅} ρ_{K}^{∗}U^{K}. Identifying 𝕋 and \overline{𝕋} via ρ_{K}, this also specifies isomorphisms U^{L} ⟶ ρ_{K/L}^{∗}(U^{L})^{K/L}. We require that these are transitive in the sense that if L ⊆ K then the composite

U ⟶ ρ_{L}^{∗}U^{L} ⟶ ρ_{L}^{∗}(ρ_{K/L}^{∗}U^{L})^{K/L} = ρ_{K}^{∗}U^{K}

is the isomorphism for K. One such cyclotomic universe is the direct sum U = ⊕_{n∈ℤ}U_{n} where U_{n} = ⨁_{i∈ℕ} z^{n} with the isomorphisms suggested by the indexing.

Suppose then that X is a 𝕋-spectrum indexed on the cyclotomic universe U. Thus, for any finite subgroup K, Φ^{K}X is a \overline{𝕋}-spectrum indexed on U^{K}; by pullback along the isomorphism ρ_{K} we obtain a 𝕋-spectrum ρ_{K}^{∗}Φ^{K}X indexed on ρ_{K}^{∗}U^{K}, which may be viewed as a 𝕋-spectrum ρ_{K}^{!}Φ^{K}X indexed on U by using the cyclotomic structure of the universe. A cyclotomic structure on X consists of a transitive system of 𝕋-equivalences

r_{K} : ρ_{K}^{!}Φ^{K}X ⟶^{≃} X.

By transitivity, the essential pieces of the structure come from the cases that K is of prime order.

Although this structure is really designed to capture profinite information, there is enough residue rationally to make it worthwhile identifying the cyclotomic objects in the algebraic model of rational 𝕋-spectra. The essential idea is that in a cyclotomic spectrum all finite subgroups behave in an analogous way, differing only in the multiplicity with which information occurs. There is no significant constraint on total fixed points Φ^{𝕋}X. The first step of our analysis was to split F-spectra into the parts over different subgroups, so it is easy to describe the cyclotomic structure in these terms. A spectrum X is cyclotomic if we have specified equivalences X(C_{1}) ≃ X(C_{2}) ≃ X(C_{3}) ≃ . . .. This uniformity itself imposes constraints on the assembly map of a 𝕋-spectrum.

Let us now describe the algebraic model for cyclotomic spectra more precisely. It is useful to bear in mind the torsion model rather than the standard model.

Definition 17.1.1. The ring of cyclotomic operations is the polynomial ring ℚ[c_{0}] on a single generator c_{0} of degree −2. The standard injective 𝕀_{0} is defined by the exact sequence 0 ⟶ ℚ[c_{0}] ⟶ ℚ[c_{0}, c_{0}^{−1}] ⟶ Σ^{2}𝕀_{0} ⟶ 0. The cyclotomic torsion category C_{t} has objects (Σ^{2}𝕀_{0} ⊗ V ⟶ T_{0}) where V is a graded vector space and T_{0} is a torsion ℚ[c_{0}]-module. The morphisms are given by commutative squares as usual.

Lemma 17.1.2. The category C_{t} is abelian and 2 dimensional. Hence we may form the derived category DC_{t}.

Proof: The proof is precisely analogous to that for the torsion model category A_{t}.

Again, it is convenient to have a 1-dimensional model; the analogue of the standard model is considerably simplified in the present context.

Definition 17.1.3. The standard cyclotomic category C has objects ℚ[c_{0}]-maps N_{0} ⟶ Σ^{2}𝕀_{0} ⊗ V with N_{0} a torsion module. The morphisms are given by commutative squares as usual.

Lemma 17.1.4. The cyclotomic category C is abelian and 1-dimensional. Hence we may form the derived category DC. Furthermore passage to fibre dgC ⟶ dgC_{t} and passage to cofibre dgC ⟶ dgC_{t} induce inverse equivalences of derived categories, so that DC ≃ DC_{t}.

Proof: The proof is similar to the case of the standard model, but with the simplification that the cofibre functor arrives in the correct category before passing to homology.

Now define a functor

Δ : C_{t} ⟶ A_{t}

as follows. For an object we define Δ(Σ^{2}𝕀_{0} ⊗ V ⟶^{s_{0}} T_{0}) to be the composite

(t_{∗}^{F} ⊗ V ⟶ Σ^{2}𝕀 ⊗ V = ⨁_{H} Σ^{2}𝕀_{0} ⊗ V_{0} \underrightarrow{⨁_{H} s_{0}} ⨁_{H} T_{0}).

Here the first map is induced by the quotient t_{∗}^{F} ⟶ t_{∗}^{F}/O_{F} = Σ^{2}𝕀, and the second is the direct sum of countably many copies of s_{0} made into a O_{F}-module in the obvious way.

The functor is obviously exact and hence induces a functor

Δ : DC_{t} ⟶ DA_{t}.

We may now state a precise theorem.

Theorem 17.1.5. A 𝕋-spectrum admits the structure of a cyclotomic spectrum if and only if it corresponds to an object of DA_{t} equivalent to one in the image of Δ.

Note that the condition in the theorem gives a rather satisfactory characterization of cyclotomic spectra. It essentially says that a cyclotomic spectrum is one that has two properties. Firstly, the structure map factors through that for its geometric fixed point spectrum (as happens for suspension spectra) and secondly, that all finite subgroups behave alike.

If a spectrum admits a cyclotomic structure then a structure is imposed by choosing particular equivalences between the idempotent pieces of the torsion part of the model. Note that for a spectrum X with torsion model t_{∗}^{F} ⊗ V ⟶ T admitting a cyclotomic structure the corresponding cyclotomic spectrum is simply Σ^{2}𝕀_{0} ⊗ V ⟶ T_{0} where T_{0} = e_{1}T = π_{∗}^{𝕋}(E𝕋_{+} ∧ X) and the map is obtained by factoring s through the projection and applying the idempotent e_{1}.

Proof: We have explained how to put a cyclotomic structure on an object in the image of Δ. Any imprecision will be eliminated in the course of the proof in the reverse direction.

Suppose then that X is a cyclotomic spectrum with cyclotomic structure maps r_{K} : ρ_{K}^{!}Φ^{K}X ⟶^{≃} X as required. We already know from Section 10.2 the effect of passage to geometric fixed points. Indeed, by Theorem 10.2.6, if M is the model of X then eM is the model for Φ^{K}X where e is the idempotent supported on the subgroups containing K. Here \overline{O}\overline{F} is identified with eO_{F} by letting a subgroup \overline{H} of \overline{𝕋} correspond to its inverse image in 𝕋. The effect of ρ_{K}^{!} simply results from identifying subgroups of 𝕋 with those of the same order in \overline{𝕋}.

Define n_{K} : \overline{F} ⟶ F by letting n(\overline{H}) be the subgroup of 𝕋 with the same order as \overline{H}, and consider the induced ring isomorphism n_{K}^{∗} : O_{F} ⟶^{≅} \overline{O}\overline{F}.

Lemma 17.1.6. The functor ρ_{K}^{!} : \overline{𝕋}−Spec ⟶ 𝕋−Spec corresponds to pullback along n_{K}^{∗} in the usual sense that the diagram

\overline{𝕋}−Spec ⟶^{ρ_{K}^{!}} 𝕋−Spec
≃↓  ↓≃
D\overline{A} ⟶^{n_{#}^{K}} DA

commutes up to natural isomorphism, and similarly for torsion models.

It is then clear (for example by using the cyclotomic structure for K itself) that the part e_{K}M of the model over any subgroup K will be the same as the piece e_{1}M over the trivial subgroup. This also forces the map to factor as specified.

If M is an object of the torsion model with zero differential, we see that the structure map must be zero on any element of form 1 ⊗ v; otherwise it would have nonzero image in e_{H}T for some H, and hence for all finite subgroups H. This contradicts F-finiteness of T. Since this argument passes to an injective resolution, it applies to all differential graded objects.

17.2. Free loop spaces.

For a based space Z, we intend to identify the place of the free loop space ΛZ in the present scheme. In particular we may consider K_{𝕋}^{∗}(ΛZ), which is a conjectural approximation to Ell^{∗}(Z).

We restrict attention to the case that Z = ΣY is a suspension. Here, Carlsson and Cohen [4] prove the splitting

E𝕋_{+} ∧_{𝕋} ΛΣY = ⋁_{n}(EC_{n})_{+} ∧_{C_{n}} Y^{∧n}.

Hence, rationally we have

π_{∗}^{𝕋}(E𝕋_{+} ∧ ΛΣY) = Σ ⨁_{n}{H_{∗}(Y)^{⊗n}}^{C_{n}}

with trivial H^{∗}(B𝕋) action. This leaves us to describe a map

Σ^{2}𝕀_{0} ⊗ H_{∗}(ΣY) ⟶ Σ^{2} ⨁_{n}{H_{∗}(Y)}^{C_{n}}.

This necessarily has zero d-invariant. Indeed, this is obvious if H_{∗}Y has even parity. In the general case we see that the map is induced by E𝕋_{+} ∧_{𝕋}ΣY ⟶ E𝕋_{+} ∧_{𝕋}ΛΣY; by duality it is sufficient to consider cohomology, and the domain has torsion free cohomology whilst the codomain has torsion cohomology. Now, exactly as in the case of suspension spectra, the e invariant is the element of Ext(Σ^{4}𝕀_{0} ⊗ H_{∗}(Y), ⨁_{n} Σ^{2}H_{∗}(Y)) corresponding to the extension obtained by applying homology to

E𝕋_{+} ∧_{𝕋} ΛΣY ⟶ E𝕋_{+} ∧_{𝕋} (ΛΣY)/ΣY ⟶ E𝕋_{+} ∧_{𝕋} Σ^{2}Y.

Since ℚ[c_{1}] acts trivially on H_{∗}(Y), one might hope the extension is always obtained by tensoring a universal extension

0 ⟶ ⨁_{n} ℚ ⟶ E ⟶ Σ^{2}𝕀_{0} ⟶ 0

with Σ^{2}H_{∗}(Y).

There is another important example of cyclotomic spectra.

Example 17.2.1. Topological Hochschild Homology:

Suppose that F is a functor with smash products in the sense of Bökstedt. One may define a cyclotomic spectrum T HH(F), which comes with a spectral sequence

HH_{∗}(F (S^{0})_{∗}) ⟹ T HH(F)_{∗}

for calculating its homotopy groups. One may then hope to calculate π_{∗}^{𝕋}(E𝕋_{+} ∧_{𝕋}T HH(F)) using the skeletal filtration of E𝕋_{+}.

It is always the case that Φ^{𝕋}T HH(F) ≃ S^{0}, and so the structure map of the cyclotomic spectrum T HH(F) takes the form

Σ^{2}𝕀_{0} ⟶ Σ^{2}T HH(F)_{∗}^{h𝕋}.

By definition, we always have a map from the identity functor to F and hence a cyclotomic map S^{0} = T HH(I) ⟶ T HH(F). Since this is an equivalence of geometric fixed points, and the structure map for S^{0} has zero d-invariant we deduce that the structure map for an arbitrary functor F has zero d-invariant. It would be interesting to understand its e-invariant more precisely.

One case of particular interest is when the FSP arises from a ring R. In this case π_{∗}^{𝕋}(E𝕋_{+} ∧ T HH(R)) = ΣHC_{∗}(R), which can be calculated by the algebraic Loday-Quillen double complex. It remains to identify the torsion model structure map, but we can obtain information by naturality from the unit ℚ ⟶ R. In fact we have the diagram

ẼF ∧ DEF_{+}  ⟶  ΣEF_{+}
≃↓  ↓≃
ẼF ∧ DEF_{+} ∧ T HH(ℚ) ⟶ ΣEF_{+} ∧ T HH(ℚ)
≃↓  ↓
ẼF ∧ DEF_{+} ∧ T HH(R) ⟶ ΣEF_{+} ∧ T HH(R)

in which we understand the top row precisely as the structure map of the sphere. Thus we only need to understand the algebraic map Σ^{2}𝕀_{0} = Σ^{2}HC_{∗}(ℚ) ⟶ Σ^{2}HC_{∗}(R). This is in fact either zero or injective: this follows from the Tate spectral sequence by naturality. Indeed t(T HH(R)) is a module over t(T HH(ℚ)), and hence over the ring spectrum t(Hℚ), whose coefficients are ℚ[c_{0}, c_{0}^{−1}]: thus the behaviour is completely determined by the image of the unit. If R is augmented, then of course the map is injective.

This special case is not too far from the general case, because any rational FSP arises from a simplicial ring, and π_{∗}^{𝕋}(E𝕋_{+}∧T HH(R_{•})) can be calculated algebraically, since there is a 𝕋-map T HH(R_{•}) ⟶ HH(R_{•}) which is a non-equviariant rational equivalence.

17.3. Topological cyclic cohomology of cyclotomic spectra.

In the first instance, the topological cyclic cohomology of a ring is designed to be the target of a refined trace map from the algebraic K-theory. Hesselholt and Madsen have shown that the cyclotomic trace is very close to being an isomorphism in many cases [17]. The construction of the topological cyclic cohomology in this case begins with the topological Hoschschild homology, and the definition of a cyclotomic spectrum abstracts precisely what is required to make the construction.

Goodwillie has identified the topolical cyclic cohomology of the topological Hoschschild homology of a rational functor with smash products [7], and we generalize this to an arbitrary cyclotomic spectrum. This is not a deep result, but it demonstrates the character of the topological cyclic cohomology and illustrates the adequacy of the present theory. The author is grateful to L. Hesselholt for many helpful discussions.

We must begin by describing the construction. For any 𝕋-spectrum X, if L ⊆ K we have an inclusion of the Lewis-May fixed points F_{L}^{K} : Ψ^{K}X ⟶ Ψ^{L}X; the letter F is chosen because it corresponds to the Frobenius map in algebraic K-theory. To avoid confusion, the reader should ignore for the duration of the present section the fact that F_{L}^{K} induces the restriction map from K-equivariant to L-equivariant homotopy groups. The cyclotomic structure supplies a second set of maps R_{L}^{K} : Ψ^{K}X ⟶ Ψ^{L}X defined as follows. First we let L^{∗} be the subgroup of K with order |K/L|. Now consider the inclusion X ⟶ X ∧Ẽ[⊉ L^{∗}]; applying Lewis-May L^{∗}-fixed points we obtain a map Ψ^{L^{∗}}X ⟶ Φ^{L^{∗}}X. Applying ρ_{L^{∗}}^{!} and the cyclotomic structure we obtain

ρ_{L^{∗}}^{!}Ψ^{L^{∗}}X ⟶ ρ_{L^{∗}}^{!}Φ^{L^{∗}}X ⟶^{r_{L^{∗}}} X;

finally we apply L-fixed points and obtain the required map

ρ_{K/L}^{!}Ψ^{K}X = ρ_{K/L}^{!}Ψ^{K/L^{∗}}Ψ^{L^{∗}}X = Ψ^{L}(ρ_{L^{∗}}^{!}Ψ^{L^{∗}}X) ⟶ Ψ^{L}X.

Again, the letter R is chosen because the induced map is the restriction map in algebraic K-theory.

To simplify notation, we index F and R simply by the order of the quotient K/L. Thus we find F_{1} = R_{1} = 1, F_{r}F_{s} = F_{rs} and R_{r}R_{s} = R_{rs}. It turns out that the Frobenius and restriction maps also commute.

The most familiar version of the topological cyclic cohomology construction is simply to take the the homotopy inverse limit of the system of non-equivariant fixed point spectra under the restriction and Frobenius maps:

T C′(X) = holim_{←}(holim_{←}(Ψ^{K}X, R)), F) = holim_{←}(holim_{←}(Ψ^{K}X, F), R).

It may help later motivation to view this as the homotopy fixed point object of an ‘action of a category’. It turns out that the intermediate object

T R′(X) = holim_{←}_{K}(Ψ^{K}X, R),

has significance of its own, so we prefer the first description T C′(X) = holim_{←}(T R′(X), F). Furthermore, we note that the above construction shows that the map R_{L}^{K} : ρ_{K}^{!}Ψ^{K}X ⟶ ρ_{L}^{!}Ψ^{L}X is a map of 𝕋-spectra so

T R(X) = holim_{←}_{K}(ρ_{K}^{!}Ψ^{K}X, R)

is a 𝕋-spectrum with underlying spectrum T R′(X). However, we warn that the identification of all terms with T R(X) means that the Frobenius maps are not maps of 𝕋-spectra. We shall identify the relevant equivariance below.

For non-profinite work, Goodwillie points out that the diagram given by the restriction and Frobenius maps should be augmented by adding in the circle action; we may now think of an action by a topological category. Since R also commutes with the Frobenius, passing to limits under R, we obtain a diagram with a copy of T R(X) for each finite subgroup, and Frobenius maps relating them; the quotient category acts on T R(X). For the present we view all objects as the same and hence we think of having an action of the monoid M occurring in a split exact sequence 1 ⟶ 𝕋 ⟶ M ⟶ ℤ_{>0} ⟶ 1; in fact if w, z ∈ 𝕋 then (wF_{r})(zF_{s}) = wz^{r}F_{rs}. This leads to the definition

T C(X) = T R(X)^{hM} ≃ (T R(X)^{h𝕋})^{hF} .

The following result may simply be regarded as evidence that the definition is a reasonable one: rationally, the topological cyclic construction is a complicated way of doing something familiar.

Theorem 17.3.1. (Goodwillie ) For any rational cyclotomic spectrum X we have an equivalence of rational spectra

T C(X) ≃ X^{h𝕋},

so that the topological cyclic cohomology agrees with the Borel cohomology.

Goodwillie proves this in the case that X = T HH(F) for a rational functor F with smash products [7, 14.2]. Proof: The first step is to note that homotopy fixed points commute with homotopy inverse limits, and that the homotopy fixed point spectrum of a non-equivariantly contractible spectrum is contractible. Thus

T R(X)^{h𝕋} = (holim_{←}_{K} ρ_{K}^{!}Ψ^{K}X)^{h𝕋}
= holim_{←}_{K}((ρ_{K}^{!}Ψ^{K}X)^{h𝕋})
= holim_{←}_{K}((E𝕋_{+} ∧ ρ_{K}^{!}Ψ^{K}X)^{h𝕋})

This shows that it is really only necessary to understand X(1) = E𝕋_{+} ∧ X. Of course the end result is simply a non-equivariant rational spectrum, so it is only necessary to calculate homotopy groups.

The main result of Part I is that the 𝕋-free spectrum E𝕋_{+} ∧ ρ^{!}Ψ^{K}X is determined by its homotopy groups as modules over ℚ[c_{1}].

Lemma 17.3.2. If X is a cyclotomic spectrum with π_{∗}^{𝕋}(E𝕋_{+} ∧ X) = T_{0} then

π_{∗}^{𝕋}(E𝕋_{+} ∧ ρ_{K}^{!}Ψ^{K}X) = ⨁_{L⊆K} T_{0}

and hence

E𝕋_{+} ∧ ρ_{K}^{!}Ψ^{K}X = ⋁_{L⊆K} X(1).

Proof: This is immediate from 17.1.6 together with our exact identification of Lewis-May fixed points in Theorem 12.2.2, (or more directly from 12.3.2).

The relevant inverse system thus has Kth term given as a wedge of copies of the spectrum X(1) indexed by the subgroups of K. It will perhaps be clearest if we think of this as the set of functions from the finite set [⊆ K] of subgroups of K to X(1). The advantage is that it permits a helpful notation for maps: any function f : A ⟶ B of finite sets induces f^{∗} : X(1)^{B} ⟶ X(1)^{A}. If f is an inclusion the map f^{∗} is simply projection.

Lemma 17.3.3. When L ⊆ K, the restriction map R_{L}^{K} induces the projection corresponding to the function λ_{L}^{K} : [⊆ L] ⟶ [⊆ K] defined by requiring λ_{L}^{K}(H) to have order |H| · |K/L|, in the sense that the diagram

E𝕋_{+} ∧ ρ_{K}^{!}Ψ^{K}X ⟶^{1∧R_{L}^{K}} E𝕋_{+} ∧ ρ_{L}^{!}Ψ^{L}X
≃↓  ↓≃
⋁_{H⊆K} X(1) ⟶^{(λ_{L}^{K})^{∗}}  ⋁_{H⊆L} X(1)

commutes.

Proof: Recall that L^{∗} denotes the subgroup of K with order |K/L|. The effect of the map Ψ^{L^{∗}}X ⟶ Ψ^{L^{∗}}(X ∧ Ẽ[⊈ L^{∗}]) = Φ^{L^{∗}}X follows from our account of the Lewis-May fixed points. Now we just need to rename subgroups using ρ_{L^{∗}}^{!} 17.1.6, and apply L-fixed points as described in 12.3.2.

Corollary 17.3.4. We have an equivalence

T R(X)^{h𝕋} ≃ ∏_{H} X(1)^{h𝕋}.

Notice that the above argument could also be used to identify the 𝕋-spectrum T R(X) exactly in the algebraic model. Indeed the maps ρ_{K}^{!}Ψ^{K}X ⟶ ρ_{L}^{!}Ψ^{L}X are all identified exactly, and we can form the homotopy inverse limit in the algebraic model. However, since inverse limits do not preserve F-free objects, the answer is not very attractive. Our present purpose requires much less; indeed, since T R(X)^{h𝕋} is just a rational spectrum, and it remains only to understand the action of F on homotopy groups.

Lemma 17.3.5. When L ⊆ K the Frobenius map F_{L}^{K} induces the projection corresponding to the inclusion i_{L}^{K} : [⊆ L] ⟶ [⊆ K] in the sense that the diagram

E𝕋_{+} ∧ Ψ^{K}X ⟶^{1∧F_{L}^{K}} E𝕋_{+} ∧ Ψ^{L}X
≃↓  ↓≃
⋁_{H⊆K} X(1) ⟶^{(i_{L}^{K})^{∗}} ⋁_{H⊆L} X(1)

commutes.

Proof: The first necessity is to understand the statement. We begin with a map Ψ^{K}X ⟶ Ψ^{L}X, which we may view as a map of 𝕋 spectra indexed on U^{K}. Once we have smashed with E𝕋_{+} the universe may be replaced by a complete one, and we obtain a map in the category for which we have a model.

To understand the map we factor Ψ^{K}X ⟶ Ψ^{L}X as Ψ^{K}X ⟶ infΨ^{K}X ⟶ Ψ^{L}X. If we view these as maps of 𝕋/L spectra, the second map is the counit of the K/L fixed point adjunction, completely understood by 12.2.2 and the contents of Section 11.3. The first map has the property that it is a nonequivariant equivalence. The result now follows from our description of the adjunction.

It remains only to index the terms so that the relevant structure is visible, and to verify that the circle action does not get in the way.

We begin by replacing subgroups by their orders, and defining a category as follows. The object set ℤ_{>0} × ℤ_{>0} consists of integer points in the strictly positive orthant. There are morphisms (ϕ_{s}, ρ_{t}) : (m, n) ⟶ (ms, nt) for s, t ∈ ℤ_{>0}. Next, consider the diagram D of divisors defined by D(m, n) = {d | d divides mn} and ϕ_{s} : D(m, n) ⟶ D(ms, n) is inclusion, ρ_{t} : D(m, n) ⟶ D(m, nt) is the multiplication by t. Finally, for an object Y we consider the contravariant functor Y^{D} defined by taking functions from D into Y; thus on objects, Y^{D}(m, n) = Y^{D(m,n)}. The connection with the restriction and Frobenius diagram is immediate from 17.3.5 and 17.3.3. The maps will be clearest if we replace D(m, n) by the set of rational numbers i/j where i divides m and j divides n. It is easy to check that these fractions are in bijective correspondence to divisors of mn: if d divides mn the relevant fraction is d/n. With this indexing, both R and F simply drop irrelevant coordinates. Now, passing to limits under restriction maps we obtain Y^{D}(m) := lim_{←}_{n} Y^{D}(m, n), which simply consists of sequences (y_{i/j}) with i dividing m. The map F_{s} : Y^{D}(ms) ⟶ Y^{D}(m) again simply drops coordinates with numerator dividing ms but not m. In other words, if we now identify Y^{D}(m) with Y^{D}(1) by dividing the coordinate indexes by m we find F_{s} is the shift map specified by mutiplying indices by s and ignoring fractions with an integer numerator bigger than 1. The system consists of surjections, so lim_{←}_{m}^{1}(Y^{D}(1), F) = 0, and evidently the only compatible families are those with all coordinates equal: lim_{←}_{m}(Y^{D}(1), F) = Y. This description suggests that we should have a means for discussing subgroups of 𝕋 with fractional orders, which suggests we should be considering the solenoid 𝕊 := lim_{←}(𝕋, ψ_{s}) which is the inverse limit of copies of the circle under the power maps ψ_{s} .

We must now check that the fact we have taken homotopy 𝕋-fixed points between the R and F stages does not invalidate the above procedure. The time has come to be precise about the equivariance of the Frobenius maps. First, note that although we have the behaviour F_{s}z = z^{s}F_{s} in the monoid M, so that F_{s} is identified with the map ψ_{s} : E𝕋_{+} ⟶ ψ_{s}^{∗}E𝕋_{+}, we expect the reverse type of behaviour for the objects acted upon.

Lemma 17.3.6. The Frobenius map induces a map of 𝕋-spectra along ψ_{s}, in the sense that F_{s} : ψ_{s}^{∗}T R(X) ⟶ T R(X) is a map of naive 𝕋-spectra.

Proof: We must remember that the map F_{s} arose from the inclusions Ψ^{K}X ⟶ Ψ^{L}X, which is a map of 𝕋-spectra. However, when we have applied ρ^{!} in the appropriate way, we must insert the power map ψ_{s} to retrieve the equivariance.

The relevant map T R(X)^{h𝕋} ⟶ T R(X)^{h𝕋} is then obtained by passage to fixed points from

F (ψ_{s}, F_{s}) : ψ_{s}^{∗}F (E𝕋_{+}, T R(X)) = F (ψ_{s}^{∗}E𝕋_{+}, ψ_{s}^{∗}T R(X)) ⟶ F (E𝕋_{+}, T R(X)).

The relevant untwisting result is as follows.

Lemma 17.3.7. The sth power map ψ_{s} : E𝕋_{+} ⟶ ψ_{s}^{∗}E𝕋_{+} is a stable rational equivalence.

Let Y = ∏_{n} X(1), and consider the map F_{s} : ψ_{s}^{∗}Y ⟶ Y of 𝕋-spectra. the commutative diagram

F (E𝕋_{+}, ψ_{s}^{∗}Y) ⟶^{F (1,F_{s})} F (E𝕋_{+}, Y)
F (ψ_{s}, 1) ↑≃  ↑=
F (ψ_{s}^{∗}E𝕋_{+}, ψ_{s}^{∗}Y) ⟶^{F (ψ_{s},F_{s})} F (E𝕋_{+}, Y)

has an equivalence in its left hand vertical. Hence we can untwist the action on ∏_{n} X(1)^{h𝕋}.

Corollary 17.3.8. Rationally, we may identify the system of copies of T R(X)^{h𝕋} under the Frobenius map with ∏_{n>0} X(1)^{h𝕋} and with the Frobenius F_{s} acting via multiplication by s shifts.

The theorem now follows.

APPENDIX A Mackey functors.

The algebra of Mackey functors for the circle group is so much simpler than that of a general compact Lie group that it is worth an elementary treatment. The general case is presented elsewhere [11]. This appendix is of interest in its own right, but parts of this section are used in justifying an assertion in Section 1.4, and it is used also in Chapter 13.

A rational Mackey functor is a contravariant additive functor on the full subcategory hSO of the rational stable homotopy category with the natural cells G/H_{+} as objects. We shall say M is an hSO-module.

A Mackey functor M extends to the full subcategory hSV O on finite wedges of natural cells. Equally if b is a retract of a natural cell n there is an idempotent self-map e : n ⟶ n with b = en and we may define M(b) = eM(n).

Now let hSB denote the full subcategory on the basic cells and note that hSV O ⊆ hSV B; we thus have functors

hSO ⟶ hSV O ⟶ hSV B ⟵ hSB,

and from the above discussion it is easy to see that they induce equivalences of module categories.

Proposition A.9. There is an equivalence of categories Mackey functors=hSO-modules ≃ hSB-modules.

As a matter of notation, if M is a Mackey functor we let V : hSB ⟶ Ab denote the corresponding hSB-module. This is somewhat redundant since V and M are equal functors on hSV B, but it allows us to follow standard usage and abbreviate M(G/H_{+}) to M(H) and V (σ_{H}^{0}) to V (H). Quite explicitly we have

A.10.

M(𝕋) = V (𝕋) and M(H) = ⨁_{K⊆H} V (K)

V (𝕋) = M(𝕋) and V (H) = e_{H}M(H).

The advantage of considering V is that hSB has very few morphisms, and in fact we see that V is specified by a diagram

[figure: V (𝕋); ⋮; V (C_{3}); V (C_{2}); V (C_{1})]

of rational vector spaces.

Example A.11. It is immediate from the Yoneda lemma that any represented Mackey functors 𝔸_{c} = [ , c]^{𝕋} is projective for any basic or natural cell c. We let P_{H} denote the hSB-module corresponding to the Mackey functor 𝔸_{σ_{H}^{0}}, and similarly P_{𝕋} corresponds to 𝔸. One readily calculates that P_{𝕋} = 𝔸 is ℚ at each basic cell, and the restriction maps are all the identity. Similarly the hSB-module P_{H} represented by σ_{H}^{0} is ℚ at H and 0 elsewhere.

We shall refer to the projectives P_{𝕋} and P_{H} as principal projectives. They are obviously indecomposable.

By the Yoneda lemma Hom(P_{H}, V) = V (H), and similarly for 𝕋.

One may make explicit the condition that an hSB-module V is injective or projective, and give canonical resolutions.

Lemma A.12. (i) An hSB-module V is projective if and only if each of the restriction maps V (𝕋) ⟶ V (H) is a monomorphism. (ii)An hSB-module V is injective if and only if the map V (𝕋) ⟶ ∏_{H} V (H) is an epimorphism.

Proof: (i) To see that any module V satisfying the specified condition is projective we suppose given a short exact sequence 0 ⟶ K ⟶ U ⟶^{f} V ⟶ 0 and construct a submodule V′ of U which is a complement of K.

We begin at 𝕋, choosing V′(𝕋) to be an arbitrary vector space complement to K(𝕋) in U(𝕋).

Now choose a finite subgroup H. Let \overline{V′(𝕋)} denote the image of V′(𝕋) in U(H), and similarly let \overline{V (𝕋)} denote the image of V (𝕋) in V (H). Since V (𝕋) ⟶ V (H) is a monomorphism by hypothesis, and since f is a map of Mackey functors, f(H) gives an isomorphism \overline{V′(𝕋)} ⟶^{≅} \overline{V (𝕋)}. It follows that \overline{V′(𝕋)} ∩ K(H) = 0 and we may choose a vector space complement V′(H) of K(H) which contains \overline{V′(𝕋)}.

With these choices the vector spaces V (𝕋) and V′(H) form a sub-hSB-module V′ of U.

The necessity of the condition will follow from the fact that it holds for the principal projectives P_{H} above. Indeed, by the Yoneda lemma, for any module V there is an epimorphism P_{V} ⟶ V with P_{V} a sum of principal projectives. If V is projective, the epimorphism must split, so the condition holds for V. (ii) The proof is similar but this time we construct the splitting at finite subgroups first. Necessity again follows using a canonical monomorphism V ⟶ I_{V} which we construct in the proof of A.13 below.

Note in particular that if V (𝕋) = 0 then V is projective, and if V (H) = 0 for all finite subgroups H then V is injective.

Lemma A.13. Any hSB-module V has a canonical projective resolution

0 ⟶ K_{V} ⟶ P_{V} ⟶ V ⟶ 0

and a canonical injective resolution

0 ⟶ V ⟶ I_{V} ⟶ C_{V} ⟶ 0,

where the modules P_{V} , K_{V} , I_{V} and C_{V} are defined in the proof. In particular, the category of hSB-modules (and hence also the category of Mackey functors) is of global dimension 1.

Proof: Note from the Yoneda lemma that Hom(P_{H}, V) = V (H). Therefore, for an arbitrary hSB-module V we may define P_{V} = (V (𝕋) ⊗ P_{𝕋}) ⊕ ⨁_{H}(V (H) ⊗ P_{H}) and a surjective map P_{V} ⟶ V. Since P_{H}(𝕋) = 0 the kernel K_{V} of this map is zero at 𝕋, and hence projective. Evidently this resolution will not usually be minimal.

Similarly we may define an injective I_{V} by taking I_{V} (𝕋) = V (𝕋) ⊕ ∏_{H} V (H) and I_{V} (H) = V (H). The restriction map I_{V} (𝕋) ⟶ I_{V} (H) is the restriction map of V on the V (𝕋) factor and projection on the ∏_{H} V (H) factor. The module I_{V} is injective and there is an obvious monomorphism V ⟶ I_{V}, whose cokernel C_{V} is concentrated at 𝕋, where we have C_{V} (𝕋) = ∏_{H} V (H).

It is obvious that any module M with M(𝕋) = 0 is a sum of projectives P_{H}, and it follows that any projective module is a direct sum of copies of the various indecomposable projectives P_{𝕋} and P_{H}.

It is appropriate to say that a Mackey functor M is finitely generated if it is a quotient of a finite sum of representable functors 𝔸_{𝕋/H}, and similarly for an hSB-module. We may thus consider the projective class group K_{0}(𝔐_{𝕋}) of rational Mackey functors, which is the Grothendieck group of finitely generated projective modules.

For topological purposes, the functors 𝔸_{𝕋/H} and 𝔸 represented by the natural cells are specially significant, and the quotient of K_{0}(𝔐_{𝕋}), by the submodule they generate is the home of the Wall finiteness obstruction. Since the Mackey functors represented by basic cells can be expressed as a difference of the representable functors corresponding to the natural cells, this quotient is zero and the Wall finiteness obstruction vanishes for all rational 𝕋-spectra.

We may be more precise.

Corollary A.14. The projective class group of rational Mackey functors is free abelian on the generators P_{𝕋} and P_{H}.

Proof: The comments above establish that the modules P_{𝕋} and P_{H} are generators. To see they are independent we define a rank function for each subgroup. Indeed we take r_{𝕋}(P) = dim_{ℚ}P (𝕋) and for a finite subgroup H we take r_{H}(P) = dim_{ℚ}(cok{P (𝕋) ⟶ P (H)}); these are additive on extensions, and hence give functions on the class group. It is clear that

[P] = r_{𝕋}(P)[P_{𝕋}] + Σ_{H}r_{H}(P)[P_{H}]

for any projective P.

We also have a structure theorem for injectives. In fact there are two types of injectives I: the F-injectives with I(𝕋) = ∏_{H} I(H) and the restriction maps being projections, and the F̃-injectives with I(H) = 0 for all finite subgroups H. An F̃-injective is determined by its value at 𝕋, and we write L[U] for the F̃-injective with value U at 𝕋. An F-injective I is determined by its values at the finite subgroups, and indeed

I ≅ ∏_{H} I(H) ⊗ R_{H}

where R_{H} is the F-injective with R_{H}(H) = R_{H}(𝕋) = ℚ and R_{H}(K) = 0 for K ≠ H.

Lemma A.15. For any injective I there is a splitting

I ≅ I_{F} ⊕ L[C(I)]

where I_{F} is an F-injective and C(V) = ker{V (𝕋) ⟶ ∏_{H} V (H)}.

Proof: Clearly L[C(I)] is a submodule of I, and we may define I_{F} by the short exact sequence

0 ⟶ L[C(I)] ⟶ I ⟶ I_{F} ⟶ 0.

This splits since L[C(I)] is injective, and I_{F} is easily seen to be a F-injective.

We conclude with some examples which occur geometrically. The most general class is that of spectra X ∧ ẼF; since these are H-contractible we find the Mackey functor \underline{π}_{n}^{𝕋}(X ∧ ẼF) is the F̃-injective L[π_{n}(Φ^{𝕋}X)].

Perhaps more important are the homotopy functors of cells.

Example A.16. By definition, \underline{π}_{0}^{𝕋}(σ_{H}^{0}) = P_{H} and similarly for 𝕋. The other homotopy functors of cells are \underline{π}_{2k+1}^{𝕋}(S^{0}) = L[ℚF] for k ≥ 0, and \underline{π}_{1}^{𝕋}(σ_{H}^{0}). Let R′_{H} = \underline{π}_{1}^{𝕋}(σ_{H}^{0}): we shall prove that R′_{H} ≅ R_{H}. It is easy to see there is an extension

0 ⟶ P_{H} ⟶ R′_{H} ⟶ L[ℚ] ⟶ 0.

Lemma A.17. (a) Ext(L[ℚ], P_{H}) ≅ ℚ, and hence all non-split extensions define isomorphic modules.

(b) The module R′_{H} is a non-trivial P_{H} extension over L[ℚ].

Proof: (a) We simply apply Hom(·, P_{H}) to the projective resolution

0 ⟶ ⨁_{K} P_{K} ⟶ P_{𝕋} ⟶ L[ℚ] ⟶ 0.

(b) It is enough to verify that the restriction map R′_{H}(𝕋) ⟶ R′_{H}(H) is nonzero. For this we recall that restriction is induced by the projection S^{1} ∧ G/H_{+} ⟶ S^{1} ∧ S^{0} in [·, σ_{H}^{0}]^{𝕋} = [·, e_{H}S^{1}]^{H}. Since the projection is H-equivariantly a retraction onto a direct factor, it is nontrivial.

200 A. MACKEY FUNCTORS.

APPENDIX B Closed model categories.

We have a number of algebraic categories modelling categories of spectra: we have in mind (0) the category of torsion ℚ[c_{H}]-modules, (1) the category of F-finite torsion O_{F}-modules,

(2) the standard model category A, and (3) the torsion module category A_{t}. Since we want to show they provide models for the homotopy theory of the categories of spectra, we need to show they admit such a structure themselves. We therefore prove that they all admit the structure of closed model categories in the sense of Quillen [21]. For reference, we recall that the structure of a closed model category consists of three classes of morphisms: the class we of weak equivalences, the class fib of fibrations, and the class cof of cofibrations. Morphisms which are weak equivalences and fibrations are known as acyclic fibrations, and similarly for cofibrations. These are required to satisfy Quillen’s axioms CM1-5 [22].

The axioms CM1-3 are formal properties, and are easily checked in the cases which concern us, but it will be convenient to record the terminology associated with axioms CM4 and CM5. Given morphisms i : A ⟶ B and p : X ⟶ Y, we say that we say that i has the left lifting property (LLP) with respect to p and that p has the right lifting property (RLP) with respect to i if, for any choice of f and g giving a commutative square

\begin{matrix}A & ⟶^{f} & X \\ i ↓ & ↗ & ↓ p \\ B & ⟶^{g} & Y, \end{matrix}

there is a diagonal arrow making both triangles commute. Axiom CM4 states that if i is a cofibration and p is a fibration then there is such a diagonal arrow if either (i) p is a weak equivalence or (ii) i is a weak equivalence. Axiom CM5 states that any morphism f : X ⟶ Y can be factorized both (i) (cofibrant approximation) as X ⟶^{i} Ŷ ⟶^{p} Y where i is a cofibration and p is an acyclic fibration and (ii) (fibrant approximation) as X ⟶^{i} X̂ ⟶^{p} Y where i is an acyclic cofibration and p is a fibration.

Our algebraic models all arise as categories of dg objects in an abelian category 𝔸. Since we aim to form the quotient category in which homology isomorphisms are inverted we take the class we of weak equivalences to be the homology isomorphisms in each case. Furthermore, the abelian categories all fail to have enough projectives. We are therefore obliged to make cofibrancy automatic, and proceed as follows.

We have a family BI of basic injectives in 𝔸; this is required to contain enough injectives that any object of 𝔸 embeds in some product of elements of BI, but it is convenient for verifications to choose BI to as small as possible. Regarding such a basic injective I as a differential graded object with zero differential, we obtain the Eilenberg-MacLane object (or co-sphere) K(I), and the path object (or co-disc) P (I) = fibre(1 : K(I) ⟶ K(I)).

Clearly P (I) is acyclic in all cases, and we use the collection of maps P (I) ⟶ 0 as the test acyclic ‘fibrations’. It will transpire they are indeed fibrations. We therefore define the class cof to consist maps i : A ⟶ B having the LLP with respect to P (I) ⟶ 0 for all basic injectives I. The class fib then consists of all maps p : X ⟶ Y having the RLP with respect to acyclic cofibrations. It is tautologous the maps P (I) ⟶ 0 are indeed fibrations for basic injectives I. Furthermore, it is immediate from the definition that the class fib is closed under base extension, and thus in particular projections Y × P (I) ⟶ Y are also fibrations.

The main result is that this procedure gives dg𝔸 the structure of a closed model category in each case. In practice, the proof proceeds as follows; we have highlighted the steps requiring properties of the particular categories in question. These follow by an identification of the group of dg maps into P (I) and K(I) for basic injectives.

Step 1: Identify the cofibrations explicitly as maps i : A ⟶ B for which i is a termwise monomorphism. Step 2: Show that for any X there is a monomorphism α : X ⟶ P (I) for some injective I.

Since we have chosen BI to include enough injectives, this is easy to prove, and the axiom CM5(i) follows quickly. Indeed for any f : X ⟶ Y we factorise it as

X ⟶^{{f,α}} Y × P (I) ⟶^{proj} Y.

Step 3: Prove that the maps P (I) ⟶ K(I) are fibrations for all basic injectives I.

Note that any injective is a retract of a product of basic injectives, so it follows that P (I) ⟶ K(I) is a fibration for any injective I.

Step 4: Prove the fibrant approximation axiom CM5 (ii) using only well understood fibrations.

More precisely, given f : X ⟶ Y we form a factorization X ⟶ Y′ ⟶ Y with X ⟶ Y′ a homology isomorphism and Y′ ⟶ Y a fibration formed by iterated pullback of fibrations P (I) ⟶ K(I). This is precisely dual to the usual argument attaching cells to make a map of spaces into a weak homotopy equivalence, but because the dual of the small object argument does not apply, we use the finiteness of injective dimension in 𝔸 to see that only finitely many steps are involved in this process. The map X ⟶ Y′ can be made into a cofibration by using an acyclic as in the proof of CM5 (ii). We may now see using the defining RLP, that an arbitrary fibration is a retract of such a standard one.

We will examine Case 0 in detail. It will make the structure of the argument clear, and there is an important point to be made about products. We let O = k[c] and we are particularly concerned with the case k = ℚ and c = c_{H}.

The set BI of basic injectives consists of suspensions of the standard torsion injective 𝕀 = Hom_{k}(O, k) = Σ^{−2}k[c, c^{−1}]/k[c]. The first lemma is elementary, and is probably clearer if we allow an arbitrary graded vector space of coefficients.

Lemma B.18. Given a graded vector space V we may form the injective 𝕀(V) = Hom_{k}(O, V).

(i) Hom_{O}(R, K(𝕀(V))) = Hom_{k}(R, V) (ii) dgHom_{O}(R, K(𝕀(V))) = Hom_{k}(R/dR, V) (iii) dgHom_{O}(R, P (𝕀(V))) = Hom_{k}(ΣR, V).

We warn that 𝕀(V) is only a torsion module if V is bounded below.

Lemma B.19. A map i : A ⟶ B is a cofibration if and only if it is monomorphic in each degree.

Proof: From the previous lemma we see that the problem

\begin{matrix}A & ⟶^{f} & P (Σ^{n+1}𝕀(k)) \\ i ↓ & ↗ & ↓ p \\ B & ⟶ & 0 \end{matrix}

in the category of dg O-modules is equivalent to the problem

\begin{matrix}A_{n} & ⟶^{f_{n}} & k \\ i_{n} ↓ & ↗ & ↓ p \\ B_{n} & ⟶ & Y \end{matrix}

in the category of k-vector spaces. Therefore i : A ⟶ B has the LLP for P (Σ^{n+1}𝕀(k)) ⟶ 0 if and only if i is monomorphic in degree n.

Lemma B.20. For any graded vector space V, the maps and P (𝕀(V)) ⟶ K(𝕀(V)) are fibrations.

Proof: We must show that P (𝕀(V)) ⟶ K(𝕀(V)) has the RLP for all termwise monomorphisms i : A ⟶ B which are homology isomorphsims. Using the first lemma, we translate the lifting problem about dg O-maps into a problem about k-maps into V. We convert the problem of dg O-modules

\begin{matrix}A & ⟶^{f} & P (𝕀(V)) \\ i ↓ & ↗ & ↓ p \\ B & ⟶^{g} & K(𝕀(V)) \end{matrix}

into the problem

\begin{matrix}A & ⟶^{f′} & Σ^{−1}V \\ i ↓ & ↗ & ↑ \overline{g}′ \\ B & ⟵^{\overline{d}} & Σ^{−1}B/dB. \end{matrix}

of k-maps, with the extra fact that i is a dg O-map. Regarding A as a sub-graded vector space of B we see how to define the diagonal on A, and on dB we must take db ⟼ g′(b). The fact that i is a homology isomorphism can be used to check this is well defined.

Now, as remarked above, the modules 𝕀(V) will not be torsion modules if V is not bounded below. However the c-power torsion functor Γ is right adjoint to the full and faithful inclusion i : torsO − mod ⟶ O − mod of the category of torsion O-modules in the category of all O-modules. Therefore, for arbitrary graded vector spaces V, the map ΓP (𝕀(V)) ⟶ ΓK(𝕀(V)) is a fibration in the category of dg torsion O-modules. It also follows that ΓP (𝕀(V)) ⟶ 0 is a fibration, and since ΓP (𝕀(V)) is the fibre of the identity map of ΓK(𝕀(V)), it is also acyclic.

It is now clear that any X admits an embedding into an acyclic fibrant object, namely X ⟶ P (𝕀(X)), and hence CM5 (i) follows. It remains to establish fibrant approximation, and we suppose given f : X ⟶ Y, with H_{∗}(X) = M, H_{∗}(Y) = N. We shall construct maps X ⟶ Y_{3} ⟶ Y_{2} ⟶ Y_{1} ⟶ Y_{0} = Y, with X ⟶ Y_{3} ⟶ Y the required factorization. To construct Y_{n+1} from Y_{n} we form a pullback diagram

X − − −  u_{n}
| ↘ f_{n+1}  ↘
|  Y_{n+1} ⟶ P (I_{n})
↘  ↓  ↓
f_{n}  Y_{n}  ⟶^{β_{n}} K(I_{n}).

This corresponds to a short exact sequence 0 ⟶ Y_{n+1} ⟶ Y_{n} ⊕ P (I_{n}) ⟶ K(I_{n}) ⟶ 0 and hence, since P (I) is acyclic, it gives a long exact sequence in homology

⋯ ⟶ Σ^{−1}I_{n} ⟶ H_{∗}(Y_{n+1}) ⟶ H_{∗}(Y_{n}) ⟶^{(β_{n})_{∗}} I_{n} ⟶ · · · .

At each stage we have to specify maps u_{n} and β_{n} making the diagram commute, and this gives f_{n+1} : X ⟶ Y_{n+1}. Equivalently, we need to specify k-maps u′_{n} and β′_{n} so that the diagram

X ⟶^{d} ΣX
f_{n} ↓  ↓ u′_{n}
Y_{n} ⟶^{β′_{n}} V_{n}

commutes. We shall usually begin by choosing β′_{n} so that β_{n} has the desired effect on homology. Provided β′_{n}f_{n} is zero on cycles, it induces a map X/ker(d) ⟶ V_{n}, and we may extend this to a k-map on ΣX without obstruction. At the stage n = 0 we must take care about u_{0}, but u_{1} and u_{2} are arbitrary extensions.

Choose an embedding η : M ⟶ Σ^{−1}I_{0} of M in an injective. First take β_{0} = 0 and u_{0} so that it induces η on homology. For example, we may choose a complement Z′ to the cyles and let u_{0} be the composite X ⟶ X/(Z′ + dX) ≅ H_{∗}(X) ⟶ Σ^{−1}I_{0}. Thus N_{1} = N ⊕ I_{0} and f_{1} : X ⟶ Y_{1} induces {f_{∗}, η} : M ⟶ N_{1} = N ⊕ I_{0}. Let C_{1} be the cokernel, and choose an injective resolution

0 ⟶ C_{1} ⟶ I_{1} ⟶ ΣI_{2} ⟶ 0.

We next choose the map β_{1} so that it induces the composite N_{1} ⟶ C_{1} ⟶ I_{1}: again we may choose a complement Z′_{1} to the cycles in Y to give a k-map Y_{1} ⟶ H_{∗}(Y_{1}) = N_{1}, and the resulting map clearly vanishes on cycles of X so that u_{1} exists. Now (β_{1})_{∗} has kernel M and cokernel ΣI_{2}. We therefore have a short exact sequence

0 ⟶ I_{2} ⟶ N_{2} ⟶ M ⟶ 0,

which necessarily splits, since I_{2} is injective.

Finally, choose the map β_{2} to kill the extraneous I_{2}, just as we chose β_{1}. The existence of u_{2} is again automatic.

Case 2: For the standard model the procedure is precisely similar. The basic injectives are suspensions of (𝕀_{H} ⟶ 0), and suspensions of (t_{∗}^{F} ⟶ t_{∗}^{F}). From this all is automatic once we have observed that the basic injectives lie in the image of the inclusions e : k−mod ⟶ A and f : torsO_{F}^{f}-mod ⟶ A, which are left adjoint to the vertex and nub functors.

Lemma B.21. (i) Hom((N ⟶ t_{∗}^{F} ⊗ V), (T ⟶ 0)) = Hom(N, T) (ii) Hom((N ⟶ t_{∗}^{F} ⊗ V), (t_{∗}^{F} ⊗ W ⟶ t_{∗}^{F} ⊗ W) = Hom(V, W).

Case 3: For the torsion model the procedure is precisely similar. The basic injectives are suspensions of (t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, 𝕀_{H}) ⟶ 𝕀_{H}), and suspensions of (t_{∗}^{F} ⟶ 0). From this all is automatic once we have observed that the basic injectives lie in the image of the inclusions e_{t} : k − mod ⟶ A and f_{t} : torsO_{F}^{f}-mod ⟶ A, which are left adjoint to the vertex and torsion functors.

Lemma B.22. (i) Hom((t_{∗}^{F} ⊗ V ⟶ S), (t_{∗}^{F} ⊗ Hom(t_{∗}^{F}, T) ⟶ T)) = Hom(S, T) (ii) Hom((t_{∗}^{F} ⊗ V ⟶ S), (t_{∗}^{F} ⊗ W ⟶ 0) = Hom(V, W).

We are frequently concerned with Quillen closed model categories of dg objects in abelian categories, and in functors between them which arise as functors between the associated abelian categories. If 𝔸 and 𝔹 are abelian categories and F : 𝔸 ⟶ 𝔹 is a functor, we obtain an induced functor on dg categories, which we continue to denote F. We shall often have cause to discuss total right derived functors RF : D𝔸 ⟶ D𝔹; the first observation is that it always exists.

Lemma B.23. Any functor F : 𝔸 ⟶ 𝔹 induces a functor on dg categories which has a total right derived functor RF : D𝔸 ⟶ D𝔹. Any fibrant approximation X ⟶ X̂ induces an equivalence RF (X) ≃ F (X̂). If F is exact then RF (X) = F (X) for any X.

Proof: Note that our categories have the property that weak equivalences between fibrant objects are homotopy equivalences. Since the functor F arises from the level of abelian categories, it preserves homotopies and hence homotopy equivalences; this is sufficient by [21, p I.4.4]. The final sentence is clear.

However we only obtain really satisfactory behaviour (such as the preservation of composites) if the functor takes fibrations to fibrations. Note that when F is the right adjoint to an exact functor, it satisfies Conditions (i) and (ii) in the following.

Lemma B.24. Suppose F : 𝔸 ⟶ 𝔹 is a functor so that

(i) F preserves injectives, (ii) F preserves limits then with closed model structures defined by homology isomorphisms and basic injectives as above, F preserves fibrations.

Proof: Note first that the functor on dg categories preserves mapping fibres. It follows that F (P (I)) = P (F (I)), and hence by (i), that F preserves fibrations P (I) ⟶ K(I) with I injective. Since any fibration is a retract of one formed from these by a finite number of pullbacks, it follows from (ii) that F preserves fibrations.

We remark that it then follows that, if F′ : 𝔹 ⟶ ℂ is a second functor satisfying Conditions (i) and (ii), then R(F′F) exists and R(F′ ∘ F) = RF′ ∘ RF.

Corollary B.25. If F : 𝔸 ⟶ 𝔹 is right adjoint to the exact functor G, RF is right adjoint to G.

Proof: The exactness of G ensures that it preserves cofibrations and weak equivalences, and hence that LG = G. Now the fact that RF is right adjoint to G follows from B.24 by [21, p I.4.5].

APPENDIX C Conventions.

C.1. Conventions for spaces and spectra.

Quite generally, when we work in several categories, we should have notation for all functors if all ambiguity is to be avoided. However, in a natural attempt to aid readability and understanding, it is common to omit notation for certain functors, particularly forgetful functors. Once this is done, notation must be chosen in the two categories so that it respects this convention. Unfortunately, this has not been done for the category of Gspectra: broadly speaking there are two notational conventions. Both give rise to formal ambiguity, although in practice authors take trouble to ensure the correct interpretation is used at all places. Each has advantages and disadvantages, but it seems worth making them explicit, together with means for resolving these difficulties. We also explain the particular convention we have found convenient to use here.

The Adams convention is based on the idea that a spectrum is a generalized space, so that if Z is a space, we should continue to write Z for its suspension spectrum, leaving context to determine which is intended. This convention is very convenient when working on the homotopy level, and is basically the one adopted here. However, it leads to difficulties if the same symbol θ is adopted for functors on spaces and spectra unless they agree under passage to suspension spectra. We need the relation

Σ^{∞}θ(Z) = θ(Σ^{∞}Z)

if the notation θ(Z) is to be unambiguous. In particular, under the Adams convention, it is inconsistent to use (·)^{H} for both the fixed point space functor and for the Lewis-May fixed point functor. Because of its historical precedence we use Z^{H} for the fixed point space functor, and Ψ^{H}X for the Lewis-May fixed point functor. To avoid confusion we use the Lewis-May notation Φ^{H}X for the geometric fixed point functor, rather than X^{H} as would be allowed by the Adams convention.

The May convention is broadly based on the idea that notation for the suspension spectrum functor should never be omitted. If fanatically implemented this convention leads to total clarity. However there are few fanatical enough to write Σ^{∞}S^{n} for the n-sphere spectrum. The symbol (S^{1})^{G} is thus formally ambiguous, although the probable meaning is clear enough.

There is a second choice of convention which has to be made. In this case both conventions are consistent, but the reader must be aware of which convention is in force.

The first convention is that functors forgetting equivariance should be omitted. Thus, if X is a G-spectrum, and H is a subgroup of G then we should continue to write X for the corresponding H-spectrum. This is in accordance with usual practice for spaces, and consistent with the suspension spectrum functor. Thus, for a G-spectrum X, we may write

X_{∗}^{G} = π_{∗}^{G}(X) = [S^{0}, X]_{∗}^{G},

and

X_{∗}^{H} = π_{∗}^{H}(X) = [G/H_{+}, X]_{∗}^{G} = [S^{0}, X]_{∗}^{H}.

However the notation does conflict with certain accepted usages.

The second convention is to display the group of equivariance as a subscript at all times. Thus we would write X_{G} for a G-spectrum, and we might write X_{H} for an H-spectrum without any implication that X_{H} was X_{G} regarded as an H-spectrum. This has the advantage of being explicit, but the above displays become

(X_{G})_{∗}^{G} = π_{∗}^{G}(X_{G}) = [(S^{0})_{G}, X_{G}]_{∗}^{G},

and

(X_{G})_{∗}^{H} = π_{∗}^{H}(X_{G}) = [(G/H_{+})_{G}, X_{G}]_{∗}^{G} = [(S^{0})_{H}, res_{H}^{G}(X_{G})]_{∗}^{H}.

For ordinary purposes we find these too unwieldy, but there are occasions when it is important to be careful. We note also that this convention leads to difficulties in the case of equivariant K-theory with reality in the sense of Atiyah.

Finally, we suppress mention of universes wherever possible, with the assumption that there is a natural choice of complete universe for the ambient group of equivariance. This has implications for certain notation for change of groups. Thus, if N is a normal subgroup of a group G, there is a natural way of regarding a G/N-spectrum as a G-spectrum. With universes explicit, we suppose given a complete G-universe U, and that our G/N-spectra are indexed on U^{N}; we let q : G ⟶ G/N denote the quotient and j : U^{N} ⟶ U denote the inclusion. Properly speaking, we have a functor q^{∗}, regarding a G/N-spectrum Y indexed on U^{N} as a G-spectrum q^{∗}Y indexed on U^{N}, and a change of universe functor j_{∗} taking a G-spectrum Z indexed on U^{N} and building in stability under the representaions not fixed by N to obtain a G-spectrum j_{∗}Z indexed on U. For work on the homotopy level, it is the composite functor Y ⟼ j_{∗}q^{∗}Y that is the important one. This was originally denoted q^{#}Y in [18], but it is now more common to abbreviate it to j_{∗}Y, often without explaining what j is. Since we want to concentrate on the groups involved, we write

inf_{G/N}^{G} : G/N − spectra ⟶ G − spectra

for this composite, and refer to it as the inflation functor. However, since the inflation of the suspension spectrum of a G/N-space Y is the suspension spectrum of the same space Y regarded as a G-space under pullback, we prefer to omit notation for the inflation functor whenever possible, in accordance with the Adams convention. This is a dangerous convention, and should only be used with care. In particular, it must not be used when it conflicts with the convention omitting the forgetful functor. For example, the inflation of the non-equivariant spectrum representing complex K-theory is different from the equivariant K-theory spectrum. In any case, we always suppress inflation in an expression of form ẼF ∧ Φ^{𝕋}X; not only is the meaning clear, but also the non-fixed representations are essentially built in by the smash product, since S^{V} ∧ ẼF ≅ ẼF at the space level.

C.2. Standing conventions.

The principal convention is that all spectra and modules are rationalized without comment. We also work at the homotopy level with topological objects unless indicated otherwise.

We regard it as natural to work equivariantly at all times, and hence we often omit the prefix ‘𝕋-’ unless required for special emphasis: thus ‘homotopy group’ unmodified means ‘equivariant homotopy group’, ‘equivalence’ unmodified means ‘equivariant equivalence’ and so forth. When we wish to refer to non-equivariant notions we shall say so explicitly. Similarly, the term ‘subgroup’ refers exclusively to closed subgroups.

We have also used homology grading throughout: in particular the cohomology of B𝕋 (arising as [E𝕋_{+}, E𝕋_{+}]_{∗}^{𝕋} and π_{∗}^{𝕋}(DE𝕋_{+})) is in negative degrees, and Chern classes are in degree −2. Of course, we also use the homology suspension: (Σ^{t}M)_{n} = M_{n−t}

There are several cases where a notational decision had to be made about suspensions. The problem is that notational simplicity in the topological and algebraic contexts suggest different decisions. The standard injective ℚ[c_{H}]-module 𝕀_{H} has its lowest non-zero group in degree 0. We were directed by mnemonic appeal, together with the fact that 𝕀_{H} = Hom(O_{F}, ℚ_{H}) and that it is the injective envelope of ℚ. This convention then extends to other standard injectives. However this means one must be careful about suspensions in certain places. Thus we have a short exact sequence 0 ⟶ ℚ[c_{H}] ⟶ ℚ[c_{H}, c_{H}^{−1}] ⟶ Σ^{2}𝕀_{H} ⟶ 0, and we have π_{∗}^{𝕋}(EF_{+}) = Σ𝕀.

A thornier issue was whether we should always write t_{∗}^{F} ⊗ V ⟶ ΣT for an object of the torsion model, so that the torsion part in the standard model, would become prominent. We decided against this on the grounds of brevity, and also because the the convention is easily forgotten, and is thereby liable to lead to confusion. Thus we write t_{∗}^{F} ⊗ V ⟶ T for an object of the torsion model, and the torsion part of the corresponding object of the standard model is Σ^{−1}T.

A Mackey functor M is a functor on the stable orbit category hSO with objects the natural cells 𝕋/H_{+}. We write M(H) for the value of this functor on 𝕋/H_{+}. However M determines a functor on the category of retracts of objects of hSO, and we let V = V_{M} denote the restriction of this to the category hSB of basic cells σ_{H}^{0}. We write V (H) for the value of this functor on σ_{H}^{0}.

We abbreviate ‘differential graded’ to ‘dg’ with few exceptions. Given an abelian category 𝔸 we have referred to the category of fractions in which the homology isomorphisms of dg𝔸 are inverted as the derived category of 𝔸, rather than as the homotopy category of dg𝔸. This is to avoid confusion with the naive meaning of homotopy and to emphasise its algebraic nature: it should lead to no confusion since it would be perverse to form the category of chain complexes of 𝔸 if 𝔸 is graded. However, after initial work with sets of naive homotopy classes, we follow the convention common in topology of using the symbol

210 C. CONVENTIONS.

[X, Y] to mean morphisms in the derived category (i.e. homotopy classes of maps from X to a fibrant approximation of Y).

APPENDIX D Indices.

D.1. Index of definitions and terminology.

A

Accessible  7.5  Constructible without indeterminacy

Adams spectral sequence Spectral sequence recovering a triangulated category from a functor to an abelian category.

B

Based map  5.3  A map of based modules with an implicit map of vertices

compatible with basing maps.

Based module 5.3 A module N with an implicit vertex and basing map. Basic injectives 4.1 A collection of injectives so that an arbitrary module embeds in a

product of suspensions of basic injectives.

Basic sphere  2.1  The principal part σ_{H}^{n} := e_{H}𝕋/H_{+} ∧ S^{n} of a natural sphere.
Basing map  5.3  A map N ⟶ t_{∗}^{F} ⊗ V which becomes an isomorphism when E is

inverted.

Brown-Comenetz 13.4 Cohomology defined by hI_{G}^{n}(X) = Hom(\underline{π}_{n}^{𝕋}(X), I) for an injective

Mackey functor I; the representing spectrum hI.

Burnside ring Ring of stable self maps of S^{0}; rationally, the ring C(ΦG, ℚ) where ΦG is the space of conjugacy classes of subgroups with finite index in their normalizer.

C

Co-Eilenberg-MacLane Spectrum JN representing ordinary homology with coefficients

in the coMackey functor N for some N.

Cofibrant B Defined by lifting property, but all objects in our algebraic categories are cofibrant.

Cofibration  B  Defined by lifting property, but all dimensionwise injective maps

in our algebraic categories are cofibrations.

CoMackey 13.3 A covariant additive functor on the stable orbit category. Complete G-universe A countable dimensional orthogonal representation of G

containing all finite dimensional representations of G.

Completion 11.1 (1) (of a O_{F}-module) the map M ⟶ ∏_{H} e_{H}M.

(2) I-adic completion for some ideal I.

Core A For a Mackey functor M: the kernel of the total

restriction map M(𝕋) ⟶ ∏_{H} M(H).

Cyclotomic 17.1 Spectrum equivalent to its geometric K-fixed points for all finite subgroups K.

D

Deflation  11.2  Pullback along q^{∗} : \overline{O}\overline{F} ⟶ O_{F}; left and right adjoint to inflation.

d-invariant The homomorphism induced by a map in the homology

theory under discussion.

Derived category 4.1 Category of dg objects with homology isomorphisms inverted.

E

Eilenberg-MacLane Spectrum HM representing ordinary cohomology with

coefficients in the Mackey functor M for some M.

e-invariant The extension class defined by the cofibre of a map with zero

d-invariant in the homology theory under discussion.

Euler classes (1) The pullback of the identity along a map e(V) : S^{0} ⟶ S^{V}

shifted into integer degrees.

8.1 (2) A power of an element c_{ϕ} = e_{ϕ}c + (1 − e_{ϕ}) for a finite set ϕ

of finite subgroups.

Euler-torsion Module annihilated by inverting all Euler classes.

F

Family A collection of subgroups closed under passage to subgroups

and conjugates.

F-contractible H-contractible for all finite H; equivalent to an object of

form ẼF ∧ Z.

F-finite O_{F}-module M so that M = ⊕_{H}e_{H}M. F-free Having isotropy groups in F Fibrant B Defined by lifting properties, but in our algebraic categories,

constructed from injectives with zero differential.

Fibration B Defined by lifting properties, but in our algebraic categories, constructed by coattaching injectives with zero differential.

Fixed A space on which all group elements act as the identity. F-space Having isotropy groups in F

F-injective  A  An injective Mackey functor with V (𝕋) ≅ ∏_{H} V (H).
F̃-injective  A  An injective Mackey functor concentrated at 𝕋.
Formal  6.2  An object of the torsion model determined up to homotopy

equivalence by its homology.

Free Having 1 as the only isotropy group.

G

Geometric fixed points The functor Φ^{K} extending the K-fixed point functor on G-spaces.

H

Hausdorff  11.1  An O_{F}-module M which embeds in ∏_{H} M(H)

Homotopy fixed points (of a based G-space X) the space of G-maps G − map(EG_{+}, X);

also the spectrum-level counterpart.

H-semifree Having isotropy in H ∪ {𝕋}.

I

inflation functor A functor from a category associated to a quotient group

G/N to that for the group G.

internal Hom 8.3 Right adjoint to tensor product. isotropy group A group H for which Φ^{H}X is non-equivariantly essential.

J

K

L

Left derived functor (of a functor) The left-universal counterpart of the functor at the

derived category level (Appendix B).

Lewis-May fixed points Right adjoint Ψ^{K} to the inflation functor.

M

Mackey functor A Contravariant additive functor on the stable orbit category. Moore spectrum Bounded below spectrum with Burnside-homology in exactly one

dimension.

N

Natural sphere 2.1 Sphere G/H_{+} ∧ S^{n} used to build CW-objects. Nub 5.3 The O_{F}-module in an object N ⟶^{β} t_{∗}^{F} ⊗ V of the standard

model.

O

Object-accessible 7.5 (of a functor on categories of spectra) all objects may be constructed from primitive objects so that the models of objects and functions are known precisely throughout.

Over H (of a 𝕋-spectrum X) Having Φ^{K}X non-equivariantly contractible if K ≠ H.

P

pure parity A dg module whose homology is only non-zero in one parity; a spectrum whose standard model has this property.

Q

R

Right derived functor (of a functor) The right-universal counterpart of the functor at the derived category level.

S

semifree Having isotropy groups 1 and 𝕋 only. sphere See ‘basic sphere’ and ‘natural sphere’. split 4.2 (1) Split one dimensional: a one dimensional abelian category in

which objects split into even and odd parts. 4.2 (2) Split linear: a triangulated category with a suitable

linearization functor to a split one dimensional abelian category.

standard model 5.3 Category with objects O_{F}-maps N ⟶ t_{∗}^{F} ⊗ V which become isomorphic when Euler classes are inverted.

T

Tate cohomology 16.2 The cohomology represented by t(k) = F (EG_{+}, k) ∧ ẼG torsion (1) A module annihilated by the inversion of all Euler classes.

(2) An object of from f(T) = (T ⟶ 0) for a torsion module T.

torsion free A module of the form e(V) = (t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V). torsion model 3.4 The algebraic counterpart of viewing 𝕋-spectra as

fibres of q_{X} : ẼF ∧ X ⟶ ΣEF_{+} ∧ X;

objects (t_{∗}^{F} ⊗ V ⟶ T) with T a torsion module.

U

V

vertex  5.3  The graded vector space V in an object N ⟶^{β} t_{∗}^{F} ⊗ V of

the standard model.

W

Weak equivalence (1) A map inducing an isomorphism of [K, ·]^{G} for all finite

complexes K.

B (2) Element of structural class for a Quillen closed model category structure.

X

Y

Z

D.2. Index of notation.

A

𝔸 An abelian category

𝔸_{c}  A  The free Mackey functor [·, c]^{𝕋} represented by c.
𝔸̂  8.3  A certain abelian category containing 𝔸.

A (i) An abelian category.

5.3 (ii) The standard model category.

A_{H} The standard model category for H-semifree spectra.

A_{t}  6.2  The torsion model category.

A(H)  2.1  The rationalized Burnside ring of H (≅ C([⊆ H], ℚ) when H is finite).

B

BI  B  Basic injectives.

C

ℂ Generic notation for a category. C Generic notation for cokernel.

C  17.1  The algebraic standard model of cyclotomic spectra.
C_{t}  17.1  The algebraic torsion model of cyclotomic spectra.

C(f) The mapping cone of f. c 4.3 The generator of O_{F} = [EF_{+}, EF_{+}]^{𝕋} in degree −2 arising as the

limit of Euler classes.

c_{H} 2.4 The part of the Euler class of V (H) lying over H:

c_{H} = e_{H}c = e_{H}χ(V (H)).

c_{ϕ} 8.1 The Euler class e_{ϕ}c + (1 − e_{ϕ}) associated to a finite subset ϕ ⊆ F. c^{v} The non-homogeneous element of t_{∗}^{F} which is c_{H}^{v(H)} at H. C(V) A The core of a Mackey functor over 𝕋: the kernel of the

restriction map V (𝕋) ⟶ ∏_{H} V (H).

C_{V} A The cokernel of the embedding V ⟶ I_{V} of a Mackey functor in the

canonical resolution.

CF[U] 13.4 The injective Mackey functor constant at U for all finite subgroups,

and U^{F} at 𝕋.

χ(V) 8.1 The Euler class of the representation V C(K, ℚ) Ring of continuous ℚ-valued functions on the (usually discrete)

space K.

cof  B  The class of cofibrations in a Quillen closed model category.
cof  6.3  The cofibre functor dgA ⟶ dgHA_{t} from the dg standard model to

the dg homology torsion model.

cof′ 6.3 The cofibre functor dgHA ⟶ dgHA_{t} from the dg homology standard model to the homology torsion model.

D

d(f) The d-invariant (ie induced map) of f. 𝔻 Generic notation for a category.

δ_{H}  15.2  (= pr_{𝕋}^{H}tr_{H}^{𝕋}) The standard generator of [S^{0}, S^{0}]_{1}^{𝕋}.
D(·)  4.1  Operation of forming derived category.
def  11.2  The deflation functor O_{F} − mod ⟶ \overline{O}\overline{F} − mod.

dg Differential graded.

Δ^{K}X  12.4  The free injective summand of the Lewis-May fixed points of X.

dgHom Differential graded homomorphisms. dgℂ Differential graded objects of ℂ. dgHA 6.3 Category of dg maps N ⟶ t_{∗}^{F} ⊗ V, whose homology is an

isomorphism when E is inverted.

dgHA_{t} 6.3 Category of dg maps t_{∗}^{F} ⊗ V ⟶ T, where the homology of T is

Euler-torsion.

Δ(X)  4.1  The fibrant approximation to X.
Δ  17.1  The functor C_{t} ⟶ A_{t} corresponding to the inclusion of cyclotomic

spectra in all spectra.

E

e (1) generic notation for an idempotent 10.2 (2) the idempotent supported at subgroups containing a fixed group K. 5.3 (3) the functor e(V) = (t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V) from graded vector spaces

to the standard model.

e_{t} The functor e_{t}(V) = (t_{∗}^{F} ⊗ V ⟶ 0) from the category of graded vector

spaces to the torsion model.

e_{H} The primitive idempotent with support H in O_{F}.

e_{ϕ}  8.1  The idempotent with support ϕ ⊆ F.
e(V)  8.1  The inclusion S^{0} ⟶ S^{V}.

e(f) The e-invariant (ie extension class) of f. eπ_{n}^{H}(·) Abbreviation of e_{H}π_{n}^{H}(·) = [σ_{H}^{n}, ·]^{𝕋}. E𝕋 The universal free 𝕋-space. EF The universal F-space. ẼF The unreduced suspension of EF.

E⟨H⟩  2.3  The part e_{H}EF_{+} of EF_{+} over H; mapping cone of E[⊃ H]_{+} ⟶ E[⊇ H]_{+}.
Ẽ⟨H⟩  2.3  The mapping cone of the natural map E⟨H⟩ ⟶ S^{0}
E  8.1  The set {c_{ϕ}^{k} | k ≥ 1, ϕ ⊆ F finite} of Euler classes.

F

F The family of finite subgroups of 𝕋. \overline{F} The family of finite subgroups of \overline{𝕋} = 𝕋/K. F (f) The mapping fibre of f. F_{H}[G, ·) The coinduction functor from H-spectra to G-spectra; right adjoint to the

forgetful map.

f 5.3 The functor f(T) = (T ⟶ 0) from the category of torsion O_{F}-modules

to the standard model.

f_{t} The functor f(T) = (0 ⟶ ΣT) from the category of torsion O_{F}-modules

to the torsion model.

F_{L}^{K} 17.3 The inclusion map Ψ^{L}X ⟶ Ψ^{K}X, which corresponds to the Frobenius map in algebraic K-theory.

218 D. INDICES.

F_{s}  17.3  The Frobenius map F_{L}^{K} when L/K is of order s.
fib  B  The class of fibrations in a Quillen closed model category.
fib  6.3  The fibre functor dgA_{t} ⟶ dgA from the dg standard model to the dg

torsion model.

fib′ 6.3 The fibre functor dgHA_{t} ⟶ dgHA from the dg homology torsion model to the dg homology standard model.

G G A compact Lie group. Γ̂ 8.3 The torsion functor: right adjoint to the inclusion i : 𝔸 ⟶ 𝔸̂. Γ_{c} The c-power torsion functor.

Γ_{f}  8.1  The E-torsion functor.

H H Generic notation for a finite subgroup of 𝕋. H A set of finite subgroups (not necessarily a family). HM The Eilenberg-MacLane spectrum with coefficient Mackey functor M.

hM  13.4  The Brown-Comenetz spectrum.
hSB  A  The category of stable maps of basic spheres.
hSB_{∗}  15.2  The category of graded stable maps of basic spheres.

hSV B A The category of finite wedges of basic spheres. hSO A The Burnside category of stable maps of natural spheres. hSV O A The category of finite wedges of natural spheres.

Hom  8.3  The Hom object in 𝔸̂.

I I Generic notation for injective or image. I 4.1 The collection of fibrant objects used to construct derived categories of

our algebraic categories.

𝕀 The standard injective F-finite O_{F}-module Σ^{−2}t_{∗}^{F}/O_{F} = ⨁_{H} 𝕀_{H}. 𝕀(H) 2.4 The standard injective ℚ[c_{H}]-module Σ^{−2}ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}]. 𝕀_{H} Variant on 𝕀(H).

I(Y)  3.4  The first stage in an injective Adams resolution Y ⟶ I(Y) ⟶ J(Y)
I_{V}  A  The canonical injective in which a Mackey functor V embeds.
I_{F}  A  The quotient of the Mackey functor I by the F̃-injective L[C(I)]

on its core.

I^{∗}  13.4  The Mackey functor \underline{π}_{−1}^{𝕋}(hI).
i  8.1  The inclusion ΓϕM ⟶ ϕM.

IntHom 8.3 The internal Hom functor.

inf_{𝕋/K}^{𝕋}  12.1  (1) The inflation functor taking a 𝕋/K-spectrum,

regarding it as a 𝕋-spectrum indexed on a K-fixed universe and then building in the other representations.

11.1 (2) The algebraic counterpart of the topological inflation functor.

J

J(Y)  3.4  The second stage in an injective Adams
resolution Y ⟶ I(Y) ⟶ J(Y)

JN 13.3 The coEilenberg-MacLane spectrum with coefficient Mackey

functor N.

j  8.1  The inclusion ϕM ⟶ M.

K

k Generic notation for a field. K (1) Generic notation for a finite subgroup of 𝕋.

(2) Generic notation for the kernel.

(3) Generic notation for a cohomology theory.

(4) Representing spectrum for complex K-theory.

K_{V} A The kernel of the map P_{V} ⟶ V

in the canonical projective resolution of a Mackey functor V.

\underline{K}_{G}^{q} The Mackey functor G/H_{+} ⟶ K_{G}^{q}(G/H_{+}) = K_{H}^{q}.

k  8.1  The inclusion Γ_{f}M ⟶ M; the composite ji.

L

L Generic notation for a finite subgroup of 𝕋.

L(·)  B  Total left derived functor operation.
L[U]  A  The F-contractible injective Mackey functor with value U at 𝕋.
λ(V)  15.4  The degree zero K-theory Euler class of the representation V.
ΛX  17.2  The space of free loops on a space X.
L_{H}  5.5  The algebraic counterpart of the basice cell σ_{H}^{0}.
Λ^{K}X  12.4  A summand of the Lewis-May fixed points of X.

M

M (1) Generic notation for a module.

(2) Generic notation for a Mackey functor regarded as

an hSO-module G/H_{+} ⟼ M(H).

M(H) (1) The summand e_{H}M of the O_{F}-module M.

(2) The value of the Mackey functor M at H.

M(F) (for a Mackey functor M) the limit lim_{→}_{H} M(H); equal to V (F) if M = M_{V}. M_{V} The Mackey functor (hSO-module) associated to the hSB-module V.

𝔐_{H}  10.1  The category of H-Mackey functors.
𝔐_{H}^{triv}  10.1  The category of H-Mackey functors M whose associated representations
e_{K}M(K) are trivial.

N

N Generic notation for a module.

O

O_{F}  4.3  The ring C(F, ℚ)[c] of operations for F-spectra.
O_{F}(v)  8.4  The v-twisted ring of operations.

O_{H} The ring C(H, ℚ)[c] of operations for H-free spectra.

O_{H}  2.4  The ring ℚ[c_{H}] = [E⟨H⟩, E⟨H⟩]_{∗}^{𝕋} of operations for 𝕋-spectra over H.

P

P Generic notation for a projective. P_{H} A The indecomposable projective Mackey functor e_{H}𝔸_{H} = [·, σ_{H}^{0}]^{𝕋}

concentrated over H.

P_{V}  A  The canonical projective mapping onto the Mackey functor V.
p  5.4  The equivalence 𝕋−Spec ⟶ DA, or one of the other equivalences

from a topological to an algebraic derived category.

pr_{𝕋}^{H}  15.2  The canonical map σ_{H}^{0} ⟶ σ_{𝕋}^{0}.
ϕ  5.3  (1) Generic notation for map of vertices.
8.1 (2) The functor ϕM = ⊕_{H}e_{H}M.

Φ^{K} 10.2 The geometric K-fixed point functor. Φ_{n} The nth cyclotomic polynomial. π_{∗}^{𝕋}(X) The stable homotopy of X. \underline{π}_{∗}^{𝕋}(X) The Mackey functor G/H_{+} ⟼ π_{∗}^{H}(X). π_{∗}^{A}(X) 5.4 The A-valued homotopy groups

(π_{∗}^{𝕋}(X ∧ DEF_{+}) ⟶ π_{∗}^{𝕋}(X ∧ DEF_{+} ∧ ẼF)) of X.

Ψ^{K} 12.1 (1) The Lewis-May K-fixed point functor.

11.3 (2) The algebraic counterpart of the Lewis-May K-fixed point functor.

\overline{Ψ}^{K} 12.4 The crude algebraic counterpart of the Lewis-May K-fixed point functor on the standard model.

Q

Q Generic notation for a projective. ℚ The rational numbers: usually suppressed. ℚ(H) The rationals viewed as a O_{F}-module via the H-augmentation

O_{F} ⟶ e_{H}O_{F} ⟶ ℚ.

ℚ_{H} Variant of ℚ(H). Q_{H} 13.3 Standard projective coMackey functor. q The quotient homomorphism 𝕋 ⟶ 𝕋/K. q_{∗} The map O_{F} ⟶ \overline{O}\overline{F} induced by the quotient homomorphism q.

q_{L}  5.5  The map e(V_{L}) ⟶ f(ΣT_{L}).
q_{X}  5.5  The map ẼF ∧ X ⟶ ΣEF_{+} ∧ X.
q̂_{X}  6.1  The map t_{∗}^{F} ⊗ π_{∗}(Φ^{𝕋}X) ⟶ π_{∗}^{𝕋}(ΣEF_{+} ∧ X) in the derived

category which classifies q_{X}.

R

R_{H}  A  The Mackey functor \underline{π}_{1}^{𝕋}(σ_{H}^{0}); the non-split extension of P_{H} over L[ℚ].
R(·)  B  Total right derived functor operation. In most cases arising

here, RF (X) = F (X̂) where X ⟶ X̂ is a fibrant approximation.

ρ_{K}  17.1  The |K|th root isomorphism 𝕋 ⟶^{≅} 𝕋/K.
ρ_{K}^{!}  17.1  The functor from 𝕋/K-spectra indexed

on U^{K} to 𝕋-spectra indexed on U arising from a cyclotomic universe.

r_{K}  17.1  The cyclotomic structure map ρ_{K}^{!}Φ^{K}X ⟶^{≃} X.
R_{L}^{K}  17.3  The ‘restriction map’ ρ_{K/L}^{!}Ψ^{K}X ⟶ Ψ^{L}X for a cyclotomic

spectrum X.

R_{s} 17.3 The restriction map R_{L}^{K} when L/K is of order s. R′Hom 8.5 Right adjoint of left derived tensor product; not the right derived

Hom functor.

S

S  4.1  (1) The class of morphisms to be inverted in forming the derived category.

(2) The multiplicative set generated by the K-theory Euler classes 1 − z^{n} for n ≥ 1.

S^{−1}ℂ 4.1 The category of fractions formed from ℂ by inverting morphisms in the

class S.

s 4.1 The fibrant approximation X ⟶ ΔX. S^{V} The one point compactification of the representation V. S^{kV (F)} 5.2 The direct limit of S^{kV (H)}. S(V) The unit sphere in the orthogonal representation V. S(V (F)) 5.2 The direct limit of S(kV (H)).

𝕊  17.3  The solenoid lim_{←}(𝕋, ψ^{s}).
σ_{H}^{n}  2.1  The basic n-sphere e_{H}(𝕋/H_{+} ∧ S^{n}) over H.

σ^{kV (H)} 2.3 The part of S^{kV (H)} which is H-semifree. Σ Suspension functor.

T T (1) Generic notation for a torsion module.

(2) Multiplicative set generated by the cyclotomic polynomials.

𝕋 The circle group. \overline{𝕋} The quotient group 𝕋/K. t_{∗}^{F} 5.2 The twisting module arising as the F-Tate cohomology of S^{0}

and also as E^{−1}O_{F}.

T C(X) 17.3 Topological cyclic homology spectrum of a cyclotomic spectrum X. T HH(F) 17.2 Topological Hochschild homology of a functor F with smash products. T R(X) 17.3 Topological restriction homology spectrum of a cyclotomic spectrum X. 𝕋 ⋉_{H} (·) 10.1 The induction functor from H-spectra to 𝕋-spectra; left adjoint to the

forgetful functor.

𝕋//F_{+} 1.4 The ‘cell with isotropy F’. 𝕋−Spec The category of rational 𝕋-spectra. 𝕋−Spec_{sf} The category of semifree rational 𝕋-spectra. 𝕋−Spec/H The category of rational 𝕋-spectra over H. 𝕋−Spec/F The category of rational 𝕋-spectra over F. torsO_{F}^{f}-mod The category of F-finite torsion O_{F}-modules.

tr_{H}^{𝕋}  15.2  The transfer map σ_{𝕋}^{1} ⟶ σ_{H}^{0} dual to the projection pr_{𝕋}^{H}.
τ_{H}  15.2  (= tr_{H}^{𝕋}pr_{𝕋}^{H}) Standard generator of [σ_{H}^{0}, σ_{H}^{0}]_{1}^{𝕋}
θ  5.3  Generic notation for a map of nubs.

U

U Generic notation for a graded vector space. U Generic notation for a universe, usually complete.

V

v  8.4  A twisting function F ⟶ ℤ.

V (1) Generic notation for a graded vector space.

A (2) Generic notation for a Mackey functor viewed as an

hSB-module σ_{H}^{0} ⟼ V (H).

V_{M} The hSB-module associated to the Mackey functor (hSO-module) M.

V (H)  A  The value of an hSB-module V on the basic cell σ_{H}^{0}.
V (F)  A  The sum ⨁_{H} V (H) for an hSB-module V; equal to M(F) if V = V_{M}.

W

W Generic notation for a graded vector space.

we  B  The class of weak equivalences in a Quillen closed model category.

X Generic notation for space or spectrum. X(𝕋) The part of a 𝕋-spectrum over 𝕋: ẼF ∧ X ≃ ẼF ∧ Φ^{𝕋}X. X(H) The part of a 𝕋-spectrum over H: E⟨H⟩ ∧ X. X(F) The part of a 𝕋-spectrum over F: EF_{+} ∧ X. X̂ A fibrant approximation to X. χ The Euler class 1 − z.

χ(V)  8.1  The Euler class of V.

Y

Y Generic notation for space or spectrum.

Z

ℤ The integers. ℤ((χ)) The ring ℤ[[χ]][χ^{−1}]. Z Generic notation for space or spectrum.

Symbols

⋉ (1) For differential graded objects, A ⋉ B denotes an object with quotient A and kernel B; the differential is given by those of A and B together with A ⟶ ΣB.

(2) Used in the induction functor 𝕋 ⋉_{H} X.

⊛  8.5  The left derived tensor product or right derived suspended torsion product.

224 D. INDICES.

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