Contents

Galois Extensions of Structured Ring Spectra 1

Abstract 2

Chapter 1. Introduction 3

Chapter 2. Galois extensions in algebra 10 §2.1. Galois extensions of fields 10 §2.2. Regular covering spaces 10 §2.3. Galois extensions of commutative rings 11

Chapter 3. Closed categories of structured module spectra 14 §3.1. Structured spectra 14 §3.2. Localized categories 15 §3.3. Dualizable spectra 17 §3.4. Stably dualizable groups 18 §3.5. The dualizing spectrum 19 §3.6. The norm map 20

Chapter 4. Galois extensions in topology 21 §4.1. Galois extensions of E-local commutative S-algebras 21 §4.2. The Eilenberg–Mac Lane embedding 22 §4.3. Faithful extensions 23

Chapter 5. Examples of Galois extensions 25 §5.1. Trivial extensions 25 §5.2. Eilenberg–Mac Lane spectra 25 §5.3. Real and complex topological K-theory 25 §5.4. The Morava change-of-rings theorem 27 §5.5. The K(1)-local case 34 §5.6. Cochain S-algebras 37

Chapter 6. Dualizability and alternate characterizations 40 §6.1. Extended equivalences 40 §6.2. Dualizability 41 §6.3. Alternate characterizations 45 §6.4. The trace map and self-duality 46 §6.5. Smash invertible modules 49

Chapter 7. Galois theory I 51 §7.1. Base change for Galois extensions 51

§7.2. Fixed S-algebras 52

Chapter 8. Pro-Galois extensions and the Amitsur complex 56 §8.1. Pro-Galois extensions 56 §8.2. The Amitsur complex 57

Chapter 9. Separable and étale extensions 61 §9.1. Separable extensions 61 §9.2. Symmetrically étale extensions 63 §9.3. Smashing maps 65 §9.4. Étale extensions 66 §9.5. Henselian maps 68 §9.6. I-adic towers 72

Chapter 10. Mapping spaces of commutative S-algebras 77 §10.1. Obstruction theory 77 §10.2. Idempotents and connected S-algebras 80 §10.3. Separable closure 82

Chapter 11. Galois theory II 85 §11.1. Recovering the Galois group 85 §11.2. The brave new Galois correspondence 86

Chapter 12. Hopf–Galois extensions in topology 89 §12.1. Hopf–Galois extensions of commutative S-algebras 89 §12.2. Complex cobordism 91

References 94

Stably Dualizable Groups 99

Abstract 100

Chapter 1. Introduction 101 §1.1. The symmetry groups of stable homotopy theory 101 §1.2. Algebraic localizations and completions 101 §1.3. Chromatic localizations and completions 103 §1.4. Applications 104

Chapter 2. The dualizing spectrum 106 §2.1. The E-local stable category 106 §2.2. Dualizable spectra 107 §2.3. Stably dualizable groups 108 §2.4. E-compact groups 109 §2.5. The dualizing and inverse dualizing spectra 111

Chapter 3. Duality theory 114 §3.1. Poincaré duality 114 §3.2. Inverse Poincaré duality 116 §3.3. The Picard group 119

Chapter 4. Computations 121 §4.1. A spectral sequence for E-homology 121 §4.2. Morava K-theories 121 §4.3. Eilenberg–Mac Lane spaces 124

Chapter 5. Norm and transfer maps 126 §5.1. Thom spectra 126 §5.2. The norm map and Tate cohomology 126 §5.3. The G-transfer map 129 §5.4. E-local homotopy classes 129

References 131

Index 133

Galois Extensions of Structured Ring Spectra

John Rognes

Author address: Department of Mathematics, University of Oslo, Norway

E-mail address: rognes@math.uio.no

Abstract

We introduce the notion of a Galois extension of commutative S-algebras (E_{∞} ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin–Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable and étale extensions of commutative S-algebras, and the Goerss–Hopkins–Miller theory for E_{∞} mapping spaces. We show that the global sphere spectrum S is separably closed, using Minkowski’s discriminant theorem, and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf–Galois extensions of commutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin–Tate Galois extensions.

Received by the editor February 8, 2005.

2000 Mathematics Subject Classification. 13B05, 13B40, 55N15, 55N22, 55P43, 55P60.

Key words and phrases. Galois theory, commutative S-algebra.

CHAPTER 1 Introduction

The present paper is motivated by (1) the “brave new rings” paradigm coined by Friedhelm Waldhausen, that structured ring spectra are an unavoidable generalization of discrete rings, with arithmetic properties captured by their algebraic K-theory, (2) the presumption that algebraic K-theory will satisfy an extended form of the étale- and Galois descent foreseen by Dan Quillen, and (3) the algebrogeometric perspective promulgated by Jack Morava, on how the height-stratified moduli space of formal group laws influences stable homotopy theory, by way of complex cobordism theory.

We here develop the arithmetic notion of a Galois extension of structured ring spectra, viewed geometrically as an algebraic form of a regular covering space, by always working intrinsically in a category of spectra, rather than at the naïve level of coefficient groups. The result is a framework that well accommodates much recent work in stable homotopy theory. We hope that this study will eventually lead to a conceptual understanding of objects like the algebraic K-theory of the sphere spectrum, which by Waldhausen’s stable parametrized h-cobordism theorem bears on such seemingly unrelated geometric objects as the diffeomorphism groups of manifolds, in much the same way that we now understand the algebraic K-theory spectrum of the ring of integers.

Let E be any spectrum and G a finite group. We say that a map A → B of E-local commutative S-algebras is an E-local G-Galois extension if G acts on B through commutative A-algebra maps in such a way that the two canonical maps

i: A → B^{hG}

and

h: B ∧_{A} B → ∏_{G} B

induce isomorphisms in E_{∗}-homology (Definition 4.1.3). When E = S this means that the maps i and h are weak equivalences, and we may talk of a global G-Galois extension. In more detail, the map i is the standard inclusion into the homotopy fixed points for the G-action on B and h is given in symbols by h(b_{1} ∧ b_{2}) = {g ↦ b_{1} · g(b_{2})}. To make the definition homotopy invariant we also assume that A is a cofibrant commutative S-algebra and that B is a cofibrant commutative A-algebra.

There are many interesting examples of such “brave new” Galois extensions.

Examples 1.1.

(a) The Eilenberg–Mac Lane functor R ↦ HR takes each G-Galois extension R → T of commutative rings to a global G-Galois extension HR → HT of commutative S-algebras (Proposition 4.2.1).

(b) The complexification map KO → KU from real to complex topological K-theory is a global ℤ/2-Galois extension (Proposition 5.3.1).

(c) For each rational prime p and natural number n the profinite extended Morava stabilizer group 𝔾_{n} = 𝕊_{n} ⋊ Gal acts on the even periodic Lubin–Tate spectrum E_{n}, with π_{0}(E_{n}) = 𝕎(𝔽_{p^{n}})[[u_{1}, …, u_{n−1}]], so that L_{K(n)}S → E_{n} is a K(n)-local pro-𝔾_{n}-Galois extension (see Notation 3.2.2 and Theorem 5.4.4(d)).

(d) For most regular covering spaces Y → X the map of cochain H𝔽_{p}-algebras F (X_{+}, H𝔽_{p}) → F (Y_{+}, H𝔽_{p}) is a Galois extension (Proposition 5.6.3(a)).

A map A → B of commutative S-algebras will be said to be faithful if for each A-module N with N ∧_{A} B ≃ ∗ we have N ≃ ∗ (Definition 4.3.1). The map A → B is separable if the multiplication map μ: B ∧_{A} B → B admits a bimodule section up to homotopy (Definition 9.1.1). A commutative S-algebra B is connected, in the sense of algebraic geometry, if its space of idempotents E(B) is weakly equivalent to the two-point space {0, 1} (Definition 10.2.1). There are analogous definitions in each E-local context.

In commutative ring theory each Galois extension is faithful, but it remains an open problem to decide whether each Galois extension of commutative S-algebras is faithful (Question 4.3.6). Rather conveniently, a commutative S-algebra B is connected if and only if the ring π_{0}(B) is connected (Proposition 10.2.2).

Here is our version of the Main Theorem of Galois theory for commutative Salgebras. The first two parts (a) and (b) of the theorem are obtained by specializing Theorem 7.2.3 and Proposition 9.1.4 to the case of a finite, discrete Galois group G. The recovery in (c) of the Galois group is Theorem 11.1.1. The converse part (d) is the less general part of Theorem 11.2.2.

Theorem 1.2. Let A → B be a faithful E-local G-Galois extension.

(a) For each subgroup K ⊂ G the map C = B^{hK} → B is a faithful E-local K-Galois extension, with A → C separable.

(b) For each normal subgroup K ⊂ G the map A → C = B^{hK} is a faithful E-local G/K-Galois extension.

If furthermore B is connected, then:

(c) The Galois group G is weakly equivalent to the mapping space C_{A}(B, B) of commutative A-algebra self-maps of B.

(d) For each factorization A → C → B of the G-Galois extension, with A → C separable and C → B faithful, there is a subgroup K ⊂ G such that C ≃ B^{hK} as an A-algebra over B.

In other words, for a faithful E-local G-Galois extension A → B with B connected there is a bijective contravariant Galois correspondence K ↔ C = B^{hK} between the subgroups of G and the weak equivalence classes of separable A-algebras mapping faithfully to B. The inverse correspondence takes C to K = π_{0}C_{C}(B, B).

The main theorem fully describes the intermediate extensions in a G-Galois extension A → B, but what about the further extensions of B? We say that a connected E-local commutative S-algebra A is separably closed if there are no connected E-local G-Galois extensions A → B for non-trivial groups G (Definition 10.3.1). The following fundamental example is a consequence of Minkowski’s discriminant theorem in number theory, and is proved as Theorem 10.3.3.

Theorem 1.3. The global sphere spectrum S is separably closed.

The absence of localization is crucial for this result. At the other extreme the Morava K(n)-local category is maximally localized, for each p and n. Here the Lubin–Tate spectrum E_{n} admits a K(n)-local pro-nℤ̂-Galois extension E_{n} → E_{n}^{nr}, with

π_{0}(E_{n}^{nr}) = 𝕎(𝔽̄_{p})[[u_{1}, …, u_{n−1}]]

given by adjoining all roots of unity of order prime to p (§5.4.6). We expect that each further G-Galois extension E_{n}^{nr} → B of such a Landweber exact even periodic spectrum must again be Landweber exact and even periodic, and such that π_{0}(E_{n}^{nr}) → π_{0}(B) will be a G-Galois extension of commutative rings. But 𝕎(𝔽̄_{p})[[u_{1}, …, u_{n−1}]] is separably closed as a commutative ring, so such a π_{0}(B) cannot be connected, and B the cannot be connected for non-trivial groups G. Therefore we expect:

Conjecture 1.4. The extension E_{n}^{nr} of the Lubin–Tate spectrum E_{n} is K(n)- locally separably closed. In particular, the Galois group 𝔾_{n}^{nr} = 𝕊_{n} ⋊ ℤ̂ of L_{K(n)}S → E_{n}^{nr} is the K(n)-local absolute Galois group of the K(n)-local sphere spectrum L_{K(n)}S.

Partial results supporting this conjecture have been obtained by Andy Baker and Birgit Richter [BR05b], for global Galois extensions that are furthermore assumed to be faithful and abelian.

The substantial supply of pro-Galois extensions in the K(n)-local category, like L_{K(n)}S → E_{n}, is not available in the E(n)-local category (see §5.5.4). This draws extra attention to the non-smashing Bousfield localizations, and thus to the distinction between the whole category of modules over L_{K(n)}S and its full subcategory of K(n)-local modules. A study of the sphere spectrum as an algebrogeometric scheme- or stack-like object, that only involves smashing localizations or only treats the whole module categories over the various Bousfield localizations, does thus not capture these very interesting examples of regular geometric covering spaces.

There are structured ring spectrum replacements for Kähler differentials, called topological Hochschild homology (= THH, see Section 9.2) and topological André– Quillen homology (= TAQ, see Section 9.4), in the context of associative and commutative S-algebras, respectively. These need not be K(n)-local when applied to K(n)-local S-algebras (see Example 9.2.3). Therefore the notions of (formally) étale extensions of associative or commutative S-algebras will again give a richer theory when considered within the K(n)-local subcategory, rather than in the whole module category over L_{K(n)}S. Thus also a study of the algebraic geometry of the sphere spectrum with respect to the étale topology will become more substantial by taking these Bousfield local subcategories into account. This phenomenon differs from that which is familiar in discrete algebraic geometry, since there all localizations are, indeed, smashing.

It therefore appears to be better to think of the algebraic geometry of the sphere spectrum as the “S-algebraic stack” of all Bousfield E-local subcategories M_{S,E} of spectra, for varying spectra E, rather than the “S-algebraic scheme” of the Bousfield E-local S-algebras L_{E}S themselves. The former stack maps to the stack of module categories of the latter scheme, but it is the former that carries the most interesting closed symmetric monoidal structures. See Definition 3.2.1 for the notations used here, and Section 3.2, Chapter 9 and Section 12.2 for more on these S-algebro-geometric ideas.

The (mono-)chromatic localizations L_{K(n)}S of the sphere are of course even more drastic than the p-localizations S_{(p)}, so that many of the principal examples studied in this paper are of an even more local nature than e.g. local number fields. But the arithmetic properties of a global number field can usefully be studied by adelic means, in terms of the system of local number fields that can be obtained from it by the various completions that are available. We are therefore also interested in finding global models for the system of naturally occurring K(n)-local Galois extensions of L_{K(n)}S, for varying p and n.

The obvious candidate, given Quillen’s discovery of the relation of formal group law theory to complex cobordism, is the unit map S → MU to the complex cobordism spectrum. The following statement is proved in Corollary 9.6.6, Proposition 12.2.1 and the discussion surrounding diagram (12.2.6). In the second part, S[BU] is the commutative Hopf S-algebra Σ^{∞}BU_{+}. In summary, MU is very close to such a global model, up to formal thickenings by Henselian maps. This makes the author inclined to think of S → MU as a kind of (large) ramified global Galois extension, with S[BU] playing the part of the functional dual of its imaginary Galois group. To make good sense of this, we introduce the notion of a Hopf–Galois extension of commutative S-algebras in Section 12.1.

Theorem 1.5. For each prime p and integer n ≥ 1 the K(n)-local pro-𝔾_{n}- Galois extension L_{K(n)}S → E_{n} factors as the composite of the following maps of commutative S-algebras

L_{K(n)}S → L̂_{K(n)}^{MU}MU ⟶^{q} { \atop E(n)} → E_{n} .

Here the first map admits the global model S → MU, by Bousfield K(n)-localization in the category of S-modules and K(n)-nilpotent completion in the category of MUmodules, respectively. The second map q is a formal thickening, or more precisely, symmetrically (and possibly commutatively) Henselian. The third map is a finite Galois extension (and can be avoided by passing to the even periodic version MUP of MU and adjoining some roots of unity).

Furthermore, the global model S → MU is an S[BU]-Hopf–Galois extension of commutative S-algebras, with coaction β: MU → MU ∧ S[BU] given by the Thom diagonal. For each element g ∈ 𝔾_{n} its Galois action on E_{n} can be directly recovered from this S[BU]-coaction, up to the adjunction of some roots of unity.

Here are some more detailed references into the body of the paper.

Chapter 2 contains a review of the basic Galois theory for fields and for commutative rings, together with some algebraic facts that we will need for our examples. We also make a comparison with the theory of regular covering spaces, for the benefit of the topologically minded reader.

As hinted at above, we sometimes consider more general Galois groups G than finite (and profinite) groups. For the initial theory, all that is required is that the unreduced suspension spectrum S[G] = L_{E}Σ^{∞}G_{+} admits a good Spanier– Whitehead dual in the E-local stable homotopy category, i.e., that G is stably dualizable (Definition 3.4.1). We review the basic properties of stably dualizable groups and their actions on spectra in Chapter 3, referring to the author’s companion paper [Rog08] for most proofs. This chapter also contains a discussion of the various categories of E-local S-modules and (commutative) S-algebras in which we work.

The precise Definition 4.1.3 of a Galois extension of commutative S-algebras is given in Chapter 4, followed by a discussion showing that the Eilenberg–Mac Lane embedding from commutative rings preserves and detects Galois extensions (Proposition 4.2.1). We also consider the elementary properties of faithful modules over structured ring spectra, flatness being implicit in our homotopy invariant work. We shall often make use of how various algebro-geometric properties of S-algebras are preserved by base change, or are detected by suitable forms of faithful base change.

Chapter 5 is devoted to the many examples of Galois extensions mentioned above, including all the intermediate K(n)-local Galois extensions between L_{K(n)}S and the maximal unramified extension E_{n}^{nr} of E_{n}. We also go through the K(1)- local case of the Lubin–Tate extensions in much detail, making explicit the close analogy with the classification of abelian extensions of the p-adic and rational fields ℚ_{p} and ℚ. Finally we extend the example of cochain algebras of regular covering spaces to cochain algebras of principal G-bundles P → X, for stably dualizable groups G.

Chapter 6 develops the formal consequences of the Galois conditions on A → B, including the basic fact that B is a dualizable A-module (Proposition 6.2.1), two useful alternate characterizations of (faithful) Galois extensions (Propositions 6.3.1 and 6.3.2), and two further characterizations of faithfulness (Proposition 6.3.3 and Lemma 6.5.4). These let us prove in Chapter 7 that faithful Galois extensions are preserved by arbitrary base change (Lemma 7.1.1) and are detected by faithful and dualizable base change (Lemma 7.1.4(b)). From these results, in turn, the “forward” part of the Galois correspondence (Theorem 7.2.3) follows rather formally, saying that for a faithful G-Galois extension A → B the homotopy fixed point spectra C = B^{hK} give rise to K-Galois extensions C → B for subgroups K ⊂ G, and to G/K-Galois extensions A → C when K is normal.

When this much of the Galois correspondence has been established, we can make sense of the notion of a pro-Galois extension, which we do somewhat informally in Section 8.1.

The “converse” part of the Galois correspondence (Theorem 11.2.2) relies on the possibility of recovering the Galois group G in a G-Galois extension A → B from the space C_{A}(B, B) of commutative A-algebra self-maps B → B, or more generally, to recover the subgroup K from the mapping space C_{C}(B, B), when C = B^{hK} is a fixed S-algebra of B (Proposition 11.2.1). This is achieved in Chapter 11, but relies on three preceding developments.

First of all, we use the commutative form of the Hopkins–Miller theory, as developed by Paul Goerss and Mike Hopkins [GH04], to study such mapping spaces. We use an extension of their work, from dealing with spaces of E_{∞} ring spectrum maps, or commutative S-algebra maps, to spaces of commutative A-algebra maps. This is discussed in Section 10.1, where we also touch on the consequences for this theory of working E-locally. The main computational tool is the Goerss–Hopkins spectral sequence (10.1.4), whose E_{2}-term involves suitable André–Quillen cohomology groups, which fortunately vanish in all relevant cases for the Galois extensions that we consider.

Second, the recovery of the Galois group G from C_{A}(B, B) only has a chance, judging from the discrete algebraic case, when B is connected in the geometric sense that it has no non-trivial idempotents. For a commutative S-algebra B there is a space E(B) of idempotents, which in turn is a commutative B-algebra mapping space of the sort that can be studied by the Goerss–Hopkins spectral sequence.

So in Section 10.2 we treat connectivity in this geometric sense for commutative S-algebras, reaching a convenient algebraic criterion in Proposition 10.2.2. This also lets us define separably closed commutative S-algebras in Section 10.3.

Thirdly, not all commutative A-algebras C mapping faithfully to B occur in the Galois correspondence as fixed S-algebras C = B^{hK}. As in the discrete algebraic case, the characteristic property is that C must be separable over A, and in Section 9.1 we develop the basic theory of separable extensions of S-algebras. As further generalizations of separable maps we have the étale maps, which we discuss in three related contexts in Sections 9.2 through 9.4, leading to the notions of symmetrically (=thh-)étale, smashing and (commutatively) étale maps of S-algebras, respectively.

Topological Hochschild homology THH controls the Kähler differentials in the associative setting, while topological André–Quillen homology TAQ takes on the same role in the purely commutative setting. Our discussion here relies heavily on the work of Maria Basterra [Bas99] and Andrej Lazarev [La01]. There is much conceptual overlap between the triviality of the topological André–Quillen homology spectrum TAQ(B/A) for (formally) étale maps A → B, and the vanishing of the Goerss–Hopkins André–Quillen cohomology groups D_{B_{∗}T}^{s} (B_{∗}^{A}(B), Ω^{t}B) for finite Galois extensions A → B, but the direct connection is not as well understood as might be desired.

The remainder of the text is concerned with the interpretation of S → MU as a Hopf–Galois extension that provides an integral model, up to Henselian maps, for all of the Lubin–Tate extensions L_{K(n)}S → E_{n}. Thus we consider square-zero extensions, singular extensions and Henselian maps as various forms of infinitesimal and formal thickenings in Section 9.5. We then obtain a good supply of relevant examples in Section 9.6, using work of Baker and Lazarev on I-adic towers. We have already cited Corollary 9.6.6 as relevant for part of Theorem 1.5.

The idea of Hopf–Galois extensions is to replace the action by the Galois group G on a commutative A-algebra B by a coaction by the functional dual DG_{+} = F (G_{+}, S) of the Galois group, which is a commutative Hopf S-algebra. In the algebraic situation such coactions have been useful, e.g. to classify inseparable Galois extensions of fields [Cha71]. In the absence of an actual Galois group the condition that i: A → B^{hG} is a weak equivalence must be rewritten, by using a cosimplicial resolution for the coaction (the Hopf cobar complex) in place of the homotopy fixed points. This rewriting can naturally go through a second cosimplicial resolution associated to A → B, which we know as the Amitsur complex. We discuss the Amitsur complex in Section 8.2, so as to have the accompanying notion of (nilpotent) completion of A along B available in Chapters 9 and 10, and give the definitions of the Hopf cobar complex and of Hopf–Galois extensions in Section 12.1.

To conclude the paper, in Section 12.2 we go through some of the details of how the inseparable extension S → MU is an S[BU]-Hopf–Galois extension, and how the Hopkins–Miller theory and the Lubin–Tate deformation theory work together to show that the global S[BU]-coaction on MU captures the Morava stabilizer group action on E_{n}, at all primes p and chromatic heights n.

Acknowledgments

The study of idempotents in Chapter 10 got going during an Oberwolfach hike with Neil Strickland, and the right way to use separability in Chapter 11 was found in a discussion with Birgit Richter. I am very grateful to them, as well as to Christian Ausoni, Andy Baker, Daniel G. Davis, Halvard Fausk, Kathryn Hess, Cornelius Greither, Jack Morava and the referee, for a number of helpful or encouraging comments.

Much of this work was done in the year 2000 and announced at various conferences. I apologize for the long delay in publication, which for most of the time was due to the unresolved Question 4.3.6, on the faithfulness of Galois extensions.

CHAPTER 2 Galois extensions in algebra

2.1. Galois extensions of fields

We first recall the basics about Galois extensions of fields. Let G be a finite group acting effectively (only the unit element acts as the identity) from the left by automorphisms on a field E, and let F = E^{G} be the fixed subfield. Let

j: E⟨G⟩ → Hom_{F} (E, E)

be the canonical associative ring homomorphism taking e_{1}g to the homomorphism e_{2} ↦ e_{1} · g(e_{2}), from the twisted group ring of G over E to the F -module endomorphisms of E. Then j is an isomorphism, for by Dedekind’s lemma j is injective, and dim_{F} (E) equals the order of G, so j is also surjective by a dimension count. See [Dr95, App.] for elementary proofs. Let

h: E ⊗_{F} E → ∏_{G} E

be the canonical commutative ring homomorphism taking e_{1} ⊗ e_{2} to the sequence {g ↦ e_{1} · g(e_{2})}, from the tensor product of two copies of E over F to the product of G copies of E. Then also h is an isomorphism, for it is the E-module dual of j, by way of the identifications Hom_{E}(E ⊗_{F} E, E) ≅ Hom_{F} (E, E) and Hom_{E}(∏_{G} E, E) ≅ E⟨G⟩ (using that G is finite).

2.2. Regular covering spaces

There is a parallel geometric theory of regular (= normal) covering spaces [Sp66, 2.6.7], [Ha02, 1.39]. Let G be a finite discrete group acting from the right on a compact Hausdorff space Y. Let X = Y/G be the orbit space, and let π: Y → X be the orbit projection. There is a canonical map

ξ: Y × G → Y ×_{X} Y

to the fiber product of π with itself, taking (y, g) to (y, y · g). This map is always surjective, by the definition of X as an orbit space, and it is injective if and only if G acts freely on Y. So ξ is a homeomorphism if and only if Y → X is a regular covering space, with G as its group of deck transformations, acting freely and transitively on each fiber. In general, the possible failure of ξ to be injective measures the extent to which G does not act freely on Y, which in turn can be interpreted as a measure of to what extent Y is ramified as a cover of X. The theory of Riemann surfaces provides numerous examples of the latter phenomenon.

Dually, let R = C(X) and T = C(Y) be the rings of continuous (real or complex) functions on X and Y, respectively. As usual the points of X can be recovered as the maximal ideals in R, and similarly for Y. The group G acts from the left on T, by the formula g(t) = g ∗ t: y ↦ t(y · g), and the natural map R → T dual to π identifies R with the invariant ring T^{G}, by the isomorphism C(Y)^{G} ≅ C(Y/G). The map ξ above is dual to the canonical homomorphism

h: T ⊗_{R} T → ∏_{G} T

taking t_{1} ⊗t_{2} to the function g ↦ t_{1} ·g(t_{2}), considered as an element in the product ∏_{G} T. Then ξ is a homeomorphism if and only if h is an isomorphism, by the categorical anti-equivalence between compact Hausdorff spaces and their function rings. The surjectivity of ξ ensures that h is always injective, and in general the possible failure of h to be surjective measures the extent of ramification in Y → X.

For a moment, let us also consider the more general case of a principal G-bundle π: P → X for a compact Hausdorff topological group G. The map ξ: P × G → P ×_{X} P is a homeomorphism, now with respect to the given topology on G. Let R = C(X), T = C(P) and H = C(G). Then H is a commutative Hopf algebra with coproduct ψ: H → H ⊗ H, if the map H → C(G × G) dual to the group multiplication G × G → G factors through the canonical map H ⊗ H → C(G × G). Likewise H coacts on T from the right by β: T → T ⊗H, if the map T → C(P ×G) induced by the group action P ×G → P factors through T ⊗H → C(P ×G). These factorizations can always be achieved by using suitably completed tensor products, but we wish to refer to the algebraic tensor products here. Then the freeness of the group action on P is expressed by saying that the composite map

h: T ⊗_{R} T ⟶^{1⊗β} T ⊗_{R} T ⊗ H ⟶^{μ⊗1} T ⊗ H

is an isomorphism. We shall return to this dualized context in Chapter 12 on Hopf–Galois extensions.

2.3. Galois extensions of commutative rings

Generalizing the two examples above, for finite Galois groups, Auslander and Goldman [AG60, App.] gave a definition of Galois extensions of commutative rings as part of their study of separable algebras over such rings. Chase, Harrison and Rosenberg [CHR65, §1] found several other equivalent definitions, and developed the Galois theory for commutative rings to also encompass the fundamental Galois correspondence. We now recall their basic results.

Let R → T be a homomorphism of commutative rings, making T a commutative R-algebra, and let G be a finite group acting on T from the left through R-algebra homomorphisms. Let

i: R → T^{G}

be the inclusion into the fixed ring, let

h: T ⊗_{R} T → ∏_{G} T

be the commutative ring homomorphism that takes t_{1} ⊗ t_{2} to the sequence {g ↦ t_{1} · g(t_{2})}, and let

j: T ⟨G⟩ → Hom_{R}(T, T)

be the associative ring homomorphism that takes t_{1}g to the R-module homomorphism t_{2} ↦ t_{1} · g(t_{2}). We give ∏_{G} T the pointwise product (t_{g})_{g} · (t′_{g})_{g} = (t_{g}t′_{g})_{g} and T ⟨G⟩ the twisted product t_{1}g_{1} · t_{2}g_{2} = t_{1}g_{1}(t_{2})g_{1}g_{2}, using the left G-action (g_{1}, t_{2}) ↦ g_{1}(t_{2}) on T.

Definition 2.3.1. Let G act on T over R, as above. We say that R → T is a G-Galois extension of commutative rings if both i: R → T^{G} and h: T ⊗_{R}T → ∏_{G} T are isomorphisms.

Here we are following Greither [Gre92, 0.1.5]. Auslander and Goldman [AG60, p. 396] instead took the condition below on i, j and T to be the defining property, but Chase, Harrison and Rosenberg [CHR65, 1.3] proved that the two definitions are equivalent.

Proposition 2.3.2. Let G act on T over R, as above. Then R → T is a G-Galois extension if and only if both i: R → T^{G} and j: T ⟨G⟩ → Hom_{R}(T, T) are isomorphisms and T is a finitely generated projective R-module.

The condition that i is an isomorphism means that we can speak of R as the fixed ring of T. The homomorphism h measures to what extent the extension R → T is ramified, and Galois extensions are required to be unramified. The injectivity of j is a form of Dedekind’s lemma, and ensures that the action by G is effective.

Example 2.3.3. If K → L is a G-Galois extension of number fields, then the corresponding extension R = O_{K} → O_{L} = T of rings of integers is a G-Galois extension of commutative rings if and only if K → L is unramified as an extension of number fields [AB59]. More generally, if Σ is a set of prime ideals in O_{K}, and Σ′ the set of primes in O_{L} above those in Σ, then the extension O_{K,Σ} → O_{L,Σ′} of rings of Σ-integers is G-Galois if and only if Σ contains all the primes that ramify in L/K [Gre92, 0.4.1]. Here O_{K,Σ} is defined as the ring of elements x ∈ K that have non-negative valuation v_{𝔭}(x) ≥ 0 for all prime ideals 𝔭 ∉ Σ. Thus O_{K} → O_{L} becomes a G-Galois extension precisely upon localizing away from (= inverting) the ramified primes.

To see this, note that if T = R{t_{1}, …, t_{n}} is a free R-module of rank n, then T ⊗_{R} T is a free T -module on the generators 1 ⊗ t_{1}, …, 1 ⊗ t_{n}, and h is represented as a T -module homomorphism by the square matrix A = (g(t_{i}))_{g,i} of rank n, with g ∈ G and i = 1, …, n. The discriminant of T/R is d = det(A)^{2}, by definition, and the prime ideals in O_{K} that ramify in L/K are precisely the prime ideals dividing the discriminant. So h is an isomorphism if and only if det(A) and d are units in R, or equivalently, if there are no ramified primes. A local version of the same argument works when T is not free over R.

Here are some further basic properties of Galois extensions of commutative rings, which will be relevant to our discussion.

Proposition 2.3.4. Let R → T be a G-Galois extension. Then:

(a) T is faithfully flat as an R-module, i.e., the functor (−)⊗_{R} T preserves and detects (=reflects) exact sequences.

(b) The trace map tr: T → R (taking t ∈ T to ∑_{g∈G} g(t) ∈ T^{G} = R) is a split surjective R-module homomorphism.

(c) T is invertible as an R[G]-module, i.e., a finitely generated projective R[G]- module of constant rank 1.

For proofs, see e.g. [Gre92, 0.1.9], [Gre92, 0.1.10] and [Gre92, 0.6.1]. Beware that part (b) does not extend well to the topological setting, as Example 6.4.4 demonstrates.

CHAPTER 3 Closed categories of structured module spectra

3.1. Structured spectra

We now adapt these ideas to the context of “brave new rings,” i.e., of commutative S-algebras. These can in Chapters 2–9 and 12 be interpreted as the commutative monoids in either one of the popular symmetric monoidal categories of structured spectra, such as the S-modules of Elmendorf, Kriz, Mandell and May [EKMM97], the symmetric spectra in simplicial sets of Hovey, Shipley and Smith [HSS00], symmetric spectra in topological spaces or orthogonal spectra of Mandell, May, Schwede and Shipley [MMSS01] or the simplicial functors of Segal and Lydakis [Ly98], according to the reader’s needs or preferences.

However, in Chapters 10 and 11 we make use of the Goerss–Hopkins obstruction theory for E_{∞} mapping spaces [GH04], which presumes that one works in a category of spectra that satisfies Axioms 1.1 and 1.4 in op. cit. In particular, this theory is needed for the proof of parts (c) and (d) of our Theorem 1.2, and for Theorem 1.3. It is known that S-modules, symmetric spectra formed in topological spaces and orthogonal spectra all satisfy the required axioms, by [GH04, 1.5]. To be concrete, and to have a convenient source for the more technical references, we shall work with the S-modules of Peter May et al.

Let S be the sphere spectrum, and let M_{S} be the category of S-modules. Among other things, it is a topological category with all limits and colimits and all topological tensors and cotensors. A map f: X → Y of S-modules is called a weak equivalence if the induced homomorphism π_{∗}(f): π_{∗}(X) → π_{∗}(Y) of stable homotopy groups is an isomorphism. The category D_{S} obtained from M_{S} by inverting the weak equivalences is called the stable homotopy category, and is equivalent to the homotopy category of spectra constructed by Boardman [Vo70].

The smash product X ∧ Y and function object F (X, Y) make M_{S} a closed symmetric monoidal category, with S as the unit object. For each topological space T the topological tensor X ∧ T_{+} equals the smash product X ∧ S[T], and the topological cotensor Y^{T} = F (T_{+}, Y) equals the function spectrum F (S[T], Y), where S[T] = Σ^{∞}T_{+} denotes the unreduced suspension S-module on T.

An (associative) S-algebra A is a monoid in M_{S}, i.e., an S-module A equipped with a unit map η: S → A and a unital and associative multiplication μ: A∧A → A. A commutative S-algebra A is a commutative monoid in M_{S}, i.e., one such that the multiplication μ is also commutative. We write A_{S} and C_{S} for the categories of S-algebras and commutative S-algebras, respectively. More generally, for a commutative S-algebra A we write M_{A}, A_{A} and C_{A} for the categories of A-modules, associative A-algebras and commutative A-algebras, respectively [EKMM97, VII.1].

3.2. Localized categories

Our first examples of Galois extensions of structured ring spectra will be maps A → B of commutative S-algebras, with a finite group G acting on B through Aalgebra maps, such that there are weak equivalences i: A ≃ B^{hG} and h: B ∧_{A} B ≃ ∏_{G} B. The formal definition appears in Section 4.1 below. However, there are interesting examples that only appear as Galois extensions to the eyes of weaker invariants than the stable homotopy groups π_{∗}(−). More precisely, for a fixed homology theory E_{∗}(−) we shall allow ourselves to work in the E-local stable homotopy category, where have arranged that each map f: X → Y such that E_{∗}(f): E_{∗}(X) → E_{∗}(Y) is an isomorphism, is in fact a weak equivalence. In particular, we will encounter situations where we only have that E_{∗}(i) and E_{∗}(h) are isomorphisms, in which case we shall interpret A → B as an E-local G-Galois extension.

Note the close analogy between the E-local theory and the case (Example 2.3.3) of rings of integers localized away from some set of primes. Doug Ravenel’s influential treatise on the chromatic filtration of stable homotopy theory [Ra84, §5], brings emphasis to the tower of cases when E = E(n), the n-th Johnson–Wilson spectrum. To us, the most interesting case is when E = K(n) is the n-th Morava K-theory spectrum. The K(n)-local stable homotopy category is studied in detail in [HSt99, §§7–8], and captures the n-th layer, or stratum, in the chromatic filtration.

Definition 3.2.1. Let E be a fixed S-module, with associated homology theory X ↦ E_{∗}(X) = π_{∗}(E ∧X). By definition, an S-module Z is said to be E-acyclic if E ∧ Z ≃ ∗, and an S-module Y is said to be E-local if F (Z, Y) ≃ ∗ for each E-acyclic S-module Z. Let M_{S,E} ⊂ M_{S} be the full subcategory of E-local Smodules. A map f: X → Y of E-local S-modules is a weak equivalence if and only if it is an E_{∗}-equivalence, i.e., if E_{∗}(f) is an isomorphism.

There is a Bousfield localization functor L_{E}: M_{S} → M_{S,E} ⊂ M_{S} [Bo79], [EKMM97, VIII.1.6], and an accompanying natural E_{∗}-equivalence X → L_{E}X for each S-module X. We may assume that this E_{∗}-equivalence is the identity when X is already E-local, so that the localization functor L_{E} is idempotent. The homotopy category D_{S,E} of M_{S,E} is the E-local stable homotopy category.

More generally, for a commutative S-algebra A we let M_{A,E} ⊂ M_{A} be the full subcategory of E-local A-modules, with homotopy category D_{A,E}. To be precise, there is an A-module 𝔽_{A}E of the homotopy type of A∧E, and a localization functor L_{𝔽_{A}E}^{A}: M_{A} → M_{A,E}, with respect to 𝔽_{A}E in the category of A-modules, which amounts to E-localization at the level of the underlying S-modules [EKMM97, VIII.1.7]. We shall allow ourselves to simply denote this functor by L_{E}.

Notation 3.2.2. We write

L_{n}X = L_{E(n)}X

for the Bousfield localization of X with respect to the Johnson–Wilson spectrum E(n) [JW73], with π_{∗}E(n) = ℤ_{(p)}[v_{1}, …, v_{n−1}, v_{n}^{±1}], for each non-negative integer n, and

L_{K(n)}X

for the Bousfield localization of X with respect to the Morava K-theory spectrum K(n) [JW75], with π_{∗}K(n) = 𝔽_{p}[v_{n}^{±1}], for each natural number n.

We will reserve the symbol L̂ for the Bousfield nilpotent completion recalled in Definition 8.2.2, and shall therefore not use this notation for the functor L_{K(n)}, unlike e.g. [HSt99].

The smash product X ∧ Y of two E-local S-modules will in general not be Elocal, although this is the case when L_{E} is a so-called smashing localization, i.e., one that commutes with direct limits [Ra84, 1.28]. The Johnson–Wilson spectra E = E(n) provide interesting examples of smashing localizations L_{n} = L_{E(n)} [Ra92, 7.5.6], while localization L_{K(n)} with respect to the Morava K-theories E = K(n) is not smashing [HSt99, 8.1]. Likewise, the unit S for the smash product is rarely E-local. So in order to work with S-algebras and related constructions internally within M_{S,E}, we first perform each construction as usual in M_{S}, and then apply the Bousfield localization functor L_{E}.

Definition 3.2.3. We implicitly give M_{S,E} all colimits, topological tensors, smash products and a unit object by applying Bousfield localization to the constructions in M_{S}. So colim_{i∈I} X_{i} means L_{E}(colim_{i∈I} X_{i}), X ∧ Y means L_{E}(X ∧ Y), S means L_{E}S and S[T] means L_{E}Σ^{∞}T_{+}. All limits, topological cotensors and function objects formed from E-local S-modules are already E-local, so no Bousfield localization is required in these cases. With these conventions, M_{S,E} is a topological closed symmetric monoidal category with all limits and colimits. The same considerations apply for M_{A,E}.

There is a natural map L_{E}X ∧ L_{E}Y → L_{E}(X ∧ Y), making L_{E} a lax monoidal functor, so that L_{E}S is always a commutative S-algebra. When E is smashing, the category M_{S,E} of E-local S-modules is equivalent (at the level of homotopy categories) to the category M_{L_{E}S} of L_{E}S-modules, so the study of E-local Smodules is a special case of the study of modules over a general commutative Salgebra A = L_{E}S. However, when E is not smashing, as is the case for E = K(n), the two homotopy categories are not equivalent, and we shall need to consider the more general notion.

When E = S, every S-module is E-local and M_{S,E} = M_{S}, etc., so the E-local context specializes to the “global”, unlocalized situation. For brevity, we shall often simply refer to the E-local S-modules as S-modules, or even as spectra, but except where we explicitly assume that E = S, the discussion is intended to encompass also the general E-local case.

Remark 3.2.4. By analogy with algebraic geometry, we may heuristically wish to view A-modules M as suitable sheaves M^{∼} over some geometric “structure space” Spec A. This structure space would come with a Zariski topology, with open subspaces U_{A,E} ⊂ Spec A corresponding to the various localization functors L_{E} on the category of A-modules, in such a way that the restriction of the sheaf M^{∼} over Spec A to the subspace U_{A,E} would be the sheaf (L_{E}M)^{∼} corresponding to the E-local A-module L_{E}M. For smashing E this would precisely amount to an L_{E}A-module, so that U_{A,E} could be identified with the structure space Spec L_{E}A.

However, for non-smashing E the condition of being an E-local A-module is strictly stronger than being an L_{E}A-module. Therefore, the geometric structure on Spec A is not simply that of an “S-algebra’ed space” carrying the (commutative) S-algebra L_{E}A over U_{A,E}, by analogy with the ringed spaces of algebraic geometry. If we wish to allow non-smashing localizations E to correspond to Zariski opens, then the geometric structure must also capture the additional restriction it is for an L_{E}A-module to be an E-local A-module. This exhibits a difference compared to the situation in commutative algebra, where localization at an ideal commutes with direct limits, and behaves as a smashing localization, while completions behave more like non-smashing localizations. It does not seem to be so common to do commutative algebra in such implicitly completed situations, however.

A continuation of this analogy would be to consider other Grothendieck-type topologies on Spec A, with coverings built from E-local Galois extensions L_{E}A → B (Definition 4.1.3) or more general étale extensions (Definition 9.4.1), subject to a combined faithfulness condition (Definition 4.3.1). In the unlocalized cases, such a (big) étale site on the opposite category of C_{S}, and associated small étale sites on the opposite category of each C_{A}, have been developed by Bertrand Toën and Gabriele Vezzosi [TV05, §5.2]. However, the rich source of K(n)-local Galois extensions of L_{K(n)}S discussed in Section 5.4 provides, by Lemma 9.4.4, an equally rich supply of K(n)-local étale maps from L_{K(n)}S. It appears, by extension from the case n = 1 discussed in Section 5.5, that these are not globally étale maps, in which case the étale topology proposed in [TV05] will be too coarse to encompass these examples. The author therefore thinks that a finer étale site, taking non-smashing localizations like L_{K(n)} into account, would lead to a stronger and more interesting theory.

3.3. Dualizable spectra

In each closed symmetric monoidal category there is a canonical natural map

ν: F (X, Y) ∧ Z → F (X, Y ∧ Z) .

It is right adjoint to a map ϵ ∧ 1: X ∧ F (X, Y) ∧ Z → Y ∧ Z, where the adjunction counit ϵ: X ∧ F (X, Y) → Y is left adjoint to the identity map on F (X, Y).

Dold and Puppe [DP80] say that an object X is strongly dualizable if the canonical map ν: F (X, Y) ∧ Z → F (X, Y ∧ Z) is an isomorphism for all Y and Z. Lewis, May and Steinberger [LMS86, III.1.1] say that a spectrum X is finite if it is strongly dualizable in the stable homotopy category, i.e., if the map ν is a weak equivalence. We shall instead follow Hovey and Strickland [HSt99, 1.5(d)] and briefly call such spectra dualizable. By [LMS86, III.1.3(ii)] it suffices to verify this condition in the special case when Y = S and Z = X, so we take this simpler criterion as our definition.

Definition 3.3.1. Let DX = F (X, S) be the functional dual of X. We say that X is dualizable if the canonical map ν: DX ∧ X → F (X, X) is a weak equivalence. More generally, for an (implicitly E-local) module M over a commutative S-algebra A, let D_{A}M = F_{A}(M, A) be the functional dual, and say that M is a dualizable A-module if the canonical map ν: D_{A}M ∧_{A} M → F_{A}(M, M) is a weak equivalence.

Lemma 3.3.2. (a) If X or Z is dualizable, then the canonical map ν: F (X, Y)∧ Z → F (X, Y ∧ Z) is a weak equivalence.

(b) If X is dualizable, then DX is also dualizable and the canonical map ρ: X → DDX is a weak equivalence.

(c) The dualizable spectra generate a thick subcategory, i.e., they are closed under passage to weakly equivalent objects, retracts, mapping cones and (de-)sus-pensions.

Here ρ: X → DDX = F (F (X, S), S) is right adjoint to F (X, S) ∧ X → S, which is obtained by twisting the adjunction counit ϵ: X ∧ F (X, S) → S. For proofs, see [LMS86, III.1.2 and III.1.3]. We sometimes also use ν to label the conjugate map Y ∧ F (X, Z) → F (X, Y ∧ Z). The corresponding results hold for E-local A-modules, by the same formal proofs.

In the unlocalized setting E = S, the following converse to Lemma 3.3.2(c) is one justification for the term “finite”.

Proposition 3.3.3. Let A be commutative S-algebra. A global A-module M is dualizable in M_{A} = M_{A,S} if and only if it is weakly equivalent to a retract of a finite cell A-module. When A is connective, this is in turn equivalent to being a retract of a finite CW A-module spectrum.

The proof [EKMM97, III.7.9] uses in an essential way that stable homotopy X ↦ π_{∗}(X) = [A, X]_{∗}^{A} commutes with coproducts, which amounts to A being small in the homotopy category D_{A} of A-modules. This fails in some E-local contexts. For example, the K(n)-local sphere spectrum L_{K(n)}S is not small in the K(n)-local category [HSt99, 8.1], and consequently π_{∗}(X) is not a homology theory on this category. So in general there will be more dualizable E-local A-modules than the semi-finite ones, i.e., the retracts of the finite cell L_{E}A-modules. In this paper we shall prefer to focus on the notion of dualizability, rather than on being semi-finite, principally because of Proposition 6.2.1 and (counter-)Example 6.2.2 below.

3.4. Stably dualizable groups

For our basic theory of G-Galois extensions of commutative S-algebras the group action by G appears through the module action by its suspension spectrum S[G] = L_{E}Σ^{∞}G_{+}, and the finiteness condition on G only enters through the property that S[G] is a dualizable spectrum. We then say that G is an E-locally stably dualizable group. Only when we turn to properties related to separability will it be relevant that G is discrete, and then usually finite. So we shall develop the basic theory in the greater generality of stably dualizable topological groups G.

Definition 3.4.1. A topological group G is E-locally stably dualizable if its suspension spectrum S[G] = L_{E}Σ^{∞}G_{+} is dualizable in M_{S,E}. Writing DG_{+} = F (G_{+}, L_{E}S) for its functional dual, the condition is that the canonical map

ν: DG_{+} ∧ S[G] → F (S[G], S[G])

is a weak equivalence in the E-local category.

Examples 3.4.2. (a) Each compact Lie group G admits the structure of a finite CW complex, so S[G] is a finite cell spectrum and G is stably dualizable, for each E.

(b) The Eilenberg–Mac Lane spaces G = K(ℤ/p, q) are loop spaces and thus admit models as topological groups. They have infinite mod p homology for each q ≥ 1, so S[G] is never dualizable in M_{S} by Proposition 3.3.3. However, the Morava K-homology K(n)_{∗}K(ℤ/p, q) is finitely generated over K(n)_{∗} by a calculation of Ravenel and Wilson [RaW80, 9.2], so G = K(ℤ/p, q) is in fact K(n)-locally stably dualizable by [HSt99, 8.6]. We are curious to see if these and similar topological Galois groups play any significant role in the K(n)-local Galois theory.

3.5. The dualizing spectrum

The weak equivalence S[G] = ⋁_{G} S → ∏_{G} S = DG_{+} for a finite group G generalizes to an E-local self-duality of the suspension spectrum S[G], when G is an E-locally stably dualizable group. The self-duality holds up to a twist by a so-called dualizing spectrum S^{adG}. When G is a compact Lie group this is the suspension spectrum on the one-point compactification of the adjoint representation adG of G, thus the notation, and so S^{adG} = S for G finite. John Klein [Kl01, §1] introduced dualizing spectra S^{adG} for arbitrary topological groups, and Tilman Bauer [Bau04, 4.1] established the twisted self-duality of S[G] in the p-complete category, when G is a p-compact group in the sense of Bill Dwyer and Clarence Wilkerson [DW94]. In [Rog08] we have extended these results to all E-locally stably dualizable groups, as we now review.

Definition 3.5.1. Let G be an E-locally stably dualizable group. The group multiplication provides the suspension spectrum S[G] = L_{E}Σ^{∞}G_{+} with mutually commuting left and right G-actions. We define the dualizing spectrum S^{adG} to be the G-homotopy fixed point spectrum

S^{adG} = S[G]^{hG} = F (EG_{+}, S[G])^{G}

of S[G], formed with respect to the right G-action [Rog08, 2.5.1]. Here EG = B(∗, G, G) is the standard free, contractible right G-space. The remaining left action on S[G] induces a left G-action on S^{adG}.

When G is finite, there is a natural weak equivalence

S^{adG} = S[G]^{hG} ≃ DG_{+}^{hG} ≃ S .

Here the last equivalence involves the collapsing homotopy equivalence c: EG → ∗, which is a G-equivariant map, but not a G-equivariant homotopy equivalence. For general stably dualizable groups G, the dualizing spectrum is indeed dualizable and smash invertible [Rog08, 3.2.3 and 3.3.4], so smashing with S^{adG} induces an equivalence of derived categories.

The left G-action on S[G] functorially dualizes to a right G-action on DG_{+}, with associated module action map α: DG_{+} ∧ S[G] → DG_{+}. The diagonal map on G induces a coproduct ψ: S[G] → S[G] ∧ S[G], using [EKMM97, II.1.2]. These combine to a shear map

sh: DG_{+} ∧ S[G] ⟶^{1∧ψ} DG_{+} ∧ S[G] ∧ S[G] ⟶^{α∧1} DG_{+} ∧ S[G] ,

which is equivariant with respect to each of three mutually commuting G-actions [Rog08, 3.1.2] and is a weak equivalence [Rog08, 3.1.3]. Taking homotopy fixed points with respect to the right action of G on S[G] in the source and the diagonal right action on DG_{+} and S[G] in the target induces a natural Poincaré duality equivalence [Rog08, 3.1.4]

(3.5.2)  DG_{+} ∧ S^{adG} ⟶^{≃} S[G] .

This identification uses the stable dualizability of G, and expresses the twisted selfduality of S[G]. The weak equivalence is equivariant with respect to both a left and a right G-action. The left G-action is by the inverse of the right action on DG_{+}, the standard left action on S^{adG} and the standard left action on S[G]. The right G-action is by the inverse of the left action on DG_{+}, the trivial action on S^{adG} and the standard right action on S[G].

3.6. The norm map

Let X be any E-local S-module with left G-action, and equip it with the trivial right G-action. The smash product X ∧S[G] then has a diagonal left G-action, and a right G-action that only affects S[G]. Consider forming homotopy orbits (−)_{hG} with respect to the left action and forming homotopy fixed points (−)^{hG} with respect to the right action, in either order. There is then a canonical colimit/limit exchange map

κ: ((X ∧ S[G])^{hG})_{hG} → ((X ∧ S[G])_{hG})^{hG} .

The source of κ receives a weak equivalence from (X ∧S^{adG})_{hG} (this uses the stable dualizability of G; see the proof of Lemma 6.4.2), and the target of κ maps by a weak equivalence to X^{hG} (this is easy). The composite of these three maps is the (homotopy) norm map [Rog08, 5.2.2]

(3.6.1)  N: (X ∧ S^{adG})_{hG} → X^{hG} .

If X = W ∧ G_{+} = W ∧ S[G] for some spectrum W with left G-action, with G acting in the standard way on S[G], then the norm map for X is a weak equivalence [Rog08, 5.2.5]. That reference only discusses the case when G acts trivially on W, but in general there is an equivariant shearing equivalence ζ: w ∧g ↦ g(w)∧g from W ∧ S[G] with G acting only on S[G] to W ∧ S[G] with the diagonal G-action.

We can define the G-Tate construction X^{tG} to be the cofiber of the norm map

(X ∧ S^{adG})_{hG} ⟶^{N} X^{hG} ⟶ X^{tG} .

Then X^{tG} ≃ ∗ if and only if N is a weak equivalence, which in turn holds if and only if the exchange map κ is a weak equivalence. From this point of view X^{tG} is the obstruction to the commutation of the G-homotopy orbit and the G-homotopy fixed point constructions, when applied to X ∧ S[G].

CHAPTER 4 Galois extensions in topology

4.1. Galois extensions of E-local commutative S-algebras

Fix an S-module E, and consider the categories M_{S,E} and C_{S,E} of E-local S-modules and E-local commutative S-algebras, respectively. These are full subcategories of the topological (closed) model categories M_{S} and C_{S}, respectively, as explained in [EKMM97, VII.4].

The reader may, if preferred, alternatively work with the “convenient” S-model structures of Jeff Smith and Brooke Shipley [Sh04], but this will not be necessary. There is another E-local model structure on M_{S}, with E_{∗}-equivalences as the weak equivalences and the E-local S-modules as the fibrant objects, see [EKMM97, VIII.1], but there does not seem to be such an E-local model structure available in the case of C_{S}.

Let A → B be a map of E-local commutative S-algebras, making B a commutative A-algebra, and let G be an E-locally stably dualizable group acting continuously on B from the left through commutative A-algebra maps. For example, G can be a finite discrete group.

Suppose that A is cofibrant as a commutative S-algebra, and that B is cofibrant as a commutative A-algebra. The commutative A-algebra B tends not to be cofibrant as an A-module, but the smash product functor B ∧_{A} (−) is still homotopically meaningful when applied to (other) cofibrant commutative A-algebras, as explained in [EKMM97, VII.6].

Let

(4.1.1)  i: A → B^{hG}

be the map to the homotopy fixed point S-algebra B^{hG} = F (EG_{+}, B)^{G} that is right adjoint to the composite G-equivariant map A ∧ EG_{+} → A → B, collapsing the contractible free G-space EG to a point. Let

(4.1.2)  h: B ∧_{A} B → F (G_{+}, B)

be the canonical map to the product (cotensor) S-algebra F (G_{+}, B) that is right adjoint to the composite map B ∧_{A} B ∧ G_{+} → B ∧_{A} B → B, induced by the action B ∧ G_{+} ≅ G_{+} ∧ B → B of G on B, followed by the A-algebra multiplication B ∧_{A} B → B in B.

We consider B ∧_{A} B and F (G_{+}, B) as B-modules by the multiplication in the first (left hand) copy of B in B ∧_{A} B, and in the target of F (G_{+}, B). Then h is a map of B-modules. The group G acts from the left on the second (right hand) copy of B in B ∧_{A} B, and by right multiplication in the source of F (G_{+}, B). Then h is also a G-equivariant map. These B- and G-actions clearly commute, and combine to a left module action by the group S-algebra B[G].

Here is our key definition, which assumes that E, A, B and G are as above, and uses the maps i and h just introduced. We introduce the related map j in Section 6.1.

Definition 4.1.3. We say that A → B is an E-local G-Galois extension of commutative S-algebras if the two canonical maps i: A → B^{hG} = F (EG_{+}, B)^{G} and h: B ∧_{A} B → F (G_{+}, B), formed in the category of E-local S-modules, are both weak equivalences.

The assumption that A and B are E-local ensures that B^{hG} and F (G_{+}, B) are E-local, without any implicit localization. But B ∧_{A} B formed in S-modules needs not be E-local, unless E is smashing. The condition that h is a weak equivalence in M_{S,E} amounts to asking that the corresponding map B ∧_{A} B → F (G_{+}, B) formed in M_{S} is an E_{∗}-equivalence, i.e., that E_{∗}(h) is an isomorphism.

Lemma 4.1.4. Subject to the cofibrancy conditions, the notion of an E-local G-Galois extension A → B is invariant under changes up to weak equivalence in A, B and the stabilized group S[G] = L_{E}Σ^{∞}G_{+}.

Proof. By [EKMM97, VII.6.7] the cofibrancy conditions ensure that the constructions A, B^{hG}, B ∧_{A} B and F (G_{+}, B) preserve weak equivalences in A and B, whether implicitly E-localized or not.

The natural E_{∗}-equivalences Σ^{∞}G_{+} → S[G] and Σ^{∞}EG_{+} → S[EG] induce a (not implicitly localized) map

F_{S[G]}(S[EG], B) → F_{Σ^{∞}G_{+}}(Σ^{∞}EG_{+}, B) ≅ F (EG_{+}, B)^{G} ,

which is a weak equivalence when B is E-local. Thus the construction B^{hG} also preserves weak equivalences in S[G].

Thus the E-local Galois conditions, that G is stably dualizable and the maps i and h are weak equivalences, are invariant under changes in A, B or G that amount to E-local weak equivalences of A, B and S[G]. □

When E = S, so there is no implicit E-localization, we may simply say that A → B is a G-Galois extension, or for emphasis, that A → B is a global G-Galois extension. However, most of the time we are implicitly working E-locally, for a general spectrum E, but omit to mention this at every turn. Hopefully no confusion will arise.

When G is discrete, we often prefer to write the target F (G_{+}, B) of h as ∏_{G} B. When G is finite and discrete, we say that A → B is a finite Galois extension.

4.2. The Eilenberg–Mac Lane embedding

The Eilenberg–Mac Lane functor H, which to a commutative ring R associates a commutative S-algebra HR with π_{∗}HR = R concentrated in degree 0, embeds the category of commutative rings into the category of commutative S-algebras. The two notions of Galois extension are compatible under this embedding. For this to make sense, we must assume that G is finite and that E = S.

Proposition 4.2.1. Let R → T be a homomorphism of commutative rings, and let G be a finite group acting on T through R-algebra homomorphisms. Then R → T is a G-Galois extension of commutative rings if and only if the induced map HR → HT is a global G-Galois extension of commutative S-algebras.

Proof. Suppose first that R → T is G-Galois. Then T is a finitely generated projective R-module by Proposition 2.3.2, hence flat, so Tor_{s}^{R}(T, T) = 0 for s ≠ 0. Furthermore, T is finitely generated projective (of constant rank 1) as an R[G]- module, by Proposition 2.3.4(c). There is an isomorphism of left R[G]-modules R[G] ≅ Hom_{R}(R[G], R), since G is finite, so Ext_{R[G]}^{s}(R, R[G]) ≅ Ext_{R}^{s}(R, R) = 0 for s ≠ 0. Therefore Ext_{R[G]}^{s}(R, T) = 0 for s ≠ 0, by the finite additivity of Ext in its second argument.

It follows that the homotopy fixed point spectral sequence

E_{s,t}^{2} = H^{−s}(G; π_{t}HT) = Ext_{R[G]}^{−s,−t}(R, T) ⟹ π_{s+t}(HT^{hG})

derived from [EKMM97, IV.4.3], and the Künneth spectral sequence

E_{s,t}^{2} = Tor_{s,t}^{R}(T, T) ⟹ π_{s+t}(HT ∧_{HR} HT)

of [EKMM97, IV.4.2], both collapse to the origin s = t = 0. So (HT)^{hG} ≃ H(T^{G}) = HR and HT ∧_{HR} HT ≃ H(T ⊗_{R} T) ≅ H(∏_{G} T) ≃ ∏_{G} HT are both weak equivalences. Thus HR → HT is a G-Galois extension of commutative Salgebras.

Conversely, suppose that HR → HT is G-Galois. Then by the same spectral sequences T^{G} ≅ π_{0}(HT^{hG}) ≅ π_{0}HR = R and T ⊗_{R} T ≅ π_{0}(HT ∧_{HR} HT) ≅ π_{0}(∏_{G} HT) ≅ ∏_{G} T, so R → T is a G-Galois extension of commutative rings. □

4.3. Faithful extensions

Galois extensions of commutative rings are always faithfully flat, and it will be convenient to consider the corresponding property for structured ring spectra. It remains an open problem whether Galois extensions of commutative S-algebras are in fact always faithful, but we shall verify that this is the case in most of our examples, with the possible exception of some cases in Section 5.6.

Definition 4.3.1. Let A be a commutative S-algebra. An A-module M is faithful if for each A-module N with N ∧_{A} M ≃ ∗ we have N ≃ ∗. An A-algebra B, or G-Galois extension A → B, is said to be faithful if B is faithful as an A-module.

A set of A-algebras {A → B_{i}}_{i} is a faithful cover of A if for each A-module N with N ∧_{A} B_{i} ≃ ∗ for every i we have N ≃ ∗. In particular, a single faithful A-algebra B covers A in this sense.

By the following lemma, this corresponds well to the algebraic notion of a faithfully flat module [Gre92, 0.1.7]. Flatness (cofibrancy) is implicit in our homotopy invariant work, so we only refer to the faithfulness in our terminology.

Lemma 4.3.2. Let M be a faithful A-module.

(a) A map f: X → Y of A-modules is a weak equivalence if and only if f ∧ 1: X ∧_{A} M → Y ∧_{A} M is a weak equivalence.

(b) A diagram of A-modules X ⟶^{f} Y ⟶^{g} Z, with a preferred null-homotopy of gf, is a cofiber sequence if and only if X ∧_{A} M → Y ∧_{A} M → Z ∧_{A} M, with the associated null-homotopy of gf ∧ 1, is a cofiber sequence.

Proof. (a) Consider the mapping cone C_{f} of f.

(b) Consider the induced map C_{f} → Z. □

Faithful modules and extensions are preserved under base change, and are detected by faithful base change.

Lemma 4.3.3. Let A → B be a map of commutative S-algebras and M a faithful A-module. Then B ∧_{A} M is a faithful B-module.

Proof. Let N be a B-module such that N ∧_{B} (B ∧_{A} M) ≃ ∗. Then N ∧_{A} M ≃ ∗, so N ≃ ∗ since M is faithful over A. □

Lemma 4.3.4. Let A → B be a faithful map of commutative S-algebras and M an A-module such that B ∧_{A} M is a faithful B-module. Then M is a faithful A-module.

Proof. Let N be an A-module such that N ∧_{A} M ≃ ∗. Then (N ∧_{A} B) ∧_{B} (B ∧_{A} M) ≅ N ∧_{A} B ∧_{A} M ≅ (N ∧_{A} M) ∧_{A} B ≃ ∗, so N ∧_{A} B ≃ ∗ since B ∧_{A} M is faithful over B, and thus N ≃ ∗ since B is faithful over A. □

Lemma 4.3.5. For each G-Galois extension R → T of commutative rings, the induced G-Galois extension HR → HT of commutative S-algebras is faithful.

Proof. Recall that T is faithfully flat over R by Proposition 2.3.4(a). For each HR-module N we have π_{∗}(N ∧_{HR} HT) ≅ π_{∗}(N) ⊗_{R} T, by the Künneth spectral sequence

E_{s,t}^{2} = Tor_{s,t}^{R}(π_{∗}(N), T) ⟹ π_{s+t}(N ∧_{HR} HT)

and the flatness of T. Therefore N ∧_{HR} HT ≃ ∗ implies π_{∗}(N) ⊗_{R} T = 0, which in turn implies that π_{∗}(N) = 0 by the faithfulness of T. Thus N ≃ ∗ and HR → HT is faithful. □

Question 4.3.6. Is every E-local G-Galois extension A → B of commutative S-algebras faithful?

By Corollary 6.3.4 (or Lemma 6.4.3) the answer is yes when the order of G is invertible in π_{0}(A), but in some sense this is the less interesting case.

In the case E = K(n), it is very easy [HSt99, 7.6] to be faithful over A = L_{K(n)}S.

Lemma 4.3.7. In the K(n)-local category, every non-trivial S-module is faithful over L_{K(n)}S.

Proof. Let M and N be K(n)-local spectra, considered as modules over L_{K(n)}S. From the Künneth formula

K(n)_{∗}(M ∧_{L_{K(n)}S} N) ≅ K(n)_{∗}(M) ⊗_{K(n)_{∗}} K(n)_{∗}(N)

it follows that if L_{K(n)}(M ∧_{L_{K(n)}S} N) ≃ ∗ then K(n)_{∗}(M) = 0 or K(n)_{∗}(N) = 0, since K(n)_{∗} is a graded field. So if M is non-trivial, we must have N ≃ ∗. Thus such an M is faithful. □

CHAPTER 5 Examples of Galois extensions

In this chapter we catalog a variety of examples of Galois extensions, some global and some local, as indicated by the section headings.

5.1. Trivial extensions

Let E be any S-module and work E-locally. For each cofibrant commutative S-algebra A and stably dualizable group G there is a trivial G-Galois extension from A to B = F (G_{+}, A), given by the parametrized diagonal map

π^{#} : A → F (G_{+}, A)

that is functionally dual to the collapse map π: G → {e}. Here G acts from the left on F (G_{+}, A) by right multiplication in the source. More precisely, B is a functorial cofibrant replacement of F (G_{+}, A) in the category of commutative Aalgebras, which inherits the G-action by functoriality of the cofibrant replacement. When G is discrete we can write this extension as Δ: A → ∏_{G} A.

It is clear that i: A → B^{hG} = F (G_{+}, A)^{hG} is a weak equivalence, since (G_{+})_{hG} ≃ {e}_{+}, and that h: B∧_{A}B = F (G_{+}, A)∧_{A}F (G_{+}, A) → F (G_{+}∧G_{+}, A) ≅ F (G_{+}, B) is a weak equivalence, since G is stably dualizable.

The trivial G-Galois extension admits an A-module retraction F (G_{+}, A) → A functionally dual to the inclusion {e} → G, so π^{#} : A → F (G_{+}, A) is always faithful.

For any G-Galois extension A → B, there is an induced G-Galois extension B ≅ B ∧_{A} A → B ∧_{A} B (see Proposition 6.2.1 and Lemma 7.1.3 below), and the map h: B ∧_{A} B → F (G_{+}, B) exhibits an equivalence between this self-induced extension and the trivial G-Galois extension π^{#} : B → F (G_{+}, B).

5.2. Eilenberg–Mac Lane spectra

Let E = S. By Proposition 4.2.1 and Lemma 4.3.5, for each finite G-Galois extension R → T of commutative rings the induced map of Eilenberg–Mac Lane commutative S-algebras HR → HT is a faithful G-Galois extension. Proposition 4.2.1 also contains a converse to this statement.

5.3. Real and complex topological K-theory

Let E = S, and let KO and KU be the real and complex topological K-theory spectra, respectively. Their connective versions ko and ku can be realized as the commutative S-algebras associated to the bipermutative topological categories of finite dimensional real and complex inner product spaces, respectively [May77, VI and VII]. The periodic commutative S-algebras KO and KU are obtained from these by Bousfield localization, in the ko- or ku-module categories, by [EKMM97, VIII.4.3].

The complexification functor from real to complex inner product spaces defines maps c: ko → ku and c: KO → KU of commutative S-algebras, and complex conjugation at the categorical level defines a ko-algebra self map t: ku → ku and a KO-algebra self map t: KU → KU. Another name for t is the Adams operation ψ^{−1}. Complex conjugation is an involution, so t^{2} = 1 is the identity in both cases. We therefore have an action by G = {e, t} ≅ ℤ/2 on KU through KO-algebra maps, and can make functorial cofibrant replacements to keep this property, while making KO cofibrant as a commutative S-algebra and KU cofibrant as a commutative KOalgebra.

Proposition 5.3.1. The complexification map c: KO → KU is a faithful ℤ/2-Galois extension, i.e., a global quadratic extension.

See also Example 6.4.4 for more about this extension.

Proof. The claim that i: KO → KU^{hℤ/2} is a weak equivalence is well-known to follow from [At66]. We outline a proof in terms of the homotopy fixed point spectral sequence

E_{s,t}^{2} = H^{−s}(ℤ/2; π_{t}KU) ⟹ π_{s+t}(KU^{hℤ/2}) .

Here π_{∗}KU = ℤ[u^{±1}] with |u| = 2, t ∈ ℤ/2 acts by t(u) = −u and

E_{∗∗}^{2} = ℤ[a, u^{±2}]/(2a)

with a ∈ E_{−1,2}^{2} = H^{1}(ℤ/2; ℤ{u}) ≅ ℤ/2. A computation with the Adams einvariant shows that i takes the generator η ∈ π_{1}KO to a class represented by a ∈ E_{−1,2}^{∞}, so η^{3} = 0 ∈ π_{3}KO implies that a^{3} ∈ E_{−3,6}^{2} is a boundary. The only possibility for this is that d^{3}(u^{2}) = a^{3}, leaving

E_{∗∗}^{4} = E_{∗∗}^{∞} = ℤ[a, b, u^{±4}]/(2a, a^{3}, ab, b^{2} = 4u^{4}) .

This abutment is isomorphic to π_{∗}KO, and the graded ring structure implies that π_{∗}(i) is indeed an isomorphism.

To show that h: KU ∧_{KO} KU → ∏_{ℤ/2} KU is a weak equivalence, we use the Bott periodicity cofiber sequence

(5.3.2)  ΣKO ⟶^{η} KO ⟶^{c} KU ⟶^{∂} Σ^{2}KO

of KO-modules and module maps, up to an implicit weak equivalence between the homotopy cofiber of c and Σ^{2}KO. It is the spectrum level version of the homotopy equivalence Ω(U/O) ≃ ℤ×BU, and is sometimes stated as an equivalence KU ≃ KO∧C_{η}. Here η is given by smashing with the stable Hopf map η: S^{1} → S^{0}, and ∂ is characterized by ∂ ∘ β ≃ Σ^{2}r: Σ^{2}KU → Σ^{2}KO, where β: Σ^{2}KU → KU is the Bott equivalence and r: KU → KO is the realification map. We could write ∂ = Σ^{2}r ∘ β^{−1} in D_{KO}.

Inducing (5.3.2) up along c: KO → KU, we obtain the upper row in the following map of horizontal cofiber sequences

(5.3.3)  \xymatrix{KU ∧_{KO} KO \ar[r]^{1∧c} \ar[d]_{≅} & KU ∧_{KO} KU \ar[r]^{1∧∂} \ar[d]^{h} & KU ∧_{KO} Σ^{2}KO \ar[d]^{β}_{≃} \\ KU \ar[r]^{Δ} & ∏_{ℤ/2} KU \ar[r]^{δ} & KU}

of KU-modules and module maps, up to another implicit identification of the homotopy cofiber of Δ with KU. Here h is the canonical map, Δ is the diagonal inclusion (so the lower row contains the trivial ℤ/2-Galois extension of KU), β is the Bott equivalence KU ∧_{KO} Σ^{2}KO ≅ Σ^{2}KU → KU, and the difference map δ is the difference of the two projections from ∏_{ℤ/2} KU, indexed by the elements of {e, t} ≅ ℤ/2, written multiplicatively.

The left hand square commutes strictly, since ℤ/2 acts on KU through KOalgebra maps. To see that the right hand square commutes up to KU-module homotopy, it suffices to prove this after precomposing with the weak equivalence 1∧β: KU ∧_{KO} Σ^{2}KU → KU ∧_{KO} KU. To show that the two resulting KU-module maps KU ∧_{KO} Σ^{2}KU → KU are homotopic, it suffices by adjunction to show that the restricted KO-module maps Σ^{2}KU → KU are homotopic. This is then the computation

β ∘ Σ^{2}c ∘ Σ^{2}r = δ ∘ h ∘ (c ∧ β)

in D_{KO}, which follows directly from δ ∘ h = μ − μ ∘ (1 ∧ t) = μ(1 ∧ (1 − t)) and the well-known relations c ∘ r = 1 + t and β ∘ Σ^{2}(1 + t) = (1 − t) ∘ β.

Finally, c: KO → KU is faithful. For if N is a KO-module such that N ∧_{KO} KU ≃ ∗, then applying N ∧_{KO} (−) to (5.3.2) gives a cofiber sequence

ΣN ⟶^{η} N ⟶ N ∧_{KO} KU → Σ^{2}N .

The assumption that N ∧_{KO} KU ≃ ∗ implies that η: ΣN → N is a weak equivalence. But η is also nilpotent, since η^{4} = 0 ∈ π_{4}(S), so we must have N ≃ ∗. Therefore KU is faithful over KO. □

The use of nilpotency in this argument may be suggestive of what could in general be required to answer Question 4.3.6. We note that the maps i: ko → ku^{hℤ/2} and h: ku ∧_{ko} ku → ∏_{ℤ/2} ku both fail to be weak equivalences. The homotopy cofiber of i is ⋁_{j<0} Σ^{4j}Hℤ/2, and the homotopy cofiber of h is Hℤ, as is easily seen by adapting the arguments above. So i: ko → ku is not Galois.

5.4. The Morava change-of-rings theorem

In this section we fix a rational prime p and a natural number n, and work locally with respect to the n-th p-primary Morava K-theory K(n). The work of Devinatz and Hopkins [DH04] reinterprets the Morava change-of-rings theorem [Mo85, 0.3.3] as giving a weak equivalence

L_{K(n)}S ≃ E_{n}^{h𝔾_{n}} .

We will regard this as a fundamentally important example of a K(n)-local pro-Galois extension L_{K(n)}S → E_{n} of commutative S-algebras. See Definition 8.1.1 for the precise notion of a pro-Galois extension, which makes most sense after some of the basic Galois theory has been developed in Chapter 7.

5.4.1. The Lubin–Tate spectra.

Recall that E_{n} is the n-th p-primary even periodic Lubin–Tate spectrum, for which

π_{0}(E_{n}) = 𝕎(𝔽_{p^{n}})[[u_{1}, …, u_{n−1}]]

(𝕎(−) denotes the ring of p-typical Witt vectors) and π_{∗}(E_{n}) = π_{0}(E_{n})[u^{±1}]. Related theories were studied by Morava [Mo79], Rudjak [Ru75] and Baker–Würgler [BW89], but in this precise form they seem to have been first considered by Hopkins and Miller [HG94], [Re98].

The height n Honda formal group law Γ_{n} is defined over 𝔽_{p} and is characterized by its p-series [p]_{n}(x) = x^{p^{n}}. Its Lubin–Tate deformation Γ̃_{n} over 𝔽_{p^{n}} is the universal formal group law over a complete local ring with residue field an extension of 𝔽_{p^{n}}, whose reduction to the residue field equals the corresponding extension of Γ_{n}. In this case the universal complete local ring equals π_{0}(E_{n}), with maximal ideal (p, u_{1}, …, u_{n−1}) and residue field 𝔽_{p^{n}}. The Lubin–Tate spectrum E_{n} is (at first) the K(n)-local complex oriented commutative ring spectrum that represents the resulting Landweber exact homology theory (E_{n})_{∗}(X) = π_{∗}(E_{n}) ⊗_{π_{∗}(MU)} MU_{∗}(X).

More generally, we can consider Γ_{n} as a formal group law over the algebraic closure 𝔽̄_{p} of 𝔽_{p}. Its universal deformation is then defined over the complete local ring

π_{0}(E_{n}^{nr}) = 𝕎(𝔽̄_{p})[[u_{1}, …, u_{n−1}]]

and there is a similar K(n)-local complex oriented commutative ring spectrum E_{n}^{nr} with π_{∗}(E_{n}^{nr}) = π_{0}(E_{n}^{nr})[u^{±1}]. The superscript “nr” is short for “non ramifiée”, indicating that 𝕎(𝔽̄_{p}) is the p-adic completion of the maximal unramified extension colim_{f} 𝕎(𝔽_{p^{f}}) of 𝕎(𝔽_{p}) = ℤ_{p}. (The infinite product defining p-typical Witt vectors only commutes with the colimit over f after completion.)

5.4.2. The extended Morava stabilizer group.

The profinite Morava stabilizer group 𝕊_{n} = Aut(Γ_{n}/𝔽_{p^{n}}) of automorphisms defined over 𝔽_{p^{n}} of the formal group law Γ_{n} (see [Ra86, §A2.2, §6.2]), and the finite Galois group Gal = Gal(𝔽_{p^{n}}/𝔽_{p}) ≅ ℤ/n of the extension 𝔽_{p} ⊂ 𝔽_{p^{n}}, both act on the universal deformation Γ̃_{n}, and thus on π_{∗}(E_{n}), by the universal property. These actions combine to one by the profinite semi-direct product 𝔾_{n} = 𝕊_{n} ⋊ Gal. By the Hopkins–Miller [Re98] and Goerss–Hopkins theory [GH04, §7] the ring spectrum E_{n} admits the structure of a commutative S-algebra, up to a contractible choice. Furthermore, the extended Morava stabilizer group 𝔾_{n} acts on E_{n} through commutative S-algebra maps, again up to contractible choice. However, these actions through commutative S-algebras do not take into account the profinite topology on 𝔾_{n}, but rather treat 𝔾_{n} as a discrete group.

It is known by recent work of Daniel G. Davis [Da06], that the profinite group 𝔾_{n} acts continuously on E_{n} in the category of K(n)-local S-modules, but only when E_{n} is reconsidered as a pro-object of discrete 𝔾_{n}-module spectra, where the terms have coefficient groups of the form π_{∗}(E_{n})/I_{k} for a suitable descending sequence of ideals {I_{k}}_{k} with ⋂_{k} I_{k} = 0. Presently, this kind of limit presentation is not available in the context of commutative S-algebras. Hopkins has suggested that a weaker form of structured commutativity, in terms of pro-spectra, may instead be available.

More generally, the Morava stabilizer group 𝕊_{n} and the absolute Galois group Gal(𝔽̄_{p}/𝔽_{p}) ≅ ℤ̂ (the Prüfer ring) of 𝔽_{p} both act on the universal deformation of Γ_{n} over the algebraic closure 𝔽̄_{p}, and thus on π_{∗}(E_{n}^{nr}) by the universal property. These combine to an action by the profinite group

𝔾_{n}^{nr} = 𝕊_{n} ⋊ ℤ̂ .

Note that the (conjugation) action by ℤ̂ on 𝕊_{n} factors through the quotient ℤ̂ → ℤ/n = Gal, since all the automorphisms of the height n Honda formal group law are already defined over 𝔽_{p^{n}} [Ra86, A2.2.20(a)]. The Goerss–Hopkins theory cited above again implies that E_{n}^{nr} is a commutative S-algebra, and the extended Morava stabilizer group 𝔾_{n}^{nr} acts on E_{n}^{nr} through commutative S-algebra maps, up to contractible choices.

5.4.3. Intermediate S-algebras.

In the Galois theory for fields, the intermediate fields F ⊂ E ⊂ F̄ correspond bijectively (via E = (F̄)^{K} and K = G_{E}) to the closed subgroups K ⊂ G_{F} of the absolute Galois group with the Krull topology, and the finite field extensions F ⊂ E correspond to the open subgroups U ⊂ G_{F}. Note that in this topology, the open subgroups are exactly the closed subgroups of finite index. Furthermore, G_{F} acts continuously on F̄ with the discrete topology, so F̄ is the union over the open subgroups U of the fixed fields (F̄)^{U}.

By analogy, it is desirable to construct intermediate K(n)-local commutative S-algebras E_{n}^{hK} for every closed subgroup K ⊂ 𝔾_{n} in the profinite topology. If E_{n} were a discrete 𝔾_{n}-module spectrum, this could be done by the usual definition E_{n}^{hK} = F (EK_{+}, E_{n})^{K}, and indeed, for finite (and thus discrete) subgroups K ⊂ 𝔾_{n} the restricted K-action is continuous, E_{n} is a discrete K-module spectrum and E_{n}^{hK} can well be defined in this way. The maximal finite subgroups M ⊂ 𝔾_{n} were classified by Hewett [He95, 1.3, 1.4]. When M is unique up to conjugacy, E_{n}^{hM} is known as the n-th higher real K-theory spectrum EO_{n} of Hopkins and Miller. Such uniqueness holds for p odd when n = (p−1)k with k prime to p, and for p = 2 when n = 2k with k an odd natural number, by loc. cit.

However, as recalled in the previous subsection, the spectrum E_{n} is not itself a discrete 𝔾_{n}-spectrum, but only an inverse limit of such, i.e., a pro-discrete 𝔾_{n}- spectrum. The homotopy invariant way to form homotopy fixed points of such objects is to take the ordinary continuous homotopy fixed points for the profinite group acting discretely at each stage in the limit system, and then to pass to the homotopy limit, if desired. Note that the formation of continuous homotopy fixed points for profinite groups acting on discrete modules involves a colimit indexed over the finite quotients of the profinite group, and does therefore not generally commute with limits. This procedure describes the approach of [Da06], but it only exhibits the homotopy fixed point spectrum E_{n}^{h𝔾_{n}} as a module spectrum, and not as an algebra spectrum, precisely because we do not know how to realize E_{n} as a pro-object of 𝔾_{n}-discrete associative or commutative S-algebras.

Devinatz and Hopkins circumvent this problem by defining E_{n}^{h𝔾_{n}}, and more generally E_{n}^{hU} for each open subgroup U ⊂ 𝔾_{n}, in a “synthetic” way [DH04, Thm. 1], as the totalization of a suitably rigidified cosimplicial diagram, to obtain a K(n)-local commutative S-algebra of the desired homotopy type. In particular, E_{n}^{h𝔾_{n}} ≃ L_{K(n)}S. (See Section 8.2 for further discussion of the kind of cosimplicial diagram involved, namely the Amitsur complex.) For closed subgroups K ⊂ 𝔾_{n} they then define [DH04, Thm. 2]

E_{n}^{hK} = L_{K(n)}(colim_{i} E_{n}^{hU_{i}K})

where {U_{i}}_{i=0}^{∞} is a fixed descending sequence of open normal subgroups in 𝔾_{n} with ⋂_{i=0}^{∞} U_{i} = {e}, and the colimit is the homotopy colimit in commutative S-algebras. For finite subgroups K ⊂ 𝔾_{n} the synthetic construction agrees [DH04, Thm. 3] with the “natural” definition of E_{n}^{hK} as F (EK_{+}, E_{n})^{K}.

Ethan Devinatz [De05] then proceeds to compare the commutative S-algebras E_{n}^{hK} and E_{n}^{hH} for closed subgroups K and H of 𝔾_{n} with H normal in K. There is a well-defined action by the quotient group K/H on E_{n}^{hH} through commutative S-algebra maps, in the K(n)-local category [De05, §3].

Theorem 5.4.4 (Devinatz–Hopkins). (a) For each pair of closed subgroups H ⊂ K ⊂ 𝔾_{n} = 𝕊_{n} ⋊ Gal with H normal and of finite index in K, the map E_{n}^{hK} → E_{n}^{hH} is a K(n)-local K/H-Galois extension.

(b) In particular, for each finite subgroup K ⊂ 𝔾_{n} the map E_{n}^{hK} → E_{n} is a K(n)-local K-Galois extension.

(c) Likewise, for each open normal subgroup U ⊂ 𝔾_{n} (necessarily of finite index) the map

L_{K(n)}S → E_{n}^{hU}

is a K(n)-local 𝔾_{n}/U-Galois extension.

(d) A choice of a descending sequence {U_{i}}_{i} of open normal subgroups of 𝔾_{n}, with ⋂_{i} U_{i} = {e}, exhibits

L_{K(n)}S → E_{n}

as a K(n)-local pro-𝔾_{n}-Galois extension, in view of the weak equivalence

L_{K(n)}(colim_{i} E_{n}^{hU_{i}}) ⟶^{≃} E_{n} .

Proof. (a) Let A = E_{n}^{hK}, B = E_{n}^{hH} and G = K/H (which is finite and discrete). By [De05, Prop. 2.3, Thm. 3.1 and Thm. A.1] the homotopy fixed point spectral sequence for π_{∗}(B^{hG}) agrees with a strongly convergent K(n)_{∗}- local Adams spectral sequence converging to π_{∗}(A). So i: A → B^{hG} is a weak equivalence. By [De05, Cor. 3.9] the natural map h: L_{K(n)}(B ∧_{A} B) → F (G_{+}, B) induces an isomorphism on homotopy groups.

Parts (b) and (c) are special cases of (a). Part (d) is contained in [DH04, Thm. 3(i)]. □

It would be nice to extend the statement of this theorem to the case when H is normal and closed, but not necessarily of finite index, in K.

For n = 2 and p = 2, the Morava stabilizer group 𝕊_{2} is the group of units in the maximal order in the quaternion algebra ℚ_{2}{1, i, j, k}, and its maximal finite subgroup is the binary tetrahedral group Â_{4} = Q_{8} ⋊ℤ/3 of order 24, containing the quaternion group Q_{8} = {±1, ±i, ±j, ±k} and the 16 other elements (±1±i±j±k)/2. See [CF67, pp. 137–138], [Ra86, 6.3.27]. The maximal finite subgroup of 𝔾_{2} is G_{48} = Â_{4} ⋊ ℤ/2, and EO_{2} = E_{2}^{hG_{48}} is the K(2)-localization of the connective spectrum eo_{2} with H^{∗}(eo_{2}; 𝔽_{2}) ≅ A//A_{2} as a module over the Steenrod algebra, which is related to the topological modular forms spectrum tmf [Hop02, §3.5].

Proposition 5.4.5. At p = 2, the natural map EO_{2} → E_{2} is a K(2)-local faithful G_{48} = Â_{4} ⋊ ℤ/2-Galois extension.

Proof. This follows from Theorem 5.4.4(b) above and Proposition 5.4.9(b) below, but we would also like to indicate a direct proof of faithfulness, using results of Hopkins and Mahowald [HM98]. There is a finite CW spectrum C_{γ} obtained as the mapping cone of a map

γ: Σ^{5}C_{η} ∧ C_{ν} → C_{η} ∧ C_{ν} ,

such that H^{∗}(C_{γ}; 𝔽_{2}) ≅ DA(1) ≅ A(2)//E(2) is the “double” of A(1) = ⟨Sq^{1}, Sq^{2}⟩. The spectrum C_{γ} and the self-map γ can be obtained by a construction analogous to that of the spectrum A_{1} and the map v_{1} : Σ^{2}Y → Y in [DM81, pp. 619–620], but replacing all the real projective spaces occurring there by complex projective spaces. Furthermore, there is a weak equivalence eo_{2} ∧ C_{γ} ≃ BP ⟨2⟩ that realizes the isomorphism A//A(2) ⊗ A(2)//E(2) ≅ A//E(2) ≅ H^{∗}(BP ⟨2⟩; 𝔽_{2}). Applying K(2)-localization yields

EO_{2} ∧ C_{γ} ≃ { \atop E(2)} ,

in the notation of 5.4.7, using that BP ⟨2⟩ → v_{2}^{−1}BP ⟨2⟩ = E(2) is a K(2)_{∗}- equivalence. Since η, ν and γ are all nilpotent (for η ∈ π_{1}(S) and ν ∈ π_{3}(S) this is well-known; for γ it can be deduced from the Devinatz–Hopkins–Smith nilpotence theorem [DHS88, Cor. 2]), it follows as in the proof of Proposition 5.3.1 that EO_{2} → { \atop E(2)} is faithful. And { \atop E(2)} → E_{2} is faithful by the elementary Proposition 5.4.9(a).

\xymatrix{ & E_{2}^{Gal} \ar[r]^{ℤ/2} & E_{2} \\ EO_{2} \ar[r] \ar[ur]^{Â_{4}} & { \atop E(2)} \ar[u]^{𝔽_{4}^{∗}} \ar@{-->}[ur]_{Σ_{3}}}

□

5.4.6. Adjoining roots of unity.

Including the maximal unramified extensions into this picture, we have the following diagram of K(n)-local extensions. The groups label Galois (or pro-Galois) extensions.

\xymatrix{E_{n}^{Gal} \ar[r]^{Gal} \ar[rr]^{ℤ̂} & E_{n} \ar[r]^{nℤ̂} & E_{n}^{nr} \\ EO_{n} \ar[u] \ar[ur]^{M} \\ L_{K(n)}S \ar[r]^{Gal} \ar[rr]_{ℤ̂} \ar[u] \ar[uur]_{𝔾_{n}} & E_{n}^{h𝕊_{n}} \ar[r]^{nℤ̂} \ar[uu]_{𝕊_{n}} & (E_{n}^{nr})^{h𝕊_{n}} \ar[uu]_{𝕊_{n}}}

The maximal extension L_{K(n)}S → E_{n}^{nr} is K(n)-locally pro-𝔾_{n}^{nr}-Galois.

The ℤ̂-extension along the bottom is that obtained by adjoining all roots of unity of order prime to p to the p-complete commutative S-algebra L_{K(n)}S. We might write E_{n}^{h𝕊_{n}} = L_{K(n)}S(μ_{p^{n}−1}) and (E_{n}^{nr})^{h𝕊_{n}} = L_{K(n)}S(μ_{∞,p}), where μ_{m} denotes the group of m-th order roots of unity and μ_{∞,p} = colim_{p∤m} μ_{m} denotes the group of all roots of unity of order prime to p. Note that in the latter case, the infinite colimit of spectra must be implicitly K(n)-completed. Similarly, E_{n} = E_{n}^{Gal}(μ_{p^{n}−1}) and E_{n}^{nr} = E_{n}^{Gal}(μ_{∞,p}).

The process of adjoining m-th roots of unity makes sense when applied to a p-local commutative S-algebra A, for p ∤ m, following Roland Schwänzl, Rainer Vogt and Waldhausen [SVW99], since A(μ_{m}) can be obtained from the group A-algebra A[C_{m}] = A ∧ C_{m+} of the cyclic group of order m by localizing with respect to a p-locally defined idempotent. Likewise, adjoining an m-th root of unity to a p-complete commutative S-algebra A, for m = p^{f} − 1, can be achieved by localizing with respect to a further idempotent. The situation is analogous to how ℚ_{p} ⊗_{ℚ} ℚ(μ_{m}) splits as a product of copies of ℚ_{p}(μ_{m}), when m = p^{f} − 1. For more on the process of adjoining roots of unity to S-algebras, see [La03, 3.4] in the associative case and [BR07, 2.2.5 and 2.2.8] in the commutative case.

These observations may justify thinking of the projection d: 𝔾_{n}^{nr} = 𝕊_{n} ⋊ℤ̂ → ℤ̂ as the degree map of a K(n)-local class field theory for structured ring spectra [Ne99, §IV.4].

5.4.7. Faithfulness.

Let { \atop E(n)} = L_{K(n)}E(n) be the K(n)-localization of the Johnson–Wilson spectrum E(n) from 3.2.2, called Morava E-theory in [HSt99]. By [BW89, 4.1] or [HSt99, §1.1, 5.2] it has coefficients

π_{∗}{ \atop E(n)} = ℤ_{(p)}[v_{1}, …, v_{n−1}, v_{n}^{±1}]_{I_{n}}^{∧} ,

where I_{n} = (p, v_{1}, …, v_{n−1}). The spectrum { \atop E(n)} was proved to be an associative S-algebra in [Bak91], and is in fact a commutative S-algebra by the homotopy fixed point description in Proposition 5.4.9(a) below.

Theorem 5.4.8 (Hovey–Strickland). L_{K(n)}S is contained in the thick sub-

category of K(n)-local spectra generated by { \atop E(n)}, so { \atop E(n)} is a faithful L_{K(n)}Smodule in the K(n)-local category.

Proof. The first claim is contained in the proof of [HSt99, 8.9], which relies heavily on the construction by Jeff Smith of a suitable finite p-local spectrum X, as explained in [Ra92, §8.3]. The second claim follows from the first, but also much more easily from Lemma 4.3.7. □

Proposition 5.4.9. (a) The K(n)-local Galois extension { \atop E(n)} → E_{n} and the K(n)-local pro-Galois extension L_{K(n)}S → E_{n} are both faithful.

(b) For each pair of closed subgroups H ⊂ K ⊂ 𝔾_{n}, with H normal and of finite index in K, the K(n)-local K/H-Galois extension E_{n}^{hK} → E_{n}^{hH} is faithful.

Proof. (a) There is a finite subgroup 𝔽_{p^{n}}^{∗} of 𝕊_{n} such that for K = 𝔽_{p^{n}}^{∗} ⋊ Gal

we have { \atop E(n)} ≃ E_{n}^{hK}. In more detail, 𝕊_{n} contains the unit group 𝕎(𝔽_{p^{n}})^{∗} [Ra86, A2.2.17], whose torsion subgroup reduces isomorphically to 𝔽_{p^{n}}^{∗}. For an element of finite order ω ∈ 𝕎(𝔽_{p^{n}})^{∗}, with mod p reduction ω̄ ∈ 𝔽_{p^{n}}^{∗}, the linear formal power series g(x) = ω̄x defines an automorphism of Γ_{n}, i.e., an element g ∈ 𝕊_{n}, which acts on π_{∗}(E_{n}) by g(u) = ωu and g(uu_{k}) = ω^{p^{k}}uu_{k} for 1 ≤ k < n by [DH95, 3.3, 4.4], leaving v_{n} = u^{1−p^{n}} and v_{k} = u^{1−p^{k}}u_{k} invariant. Thus π_{∗}E_{n}^{Gal} = ℤ_{p}[[u_{1}, …, u_{n−1}]][u^{±1}] and π_{∗}E_{n}^{hK} is the I_{n}-adic completion of π_{∗}E(n).

Then for any spectrum X,

(E_{n})_{∗}^{∨}(X) ≅ π_{∗}E_{n} ⊗_{π_{∗}{ \atop E(n)}} { \atop E(n)}_{∗}^{∨}(X)

with π_{∗}E_{n} a free module of rank |K| = (p^{n}−1)n over π_{∗}{ \atop E(n)}. Here we are using the notation (E_{n})_{∗}^{∨}(X) = π_{∗}L_{K(n)}(E_{n} ∧ X), and similarly for { \atop E(n)}, of [HSt99, 8.3].

It follows easily from this formula that { \atop E(n)} → E_{n} is faithful in the K(n)-local category.

In combination with 5.4.8 this shows that the composite extension L_{K(n)}S →

{ \atop E(n)} → E_{n} is faithful, but Lemma 4.3.7 provides a much easier argument.

(b) The second result follows by faithful base change along ϕ: L_{K(n)}S → E_{n}. There is a commutative diagram (for H and K as in the statement)

\xymatrix{E_{n}^{hH} \ar[r] & L_{K(n)}(E_{n} ∧ E_{n}^{hH}) \\ E_{n}^{hK} \ar[r]^{1∧ϕ} \ar[u]^{ψ} & L_{K(n)}(E_{n} ∧ E_{n}^{hK}) \ar[u]_{1∧ψ} \\ L_{K(n)}S \ar[r]^{ϕ} \ar[u] & E_{n} \ar[u]}

where the squares are pushouts in the category of K(n)-local commutative Salgebras. By the Morava change-of-rings theorem and [DH04, Thm. 1(iii)],

π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{hH}) ≅ Map(𝔾_{n}/H, π_{∗}E_{n})

and

π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{hK}) ≅ Map(𝔾_{n}/K, π_{∗}E_{n}) .

See also the proof of Theorem 7.2.3 below. Here Map denotes the unbased continuous maps with respect to the profinite topologies on 𝔾_{n}/H, 𝔾_{n}/K and π_{∗}E_{n} (in each degree). Note that K/H is a finite group acting freely on the Hausdorff space 𝔾_{n}/H, with orbit space 𝔾_{n}/K, so π: 𝔾_{n}/H → 𝔾_{n}/K is a regular K/H-covering space. We claim that it admits a continuous section σ: 𝔾_{n}/K → 𝔾_{n}/H, so that there is a homeomorphism K/H × 𝔾_{n}/K → 𝔾_{n}/H, and

Map(𝔾_{n}/H, π_{∗}E_{n}) ≅ ∏_{K/H} Map(𝔾_{n}/K, π_{∗}E_{n}) .

Thus π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{hH}) is a free module of rank |K/H| over π_{∗}L_{K(n)}(E_{n} ∧ E_{n}^{hK}), so that L_{K(n)}(E_{n} ∧ E_{n}^{hH}) is faithful over L_{K(n)}(E_{n} ∧ E_{n}^{hK}). The map 1 ∧ ϕ is obtained by base change from ϕ, which is faithful by (a), and is therefore faithful by Lemma 4.3.3, so ψ: E_{n}^{hK} → E_{n}^{hH} is faithful by Lemma 4.3.4.

It remains to verify the claim. Let {U_{i}}_{i=0}^{∞} be a descending sequence of open normal subgroups of 𝔾_{n}, with trivial intersection as above. Then U_{i}H is normal of finite index in U_{i}K, K/H surjects to U_{i}K/U_{i}H and there is a regular covering space π_{i}: 𝔾_{n}/U_{i}H → 𝔾_{n}/U_{i}K, for each i. We have the following commutative diagram for i < j:

\xymatrix{K/H \ar[r] \ar[d] & U_{j}K/U_{j}H \ar[r] \ar[d] & U_{i}K/U_{i}H \ar[d] \\ 𝔾_{n}/H \ar[r] \ar[d]_{π} & 𝔾_{n}/U_{j}H \ar[r] \ar[d]^{π_{j}} & 𝔾_{n}/U_{i}H \ar[d]^{π_{i}} \\ 𝔾_{n}/K \ar[r] & 𝔾_{n}/U_{j}K \ar[r] & 𝔾_{n}/U_{i}K}

Since K/H is finite, the surjections U_{j}K/U_{j}H → U_{i}K/U_{i}H are isomorphisms for all sufficiently large i and j, say for i, j ≥ i_{0}, and then π_{j} is the pullback of π_{i} along 𝔾_{n}/U_{j}K → 𝔾_{n}/U_{i}K. Thus any choice of section σ_{i} to π_{i} pulls back to a section σ_{j} of π_{j}, so that the composite maps 𝔾_{n}/K → 𝔾_{n}/U_{i}K → 𝔾_{n}/U_{i}H are compatible for all i ≥ i_{0}. Their limit defines the continuous section σ: 𝔾_{n}/K → 𝔾_{n}/H. □

5.5. The K(1)-local case

When n = 1, the discussion in Section 5.4 reduces to more classical statements about variants of topological K-theory, which we now make explicit, together with a comparison to the even more classical arithmetic theory of abelian extensions of ℚ_{p} and ℚ.

5.5.1. p-complete topological K-theory.

Mod p complex topological K-theory, with π_{∗}(KU/p) = 𝔽_{p}[u^{±1}], splits as

KU/p ≃ ⋁_{i=0}^{p−2} Σ^{2i}K(1)

where π_{∗}K(1) = 𝔽_{p}[v_{1}^{±1}]. Bousfield K(1)-localization equals Bousfield KU/plocalization, which in turn equals Bousfield KU-localization followed by p-adic completion: L_{K(1)}X = L_{KU/p}X = (L_{KU}X)_{p}^{∧} [Bo79, 2.11].

The height 1 Honda formal group law over 𝔽_{p} is isomorphic to the multiplicative one:

Ĝ_{m}(x, y) = x + y + xy ,

its universal deformation Γ̃_{1} is isomorphic to the multiplicative formal group law over ℤ_{p}, and the Lubin–Tate spectrum E_{1} is weakly equivalent to p-completed complex topological K-theory KU_{p}^{∧} with π_{∗}(KU_{p}^{∧}) = ℤ_{p}[u^{±1}]. The Morava stabilizer group 𝔾_{1} = 𝕊_{1} is the group of p-adic units ℤ_{p}^{∗}, with its profinite topology, and k ∈ ℤ_{p}^{∗} acts on the commutative S-algebra KU_{p}^{∧} by the p-adic Adams operation

ψ^{k}: KU_{p}^{∧} → KU_{p}^{∧} .

On homotopy, ψ^{k}(u) = ku.

5.5.2. Subalgebras.

The homotopy fixed point spectrum E_{1}^{h𝔾_{n}} = (KU_{p}^{∧})^{hℤ_{p}^{∗}} is the p-complete (nonconnective) image-of-J spectrum L_{K(1)}S = J_{p}^{∧}, defined for p = 2 by the fiber sequence

J_{2}^{∧} → KO_{2}^{∧} ⟶^{ψ^{3}−1} KO_{2}^{∧}

and for p odd by the fiber sequence

J_{p}^{∧} → KU_{p}^{∧} ⟶^{ψ^{r}−1} KU_{p}^{∧}

for r a topological generator of ℤ_{p}^{∗}. These identifications of the p-completed KUlocalization of S with J_{p}^{∧} are basically due to Mark Mahowald and Haynes Miller [Bo79, 4.2], respectively. (Adams–Baird and Ravenel went on to identify the p-local KU-localization of S, see [Bo79, 4.3].)

The Morava stabilizer group 𝕊_{1} = ℤ_{p}^{∗} is isomorphic to the Galois group of the maximal (totally ramified) p-cyclotomic extension ℚ_{p} ⊂ ℚ_{p}(μ_{p^{∞}}), so the classification of intermediate commutative S-algebras J_{p} → C → KU_{p}^{∧} of the form

5.5. THE K(1)-LOCAL CASE 35

C = (KU_{p}^{∧})^{hK} for K closed in ℤ_{p}^{∗} is identical to the classification of intermediate fields ℚ_{p} ⊂ E ⊂ ℚ_{p}(μ_{p^{∞}}). In this way J_{p}^{∧} → KU_{p}^{∧} provides a K(1)-local “realization” of the K(0)-local extension ℚ_{p} → ℚ_{p}(μ_{p^{∞}}). There are similar K(n)-local realizations of the form L_{K(n)}S → E_{n}^{hK}, when K is the kernel of the determinant/abelianization homomorphism 𝔾_{n} → 𝔾_{n}^{ab} → ℤ_{p}^{∗} [Ra86, 6.2.6(b)].

When p = 2, ℤ_{2}^{∗} ≅ ℤ_{2} × ℤ/2, where ℤ_{2} ≅ 1 + 4ℤ_{2} is open of index 2, and ℤ/2 ≅ {±1} ⊂ ℤ_{2}^{∗} is closed. There are three different subgroups of index 2, namely the topologically generated subgroups ⟨3⟩, ⟨5⟩ and ⟨−1, 9⟩. The first of these corresponds to the complex image-of-J spectrum JU_{2}^{∧} = (KU_{2}^{∧})^{h⟨3⟩} given by the fiber sequence

JU_{2}^{∧} → KU_{2}^{∧} ⟶^{ψ^{3}−1} KU_{2}^{∧} ,

and there is a K(1)-local (quadratic) ℤ/2-Galois extension c: J_{2}^{∧} → JU_{2}^{∧}, which is compatible with the complexification map c: KO_{2}^{∧} → KU_{2}^{∧}. See Example 6.2.2 for more on this quadratic extension. The closed subgroup ℤ/2 of ℤ_{2}^{∗} corresponds to 2-complete real K-theory: (KU_{2}^{∧})^{hℤ/2} ≃ KO_{2}^{∧}.

When p is odd, ℤ_{p}^{∗} ≅ ℤ_{p}×𝔽_{p}^{∗} is pro-cyclic. Let r ∈ ℤ_{p}^{∗} be a topological generator, chosen to be a natural number. Then ℤ_{p}^{∗} has a unique open subgroup ⟨r^{n}⟩ of index n, for each integer n of the form n = p^{e}d with e ≥ 0 and d | p − 1. In addition, it has the closed subgroups that appear as subgroups of 𝔽_{p}^{∗}. In particular, ℤ_{p}^{∗} has an open subgroup ℤ_{p} ≅ 1 + pℤ_{p} of index (p − 1), and a closed subgroup 𝔽_{p}^{∗} ⊂ ℤ_{p}^{∗}. The latter corresponds to the p-complete Adams summand L_{p}^{∧} = (KU_{p}^{∧})^{h𝔽_{p}^{∗}} with π_{∗}(L_{p}^{∧}) = ℤ_{p}[v_{1}^{±1}]. There are K(1)-local 𝔽_{p}^{∗}-Galois extensions J_{p}^{∧} → (KU_{p}^{∧})^{hℤ_{p}} and L_{p}^{∧} → KU_{p}^{∧}. Let us write F ψ^{r^{n}} = (KU_{p}^{∧})^{h⟨r^{n}⟩} for the homotopy fixed point spectrum of ψ^{r^{n}}, which is equivalent to the homotopy fiber of ψ^{r^{n}} − 1. Then there is a K(1)-local ℤ/n-Galois extension

J_{p}^{∧} = F ψ^{r} → F ψ^{r^{n}}

for each integer n = p^{e}d with d | p − 1, as above.

5.5.3. Extensions.

Incorporating the roots of unity of order prime to p, we have the following diagram

\xymatrix{KU_{p}^{∧} \ar[r]^{ℤ̂} & KU_{p}^{∧}(μ_{∞,p}) \\ J_{p}^{∧} \ar[r]^{ℤ̂} \ar[u]^{ℤ_{p}^{∗}} \ar[ur]^{𝔾_{1}^{nr}} & J_{p}^{∧}(μ_{∞,p}) \ar[u]_{ℤ_{p}^{∗}}}

with E_{1}^{nr} = KU_{p}^{∧}(μ_{∞,p}). Here the maximal Galois group 𝔾_{1}^{nr} = ℤ_{p}^{∗} × ℤ̂ is abelian, since ℤ̂ acts trivially on 𝕊_{1} = ℤ_{p}^{∗}. It provides a K(1)-local realization of the Galois group of the maximal abelian extension ℚ_{p} → ℚ_{p}(μ_{∞}).

It also appears to be possible to fit the various rational primes together, so as to obtain KU-local realizations of the abelian extensions of the rational field ℚ itself. The Galois group G = ℤ̂^{∗} of the maximal abelian extension ℚ → ℚ(μ_{∞}) contains the Galois group of ℚ_{p} → ℚ_{p}(μ_{∞}) as the decomposition group D_{p} of the prime ideal (p). Let Z_{p} = ℚ(μ_{∞})^{D_{p}} be the corresponding decomposition field [Ne99, I.9.2].

ℚ ⟶^{G/D_{p}} Z_{p} ⟶^{D_{p}} ℚ(μ_{∞}) .

After base change along ℚ → ℚ_{p} there are weak product splittings [Ne99, II.8.3]

ℚ_{p} ⊗_{ℚ} Z_{p} ≅ \begin{matrix}∏ & ′ \\ G/D_{p} \end{matrix} ℚ_{p}  and  ℚ_{p} ⊗_{ℚ} ℚ(μ_{∞}) ≅ \begin{matrix}∏ & ′ \\ G/D_{p} \end{matrix} ℚ_{p}(μ_{∞}) ,

i.e., as colimits of the products over the finite quotients of

G/D_{p} = ℤ̂^{∗}/(ℤ_{p}^{∗} × ℤ̂) ≅ (∏_{ℓ≠p} ℤ_{ℓ}^{∗})/ℤ̂ .

In the latter profinite quotient, the unit of ℤ̂ maps diagonally to the class of p in each ℤ_{ℓ}^{∗}. Hence G = G_{ℚ}^{ab} is realized as the Galois group of

ℚ_{p} ⟶^{G/D_{p}} \begin{matrix}∏ & ′ \\ G/D_{p} \end{matrix} ℚ_{p} ⟶^{D_{p}} \begin{matrix}∏ & ′ \\ G/D_{p} \end{matrix} ℚ_{p}(μ_{∞}) ,

where the first map is a (pro-)trivial Galois extension.

We can realize the same groups in the K(1)-local category, by the two pro-Galois extensions

J_{p}^{∧} ⟶^{G/D_{p}} \begin{matrix}∏ & ′ \\ G/D_{p} \end{matrix} J_{p}^{∧} ⟶^{D_{p}} \begin{matrix}∏ & ′ \\ G/D_{p} \end{matrix} E_{1}^{nr} .

Here the first is the implicitly K(1)-localized colimit of the trivial Galois extensions of J_{p}^{∧}, indexed over the finite quotients of G/D_{p}.

For brevity, let B_{p} = ∏′_{G/D_{p}} E_{1}^{nr}. Then J_{p}^{∧} → B_{p} is a K(1)-local realization of the maximal abelian extension of ℚ. It seems plausible to find arithmetic pullback squares

\xymatrix{L_{KU}S \ar[r] \ar[d] & ∏_{p} J_{p}^{∧} \ar[d] & B \ar[r] \ar[d] & ∏_{p} B_{p} \ar[d] \\ L_{0}S \ar[r] & L_{0} ∏_{p} J_{p}^{∧} & L_{0}B \ar[r] & L_{0} ∏_{p} B_{p}}

of commutative S-algebras, so as to get an integral KU-local realization L_{KU}S → B of the same Galois group. It would be wonderful if analogous (non-abelian) K(n)- local constructions for n ≥ 2 turn out to detect more of the absolute Galois group of ℚ_{p} in 𝔾_{n}^{nr}, or of the absolute Galois group of ℚ. The paper [Mo05] may be relevant.

5.5.4. p-local topological K-theory.

The p-local complex K-theory spectrum KU_{(p)} is also a commutative S-algebra, and admits an action by the Adams operation ψ^{r} and its powers through commutative S-algebra maps [BR05a, 9.2]. However, in this case the E(1)-local extension

KU_{(p)}^{h⟨r⟩} → KU_{(p)}^{h⟨r^{n}⟩}

is not a ℤ/n-Galois extension. It even fails to be one rationally, i.e., K(0)-locally. For

π_{∗}(KU_{(p)}^{h⟨r^{n}⟩}) ⊗ ℚ ≅ E_{ℚ}(ζ_{r^{n}})

5.6. COCHAIN S-ALGEBRAS 37

is an exterior algebra over ℚ on one generator, and E_{ℚ}(ζ_{r}) → E_{ℚ}(ζ_{r^{n}}) is an isomorphism, so {e}-Galois, but not ℤ/n-Galois. In spite of the relatively rich source of K(n)-local Galois extensions, there are ramification phenomena that frequently enter when several chromatic strata are involved.

The idempotent operation (p − 1)^{−1} ∑_{k∈𝔽_{p}^{∗}} ψ^{k} on KU_{p}^{∧} that defines the pcomplete Adams summand L_{p}^{∧} is in fact p-locally defined [Ad69, p. 85], so as to split off the p-local Adams summand L_{(p)} in

KU_{(p)} ≃ ⋁_{i=0}^{p−2} Σ^{2i}L_{(p)} .

However, the p-adic Adams operations ψ^{k} of finite order, for k in the torsion subgroup 𝔽_{p}^{∗} ⊂ ℤ_{p}^{∗}, are not defined over ℤ_{(p)}, since ψ^{k}(u) = ku on homotopy. Therefore the extension L_{(p)} → KU_{(p)} only becomes Galois after p-adic completion. This provides an example of an E(1)-local étale extension (in the sense of Section 9.4) that does not extend to a Galois extension. Again, this is an instance of K(0)-local ramification of the E(1)-local prolongation of a, by definition unramified, K(1)-local Galois extension. These examples are meant as partial justification for the last paragraph of Remark 3.2.4.

5.6. Cochain S-algebras

Let G be a topological group and consider a principal G-bundle π: P → X. Fix a rational prime p and let A = F (X_{+}, H𝔽_{p}) and B = F (P_{+}, H𝔽_{p}) be the mod p cochain H𝔽_{p}-algebras on X and P, respectively. Note that π_{∗}(A) = H^{−∗}(X; 𝔽_{p}) and π_{∗}(B) = H^{−∗}(P; 𝔽_{p}). We think of A and B as models for the singular cochain algebras C^{∗}(X; 𝔽_{p}) and C^{∗}(P; 𝔽_{p}), in conformance with [DGI06, §3]. The direct relation between the differential graded E_{∞} structure on C^{∗}(X; 𝔽_{p}) and the commutative S-algebra structure on A = F (X_{+}, H𝔽_{p}) seems not to have been made explicit, however.

The projection π induces a map of commutative H𝔽_{p}-algebras A → B, the right action of G on P induces a left action of G on B through commutative Aalgebra maps, and the weak equivalence P ×_{G} EG → X makes its cochain dual i: A → F ((P ×_{G}EG)_{+}, H𝔽_{p}) ≅ B^{hG} a weak equivalence. We now investigate when h: B ∧_{A} B → F (G_{+}, B) is a weak equivalence.

The Künneth spectral sequence

(5.6.1)  E_{s,t}^{2} = Tor_{s,t}^{π_{∗}(A)}(π_{∗}(B), π_{∗}(B)) ⟹ π_{s+t}(B ∧_{A} B)

can be derived from the skeleton filtration of the (simplicial) two-sided bar construction

B^{H𝔽_{p}}(B, A, B): [q] ↦ B ∧ A^{∧q} ∧ B

with all smash products formed over H𝔽_{p} [EKMM97, IV.7.7]. Dually, let

Ω(P, X, P): [q] ↦ P × X^{q} × P

be the (cosimplicial) two-sided cobar construction, with totalization equal to the fiber product P ×_{X} P. There is a natural simplicial map

∧: B^{H𝔽_{p}}(B, A, B) → F (Ω(P, X, P), H𝔽_{p})

which is a degreewise weak equivalence by the Künneth formula in mod p cohomology, under the assumption that H_{∗}(X; 𝔽_{p}) and H_{∗}(P; 𝔽_{p}) are finite in each degree. So the Künneth spectral sequence equals the one obtained by applying mod p cohomology to the cobar construction, i.e., the mod p Eilenberg–Moore spectral sequence

(5.6.2) E_{s,t}^{2} = Tor_{s,t}^{H^{∗}(X;𝔽_{p})}(H^{∗}(P; 𝔽_{p}), H^{∗}(P; 𝔽_{p})) ⟹ H^{−(s+t)}(P ×_{X} P; 𝔽_{p})

[EM66]. By [Dw74], [Sh96, 3.1], the Eilenberg–Moore spectral sequence converges strongly if, for example, π_{0}(G) is finite, X is path-connected, and π_{1}(X) acts nilpotently on H_{∗}(G; 𝔽_{p}). The Künneth spectral sequence is always strongly convergent, so this comparison implies that the upper horizontal map in

\xymatrix{B ∧_{A} B \ar[r]^{∧} \ar[d]_{h} & F ((P ×_{X} P)_{+}, H𝔽_{p}) \ar[d]^{ξ^{∗}} \\ F (G_{+}, B) \ar[r]^{≅} & F ((P × G)_{+}, H𝔽_{p})}

is a weak equivalence. The right hand vertical map is induced by the homeomorphism ξ: P × G → P ×_{X} P, hence is an isomorphism, as is the lower horizontal map. Therefore these hypotheses ensure that the left hand vertical map h is a weak equivalence.

Proposition 5.6.3. Let G be a stably dualizable group and P → X a principal G-bundle.

(a) Suppose that π_{0}(G) is finite, X is path-connected, π_{1}(X) acts nilpotently on H_{∗}(G; 𝔽_{p}), and that H_{∗}(X; 𝔽_{p}) and H_{∗}(P; 𝔽_{p}) are finite in each degree. Then the map of cochain H𝔽_{p}-algebras

F (X_{+}, H𝔽_{p}) → F (P_{+}, H𝔽_{p})

is a G-Galois extension.

(b) In particular, when G is a finite discrete group acting nilpotently on 𝔽_{p}[G] (this includes all finite p-groups), then there is a G-Galois extension

F (BG_{+}, H𝔽_{p}) → F (EG_{+}, H𝔽_{p}) ≃ H𝔽_{p}

that exhibits H𝔽_{p} as a Galois extension by each such group.

A similar argument applies for the map of rational cochain algebras

F (X_{+}, Hℚ) → F (P_{+}, Hℚ) ,

when H^{∗}(X; ℚ) and H^{∗}(P; ℚ) are finite dimensional over ℚ in each degree.

For each natural number n the Morava K-theory spectrum K(n) admits uncountably many associative S-algebra structures [Rob89, 2.5], none of which are strictly commutative (cf. Lemma 5.6.4). Therefore

F (X_{+}, K(n)) → F (P_{+}, K(n))

is at best a kind of non-commutative G-Galois extension. As a further complication, the convergence of the K(n)-based Eilenberg–Moore spectral sequence, analogous to (5.6.2), is not yet well understood.

Lemma 5.6.4. K(n) does not admit the structure of a commutative S-algebra.

Proof. Suppose that K(n) is a commutative S-algebra. Then so is its connective cover k(n), and there is a 1-connected commutative S-algebra map u: k(n) → H𝔽_{p}. Then u_{∗}: H_{∗}(k(n); 𝔽_{p}) → H_{∗}(H𝔽_{p}; 𝔽_{p}) is an injective algebra homomorphism, that commutes with the Dyer–Lashof operations on both sides [BMMS86, III.2.3]. The target equals the dual Steenrod algebra A_{∗} = E(χτ_{k} | k ≥ 0)⊗P (χξ_{k} | k ≥ 1), and the image of u_{∗} contains χτ_{n−1}, but not χτ_{n}. This contradicts the operation Q^{p^{k}}(χτ_{k}) = χτ_{k+1} in A_{∗}, in the case k = n−1. □

CHAPTER 6 Dualizability and alternate characterizations

6.1. Extended equivalences

Let A → B be a map of E-local commutative S-algebras, and let G be a topological group acting from the left on B through A-algebra maps, say by α: G_{+}∧ B → B. For example, A → B could be a G-Galois extension.

The twisted group S-algebra B⟨G⟩ is defined to be B ∧ G_{+} (implicitly Elocalized, like B[G]), with the multiplication B⟨G⟩ ∧ B⟨G⟩ → B⟨G⟩ obtained from the composite map

G_{+} ∧ B ⟶^{Δ∧1} G_{+} ∧ G_{+} ∧ B ⟶^{1∧α} G_{+} ∧ B ≅ B ∧ G_{+}

and the multiplications on B and G. As usual, Δ is the diagonal map. The map A → B and the unit inclusion {e} → G induce a central map η: A → B⟨G⟩, which makes B⟨G⟩ an associative A-algebra. Likewise, the endomorphism algebra F_{A}(B, B) of B over A is an associative A-algebra with respect to the composition pairing.

Let

(6.1.1)  j: B⟨G⟩ → F_{A}(B, B)

be the canonical map of A-algebras that is right adjoint to the composite map

B ∧ G_{+} ∧_{A} B ⟶^{1∧α} B ∧_{A} B ⟶^{μ} B ,

induced by the (A-linear) action of G on B and the multiplication on B. Note that B⟨G⟩ and F_{A}(B, B) are left B-modules, with respect to the action on the target in the latter case, and that j is a map of B-modules. There is also a diagonal left action by G on B ∧ G_{+} and on the target in F_{A}(B, B), and j is G-equivariant with respect to these actions. These B- and G-actions do not commute, but combine to a left module action by B⟨G⟩.

For a map f of spectra, we will write f_{#} and f^{#} for various maps induced by left and right composition with f, respectively.

Lemma 6.1.2. Let A → B be a map of commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps, such that h: B ∧_{A} B → F (G_{+}, B) is a weak equivalence. For example, A → B could be a G-Galois extension. Then:

(a) For each B-module M there is a natural weak equivalence

h_{M}: M ∧_{A} B → F (G_{+}, M) .

(b) The canonical map

j: B⟨G⟩ → F_{A}(B, B)

is a weak equivalence.

(c) For each B-module M there is a natural weak equivalence

j_{M}: M ∧ G_{+} → F_{A}(B, M) .

Proof. (a) By definition, h_{M} is the composite map

M ∧_{A} B ≅ M ∧_{B} B ∧_{A} B ⟶^{1∧h} M ∧_{B} F (G_{+}, B) ⟶^{ν} F (G_{+}, M) ,

which is a weak equivalence because h is a weak equivalence and G is stably dualizable.

(b) This is the special case of (c) below when M = B.

(c) By definition, j_{M} is right adjoint to the composite map M ∧ G_{+} ∧_{A} B → M ∧_{A} B → M induced by the group action of G on B and the module action of B on M. We can factor j_{M} in the stable homotopy category as the following chain of weak equivalences:

M ∧ G_{+} ⟶^{1∧ρ} M ∧ DDG_{+} ⟶^{ν} F (DG_{+}, M) ≅ F_{B}(B ∧ DG_{+}, M)

⟵^{ν^{#}} F_{B}(F (G_{+}, B), M) ⟶^{h^{#}} F_{B}(B ∧_{A} B, M) ≅ F_{A}(B, M) .

Here the map h^{#} makes sense because h is a map of B-modules, and similarly for ν^{#}. Algebraically, m∧g lifts over ν^{#} to the map f ↦ f(g)·m in F_{B}(F (G_{+}, B), M), which h^{#} takes to j_{M}(m ∧ g). □

Lemma 6.1.3. Let A → B be a G-Galois extension. For each B-module M the canonical map

ν′: M ∧_{A} B^{hG} → (M ∧_{A} B)^{hG}

is a weak equivalence.

Proof. The weak equivalence M ∧_{A} A ≅ M → F (G_{+}, M)^{hG} factors as the composite

M ∧_{A} A ⟶_{≃}^{1∧i} M ∧_{A} B^{hG} ⟶^{ν′} (M ∧_{A} B)^{hG} ⟶_{≃}^{h_{M}^{hG}} F (G_{+}, M)^{hG}

where i and h_{M} are weak equivalences by hypothesis and the previous lemma, respectively. The G-equivariance of h_{M}, needed to make sense of h_{M}^{hG}, follows like that of h. □

6.2. Dualizability

For each G-Galois extension R → T of commutative rings, T is a finitely generated projective R-module. The following is the analogous statement for Elocal commutative S-algebras.

Proposition 6.2.1. Let A → B be a G-Galois extension. Then B is a dualizable A-module.

Proof. We must show that the canonical map ν: D_{A}B ∧_{A} B → F_{A}(B, B) is a weak equivalence. To keep the different B’s apart, we observe more generally that for each B-module M there is a commutative diagram

\xymatrix{M ∧_{A} F_{A}(B, A) \ar[d]_{1∧i_{#}} \ar[r]^{ν} & F_{A}(B, M ∧_{A} A) \ar[d]^{(1∧i)_{#}} \\ M ∧_{A} F_{A}(B, B^{hG}) \ar[d]_{≅} \ar[r]^{ν} & F_{A}(B, M ∧_{A} B^{hG}) \ar[d]^{ν′_{#}} \\ M ∧_{A} F_{A}(B, B)^{hG} \ar[r]^{ν} & F_{A}(B, M ∧_{A} B)^{hG} \\ M ∧_{A} (B ∧ G_{+})^{hG} \ar[u]^{1∧j^{hG}} \ar[r]^{ν} & (M ∧_{A} B ∧ G_{+})^{hG} \ar[u]_{j_{M∧_{A}B}^{hG}} \\ M ∧_{A} (B ∧ G_{+} ∧ S^{adG})_{hG} \ar[u]^{1∧N} \ar[r]^{≅} & (M ∧_{A} B ∧ G_{+} ∧ S^{adG})_{hG} \ar[u]_{N}}

where ν′_{#} is a weak equivalence by Lemma 6.1.3, the maps induced by i: A → B^{hG} are weak equivalences by hypothesis, the maps involving j are well-defined by the G-equivariance of j (and j_{M∧_{A}B}), and are weak equivalences by Lemma 6.1.2, and finally the norm maps N from (3.6.1) are weak equivalences because the spectra with G-action in question have the form W ∧ G_{+}, with G acting freely on itself [Rog08, 5.2.5]. Thus all maps in this diagram are weak equivalences.

The special case when M = B then verifies that B is dualizable over A. □

In the global case E = S it follows from Propositions 3.3.3 and 6.2.1 that in any G-Galois extension A → B, B is a semi-finite A-module, i.e., it is weakly equivalent to a retract of a finite cell A-module. For example, by Proposition 5.3.1 the complexification map KO → KU is a global quadratic extension, and indeed, KU ≃ KO ∧ C_{η} is a finite 2-cell KO-module. However, in the localized cases the following counterexample shows that dualizability is probably the best one can hope for.

Example 6.2.2. Let p = 2, recall that L_{K(1)}S = J_{2}^{∧}, and consider the K(1)- local quadratic Galois extension c: J_{2}^{∧} → JU_{2}^{∧} from 5.5.2. We claim that JU_{2}^{∧} is not a semi-finite J_{2}^{∧}-module, even if it is a dualizable J_{2}^{∧}-module, in the K(1)-local category. There is a diagram of horizontal and vertical fiber sequences:

\xymatrix{J_{2}^{∧} \ar[r] \ar[d]_{c} & KO_{2}^{∧} \ar[r]^{ψ^{3}−1} \ar[d]^{c} & KO_{2}^{∧} \ar[d]^{c} \\ JU_{2}^{∧} \ar[r] \ar[d] & KU_{2}^{∧} \ar[r]^{ψ^{3}−1} \ar[d]^{∂} & KU_{2}^{∧} \ar[d]^{∂} \\ Σ^{2}X_{3} \ar[r] & Σ^{2}KO_{2}^{∧} \ar[r]^{Σ^{2}(3^{−1}ψ^{3}−1)} & Σ^{2}KO_{2}^{∧}}

The factor 3^{−1} in the lower row comes from the appearance of the inverse of the Bott equivalence β: Σ^{2}KU → KU in the connecting map ∂, and the relation ψ^{k}β = kβψ^{k}. By definition, following [HMS94, 2.6], but using real K-theory, X_{3} is the homotopy fiber of 3^{−1}ψ^{3} − 1: KO_{2}^{∧} → KO_{2}^{∧}.

We can compute the zero-th E_{1} = KU_{2}^{∧}-cohomology of the spectra in the upper left hand square, as modules over the group 𝕊_{1} = ℤ_{2}^{∗} of stable Adams operations, with k ∈ ℤ_{2}^{∗} acting by ψ^{k}. First, E_{1}^{0}(KU_{2}^{∧}) ≅ ℤ_{2}[[ℤ_{2}^{∗}]] (see also Example 8.1.4), and the remaining modules are the following quotients:

\xymatrix{ℤ_{2} & ℤ_{2}[[ℤ_{2}^{∗}/⟨−1⟩]] \ar[l] \\ ℤ_{2}[[ℤ_{2}^{∗}/⟨3⟩]] \ar[u]^{c^{∗}} & ℤ_{2}[[ℤ_{2}^{∗}]] \ar[l] \ar[u]}

Here ⟨3⟩ ⊂ ℤ_{2}^{∗} is the subgroup topologically generated by 3. The map c^{∗} takes E_{1}^{0}(JU_{2}^{∧}) ≅ ℤ_{2}[[ℤ_{2}^{∗}/⟨3⟩]] ≅ ℤ_{2}{1, ψ^{−1}} to E_{1}^{0}(J_{2}^{∧}) ≅ ℤ_{2}{1} by mapping both 1 and ψ^{−1} to the generator. Thus E_{1}^{0}(Σ^{2}X_{3}) = ker(c^{∗}) ≅ ℤ_{2}{1 − ψ^{−1}} is such that ψ^{3} acts as the identity, but ψ^{−1} acts by reversing the sign.

We claim that there is no semi-finite spectrum with this Morava module, i.e., this E_{1}-cohomology as an 𝕊_{1}-module. For each finite cell spectrum X the Atiyah– Hirzebruch spectral sequence

E_{2}^{s,t} = H^{s}(X; π_{−t}(E_{1})) ⟹ E_{1}^{s+t}(X)

is strongly convergent. After rationalization (inverting 2) it collapses at the E_{2}- term, yielding the Chern character isomorphism

ch: E_{1}^{0}(X)[2^{−1}] ≅ ⨁_{i∈ℤ} H^{2i}(X; ℚ_{2})

in degree zero. Here the i-th summand appears as the eigenspace of weight i, where ψ^{k} acts by multiplication by k^{i} for each k ∈ ℤ_{2}^{∗}. By naturality, there is also such an eigenspace decomposition of E_{1}^{0}(X)[2^{−1}] for each semi-finite J_{2}^{∧}-module X. (For general spectra X, the Atiyah–Hirzebruch spectral sequence needs not converge.)

Now note that E_{1}^{0}(Σ^{2}X_{3})[2^{−1}] ≅ ℚ_{2} has ψ^{3} acting as the identity, and ψ^{−1} acting by sign, which means that it should lie both in the weight 0 eigenspace and in an eigenspace of odd weight. This contradicts the possibility that Σ^{2}X_{3} is semi-finite. It follows that also JU_{2}^{∧} cannot be K(1)-locally semi-finite.

Dualizable modules are preserved under base change, and are detected by faithful and dualizable base change.

Lemma 6.2.3. Let A → B be a map of commutative S-algebras and M a dualizable A-module. Then B ∧_{A} M is a dualizable B-module.

Proof. We must verify that the canonical map

ν: F_{B}(B ∧_{A} M, B) ∧_{B} (B ∧_{A} M) → F_{B}(B ∧_{A} M, B ∧_{A} M)

is a weak equivalence. It factors as the composite

F_{B}(B ∧_{A} M, B) ∧_{B} (B ∧_{A} M) ≅ F_{A}(M, B) ∧_{A} M

⟶^{ν} F_{A}(M, B ∧_{A} M) ≅ F_{B}(B ∧_{A} M, B ∧_{A} M) ,

where the middle map is a weak equivalence by Lemma 3.3.2(a), since M is a dualizable A-module. □

Lemma 6.2.4. Let A → B be a faithful map of commutative S-algebras, with B dualizable over A, and let M be an A-module such that B ∧_{A} M is a dualizable B-module. Then M is a dualizable A-module.

Proof. We must verify that ν: F_{A}(M, A)∧_{A}M → F_{A}(M, M) is a weak equivalence. It suffices to show that the map 1 ∧ ν in the commutative square below is a weak equivalence, since B is assumed to be faithful over A.

\xymatrix{B ∧_{A} F_{A}(M, A) ∧_{A} M \ar[r]^{1∧ν} \ar[d]_{ν∧1} & B ∧_{A} F_{A}(M, M) \ar[d]^{ν} \\ F_{A}(M, B) ∧_{A} M \ar[r] & F_{A}(M, B ∧_{A} M)}

Here the lower horizontal map is isomorphic to

ν: F_{B}(B ∧_{A} M, B) ∧_{B} (B ∧_{A} M) → F_{B}(B ∧_{A} M, B ∧_{A} M) ,

which is a weak equivalence because B ∧_{A} M is assumed to be dualizable over B. The vertical maps are weak equivalences because B is dualizable over A, in view of Lemma 3.3.2(a). Therefore the upper horizontal map 1 ∧ ν is also a weak equivalence. □

Corollary 6.2.5. If A is a commutative S-algebra and G is a stably dualizable group, so S[G] is dualizable over S, then A[G] is dualizable over A.

Conversely, if A is a faithful commutative S-algebra, with A dualizable over S, and G is a topological group such that A[G] is dualizable over A, then G is stably dualizable.

The following lemma gives the same conclusion as Lemma 6.1.3, but under different hypotheses, and will be often used.

Lemma 6.2.6. Let A → B be a map of commutative S-algebras, let G be a topological group acting on B through A-algebra maps, and let M be a dualizable A-module. Then the canonical map

ν′: M ∧_{A} B^{hG} → (M ∧_{A} B)^{hG}

is a weak equivalence.

Proof. In the commutative diagram

\xymatrix{M ∧_{A} B^{hG} \ar[r]^{ρ∧1} \ar[d]_{ν′} & D_{A}D_{A}M ∧_{A} B^{hG} \ar[r]^{ν} \ar[d] & F_{A}(D_{A}M, B^{hG}) \ar[d]^{≅} \\ (M ∧_{A} B)^{hG} \ar[r]^{(ρ∧1)^{hG}} & (D_{A}D_{A}M ∧_{A} B)^{hG} \ar[r]^{ν^{hG}} & F_{A}(D_{A}M, B)^{hG}}

the horizontal maps derived from ν and ρ are weak equivalences because M is dualizable over A, and the right hand vertical map is an isomorphism. Thus the left hand vertical map ν′ is a weak equivalence. □

6.3. Alternate characterizations

The following alternate characterization of Galois extensions corresponds to the Auslander–Goldman definition. Compare Proposition 2.3.2. Implicit cofibrancy and localization at some S-module E is to be understood.

Proposition 6.3.1. Let A → B be a map of commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps. Then A → B is a G-Galois extension if and only if both i: A → B^{hG} and j: B⟨G⟩ → F_{A}(B, B) are weak equivalences and B is a dualizable A-module.

Proof. Lemma 6.1.2(b) and Proposition 6.2.1 establish one implication. For the converse, suppose that i and j are weak equivalences and that B is dualizable over A. We must show that h: B∧_{A}B → F (G_{+}, B) is a weak equivalence. Again, to keep the B’s apart we shall observe that for each B-module M the map h_{M} factors in the stable homotopy category as the following chain of weak equivalences:

M ∧_{A} B ⟶^{1∧ρ} M ∧_{A} D_{A}D_{A}B ⟶^{ν} F_{A}(D_{A}B, M) ≅ F_{B}(D_{A}B ∧_{A} B, M)

⟵^{ν^{#}} F_{B}(F_{A}(B, B), M) ⟶^{j^{#}} F_{B}(B⟨G⟩, M) ≅ F (G_{+}, M) .

Algebraically, the forward image of m ∧ b lifts over ν^{#} to f ↦ f(b) · m, which maps by j^{#} to h_{M}(m ∧ b) = {g ↦ g(b) · m}. The hypotheses that B is dualizable over A and j is a weak equivalence thus imply that h_{M} is a weak equivalence. The special case M = B lets us conclude that A → B is G-Galois. □

In the presence of faithfulness we have a third characterization of Galois extensions. See also Propositions 8.2.8 and 12.1.8.

Proposition 6.3.2. Let A → B be a map of commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps. Then A → B is a faithful G-Galois extension if and only if h: B ∧_{A} B → F (G_{+}, B) is a weak equivalence and B is faithful and dualizable as an A-module.

Proof. Proposition 6.2.1 provides one implication. For the converse, suppose that h is a weak equivalence and that B is dualizable and faithful over A. We must show that i: A → B^{hG} is a weak equivalence, and by faithfulness it suffices to show that 1 ∧ i: B ≅ B ∧_{A} A → B ∧_{A} B^{hG} is a weak equivalence. In the stable homotopy category we can identify this map with the chain of weak equivalences

B ⟶^{≃} F (G_{+}, B)^{hG} ⟵^{h^{hG}} (B ∧_{A} B)^{hG} ⟵^{ν′} B ∧_{A} B^{hG} .

Here ν′ is a weak equivalence by Lemma 6.2.6, because B is dualizable over A. We are viewing h as a G-equivariant map with respect to the left G-actions specified in Section 4.1. □

Here is a characterization of faithfulness in terms of the norm map.

Proposition 6.3.3. A G-Galois extension A → B is faithful if and only if the norm map N: (B ∧ S^{adG})_{hG} → B^{hG} is a weak equivalence, or equivalently, if the Tate construction B^{tG} is contractible.

Proof. If the norm map is a weak equivalence, and Z is an A-module so that Z ∧_{A} B ≃ ∗, then Z ≃ Z ∧_{A} B^{hG} ≃ Z ∧_{A} (B ∧ S^{adG})_{hG} ≅ (Z ∧_{A} B ∧ S^{adG})_{hG} ≃ ∗. Thus A → B is faithful.

For the converse, consider B ∧_{A} (−) applied to the norm map, appearing as the left hand vertical map in the following commutative diagram.

\xymatrix{B ∧_{A} (B ∧ S^{adG})_{hG} \ar[r]^{≅} \ar[d]_{1∧N} & (B ∧_{A} B ∧ S^{adG})_{hG} \ar[r]^{(h∧1)_{hG}} \ar[d]^{N} & (F (G_{+}, B) ∧ S^{adG})_{hG} \ar[d]^{N} \\ B ∧_{A} B^{hG} \ar[r]^{ν′} & (B ∧_{A} B)^{hG} \ar[r]^{h^{hG}} & F (G_{+}, B)^{hG}}

The map ν′ is a weak equivalence because B is dualizable over A, by Lemma 6.2.6. The upper and lower right hand horizontal maps are weak equivalences since h is G-equivariant and a weak equivalence.

The right hand vertical map is the norm map for the spectrum with G-action F (G_{+}, B). In the source,

(F (G_{+}, B) ∧ S^{adG})_{hG} ≃ (B ∧ DG_{+} ∧ S^{adG})_{hG} ≃ (B ∧ S[G])_{hG} ≃ B

by the stable dualizability of G and the Poincaré duality equivalence (3.5.2). In the target, F (G_{+}, B)^{hG} ≃ B. A direct inspection (inducing up from the case B = S, where it suffices to check on π_{0}) verifies that these identifications are compatible under the norm map. Therefore the right hand vertical map N is a weak equivalence, and so the norm map for B must be a weak equivalence, assuming that B is faithful over A.

The second equivalence is obvious from the definition of B^{tG} as the homotopy cofiber of the norm map. □

Corollary 6.3.4. Any finite G-Galois extension A → B is faithful if the order |G| of G is invertible in π_{0}(B).

Proof. Under these hypotheses π_{∗}(B_{hG}) ≅ π_{∗}(B)/G, π_{∗}(B^{hG}) ≅ π_{∗}(B)^{G} and the composite

π_{∗}(B) → π_{∗}(B)/G ⟶^{N_{∗}} π_{∗}(B)^{G} → π_{∗}(B)

is multiplication by |G|, so the norm map N must induce an isomorphism in homotopy. □

The same conclusion, under different hypotheses (allowing ramification) appears in Lemma 6.4.3.

6.4. The trace map and self-duality

In this section we work principally in the derived category, i.e., in the stable homotopy category D_{A,E}.

Let A → B be a map of E-local commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps. Suppose that i: A → B^{hG} is a weak equivalence.

Definition 6.4.1. The trace map tr: B ∧ S^{adG} → A in D_{A,E} is defined by the natural chain of maps

B ∧ S^{adG} ⟶^{in} (B ∧ S^{adG})_{hG} ⟶^{N} B^{hG} ⟵_{≃}^{i} A ,

where in denotes the inclusion induced by G ⊂ EG, and the wrong-way map i is a weak equivalence.

When G is finite, the dualizing spectrum S^{adG} = S can of course be ignored.

Lemma 6.4.2. The trace map tr: B ∧ S^{adG} → A equals the composite map

B ∧ S^{adG} = B ∧ S[G]^{hG} ⟶_{≃}^{ν} (B ∧ S[G])^{hG} ⟶^{(α′)^{hG}} B^{hG} ⟵_{≃}^{i} A

where α′: B ∧ S[G] → B is the right action derived from α: G_{+} ∧ B → B by way of the group inverse.

Proof. The canonical map ν: B ∧ S^{adG} → (B ∧ S[G])^{hG} can be identified with the chain of weak equivalences

B ∧ S^{adG} ⟶^{≃} F (G_{+}, B ∧ S^{adG})^{hG} ⟵_{≃}^{ν^{hG}} (B ∧ DG_{+} ∧ S^{adG})^{hG} ⟶^{≃} (B ∧ S[G])^{hG} ,

using that G is stably dualizable and the (right G-equivariant) Poincaré duality equivalence (3.5.2). In particular, ν itself is a weak equivalence.

The claim is then clear from the commutative diagram

\xymatrix{B ∧ S^{adG} \ar[r]^{ν}_{≃} \ar[d]_{in} & (B ∧ S[G])^{hG} \ar[r]^{=} \ar[d]_{in} & (B ∧ S[G])^{hG} \ar[d]_{(in)^{hG}} \ar[dr]^{(α′)^{hG}} \\ (B ∧ S^{adG})_{hG} \ar[r]^{ν_{hG}}_{≃} & ((B ∧ S[G])^{hG})_{hG} \ar[r]^{κ} & ((B ∧ S[G])_{hG})^{hG} \ar[r]_{≃} & B^{hG}}

where κ is the canonical hocolim/holim exchange map and the bottom row defines the norm map N, as in [Rog08, 5.2.2]. The right hand triangle uses that the homotopy orbits (B ∧ S[G])_{hG} are formed with respect to the diagonal left Gaction, so the identification with B extends the right action map α′. Algebraically, b ∧ g in B ∧ S[G] is identified with g^{−1}b ∧ e in the homotopy orbits, which maps to α′(b ∧ g) = g^{−1}b in B. □

Lemma 6.4.3. When G is finite the composite B ⟶^{tr} A → B is homotopic to the sum over all g ∈ G of the group action maps g: B → B, and the composite A → B ⟶^{tr} A is homotopic to the map multiplying by the order |G| of G.

Thus, if |G| is invertible in π_{0}(A) then tr is a split surjective map of A-modules, up to homotopy, and B is a faithful A-module. In particular, every G-Galois extension A → B with |G| invertible in π_{0}(A) is faithful.

Proof. When G is finite, the composite B ⟶^{tr} A → B can be expressed by continuing the factorization in Lemma 6.4.2 with the map B^{hG} → B that forgets homotopy invariance, and therefore factors as

B ⟶^{1∧Δ} B ∧ S[G] ⟶^{α′} B ,

where Δ: S → S[G] ≃ ∏_{G} S is the diagonal map. Clearly this is the sum over the elements g ∈ G of the group action maps g: B → B, up to homotopy.

On the other hand, the composite A → B ⟶^{tr} A is the map of G-homotopy fixed points induced by the same composite displayed above. Since the action of each group element is homotopic to the identity when restricted to the homotopy fixed points, their sum equals multiplication by the group order |G|, up to homotopy. □

Example 6.4.4. In the ℤ/2-Galois extension c: KO → KU the trace map tr is homotopic to the realification map r: KU → KO, as a KO-module map, and therefore also as an S-module map. For c^{#} : D_{KO}(KU, KO) → D_{KO}(KO, KO) is injective, and both tr ∘ c and r ∘ c are homotopic to the multiplication by 2 map KO → KO, by Lemma 6.4.3.

To justify the claim just made, that c^{#} is injective, we use the equivalence KU ≃ KO ∧ C_{η} and adjunction to identify c^{#} with i^{#} in the exact sequence

π_{1}(KO) ⟶^{η^{#}} π_{2}(KO) ⟶^{j^{#}} [C_{η}, KO] ⟶^{i^{#}} π_{0}(KO)

induced by the cofiber sequence S^{0} ⟶^{i} C_{η} ⟶^{j} S^{2} ⟶^{η} S^{1}. Here i^{#} is injective because η^{#} is well-known to be surjective.

In particular, the trace map tr = r: KU → KO is not split surjective up to homotopy (it is not even surjective on homotopy groups), so the analog of the algebraic Proposition 2.3.4(b) does not hold in topology.

Recall from Section 3.6 the shearing equivalence ζ: B ∧ S[G] → B ∧ S[G] that takes the left action on S[G] to the diagonal left action on B and S[G].

Definition 6.4.5. The trace pairing B ∧_{A} B ∧ S^{adG} → A in D_{A,E} is defined as the composite

B ∧_{A} B ∧ S^{adG} ⟶^{μ∧1} B ∧ S^{adG} ⟶^{tr} A .

The discriminant map 𝔡_{B/A}: B ∧S^{adG} → D_{A}B in D_{A,E} is defined as the composite

B ∧ S^{adG} = B ∧ S[G]^{hG} ⟶_{≃}^{ν} (B ∧ S[G])^{hG} ⟶_{≃}^{ζ^{hG}} (B ∧ S[G])^{hG}

⟶^{j^{hG}} F_{A}(B, B)^{hG} ≅ F_{A}(B, B^{hG}) ⟵_{≃}^{i_{#}} F_{A}(B, A) = D_{A}B .

Here j is G-equivariant with respect to the left G-action from Section 6.1.

We define Pic_{E} = Pic_{E}(S) in Definition 6.5.1 below to be the group of weak equivalence classes of E-locally smash invertible spectra. The dualizing spectrum S^{adG} is one such [Rog08, 3.3.4]. By the Pic_{E}-graded homotopy groups π_{∗}(Y) of a spectrum Y we mean the collection of groups π_{X}(Y) = [X, Y], where X ranges through Pic_{E}. See [HSt99, 14.1]. This includes the ordinary stable homotopy groups as the cases X = S^{n}, n ∈ ℤ, as well as the possibly exceptional case X = S^{adG}.

Lemma 6.4.6. The trace pairing B ∧_{A} B ∧ S^{adG} → A is left adjoint to the discriminant map 𝔡_{B/A}: B ∧ S^{adG} → D_{A}B. Thus 𝔡_{B/A} is in fact a map in D_{B,E}, and represents a Pic_{E}-graded class in π_{∗}D_{A}(B).

Proof. The first claim is a chase of definitions. The multiplications by B in the two copies of B in the source of the trace pairing get equalized by μ, so the adjoint (weak) map 𝔡_{B/A} commutes with the obvious B-module actions on B∧S^{adG} and D_{A}B. □

Proposition 6.4.7. If A → B is a G-Galois extension, then the discriminant map 𝔡_{B/A}: B ∧ S^{adG} → D_{A}B is a weak equivalence. In particular, B is self-dual as an A-module, up to an invertible shift by S^{adG}.

Proof. When A → B is G-Galois, j: B ∧ G_{+} → F_{A}(B, B) is a weak equivalence by Lemma 6.1.2(b), so the discriminant map is defined as a composite of weak equivalences. □

In general, we think of the discriminant map 𝔡_{B/A} as a measure of the extent to which A → B is ramified. When it is an equivalence, we think of the trace pairing as a perfect pairing.

6.5. Smash invertible modules

The K(n)-local Picard group Pic_{n} = Pic_{K(n)}(S) was introduced in [HMS94]. Here is a slight generalization.

Definition 6.5.1. Let A be a commutative S-algebra, and work locally with respect to the fixed spectrum E. An A-module M is smash invertible if there exists an A-module N such that N ∧_{A} M ≃ A as (implicitly E-local) A-modules.

Let Pic_{E}(A) be the class of weak equivalence classes of E-locally smash invertible A-modules. When Pic_{E}(A) is a set we call it the E-local Picard group of A, with the group structure induced by the (implicitly E-local) smash product of A-modules.

The following proof of the analog of Proposition 2.3.4(c) is close to one found by Andy Baker and Birgit Richter in the case of a finite abelian group G.

Proposition 6.5.2. Let A → B be a faithful abelian G-Galois extension, i.e., one with G an (E-locally stably dualizable) abelian group. Then B is smash invertible as an A[G]-module.

Proof. We consider B as an A[G]-module by way of the given left G-action. The smash inverse of B over A[G] will be its functional dual

D_{A[G]}(B) = F_{A[G]}(B, A[G])

in the category M_{A[G],E}. There is a natural counit map

ϵ: F_{A[G]}(B, A[G]) ∧_{A[G]} B → A[G] ,

that is left adjoint to the identity map on F_{A[G]}(B, A[G]) in the category of A[G]- modules. In symbols, ϵ: f ∧x ↦ f(x). The claim is that ϵ is a weak equivalence. By assumption B is faithful over A, so it suffices to verify that ϵ becomes an equivalence after inducing up along A → B. We factor the resulting map 1 ∧ ϵ as

B ∧_{A} F_{A[G]}(B, A[G]) ∧_{A[G]} B ⟶^{ν′} F_{A[G]}(B, B[G]) ∧_{A[G]} B

≅ F_{B[G]}(B ∧_{A} B, B[G]) ∧_{B[G]} (B ∧_{A} B) ⟶^{ϵ_{1}} B[G] .

Here ν′ is a weak equivalence because B is dualizable over A (cf. Lemma 6.2.6), the middle isomorphism is a composite of two standard adjunctions, and ϵ_{1} is a counit of the same sort as ϵ, now in the category of B[G]-modules. We have left to prove that ϵ_{1} is a weak equivalence.

There is a chain of left B[G]-module maps

(6.5.3) (B ∧_{A} B) ∧ S^{adG} ⟶^{h∧1} F (G_{+}, B) ∧ S^{adG}

⟵^{ν∧1} B ∧ DG_{+} ∧ S^{adG} ⟶^{≃} B[G] ⟶^{χ} B[G] ,

each of which is a weak equivalence. Here h is a weak equivalence because A → B is G-Galois, ν is a weak equivalence because G is stably dualizable, and the unnamed weak equivalence is the identity on B smashed with the Poincaré duality equivalence from (3.5.2). The latter is left G-equivariant with respect to the inverse of the right G-action mentioned in Section 3.5, i.e., with respect to the left action on DG_{+} given by right multiplication in the source, the trivial action on S^{adG}, and the inverse of the standard right action on B[G]. The map χ is induced by the group inverse in G, and takes the inverse of the standard right action on B[G] to the standard left action on B[G].

(When G is finite, the chain simplifies to

B ∧_{A} B ⟶^{h} F (G_{+}, B) ⟵^{κ} B[G] ⟶^{χ} B[G] ,

where κ is the usual inclusion and weak equivalence B[G] ≅ ⋁_{G} B → ∏_{G} B = F (G_{+}, B). Again, the right hand B[G] has the standard left B[G]-module structure.)

By [Rog08, 3.3.4, 3.2.3] the dualizing spectrum S^{adG} is smash invertible (in the E-local stable homotopy category), with smash inverse its functional dual S^{−adG} = (DG_{+})_{hG}. It follows that the counit map ϵ_{1} for the B[G]-module B ∧_{A} B is the composite of a weak equivalence and the counit map ϵ_{2} for (B ∧_{A} B) ∧ S^{adG}. Furthermore, it follows by naturality with respect to the chain (6.5.3) of B[G]- module weak equivalences that the counit map ϵ_{2} is related by a chain of weak equivalences to the counit map

ϵ_{3} : F_{B[G]}(B[G], B[G]) ∧_{B[G]} B[G] → B[G] ,

for B[G] considered as a left B[G]-module in the standard way. The latter map ϵ_{3} is obviously an isomorphism. □

So each (implicitly E-local) abelian G-Galois extension A → B exhibits B as a possibly interesting element in the Picard group Pic_{E}(A[G]).

The following converse to Proposition 6.5.2 does not require that G is abelian, but for abelian G it follows that the smash invertibility of B over A[G] is equivalent to B being faithful over A.

Lemma 6.5.4. Let A → B be a (not necessarily abelian) G-Galois extension. If B is smash invertible as an A[G]-module, i.e., if there exists an A[G]-module C and a weak equivalence B ∧_{A[G]} C ≃ A[G] of A-modules, then B is faithful over A.

Proof. If N ∧_{A} B ≃ ∗ then N[G] ≅ N ∧_{A} A[G] ≃ N ∧_{A} B ∧_{A[G]} C ≃ ∗, and N is a retract of N[G], so N ≃ ∗. □

CHAPTER 7 Galois theory I

We continue to work locally with respect to some S-module E.

7.1. Base change for Galois extensions

Faithful G-Galois extensions A → C are preserved by base change along arbitrary maps A → B,

\xymatrix{C \ar[r] & B ∧_{A} C \\ A \ar[u] \ar[r] & B \ar[u]}

and all Galois extensions are preserved by dualizable base change. Conversely, (faithful) Galois extensions are detected by faithful and dualizable base change. We do not know whether these dualizability hypotheses are necessary.

Lemma 7.1.1. Let A → B be a map of commutative S-algebras and A → C a faithful G-Galois extension. Then B → B ∧_{A} C is a faithful G-Galois extension.

Proof. The action by G on C through A-algebra maps extends uniquely to an action on B ∧_{A} C through B-algebra maps, taking g: C → C to 1∧g: B ∧_{A} C → B ∧_{A} C on the point set level, for g ∈ G. The group G remains stably dualizable, irrespective of whether it is being regarded as acting on C or B ∧_{A} C.

We show that B → B ∧_{A} C is a faithful G-Galois extension by appealing to Proposition 6.3.2. We know that C is a dualizable A-module by Proposition 6.2.1, and it is faithful by hypothesis. Therefore B ∧_{A} C is a dualizable and faithful Bmodule by the base change lemmas 6.2.3 and 4.3.3. It remains to verify that the canonical map h: (B ∧_{A} C) ∧_{B} (B ∧_{A} C) → F (G_{+}, B ∧_{A} C) is a weak equivalence. It is the lower horizontal map in the commutative square

(7.1.2)  \xymatrix{B ∧_{A} C ∧_{A} C \ar[r]^{1∧h} \ar[d]_{≅} & B ∧_{A} F (G_{+}, C) \ar[d]^{ν} \\ (B ∧_{A} C) ∧_{B} (B ∧_{A} C) \ar[r]^{h} & F (G_{+}, B ∧_{A} C) ,}

where the upper horizontal map 1 ∧ h is a weak equivalence because A → C is G-Galois, and the right hand vertical map ν is a weak equivalence because G is stably dualizable. This verifies the hypotheses of Proposition 6.3.2, so B → B ∧_{A} C is a faithful G-Galois extension. □

Lemma 7.1.3. Let A → B be a map of commutative S-algebras, with B dualizable over A, and let A → C be a G-Galois extension. Then B → B ∧_{A} C is a G-Galois extension.

Proof. The group G is stably dualizable, acts on B ∧_{A} C through B-algebra maps, and makes the canonical map h: (B ∧_{A} C) ∧_{B} (B ∧_{A} C) → F (G_{+}, B ∧_{A} C) a weak equivalence, just as in the previous proof. In order to verify the conditions in Definition 4.1.3 of a G-Galois extension, it remains to show that the canonical map i: B → (B ∧_{A} C)^{hG} is a weak equivalence. But B ≅ B ∧_{A} A ≃ B ∧_{A} C^{hG}, so we can identify i with ν′: B ∧_{A} C^{hG} → (B ∧_{A} C)^{hG}, which is a weak equivalence by Lemma 6.2.6 because B is dualizable over A. □

Lemma 7.1.4. Let A → B and A → C be maps of commutative S-algebras, with B a faithful and dualizable A-module, and let G be a stably dualizable group acting on C through A-algebra maps.

(a) If B → B ∧_{A} C is a G-Galois extension, then A → C is a G-Galois extension.

(b) If B → B ∧_{A} C is a faithful G-Galois extension, then A → C is a faithful G-Galois extension.

Proof. We must verify that the two maps i: A → C^{hG} and h: C ∧_{A} C → F (G_{+}, C) are weak equivalences. For the first map we factor the weak equivalence i: B → (B ∧_{A} C)^{hG} for the G-Galois extension B ≅ B ∧_{A} A → B ∧_{A} C as the composite

B ∧_{A} A ⟶^{1∧i} B ∧_{A} C^{hG} ⟶^{ν′} (B ∧_{A} C)^{hG} .

Here the right hand map ν′ is a weak equivalence because B is dualizable over A, by Lemma 6.2.6. Therefore the left hand map 1 ∧ i is a weak equivalence, and so i: A → C^{hG} is a weak equivalence because B is faithful over A.

For the second map we use the commutative square (7.1.2) again. The right hand vertical map ν is a weak equivalence because G is stably dualizable, and the lower horizontal map h is a weak equivalence because B → B ∧_{A} C is assumed to be G-Galois. So the upper horizontal map 1 ∧ h is a weak equivalence, and so h: C ∧_{A} C → F (G_{+}, C) is a weak equivalence because B is faithful over A.

Finally, if B → B ∧_{A} C is faithful, then we know that A → C is faithful by Lemma 4.3.4. □

7.2. Fixed S-algebras

Let G be a stably dualizable group and let A → B be a G-Galois extension. We consider the sub-extensions that occur as the homotopy fixed points C = B^{hK}, for suitable subgroups K of G.

Definition 7.2.1. Let K ⊂ G be a topological subgroup. We say that K is an allowable subgroup if (a) K is stably dualizable, (b) the collapse map c: G×_{K} EK → G/K induces a stable equivalence

S[G ×_{K} EK] ⟶^{≃} S[G/K] ,

and (c) as a continuous map of spaces, the projection π: G → G/K admits a section up to homotopy.

We consider two allowable subgroups K and K′ to be equivalent if K ⊂ K′ and S[K] → S[K′] is a weak equivalence, or more generally, if K and K′ are related by a chain of such (elementary) equivalences. We say that K is an allowable normal subgroup if, furthermore, K is a normal subgroup of G.

7.2. FIXED S-ALGEBRAS 53

It follows immediately from (c) above that the orbit space G/K is stably dualizable, since S[G/K] is a retract up to homotopy of S[G], and that there is a homotopy equivalence G ≃ K × G/K compatible with the obvious projections π and pr_{2} to G/K. If K is an allowable normal subgroup then G/K is a stably dualizable group.

Example 7.2.2. When G is discrete the allowable subgroups of G are just the subgroups of G in the usual sense, for then G is a disjoint union of free K-orbits, so c: G ×_{K} EK → G/K is already a weak equivalence, and there is no difficulty in finding a continuous section to π: G → G/K.

For A → B a G-Galois extension and K ⊂ G an allowable subgroup, we can form the following maps of commutative A-algebras

F (EG_{+}, B)^{G} → F (EG_{+}, B)^{K} → F (EG_{+}, B) .

In view of the natural weak equivalences A → F (EG_{+}, B)^{G} and F (EG_{+}, B) → B, we will keep the notation simple by writing the maps above as

A → B^{hK} → B .

So to be precise, we interpret B as F (EG_{+}, B), which then admits a K-action through B^{hK}-algebra maps. Likewise, if K is normal in G then B^{hK} admits a G/K-action through B^{hG}-algebra maps, which in turn are A-algebra maps. An implicit cofibrant replacement is also necessary at this stage.

Here is the forward part of the Galois correspondence for E-local commutative S-algebras.

Theorem 7.2.3. Let A → B be a faithful G-Galois extension and K ⊂ G any allowable subgroup. Then C = B^{hK} → B is a faithful K-Galois extension.

If furthermore K ⊂ G is an allowable normal subgroup, then A → C = B^{hK} is a faithful G/K-Galois extension.

Proof. We shall detect that C → B (resp. A → C) is faithfully Galois by applying Lemma 7.1.4 to the case of faithful and dualizable base change along C → B ∧_{A} C (resp. A → B). Here B is faithful and dualizable as an A-module by hypothesis and Proposition 6.2.1, so B∧_{A}C is faithful and dualizable as a C-module by Lemma 4.3.3 and Lemma 6.2.3. In the commutative diagram

(7.2.4)  \xymatrix{B \ar[r] & B ∧_{A} B \ar[r]^{h}_{≃} & F (G_{+}, B) & F (G_{+}, B) \ar[l]^{=} \\ C \ar[u] \ar[r] & B ∧_{A} C \ar[u] \ar[r]^{h′}_{≃} & F (G_{+}, B)^{hK} \ar[u] & F (G/K_{+}, B) \ar[l]_{c^{#}}^{≃} \ar[u]_{π^{#}} \\ A \ar[u] \ar[r] & B \ar[u] \ar[r]_{=} & B \ar[u] & B \ar[l]^{=} \ar[u]_{π^{#}}}

the left hand squares are base change pushouts in the category of commutative S-algebras.

The middle horizontal maps are weak equivalences. For h is a weak equivalence by the assumption that A → B is G-Galois. The map h′: B ∧_{A} C = B ∧_{A} B^{hK} → F (G_{+}, B)^{hK} factors as a composite weak equivalence

B ∧_{A} B^{hK} ⟶_{≃}^{ν′} (B ∧_{A} B)^{hK} ⟶_{≃}^{h^{hK}} F (G_{+}, B)^{hK}

using that B is dualizable over A (and Lemma 6.2.6) and that h is a weak equivalence. Here K acts from the left on B ∧_{A} B and F (G_{+}, B) by restriction of the actions by G, i.e., on the second copy of B in B ∧_{A} B and by right multiplication in the source in F (G_{+}, B), so in particular h is K-equivariant.

Likewise, the right hand horizontal maps are weak equivalences. For c^{#} is the composite map

F (G/K_{+}, B) ⟶_{≃}^{c^{#}} F ((G_{+})_{hK}, B) ≅ F (G_{+}, B)^{hK}

functionally dual to the collapse map c: (G_{+})_{hK} = (G ×_{K} EK)_{+} → G/K_{+}, which is a stable equivalence by part (b) of the hypothesis that K is allowable.

Therefore, the induced extension B ∧_{A} C → B ∧_{A} B is weakly equivalent to the map π^{#} : F (G/K_{+}, B) → F (G_{+}, B) functionally dual to the projection π: G → G/K. By part (c) of the hypothesis that K is allowable there is a weak equivalence

F (G_{+}, B) ≃ F ((K × G/K)_{+}, B) ≅ F (K_{+}, F (G/K_{+}, B)) ,

compatible with the commutative S-algebra maps π^{#} and pr_{2}^{#} from F (G/K_{+}, B), so that π^{#} is indeed weakly equivalent to the trivial K-Galois extension (Section 5.1) of F (G/K_{+}, B). In particular, B ∧_{A} C → B ∧_{A} B is faithfully K-Galois, and so by the faithful and dualizable detection result Lemma 7.1.4 it follows that C → B is faithfully K-Galois.

If furthermore K is normal in G, then the induced extension B → B ∧_{A} C is weakly equivalent to the map π^{#} : B → F (G/K_{+}, B) functionally dual to the collapse map π: G/K → {e}, i.e., to the trivial G/K-Galois extension of B. So B → B ∧_{A} C is faithfully G/K-Galois, and by Lemma 7.1.4 we can conclude that A → C is faithfully G/K-Galois. □

The following lemma will be applied in Section 9.1, when we discuss separable extensions.

Lemma 7.2.5. Let A → B be a faithful G-Galois extension and K ⊂ G an allowable subgroup. Then C = B^{hK} is faithful and dualizable over A, and the canonical map κ: B^{hK} ∧_{A} B^{hK} → (B ∧_{A} B)^{h(K×K)} is a weak equivalence.

Proof. It is formal that A → C is faithful when the composite A → C → B is faithful. For if N ∈ M_{A} has N ∧_{A} C ≃ ∗ then N ∧_{A} B ≅ N ∧_{A} C ∧_{C} B ≃ ∗, so N ≃ ∗.

The extension A → B is faithful with B dualizable over A by Proposition 6.2.1, and B ∧_{A} C ≃ F (G/K_{+}, B) as in (7.2.4) is dualizable over B, since S[G/K] is assumed to be a retract up to homotopy of S[G] and therefore is dualizable over S. Thus C is dualizable over A by Lemma 6.2.4.

The map κ factors as the composite of two weak equivalences

B^{hK} ∧_{A} B^{hK} ⟶_{≃}^{ν′} (B ∧_{A} B^{hK})^{hK} ⟶_{≃}^{(ν′)^{hK}} (B ∧_{A} B)^{h(K×K)}

derived from Lemma 6.2.6, where the first uses that C = B^{hK} (on the right hand side of the smash product) is dualizable over A, and the second uses that B (on the left hand side of the smash product) is dualizable over A. □

CHAPTER 8 Pro-Galois extensions and the Amitsur complex

We continue to let E be a fixed S-module and to work entirely in the E-local category.

8.1. Pro-Galois extensions

Definition 8.1.1. Let A be an E-local cofibrant commutative S-algebra, and consider a directed system of E-local finite G_{α}-Galois extensions A → B_{α}, such that B_{α} → B_{β} is a cofibration of commutative A-algebras for each α ≤ β. Suppose further that each A → B_{α} is an E-local sub-Galois extension of A → B_{β}, so/such that there is a preferred surjection G_{β} → G_{α} with kernel K_{αβ}, and a natural weak equivalence B_{α} ≃ B_{β}^{hK_{αβ}}. Let B = colim_{α} B_{α}, where the colimit is formed in C_{A,E}, and let G = lim_{α} G_{α}, with the (profinite) limit topology. Then, by definition, A → B is an E-local pro-G-Galois extension.

More generally, one might consider a directed system of (E-local) Galois extensions with stably dualizable (rather than finite) Galois groups G_{α}, arranging that each normal subgroup K_{αβ} is stably dualizable. We prefer to wait for some relevant examples before discussing the analog of the Krull topology on the resulting limit group G, but compatibility with the “natural topology” on E-local Hom-sets (see [HPS97, §4.4] and [HSt99, §11]) is certainly desirable.

For each α the weak equivalence h_{α}: B_{α} ∧_{A} B_{α} → F (G_{α+}, B_{α}) extends by Lemma 6.1.2(a) to a weak equivalence h_{α,B}: B ∧_{A} B_{α} → F (G_{α+}, B). The colimit of these over α is a weak equivalence

(8.1.2)  h: B ∧_{A} B → F ((G_{+}, B)) ,

where by definition F ((G_{+}, B)) = colim_{α} F (G_{α+}, B) is the “continuous” mapping spectrum with respect to the Krull topology, and colim_{α} B ∧_{A} B_{α} = B ∧_{A} colim_{α} B_{α} = B ∧_{A} B, since pushout with B commutes with colimits in the category of commutative A-algebras.

Likewise, for each α the weak equivalence j_{α}: B_{α}⟨G_{α}⟩ → F_{A}(B_{α}, B_{α}) extends by Lemma 6.1.2(c) to a weak equivalence j_{α,B}: B⟨G_{α}⟩ → F_{A}(B_{α}, B). The limit of these over α is a weak equivalence

(8.1.3)  j: B⟨⟨G⟩⟩ → F_{A}(B, B) ,

where by definition B⟨⟨G⟩⟩ = lim_{α} B⟨G_{α}⟩ is the “completed” twisted group Aalgebra, and lim_{α} F_{A}(B_{α}, B) ≅ F_{A}(colim_{α} B_{α}, B) = F_{A}(B, B).

Example 8.1.4. In the case of the K(n)-local pro-𝔾_{n}-Galois extension

L_{K(n)}S → E_{n}

these weak equivalences induce the isomorphism

Φ: E_{n}_{∗}^{∨}(E_{n}) ≅ Map(𝔾_{n}, π_{∗}(E_{n}))

that is implicit in [Mo85] and explicit in [St00, Thm. 12] and [Hov04, 4.11], and the isomorphism

Ψ: E_{n∗}⟨⟨𝔾_{n}⟩⟩ ≅ E_{n}^{∗}(E_{n})

from [St00, p. 1029] and [Hov04, 5.1]. The appearance of the continuous mapping space and the completed twisted group ring corresponds to the spectrum level colimits and limits above, combined with the I_{n}-adic completion at the level of homotopy groups induced by the implicit K(n)-localization [HSt99, 7.10(e)].

The pro-Galois formalism thus accounts for the first steps in a proof of Gross– Hopkins duality [HG94], following [St00]. The next step would be to study the K(n)-local functional dual of E_{n} as the continuous homotopy fixed point spectrum

L_{K(n)}DE_{n} = F (E_{n}, L_{K(n)}S) ≃ F (E_{n}, E_{n})^{h𝔾_{n}} ≃ (E_{n}⟨⟨𝔾_{n}⟩⟩)^{h𝔾_{n}} ,

but here technical issues related to the continuous cohomology of profinite groups arise, which are equivalent to those handled by Strickland.

8.2. The Amitsur complex

As usual, let A be a cofibrant commutative S-algebra and B a cofibrant commutative A-algebra.

Definition 8.2.1. The (additive) Amitsur complex [Am59, §5], [KO74, §II.2] is the cosimplicial commutative A-algebra

C^{•}(B/A): [q] ↦ B ⊗_{A} [q] = B ∧_{A} ⋯ ∧_{A} B

((q + 1) copies of B), coaugmented by A → B = C^{0}(B/A). Here B ⊗_{A} [q] refers to the tensored structure in C_{A,E}, and the cosimplicial structure is derived from the functoriality of this construction. In particular, the i-th coface map is induced by smashing with A → B after the i first copies of B, and the j-th codegeneracy map is induced by smashing with B ∧_{A} B → B after the j first copies of B.

Let the completion of A along B be the totalization A_{B}^{∧} = Tot C^{•}(B/A) of this cosimplicial resolution. Here, and everywhere, an implicit fibrant replacement is needed to make the totalization homotopy invariant. The coaugmentation induces a natural completion map η: A → A_{B}^{∧} of commutative A-algebras.

Gunnar Carlsson has considered this form of completion in his work on the descent problem for the algebraic K-theory of fields [Ca:d, §3], and Bendersky– Thompson have considered an unstable analog in [BT00]. It compares perfectly with Bousfield’s B-nilpotent completion [Bo79, §5], as extended from spectra to the context of A-modules.

Definition 8.2.2. The canonical B-based Adams resolution of A, formed in the category of A-modules, is the diagram below, inductively defined from D_{0} = A by letting D_{s+1} be the homotopy fiber of the natural map D_{s} = A∧_{A}D_{s} → B∧_{A}D_{s} for all s ≥ 0.

\xymatrix{A \ar[d] & D_{1} \ar[d] \ar[l] & D_{2} \ar[d] \ar[l] & … \ar[l] \\ B \ar@{-->}[ur] & B ∧_{A} D_{1} \ar@{-->}[ur] & B ∧_{A} D_{2} \ar@{-->}[ur]}

Continuing, K_{s} is defined to be the homotopy cofiber of the composite map D_{s} → A, and the B-nilpotent completion of A in A-modules is the homotopy limit L̂_{B}^{A}A = holim_{s} K_{s}.

Lemma 8.2.3. The completion A_{B}^{∧} of A along B is weakly equivalent to the Bousfield B-nilpotent completion L̂_{B}^{A}A of A formed in A-modules.

Proof. One proof uses Bousfield’s paper [Bo03] on cosimplicial resolutions. The functor Γ(M) = B ∧_{A} M defines a triple, or monad, on M_{A}, and A → C^{•}(B/A) is the corresponding triple resolution of A [Bo03, §7]. The B-module spectra define a class G of injective models in D_{A}, whose G-completion is the Bnilpotent completion in A-modules, by [Bo79, 5.8] and [Bo03, 5.7]. It agrees with the totalization of the triple resolution by [Bo03, 6.5], which by definition is the completion of A along B, in the sense above.

A more computational proof follows the unstable case of [BK73, §3–5], especially 5.3. There is an “iterated boundary isomorphism” from the E_{1}-term of the Bousfield–Kan spectral sequence associated to the Tot-tower of the cosimplicial spectrum C^{•}(B/A), to the E_{1}-term of the Adams spectral sequence associated to the tower of derived spectra {D_{s}}_{s}. The isomorphisms persist, with a shift in indexing, upon passage to the tower of cofibers {K_{s}}_{s}. Since Tot_{0} C^{•}(B/A) = B ≃ K_{1}, it follows that the homotopy limits A_{B}^{∧} and L̂_{B}^{A}A are also weakly equivalent. □

More generally, for each functor F from commutative A-algebras to a category of spaces or spectra, like the units functor U = GL_{1}, the Amitsur complex C^{•}(B/A; F) is the cosimplicial object [q] ↦ F (B ⊗_{A} [q]). It is natural to consider the colimit of its totalization, as B ranges over a class of A-algebras. When F is the identity functor, this is the completion defined above. When A → B is Galois, or ranges through all Galois extensions, we obtain forms of Amitsur cohomology [Am59] and Galois cohomology [CHR65, §5]. Note that if Spec B is thought of as a covering of Spec A, then Spec(B ∧_{A} B) consists of the covering of Spec A by double intersections, or fiber products, from the first covering, and likewise for Spec C^{q}(B/A) and (q + 1)-fold intersections. We are therefore recovering a form of Čech cohomology. In general, the appropriate context for what classes of extensions A → B to consider is that of a Grothendieck model topology on the category of commutative A-algebras, or a model site. We simply refer to [TV05] for a detailed exposition on this matter.

The following is a form of faithfully projective descent.

Lemma 8.2.4. If B is faithful and dualizable over A, then η: A → A_{B}^{∧} is a weak equivalence, i.e., A is complete along B.

Proof. It suffices to prove that 1∧η: B∧_{A}A → B∧_{A}A_{B}^{∧} is a weak equivalence. Here B ∧_{A} A_{B}^{∧} ≃ F_{A}(D_{A}B, Tot C^{•}(B/A)) ≅ Tot F_{A}(D_{A}B, C^{•}(B/A)) ≃ Tot B ∧_{A} C^{•}(B/A), and

B ∧_{A} C^{•}(B/A): [q] → B ∧_{A} (B ⊗_{A} [q]) ≅ B ⊗_{A} [q]_{+}

admits a cosimplicial contraction to B, so 1 ∧ η is indeed a weak equivalence. □

Let G be a topological group acting from the left on an S-module M, and let

EG_{•} = B(G, G, ∗): [q] ↦ Map([q], G) ≅ G^{q+1}

be the usual free contractible simplicial left G-space.

Definition 8.2.5. The (group) cobar complex for G acting on M is the cosimplicial S-module

C^{•}(G; M) = F (EG_{•+}, M)^{G}: [q] ↦ F (G_{+}^{q+1}, M)^{G} ≅ F (G_{+}^{q}, M) .

Its totalization is the homotopy fixed point spectrum M^{hG} = Tot C^{•}(G; M).

Here the standard identification F (G_{+}^{q+1}, M)^{G} ≅ F (G_{+}^{q}, M) takes the left Gmap f: G_{+}^{q+1} → M to the map ϕ: G_{+}^{q} → M that satisfies

f(g_{0}, …, g_{q}) = g_{0} · ϕ([g_{0}^{−1}g_{1}| … |g_{q−1}^{−1}g_{q}])
ϕ([h_{1}| … |h_{q}]) = f(e, h_{1}, …, h_{1} … h_{q})

(adapted as needed to make sense when the target is a spectrum).

In the presence of a left G-action on B through commutative A-algebra maps, these two cosimplicial constructions can be compared.

Definition 8.2.6. There is a natural map of cosimplicial commutative Aalgebras h^{•}: C^{•}(B/A) → C^{•}(G; B) given in codegree q by the map

h^{q}: B ∧_{A} ⋯ ∧_{A} B → F (G_{+}^{q+1}, B)^{G} ≅ F (G_{+}^{q}, B)

given symbolically by

b_{0} ∧ ⋯ ∧ b_{q} ↦ (f: (g_{0}, …, g_{q}) ↦ g_{0}(b_{0}) · . . . · g_{q}(b_{q}))
≅ (ϕ: [h_{1}| … |h_{q}] ↦ b_{0} · h_{1}(b_{1}) · . . . · (h_{1} … h_{q})(b_{q})) .

On totalizations, h^{•} induces a natural map of commutative A-algebras h′: A_{B}^{∧} → B^{hG}.

In codegree 1, we can recognize h^{1} : B ∧_{A} B → F (G_{+}, B) as the canonical map h from (4.1.2). It is not hard to give a formal definition of h^{q} as the right adjoint of a G-equivariant map B ⊗_{A} [q] ∧ Map([q], G)_{+} → B.

Lemma 8.2.7. Let G be a stably dualizable group acting on B through A-algebra maps, and suppose that h: B ∧_{A} B → F (G_{+}, B) is a weak equivalence. Then h^{•} is a codegreewise weak equivalence that induces a weak equivalence h′: A_{B}^{∧} → B^{hG}.

Proof. In each codegree q, the map h^{q} factors as a composite of weak equivalences of the form

B^{∧_{A}i} ∧_{A} F (G_{+}^{j}, B) ⟶^{≃} B^{∧_{A}(i−1)} ∧_{A} F (G_{+}^{j}, B ∧_{A} B)

⟶^{≃} B^{∧_{A}(i−1)} ∧_{A} F (G_{+}^{j}, F (G_{+}, B)) ≅ B^{∧_{A}(i−1)} ∧_{A} F (G_{+}^{(j+1)}, B)

with j = 0, …, q − 1 and i + j = q. Here the first map is a weak equivalence because G, and thus G^{j}, is stably dualizable, and the second map is a weak equivalence because h: B ∧_{A} B → F (G_{+}, B) is assumed to be one. The claim follows by induction. □

The following is close to Proposition 6.3.2. See also Proposition 12.1.8.

Proposition 8.2.8. Let G be a stably dualizable group acting on B through commutative A-algebra maps, and suppose that h: B ∧_{A} B → F (G_{+}, B) is a weak equivalence. Then A → B is G-Galois if and only if A is complete along B

Proof. We have i = h′ ∘ η, with h′ a weak equivalence, so i: A → B^{hG} is a weak equivalence if and only if η: A → A_{B}^{∧} is a weak equivalence. □

CHAPTER 9 Separable and étale extensions

We now address structured ring spectrum analogs of the unique lifting properties in covering spaces, continuing to work implicitly in some E-local category. Throughout, we let A be a cofibrant commutative S-algebra and B a cofibrant associative or cofibrant commutative A-algebra. (There appear to be interesting intermediate theories of E_{n} A-ring spectra for 1 ≤ n ≤ ∞, in the operadic sense, but we shall focus on the extreme cases of E_{1} = A_{∞} A-ring spectra, i.e., associative A-algebras, and E_{∞} A-ring spectra, i.e., commutative A-algebras.)

Our main observations are that G-Galois extensions A → B with G discrete are necessarily separable and dualizable, hence symmetrically étale (= thh-étale) and étale (= taq-étale). In most cases of current interest, including E = S and E = K(n) for 0 ≤ n ≤ ∞, a discrete group G is stably dualizable if and only if it is finite.

9.1. Separable extensions

The algebraic definition [KO74, p. 74] of a separable extension of commutative rings can be adapted to stable homotopy theory as follows.

Definition 9.1.1. We say that A → B is separable if the A-algebra multiplication map μ: B∧_{A}B^{op} → B, considered as a map in the stable homotopy category D_{B∧_{A}B^{op}} of B-bimodules relative to A, admits a section σ: B → B ∧_{A} B^{op}. Equivalently, there is a map σ: B′ → B ∧_{A} B^{op} of B-bimodules relative to A, such that the composite μσ: B′ → B is a weak equivalence.

Here B^{op} is B with the opposite A-algebra multiplication μγ: B ∧_{A} B ≅ B ∧_{A} B → B. It equals B precisely when B is commutative. Since B will rarely be cofibrant as a B-bimodule relative to A, it is only reasonable to ask for the existence of a bimodule section σ in the stable homotopy category. The condition for A → B to be separable only involves the bimodule structure on B, so it is quite accessible to verification by calculation. For example, it is equivalent to the condition that the algebra multiplication μ induces a surjection

μ_{#} : THH_{A}^{0}(B, B ∧_{A} B^{op}) → THH_{A}^{0}(B, B)

of zero-th topological Hochschild cohomology groups. See [La01, 9.3] for a spectral sequence computing the latter in many cases.

Lemma 9.1.2. Let A → B be a G-Galois extension, with G a discrete group. Then A → B is separable.

Proof. Let d: G_{+} → {e}_{+} be the continuous (Kronecker delta) map given by d(e) = e (the unit element in G) and d(g) = ∗ (the base point) for g ≠ e. Its functional dual

in_{e} = d^{#} : B ≅ F ({e}_{+}, B) → F (G_{+}, B)

and the canonical weak equivalence h define the required weak B-bimodule section σ = h^{−1} ∘ in_{e} to μ, as a morphism in the stable homotopy category.

(9.1.3)  \xymatrix{B \ar[dr]_{in_{e}} \ar@{-->}[r]^{σ} & B ∧_{A} B \ar[d]^{h}_{≃} \ar[r]^{μ} & B \\  & ∏_{G} B \ar[ur]_{pr_{e}}}

□

Proposition 9.1.4. Let A → B be a faithful G-Galois extension, with G a discrete group and K ⊂ G any subgroup. Then A → C = B^{hK} is separable.

Proof. By Example 7.2.2, any subgroup K of G is allowable. We are therefore in the situation of Lemma 7.2.5.

The map h: B ∧_{A} B → ∏_{G} B is (K ×K)-equivariant with respect to the action (k_{1}, k_{2})·(b_{1}∧b_{2}) = k_{1}(b_{1})∧k_{2}(b_{2}) in the source, and the action that takes a sequence {g ↦ ϕ(g)} to the sequence {g ↦ k_{1}(ϕ(k_{1}^{−1}gk_{2}))} in the target. There are maps

∏_{K} B ⟶^{in_{K}} ∏_{G} B ⟶^{pr_{K}} ∏_{K} B

functionally dual to a characteristic map d_{K}: G_{+} → K_{+} (taking G \ K to the base point) and the inclusion K_{+} ⊂ G_{+}, whose composite is the identity. We give ∏_{K} B the (K × K)-action that takes {k ↦ ϕ(k)} to {k ↦ k_{1}(ϕ(k_{1}^{−1}kk_{2}))}, so that in_{K} and pr_{K} are (K ×K)-equivariant. The weak equivalence B → (∏_{K} B)^{hK} induces a natural weak equivalence B^{hK} → (∏_{K} B)^{h(K×K)} that makes the following diagram commute:

\xymatrix{B^{hK} \ar[d]_{=} \ar@{-->}[r] & B^{hK} ∧_{A} B^{hK} \ar[d]^{κ}_{≃} \ar[r]^{μ} & B^{hK} \ar[d]^{=} \\ B^{hK} \ar[d]_{≃} \ar@{-->}[r] & (B ∧_{A} B)^{h(K×K)} \ar[d]^{h_{#}}_{≃} \ar[r] & B^{hK} \ar[d]^{≃} \\ (∏_{K} B)^{h(K×K)} \ar[r]^{in_{K#}} & (∏_{G} B)^{h(K×K)} \ar[r]^{pr_{K#}} & (∏_{K} B)^{h(K×K)}}

The vertical map κ is a weak equivalence by Lemma 7.2.5, and the maps h_{#} and pr_{K#} ∘in_{K#} are obtained from weak equivalences by passage to (K ×K)-homotopy fixed points, so a little diagram chase shows that μ: B^{hK} ∧_{A} B^{hK} → B^{hK} does indeed admit a weak bimodule section. □

Remark 9.1.5. It is easy to see that separable extensions are preserved by base change. To detect separable extensions by faithful base change will require some additional hypotheses, as in [KO74, III.2.2].

9.2. Symmetrically étale extensions

The topological Hochschild homology THH^{A}(B) of B relative to A is the geometric realization of a simplicial A-module

\xymatrix{B \ar[r] & B ∧_{A} B \ar[l] \ar[l] \ar[r] \ar[r] & B ∧_{A} B ∧_{A} B \ar[l] \ar[l] \ar[l] \ar[r] \ar[r] \ar[r] & … \ar[l] \ar[l] \ar[l] \ar[l]}

with the smash product of (q + 1) copies of B in degree q. See [EKMM97, IX.2]. Alternatively, THH^{A}(B) can be computed in the stable homotopy category as

Tor^{B∧_{A}B^{op}}(B, B) = B ∧_{B∧_{A}B^{op}}^{L} B .

In the case A = S, we will often write THH(B) for THH^{S}(B), which agrees with the topological Hochschild homology introduced by Marcel Bökstedt [BHM93]. The inclusion of 0-simplices defines a natural map ζ: B → THH^{A}(B). When B is commutative, THH^{A}(B) can be expressed in terms of the topologically tensored structure on C_{A} as B ⊗_{A} S^{1}.

It is also possible to define THH^{A}(B) for non-commutative A, by analogy with the definition of Hochschild homology over a non-commutative ground ring [Lo98, 1.2.11], but we have found no occasion to make use of this more general definition.

Definition 9.2.1. We say that A → B is formally symmetrically étale (= formally thh-étale) if the map ζ: B → THH^{A}(B) is a weak equivalence. If furthermore B is dualizable as an A-module, then we say that A → B is symmetrically étale (= thh-étale).

Remark 9.2.2. This definition of an (symmetrically) étale map does not quite conform to the algebraic case, in that it may be too restrictive to ask that B is dualizable as an A-module. Instead, it is likely to be more appropriate to only impose the dualizability condition locally with respect to some Zariski open cover of Spec A. This may be taken to mean that for some set of (smashing, Bousfield) localization functors {L_{E_{i}}}_{i}, such that the collection {A → L_{E_{i}}A}_{i} is a faithful cover in the sense of Definition 4.3.1, each localization L_{E_{i}}B is dualizable as an L_{E_{i}}A-module. The author is undecided about exactly which localization functors to allow. However, for Galois extensions the stronger (global) dualizability hypothesis will always be satisfied, and this may permit us to leave the issue open.

Example 9.2.3. Note that Definition 9.2.1 implicitly takes place in an E-local category. By McClure–Staffeldt [MS93, 5.1] at odd primes p, and Angeltveit– Rognes [AnR05, 8.10] at p = 2, the inclusion ζ: ℓ → THH(ℓ) is a K(1)-local equivalence, where ℓ = BP ⟨1⟩ is the p-local connective Adams summand of topological K-theory, so S → ℓ is K(1)-locally formally symmetrically étale. It also follows that the localization of this map, J_{p}^{∧} = L_{K(1)}S → L_{K(1)}ℓ = L_{p}^{∧} is K(1)- locally formally symmetrically étale. Here L_{p}^{∧} is the p-complete periodic Adams summand, as in 5.5.2.

These maps are not K(1)-locally symmetrically étale, because L_{p}^{∧} is not dualizable as a J_{p}^{∧}-module. More globally, S → L_{p}^{∧} fails to be E(1)-locally formally symmetrically étale. For by [MS93, 8.1], THH(L_{p}^{∧}) ≃ L_{p}^{∧} ∨ L_{0}(ΣL_{p}^{∧}), so ζ has a rationally non-trivial cofiber.

Similarly, ζ: ku → THH(ku) is a K(1)-homology equivalence by Christian Ausoni’s calculation [Au05, 6.5] for p odd, and [AnR, 8.10] again for p = 2, so the map S → ku to connective topological K-theory, and its K(1)-localization J_{p}^{∧} → KU_{p}^{∧}, are K(1)-locally formally symmetrically étale. The map L_{p}^{∧} → KU_{p}^{∧} is K(1)- locally 𝔽_{p}^{∗}-Galois, as noted in 5.5.2, so by Lemma 9.2.6 below L_{p}^{∧} → KU_{p}^{∧} is K(1)- locally symmetrically étale. In other words, ζ: ku → THH^{ℓ}(ku) and ζ: KU_{p}^{∧} → THH^{L_{p}^{∧}}(KU_{p}^{∧}) are K(1)-local equivalences.

The terminology “thh-étale” is that of Randy McCarthy and Vahagn Minasian [MM03, 3.2], except that for brevity they suppress the distinction between the formal and non-formal cases. The author’s lengthier term “symmetrically étale” was motivated by the following definitions and result.

Definition 9.2.4. Let M be a B-bimodule relative to A, i.e., a B ∧_{A} B^{op}- module. The space of associative A-algebra derivations of B with values in M is defined to be the derived mapping space

ADer_{A}(B, M) := (A_{A}/B)(B, B ∨ M)

in the topological model category of associative A-algebras over B, where pr_{1} : B ∨ M → B is the square-zero A-algebra extension of B with fiber M. We say that a B-bimodule relative to A is symmetric if it has the form μ^{!}N for some B-module N, i.e., if the bimodule action is obtained by composing with the A-algebra multiplication map μ: B ∧_{A} B^{op} → B.

Proposition 9.2.5. A → B is formally symmetrically étale if and only if the space of associative derivations ADer_{A}(B, M) is contractible for each symmetric B-bimodule M.

Proof. Let Ω_{B/A} be a cofibrant replacement of the homotopy fiber of μ: B∧_{A} B^{op} → B in the category of B-bimodules relative to A. There is a cofiber sequence

B ∧_{B∧_{A}B^{op}} Ω_{B/A} → B ⟶^{ζ} THH^{A}(B)

and for each B-module N, with associated symmetric B-bimodule M = μ^{!}N, there is an adjunction equivalence

M_{B∧_{A}B^{op}}(Ω_{B/A}, M) ≃ M_{B}(B ∧_{B∧_{A}B^{op}} Ω_{B/A}, N) .

Furthermore, there is an equivalence (for each B ∧_{A} B^{op}-module M)

ADer_{A}(B, M) = (A_{A}/B)(B, B ∨ M) ≃ M_{B∧_{A}B^{op}}(Ω_{B/A}, M)

obtained by Lazarev [La01, 2.2]. So ζ is an equivalence if and only if B ∧_{B∧_{A}B^{op}} Ω_{B/A} ≃ ∗, which is equivalent to ADer_{A}(B, M) ≃ M_{B}(B ∧_{B∧_{A}B^{op}} Ω_{B/A}, N) being contractible for each symmetric B-bimodule M = μ^{!}N.

In the E-local context, this argument shows that E_{∗}(ζ) is an isomorphism if and only if ADer_{A}(B, M) ≃ ∗ for each E-local symmetric B-module M. For A_{A,E}/B is a full subcategory of A_{A}/B, and likewise for the homotopy categories. □

Lemma 9.2.6. Each separable extension A → B of commutative S-algebras is formally symmetrically étale. In particular, each G-Galois extension A → B with G discrete is symmetrically étale.

Proof. By assumption there is a bimodule section σ so that the composite B ⟶^{σ} B ∧_{A} B^{op} ⟶^{μ} B is homotopic to the identity. Smashing with B over B ∧_{A} B^{op} tells us that the composite

THH^{A}(B) ⟶^{σ∧1} B ⟶^{ζ} THH^{A}(B)

is an equivalence. Furthermore, there is a retraction ρ: THH^{A}(B) → B given in simplicial degree q by the iterated multiplication map μ^{(q)} : B ∧_{A} ⋯ ∧_{A} B → B, since we are assuming that B is commutative. Therefore ζ admits a right and a left inverse, up to homotopy, and is therefore a weak equivalence.

When A → B is G-Galois with G discrete, we showed in Lemma 9.1.2 that A → B is separable and in Proposition 6.2.1 that B is a dualizable A-module. The above argument then implies that A → B is symmetrically étale. □

9.3. Smashing maps

Maps A → B having the corresponding property to the conclusion of Proposition 9.2.5 for associative derivations into arbitrary (not necessarily symmetric) B-bimodules relative to A, also have a familiar characterization. This material is not needed for our Galois theory, but nicely illustrates the relation of smashing localizations (and Zariski open sub-objects) to étale and symmetrically étale maps.

Definition 9.3.1. We say that A → B is smashing if the algebra multiplication map μ: B ∧_{A} B^{op} → B is a weak equivalence.

In view of the following proposition, smashing maps could also be called formally associatively étale extensions.

Proposition 9.3.2. A → B is smashing if and only if ADer_{A}(B, M) is contractible for each B-bimodule M relative to A.

Proof. This is immediate from the equivalence

ADer_{A}(B, M) ≃ M_{B∧_{A}B^{op}}(Ω_{B/A}, M)

from [La01], since A → B is smashing if and only if Ω_{B/A} ≃ ∗. □

The terminology is explained by the following result, one part of which the author learned from Mark Hovey.

Proposition 9.3.3. A → B is smashing if and only if LM = B ∧_{A} M defines a smashing Bousfield localization functor on M_{A}, in which case B = LA. In particular, B will be a commutative A-algebra.

Proof. Let B_{∗}^{A}(−) be the homotopy functor on M_{A} defined by B_{∗}^{A}(M) = π_{∗}(B ∧_{A} M). The natural map M → B ∧_{A} M is a B_{∗}^{A}-equivalence, since A → B is smashing, and B ∧_{A} M is B_{∗}^{A}-local by the prototypical ring spectrum argument of Adams [Ad71]: if B ∧_{A} Z ≃ ∗ then any map f: Z → B ∧_{A} M factors as

Z → B ∧_{A} Z ⟶^{1∧f} B ∧_{A} B ∧_{A} M ⟶^{μ∧1} B ∧_{A} M

and is therefore null-homotopic. So LM = B∧_{A}M defines a (Bousfield) localization functor L on M_{A}.

Conversely, a smashing localization functor L on M_{A} produces an associative A-algebra B = LA, by [EKMM97, VIII.2.1], such that LM ≃ B ∧_{A} M (since L is assumed to be smashing). The idempotency of L then ensures that the multiplication map B ∧_{A} B^{op} → B is a weak equivalence. □

Lemma 9.3.4. Each smashing map A → LA is separable, hence formally symmetrically étale.

Proof. If A → B = LA is smashing, then μ: B ∧_{A} B^{op} → B is an equivalence. It therefore admits a bimodule section σ up to homotopy, so A → B is separable.□

In general, LA is not dualizable as an A-module, as easy algebraic examples illustrate (ℤ ⊂ ℤ_{(p)}). Instead, the local dualizability of Remark 9.2.2 is more appropriate.

9.4. Étale extensions

We keep on working implicitly in an E-local category, now with B a cofibrant commutative A-algebra.

For a map A → B of commutative S-algebras, the topological André–Quillen homology TAQ(B/A) is defined in [Bas99, 4.1] as

TAQ(B/A) := (LQ_{B})(RI_{B})(B ∧_{A}^{L} B) ,

i.e., as the B-module of (left derived) indecomposables in the non-unital B-algebra given by the (right derived) augmentation ideal in the augmented B-algebra defined by the (left derived) smash product B ∧_{A}^{L} B, augmented over B by the A-algebra multiplication μ.

Definition 9.4.1. Let A → B be a map of commutative S-algebras. We say that A → B is formally étale (= formally taq-étale) if TAQ(B/A) is weakly equivalent to ∗. If furthermore B is dualizable as an A-module, then we say that A → B is étale (= taq-étale).

Like in Remark 9.2.2, the condition that B is dualizable over A is likely to be stronger than necessary for B to qualify as étale over A, and should eventually be replaced with a local condition over each subobject in an open cover of A. The apologetic discussion from the associative/symmetric case applies in the same way here.

The terminology is justified by the following definition and result from [Bas99]. The vanishing of TAQ(B/A) gives a unique infinitesimal lifting property, up to contractible choice, for geometric maps into the affine covering represented (in the opposite category) by a formally étale map A → B.

\xymatrix{B \ar[r]^{=} \ar@{-->}[dr] & B \\ A \ar[u] \ar[r] & B ∨ M \ar[u]}

Compare [Mil80, I.3.22].

Definition 9.4.2. Let A → B be a map of commutative S-algebras and let M be a B-module. The space of commutative A-algebra derivations of B with values in M is defined to be the derived mapping space

CDer_{A}(B, M) := (C_{A}/B)(B, B ∨ M)

in the topological model category of commutative A-algebras over B, where pr_{1} : B∨ M → B is the square-zero extension of B with fiber M.

Proposition 9.4.3. A map A → B of commutative S-algebras is formally étale if and only if CDer_{A}(B, M) is contractible for each B-module M.

Proof. There is an equivalence

CDer_{A}(B, M) = (C_{A}/B)(B, B ∨ M) ≃ M_{B}(TAQ(B/A), M)

for each B-module M, by [Bas99, 3.2]. By considering the universal example M = TAQ(B/A), we conclude that TAQ(B/A) ≃ ∗ if and only if CDer_{A}(B, M) ≃ ∗ for each B-module M. In the implicitly local context only E-local M occur, so we can conclude that TAQ(B/A) is E-acyclic, i.e., E-locally weakly equivalent to ∗.□

For a finite commutative R-algebra T, the two conditions T ≅ HH_{∗}^{R}(T) and D_{∗}(T/R) = AQ_{∗}(T/R) = 0 are logically equivalent [Gro67, 18.3.1(ii)], where HH_{∗}^{R} denotes Hochschild homology and D_{∗} = AQ_{∗} denotes André–Quillen homology. In the context of commutative S-algebras this is only true subject to a connectivity hypothesis [Min03, 2.8], due to a convergence issue in the analog of the Quillen spectral sequence from André–Quillen homology to Hochschild homology. However, one implication (from symmetrically étale to étale) does not depend on the connectivity hypothesis stated there. In other words, if ζ: B → THH^{A}(B) is a weak equivalence, then TAQ(B/A) ≃ ∗. We discuss a proof below, based on [BMa05].

There is a counterexample to the opposite implication, due to Mike Mandell, which is discussed in [MM03, 3.5]. For n ≥ 2 let X = K(ℤ/p, n) be an Eilenberg– Mac Lane space and let B = F (X_{+}, H𝔽_{p}) be its mod p cochain H𝔽_{p}-algebra, with π_{∗}(B) = H^{−∗}(K(ℤ/p, n); 𝔽_{p}). Then H𝔽_{p} → B is formally étale, but not symmetrically (=thh-)étale. So, any converse statement deducing that an étale map is symmetrically étale must contain additional hypotheses to exclude this example.

Lemma 9.4.4. Each (formally) symmetrically étale extension A → B of commutative S-algebras is (formally) étale. In particular, each G-Galois extension A → B with G discrete is étale, and each smashing localization A → LA = B is formally étale.

Proof. Recall that THH^{A}(B) ≃ B ⊗_{A} S^{1} as commutative A-algebras. Here ⊗_{A} denotes the tensored structure on C_{A} over unbased topological spaces. To describe the commutative B-algebra structure on THH^{A}(B) in similar terms, and to relate it to the B-module TAQ(B/A), we will need a tensored structure over based topological spaces. This makes sense when we replace C_{A} by the pointed category C_{B}/B of commutative B-algebras augmented over B. There is then a (reduced) tensor structure (−)˜⊗_{B}X on C_{B}/B over based topological spaces X, with

(C_{B}/B)(C ˜⊗_{B}X, C′) ≅ Map_{∗}(X, (C_{B}/B)(C, C′)) ,

where Map_{∗} denotes the base-point preserving mapping space. It follows that (C ˜⊗_{B}X)˜⊗_{B}Y ≅ C ˜⊗_{B}(X ∧ Y). The unbased and based tensored structures are related by C ˜⊗_{B}X ≅ B∧_{C} (C ⊗_{B} X) and C ⊗_{B} T ≅ C ˜⊗_{B}(T_{+}), for unbased spaces T.

There is a pointed model structure on C_{B}/B, and the associated Quillen suspension functor E is given on cofibrant objects by the reduced tensor E(C) = C ˜⊗_{B}S^{1} with the based circle. For each n ≥ 0 we can form the n-fold iterated suspension

E^{n}(C) = C ˜⊗_{B}S^{n}

in C_{B}/B, so that E(E^{n}(C)) ≅ E^{n+1}(C), and these objects assemble to a sequential suspension spectrum E^{∞}(C), in this category. By [BMa05, Thm. 3], the homotopy category of such spectra, up to stable equivalence, is equivalent to the homotopy category D_{B} of B-modules, up to weak equivalence.

Base change along A → B takes B to B ∧_{A} B, which is a cofibrant commutative B-algebra, augmented over B by the multiplication map μ: B∧_{A}B → B. Hereafter, write C = B ∧_{A} B for brevity. By [BMa05, Thm. 4], the cited equivalence takes E^{∞}(C) to the topological André–Quillen homology spectrum TAQ(B/A). So E^{∞}(C) is stably trivial if and only if TAQ(B/A) ≃ ∗, i.e., if and only if A → B is formally étale.

On the other hand,

E(C) = C ˜⊗_{B}S^{1} ≃ B ∧_{C} THH^{B}(C) ≅ THH^{A}(B) ,

now as commutative B-algebras. So E(C) is weakly trivial, i.e., weakly equivalent to the base point B in C_{B}/B, if and only if ζ: B → THH^{A}(B) is a weak equivalence.

The proof of the lemma is now straightforward. If A → B is formally symmetrically étale, then E(C) is weakly trivial, and therefore so is each of its suspensions E^{n}(C) = E^{n−1}(E(C)) for n ≥ 1. Thus the suspension spectrum E^{∞}(C) is stably trivial (in a very strong sense), and so TAQ(B/A) is weakly equivalent to the trivial B-module. □

In the notation of the above proof: C = B ∧_{A} B is weakly trivial in C_{B}/B if and only if A → B is smashing, E(C) = THH^{A}(B) is weakly trivial if and only if A → B is formally symmetrically étale, and E^{∞}(C) is stably trivial if and only if A → B is formally étale.

9.5. Henselian maps

By definition, an étale map A → B has the unique lifting property up to contractible choice for each square-zero extension of commutative A-algebras B ∨ M → B, and satisfies a finiteness condition. In this chapter we conversely ask which extensions D → C of commutative A-algebras are such that each étale map A → B, with B mapping to C, has this homotopy unique lifting property with respect to D → C.

\xymatrix{B \ar[r] \ar@{-->}[dr] & C \\ A \ar[u] \ar[r] & D \ar[u]}

We shall refer to such D → C as Henselian maps. Section 9.6 will exhibit some interesting examples of Henselian maps.

In the opposite category to that of commutative A-algebras, of affine algebrogeometric objects in a homotopy-theoretic sense [TV05, §5.1], we can view the square-zero extensions as infinitesimal thickenings of a special kind, forming a generating class of acyclic cofibrations. The étale extensions then correspond to smooth and unramified covering maps, and constitute a class of fibrations characterized by their right lifting property with respect to these generating acyclic cofibrations, together with a finiteness hypothesis. The Henselian maps, in turn characterized by their left lifting property with respect to these fibrations, then form a class of thickenings that contains all composites of the generating acyclic cofibrations of the theory, i.e., all infinitesimal thickenings, but which also encompasses many other maps. By comparison, in the algebraic context Hensel’s lemma applies to a complete local ring mapping to its residue field, but also to many other cases.

For a fixed commutative S-algebra A, this discussion could take place as above in the context of commutative A-algebras, with maps from (taq-)étale extensions A → B, but also in the alternate context of associative A-algebras, with maps from symmetrically (= thh-)étale extensions. To be concrete we shall focus on the commutative case, although all of the formal arguments carry over to the associative category and extensions by symmetric bimodules.

Throughout this section we continue to work E-locally, and let A be a cofibrant commutative S-algebra, B → C a map of commutative A-algebras and M any Cmodule. We sometimes consider M as a B-module by pull-back along B → C. We always make the cofibrant and fibrant replacements required for homotopy invariance, implicitly.

Lemma 9.5.1. The square-zero extension B ∨ M → B is the pull-back in C_{A} of the square-zero extension C ∨ M → C along B → C,

\xymatrix{B \ar[r]^{=} \ar@{-->}[dr] \ar@{-->}[drr] & B \ar[r] & C \\ A \ar[r] \ar[u] & B ∨ M \ar[r] \ar[u] & C ∨ M \ar[u]_{pr_{1}}}

so there is a weak equivalence

(C_{A}/B)(B, B ∨ M) ≃ (C_{A}/C)(B, C ∨ M) .

In particular, both of these spaces are contractible whenever A → B is formally étale.

Proof. The pullback along B → C of a fibrant replacement for C ∨M → C is a fibrant replacement for B∨M → B, and forming mapping spaces from a cofibrant replacement for B in C_{A} has a left adjoint given by the tensored structure, hence commutes with pullbacks and other limits. So the homotopy fiber at the identity of B of C_{A}(B, B ∨ M) → C_{A}(B, B) is weakly equivalent to the homotopy fiber at B → C of C_{A}(B, C ∨ M) → C_{A}(B, C). □

Lemma 9.5.2. The commutative diagram

\xymatrix{B \ar[rr] \ar@{-->}[drr] \ar@{-->}[dr] &  & C \\ A \ar[u] \ar[r] & C \ar[r]_{in_{1}} \ar[ur]^{=} & C ∨ M \ar[u]_{pr_{1}}}

yields a homotopy fiber sequence

(C_{A}/C ∨ M)(B, C) → (C_{A}/C)(B, C) → (C_{A}/C)(B, C ∨ M)

for which the middle space is contractible. In particular, all three spaces are contractible whenever A → B is formally étale.

Proof. After replacing first pr_{1} and then in_{1} by fibrations, the mapping spaces in C_{A} from a cofibrant replacement for B to these fibrations sit in two fibrations p and i, whose composite p ∘ i is also a fibration. The fibers of the i, p ∘ i and p above B → C then form the desired fiber sequence. □

The following definition is the commutative analog of that in [La01, 3.3].

Definition 9.5.3. A map π: D → C of commutative A-algebras is a singular extension if there is an A-linear derivation of C with values in M, i.e., a commutative A-algebra map d: C → C ∨ M over C, and a homotopy pull-back square

\xymatrix{C \ar[r]^{d} & C ∨ M \\ D \ar[r] \ar[u]^{π} & C \ar[u]_{in_{1}}}

of commutative A-algebras.

For example, the square-zero extension C∨Σ^{−1}M → C is the singular extension pulled back from the trivial derivation d = in_{1} : C → C ∨M. So the class of singular extensions contains the class of square-zero extensions.

Lemma 9.5.4. For each singular extension π: D → C the commutative diagram

\xymatrix{B \ar[r] \ar@{-->}[dr] \ar@{-->}[drr] & C \ar[r]^{d} & C ∨ M \\ A \ar[u] \ar[r] & D \ar[r] \ar[u]^{π} & C \ar[u]_{in_{1}}}

induces a weak equivalence

(C_{A}/C)(B, D) ≃ (C_{A}/C ∨ M)(B, C) .

In particular, both of these spaces are contractible whenever A → B is formally étale.

Proof. The first part of the proof is like that of Lemma 9.5.1. The second claim follows by using the definition of formally étale maps to deduce that both mapping spaces are contractible for all formally étale maps A → B, in the special case when π: D = C ∨ Σ^{−1}M → C is the square-zero extension pulled back from the trivial derivation d = in_{1} : C → C ∨ M. The right hand mapping space does not depend on the particular singular extension, so it follows from the first claim applied to a general singular extension π: D → C that also the left hand mapping space is contractible for arbitrary singular extensions D → C and formally étale maps A → B. □

In view of [Gro67, 18.5.5] or [Mil80, I.4.2(d)] we can make the following definition.

Definition 9.5.5. Let D → C be a map of commutative S-algebras. We say that D → C is Henselian if for each étale map A → B, with B and D commutative A-algebras over C,

\xymatrix{B \ar[r] \ar@{-->}[dr] & C \\ A \ar[u] \ar[r] & D \ar[u]}

the derived mapping space

(C_{A}/C)(B, D) ≃ ∗

is contractible, i.e., if A → B has the unique lifting property up to contractible choice with respect to D → C.

If D is a commutative S-algebra and C is an associative D-algebra, we say that D → C is symmetrically (= thh-)Henselian if for each symmetrically (= thh- )étale map A → B, in a diagram as above, the associative A-algebra mapping space (A_{A}/C)(B, D) is contractible.

By the following lemma it suffices (in the commutative case) to verify the homotopy unique lifting property for the étale maps A → B with A = D. For A → B étale implies D → B ∧_{A} D étale by the base change formula TAQ(B ∧_{A} D/D) ≃ TAQ(B/A) ∧_{A} D [Bas99, 4.6] and Lemma 6.2.3.

Lemma 9.5.6. Let B → C and D → C be maps of commutative A-algebras, with pushout B ∧_{A} D → C. The commutative diagram

\xymatrix{B \ar[r] \ar@{-->}[drr] & B ∧_{A} D \ar[r] \ar@{-->}[dr] & C \\ A \ar[r] \ar[u] & D \ar[r]_{=} \ar[u] & D \ar[u]}

induces a weak equivalence

(C_{A}/C)(B, D) ≃ (C_{D}/C)(B ∧_{A} D, D) .

Proof. Dual to the proof of Lemma 9.5.1. □

Proposition 9.5.7. The class of Henselian maps D → C contains the squarezero extensions C ∨ M → C and the singular extensions π: D → C. It is closed under weak equivalences, compositions, retracts and filtered homotopy limits (for diagrams of maps to a fixed C).

Proof. The first claims follow from Lemma 9.5.4 and the remark that squarezero extensions are trivial examples of singular extensions. The closure claims are clear, perhaps except for the the last one. If α ↦ (D_{α} → C) is a diagram of Henselian maps to C, then let D = holim_{α} D_{α}. For each étale map A → B (mapping to D → C as above) there is a weak equivalence

(C_{A}/C)(B, D) ≃ holim_{α}(C_{A}/C)(B, D_{α}) ≃ ∗ ,

since each D_{α} → C is Henselian and the limit category is assumed to be filtering.□

In fact, the Henselian maps that we will encounter in the following section are sequential homotopy limits of towers of singular extensions, and thus of a rather special form. If desired, the reader can view them as the residue maps of complete local rings, and refer to them as formal thickenings, rather than as general Henselian maps.

9.6. I-adic towers

In this section we let R be a commutative S-algebra and R/I an R-ring spectrum, i.e., an R-module with homotopy unital and homotopy associative maps R → R/I and R/I ∧_{R} R/I → R/I. Define the R-module I by the cofiber sequence I → R → R/I, and let

I^{(s)} = I ∧_{R} ⋯ ∧_{R} I

(s copies of I) be its s-fold smash power over R, for each s ≥ 1. Define the Rmodule R/I^{(s)} by the cofiber sequence I^{(s)} → R → R/I^{(s)}. There is then a tower of R-modules

(9.6.1)  R → ⋯ → R/I^{(s)} → ⋯ → R/I

that Baker and Lazarev [BL01, §4] refer to as the external I-adic tower.

Angeltveit [An:t, Cor. 3.7] recently showed that when R is even graded, i.e., the homotopy ring π_{∗}(R) is concentrated in even degrees, and R/I is a regular quotient, i.e., π_{∗}(I) is an ideal in π_{∗}(R) that can be generated by a regular sequence, then each R-ring spectrum multiplication on R/I can be rigidified to an associative R-algebra structure.

Given that R/I is an associative R-algebra, Lazarev [La01, 7.1] proved earlier on that the whole I-adic tower can be given the structure of a tower of associative R-algebras, and that each cofiber sequence

I^{(s)}/I^{(s+1)} → R/I^{(s+1)} ⟶^{π} R/I^{(s)}

is a singular extension of associative R-algebras.

It remains an open problem to decide when the diagram (9.6.1) can be realized as a tower of commutative R-algebras, and whether each map R/I^{(s+1)} → R/I^{(s)} can be taken to be a singular extension in the commutative context. See [La04, 4.5] for a remark on a similar problem for square-zero extensions.

The homotopy limit

L̂_{R/I}^{R}R = holim_{s} R/I^{(s)}

of the I-adic tower is the Bousfield R/I-nilpotent completion of R, formed in the category of R-modules (or R-algebras), which we introduced in Definition 8.2.2. It is in general not the same as the Bousfield R/I-localization of R, formed in the category of R-modules, which we denote by L_{R/I}^{R}R.

However, Baker and Lazarev [BL01, 6.3] use an internal I-adic tower to prove that when R is even graded and R/I is a homotopy commutative regular quotient R-algebra, then the R/I-nilpotent completion has the expected homotopy ring

π_{∗}L̂_{R/I}^{R}R ≅ π_{∗}(R)_{π_{∗}(I)}^{∧} .

If the regular sequence generating π_{∗}(I) is finite, then they also show that the R/Ilocalization and the R/I-nilpotent completion of R, both formed in R-modules, do in fact agree

L_{R/I}^{R}R ≃ L̂_{R/I}^{R}R ,

9.6. I-ADIC TOWERS 73

but we shall most be interested in cases when the regular ideal π_{∗}(I) is not finitely generated.

Proposition 9.5.7 therefore has the following consequence, which admits some fairly obvious algebraically localized generalizations that we shall also make use of.

Proposition 9.6.2 (Baker–Lazarev). Let R be an even graded commutative S-algebra, and R/I a homotopy commutative regular quotient R-algebra. Then the limiting map

L̂_{R/I}^{R}R = holim_{s} R/I^{(s)} → R/I

is symmetrically (=thh-)Henselian, and induces the canonical surjection

π_{∗}(R)_{π_{∗}(I)}^{∧} → π_{∗}(R)/π_{∗}(I)

of homotopy rings. In particular, if π_{∗}(R) is already π_{∗}(I)-adically complete, so that R ≃ L̂_{R/I}^{R}R, then R → R/I is symmetrically Henselian. □

We now claim that the complex cobordism spectrum MU can be viewed as a global model, up to Henselian maps, of each of the commutative S-algebras { \atop E(n)} = L_{K(n)}E(n) that occur as fixed S-algebras in the p-primary K(n)-local pro-Galois extensions L_{K(n)}S → E_{n} → E_{n}^{nr}. So, even if there is ramification between the expected maximal unramified Galois extensions (covering spaces) over the different chromatic strata, reflected in the changing pro-Galois groups 𝔾_{n} and 𝔾_{n}^{nr} for varying n and p, these can all be compensated for by appropriate Henselian maps (formal thickenings), and unified into one global model, namely MU.

For the sphere spectrum S, the chromatic stratification we have in mind is first branched over the rational primes p, and then S_{(p)} is filtered by the Bousfield localizations L_{n}S = L_{E(n)}S for each n ≥ 0. The associated (Zariski) stack has the category M_{S,E(n)} of E(n)-local S-modules over the n-th open subobject in the filtration, and the n-th monochromatic category of E(n)-local E(n−1)-acyclic S-modules over the n-th half-open stratum. The latter category is equivalent to the category M_{S,K(n)} of K(n)-local S-modules, at least in the sense that their homotopy categories are equivalent [HSt99, 6.19].

The latter K(n)-local module category is in turn equivalent to the category of K(n)-local L_{K(n)}S-modules, and we propose to understand it better by way of Galois descent from the related categories M_{B,K(n)} of K(n)-local B-modules, for the various K(n)-local Galois extensions L_{K(n)}S → B. The limiting case of pro-Galois descent from K(n)-local modules over B = E_{n}^{nr}, or over the separable closure B = Ē_{n} (cf. Section 10.3), can optimistically be hoped to be particularly transparent.

This decomposition of the sphere spectrum, appearing in the lower row in the diagram below, can be paralleled for MU by applying the same localization functors in spectra. However, the proposition above indicates that it may be more appropriate to nilpotently complete MU, in the category of MU-modules. In other words, we are led to focus attention on the upper row, rather than the middle row, in the following commutative diagram.

(9.6.3)  \xymatrix{MU \ar[r] & L̂_{E(n)}^{MU}MU \ar[r] & L̂_{K(n)}^{MU}MU \\ MU \ar[r] \ar[u]^{=} & L_{E(n)}MU \ar[r] \ar[u] & L_{K(n)}MU \ar[u] \\ S \ar[r] \ar[u] & L_{E(n)}S \ar[r] \ar[u] & L_{K(n)}S \ar[u]}

In the middle column we have L_{E(n)}S ≃ L̂_{E(n)}S, since every E(n)-local spectrum is E(n)-nilpotent [HSa99, 5.3]. However, in the right hand column L_{K(n)}S ≄ L̂_{K(n)}S, since K(n)-localization is not smashing [Ra92, 8.2.4] and [HSt99, 8.1].

The coefficient rings of the various localizations and nilpotent completions of MU occurring in the diagram above are mostly understood. See [Ra92, 8.1.1] for π_{∗}L_{E(n)}MU (or rather, its BP -version). Let J_{n} ⊂ π_{∗}MU_{(p)} be the kernel of the ring homomorphism π_{∗}MU_{(p)} → π_{∗}E(n), i.e., the regular ideal generated by the kernel of π_{∗}MU_{(p)} → π_{∗}BP and the infinitely many classes v_{k} for k > n. Let I_{n} = (p, v_{1}, …, v_{n−1}), also considered as an ideal in π_{∗}MU_{(p)}, so that the sum of ideals I_{n} + J_{n} is the kernel of the ring homomorphism π_{∗}MU_{(p)} → π_{∗}K(n). Then

π_{∗}L_{K(n)}MU = π_{∗}MU_{(p)}[v_{n}^{−1}]_{I_{n}}^{∧}

by [HSt99, 7.10(e)]. By [HSa99, Thm. B], L_{K(n)}BP splits as the K(n)-

localization of an explicit countable wedge sum of suspensions of { \atop E(n)}. It follows that L_{K(n)}MU splits in a similar way.

By Proposition 9.6.2, applied to R = MU_{(p)}[v_{n}^{−1}] and R/I = E(n), we find that L̂_{E(n)}^{MU}MU ≃ L̂_{R/I}^{R}R → E(n) is symmetrically Henselian, with

π_{∗}L̂_{E(n)}^{MU}MU = π_{∗}MU_{(p)}[v_{n}^{−1}]_{J_{n}}^{∧} .

By the same proposition applied to R = MU_{(p)}[v_{n}^{−1}] and R/I = K(n), at least for p ≠ 2 to ensure that K(n) is homotopy commutative, we also find that L̂_{K(n)}^{MU}MU ≃ L̂_{R/I}^{R}R → K(n) is symmetrically Henselian, with

(9.6.4)  π_{∗}L̂_{K(n)}^{MU}MU = π_{∗}MU_{(p)}[v_{n}^{−1}]_{I_{n}+J_{n}}^{∧} .

This differs from the K(n)-localization of MU in S-modules by the additional completion along J_{n}.

This K(n)-nilpotently complete part, in MU-modules, of the global commutative S-algebra MU, can now be related by a symmetrically Henselian map to the extension L_{K(n)}S → { \atop E(n)}, which is closely related to the K(n)-locally pro-Galois extension L_{K(n)}S → E_{n}.

(9.6.5)  \xymatrix{E_{n} \ar[r]^{≃?} & L̂_{K(n)}^{MU}E_{n} \\ { \atop E(n)} \ar[r]^{≃?} \ar[u] & L̂_{K(n)}^{MU}E(n) \ar[u] \\ L_{K(n)}MU \ar[r] \ar[u] & L̂_{K(n)}^{MU}MU \ar[u] \ar@{-->}[ul]_{q} \\ L_{K(n)}S \ar[r]_{=} \ar[u] & L_{K(n)}S \ar[u]}

Here the horizontal map { \atop E(n)} = L_{K(n)}E(n) → L̂_{K(n)}^{MU}E(n), and its analog for E_{n}, are both plausibly weak equivalences. For instance, the corresponding map of nilpotent completions of MU induces completion along J_{n} at the level of homotopy groups, and π_{∗}{ \atop E(n)} and π_{∗}E_{n} are already J_{n}-adically complete in a trivial way.

We shall now apply Proposition 9.6.2 with R = L̂_{K(n)}^{MU}MU. Formula (9.6.4) exhibits R as an even graded commutative S-algebra. Considering J_{n} as an ideal in π_{∗}R, it is still generated by a regular sequence and (π_{∗}R)/J_{n} ≅ π_{∗}{ \atop E(n)}. So we can form R/I ≃ { \atop E(n)} as a homotopy commutative regular quotient R-algebra. Then π_{∗}(I) = J_{n}, and π_{∗}(R) is J_{n}-adically complete, so by the last clause of Proposition 9.6.2 the map q: R → R/I, labeled q in the diagram (9.6.5) above, is symmetrically Henselian.

Corollary 9.6.6. Each K(n)-local pro-Galois extension L_{K(n)}S → E_{n} factors as the composite map of commutative S-algebras

L_{K(n)}S → L̂_{K(n)}^{MU}MU ⟶^{q} { \atop E(n)} → E_{n} ,

where the first map admits the global model S → MU, the second map is symmetrically (= thh-)Henselian, and the third map is a K(n)-local pro-Galois extension.

In other words, each K(n)-local stratum of S is related by a chain of pro-Galois covers L_{K(n)}S → E_{n} ← { \atop E(n)} to a formal thickening q: L̂_{K(n)}^{MU}MU → { \atop E(n)} of the corresponding K(n)-nilpotently complete stratum of MU, formed in MU-modules.

We shall argue in Section 12.2 that there is a Hopf–Galois structure on this global model S → MU that also encapsulates all the known Galois symmetries over L_{K(n)}S, at least up to the adjunction of roots of unity, i.e., up to the passage

from { \atop E(n)} to E_{n} (or to E_{n}^{nr}). The question remains whether q is (commutatively) Henselian, which would follow if the diagram (9.6.1) could be realized by singular extensions of commutative S-algebras.

After this discussion of K(n)-localization and K(n)-nilpotent completion in MU-modules, we make some remarks on the chromatic filtration in MU-modules. The study of the chromatic filtration and the monochromatic category of S-modules relies on the basic fact [JY80, 0.1] that E(n)_{∗}(X) = 0 implies E(n−1)_{∗}(X) for S-modules X, so that there is a natural map L_{E(n)}X → L_{E(n−1)}X. The analogous claim in the context of MU-modules is false, i.e., that E(n)_{∗}^{MU}(X) = 0 implies E(n−1)_{∗}^{MU}(X) = 0, as the easy example X = MU_{(p)}/(v_{n}) illustrates. Thus there is no natural map L_{E(n)}^{MU}X → L_{E(n−1)}^{MU}X.

For brevity, let K[0, n] = K(0) ∨ ⋯ ∨ K(n). It is well-known that L_{K[0,n]} = L_{E(n)} in the category of S-modules [Ra84, 2.1(d)]. For any MU-module X it is obvious that K[0, n]_{∗}^{MU}(X) = 0 implies K[0, n−1]_{∗}^{MU}(X) = 0, so that there is a natural map L_{K[0,n]}^{MU}X → L_{K[0,n−1]}^{MU}X. Therefore the example above shows that the two localization functors L_{K[0,n]}^{MU} and L_{E(n)}^{MU} in MU-modules cannot be equivalent.

We therefore think that it will be more appropriate to filter the category of MU-modules by the essential images

M_{MU} ⊃ ⋯ ⊃ M_{MU,K[0,n]}^{MU} ⊃ M_{MU,K[0,n−1]}^{MU} ⊃ …

of the Bousfield localization functors L_{K[0,n]}^{MU}, i.e., the full subcategories of K[0, n]- local MU-modules, within MU-modules, or the corresponding essential images

M_{MU} ⊃ ⋯ ⊃ M̂_{MU,K[0,n]}^{MU} ⊃ M̂_{MU,K[0,n−1]}^{MU} ⊃ …

of the nilpotent completion functors L̂_{K[0,n]}^{MU}, i.e., the full subcategories of K[0, n]- nilpotently complete MU-modules, within MU-modules. Then we can consider the MU-chromatic towers

X → ⋯ → L_{K[0,n]}^{MU}X → L_{K[0,n−1]}^{MU}X → …

and

X → ⋯ → L̂_{K[0,n]}^{MU}X → L̂_{K[0,n−1]}^{MU}X → …

for each MU-module X. We then suspect that L_{K[0,n]}^{MU} is a smashing localization, and that there is an equivalence of homotopy categories between the n-th monochromatic category of MU-modules and the K(n)-local category of MU-modules, like that of [HSt99, 6.19], but we have not verified this expectation. To be precise, the monochromatic category in question has objects the MU-modules that are L_{K[0,n]}^{MU}- local and L_{K[0,n−1]}^{MU}-acyclic. The K(n)-local category has objects the MU-modules that are L_{K(n)}^{MU}-local.

The thrust of Corollary 9.6.6 is now that the chromatic filtration on S-modules is related to a chromatic filtration on MU-modules, by a chain of pro-Galois extensions and Henselian maps with geometric content. The chromatic filtration on MU-modules is likely to be much easier to understand algebraically, in terms of the theory of formal group laws. Taken together, these two points of view may clarify the chromatic filtration on S-modules.

CHAPTER 10 Mapping spaces of commutative S-algebras

We turn to the computation of the mapping space C_{A}(B, B) for a G-Galois extension A → B, and related mapping spaces of commutative S-algebras, using the Hopkins–Miller obstruction theory in the commutative form presented by Goerss and Hopkins [GH04]. For the more restricted problem of the classification of commutative S-algebra structures, the related obstruction theory of Alan Robinson [Rob03] is also relevant.

10.1. Obstruction theory

Let A be a cofibrant commutative S-algebra and let E be an S-module. We shall need an extension of the Goerss–Hopkins theory to the context of (simplicial algebras over simplicial operads in) the category M_{A,E} of E-local A-modules. The base change to A-modules is harmless, but in working E-locally we may loose the identification of the dualizable A-modules with the (homotopy retracts of) finite cell A-modules, recalled in Proposition 3.3.3 above. It seems clear that only the formal properties of dualizable modules are important to the Goerss–Hopkins theory, so that the whole extension can be carried through in full generality. However, for our specific purposes the only dualizable A-modules we must consider will in fact be finite cell A-modules, so we do not actually need to carry the generalization through.

Next, consider a fixed (cofibrant, E-local) commutative A-algebra B. The Goerss–Hopkins spectral sequence [GH04, Thm. 4.3 and Thm. 4.5] for the computation of the homotopy groups of commutative A-algebra mapping spaces like C_{A}(C, B), for various commutative A-algebras C, is based on working with a fixed homology theory given by a commutative A-algebra that they call E, but which we will take to be B. In particular, the target B in the mapping space is then equivalent to its completion along the given homology theory (cf. Definition 8.2.1), as required for the convergence of the spectral sequence.

This commutative A-algebra B is required to satisfy the so-called Adams conditions [Ad69, p. 28], [GH04, Def. 3.1], which in our notation asks that B is weakly equivalent to a homotopy colimit of finite cell A-module spectra B_{α}, satisfying two conditions. For our purposes it will suffice that B itself satisfies the two conditions, i.e., that there is only a trivial colimit system. The conditions are then:

Adams conditions 10.1.1. The commutative A-algebra B is weakly equivalent to a finite cell A-module, such that

(a) B_{∗}^{A}(D_{A}B) is finitely generated and projective as a B_{∗}-module.

(b) For each B-module M the Künneth map

[D_{A}B, M]_{∗}^{A} → Hom_{B_{∗}}(B_{∗}^{A}(D_{A}B), M_{∗})_{∗}

78 10. MAPPING SPACES OF COMMUTATIVE S-ALGEBRAS

is an isomorphism.

In the E-local situation we expect that it suffices to assume that B is a dualizable A-module, but in our applications the stronger finite cell hypothesis will always be satisfied.

Lemma 10.1.2. The Adams conditions (a) and (b) are satisfied when A → B is an E-local G-Galois extension, with G a finite discrete group.

Proof. From Lemma 6.1.2 we know that j: B⟨G⟩ → F_{A}(B, B) is a weak equivalence, and that h_{M}: B ∧_{A} M → F (G_{+}, M) is a weak equivalence for each Bmodule M. By Proposition 6.2.1, B is dualizable over A, so B∧_{A}D_{A}B ≃ F_{A}(B, B). So B_{∗}^{A}(D_{A}B) ≅ π_{∗}F_{A}(B, B) ≅ B_{∗}⟨G⟩ is a finitely generated free B_{∗}-module, and B_{∗}^{A}M ≅ [D_{A}B, M]_{∗}^{A} is isomorphic to

Hom_{B_{∗}}(B_{∗}^{A}(D_{A}B), M_{∗}) ≅ Hom_{B_{∗}}(B_{∗}⟨G⟩, M_{∗}) ≅ ∏_{G} M_{∗} ≅ π_{∗}F (G_{+}, M) .

A diagram chase verifies that the Künneth map equals the composite of this chain of isomorphisms. □

The more general situation, with G an indiscrete stably dualizable group, will lead to much more complicated spectral sequence calculations, which we will not try to address here.

Goerss and Hopkins proceed to consider an E_{2}- or resolution model structure on spectra, which is suitably generated by a class P of finite cellular spectra. This class is required to satisfy a list of conditions [GH04, Def. 3.2.(1)–(5)]. Following the proof of [BR07, 2.2.4], by Baker and Richter, we take P to be the smallest set of dualizable A-modules that contains A and B, and is closed under (de-)suspensions and finite wedge sums. This immediately takes care of conditions (3) and (4).

Lemma 10.1.3. The resolution model category conditions, listed in [GH04, Def. 3.2.(1)–(5)], are satisfied when A → B is a finite E-local G-Galois extension.

Proof. (1) B_{∗}^{A}(X) is a finite sum of shifted copies of B_{∗}^{A}(A) = B_{∗} and B_{∗}^{A}(B) ≅ ∏_{G} B_{∗}, for each A-module X ∈ P, hence is projective as a B_{∗}-module.

(2) D_{A}B is represented in P, since B is self-dual as an A-module by Proposition 6.4.7. (5) The Künneth map

[X, M]_{∗}^{A} → Hom_{B_{∗}}(B_{∗}^{A}(X), M_{∗})_{∗}

is an isomorphism for all B-module spectra M when X = D_{A}B, by the Adams condition (b), and trivially for X = A, so the same follows for all X ∈ P by passage to (de-)suspensions and finite wedge sums. □

To sum up, a finite Galois extension A → B satisfies the Adams conditions and has an associated resolution model structure on A-modules, as required by [GH04, §3], whenever B is weakly equivalent to a finite cell A-module. It seems likely that the cited theory also extends to cover all finite Galois extensions, by replacing all references to finite cell objects by dualizable objects. However, in the following applications we shall always make use of the identification

C_{A}(C, B) ≅ C_{B}(B ∧_{A} C, B)

and only apply the Goerss–Hopkins spectral sequence in the case of commutative B-algebra maps to B. This is the very special case of Lemmas 10.1.2 and 10.1.3 when A = B and G is the trivial group, in which case B is certainly a finite cell A-module. So we are only using the straightforward extension of [GH04] to a more general (cofibrant, commutative) ground S-algebra, namely B. Note also that B_{∗}^{B}(B ∧_{A} C) ≅ B_{∗}^{A}(C), so the two equivalent mapping spaces above will have the same associated spectral sequences, which we now review.

Goerss and Hopkins define André–Quillen cohomology groups D^{s} of algebras and modules over a simplicially resolved E_{∞}-operad [GH04, (4.1)], as non-abelian right derived functors of algebra derivations. They then construct a convergent spectral sequence of Bousfield–Kan type [GH04, Thm. 4.5], which in our notation appears as

(10.1.4)  E_{2}^{s,t} ⟹ π_{t−s}C_{A}(C, B)

(based at a given commutative A-algebra map C → B), with E_{2}-term

E_{2}^{0,0} = Alg_{B_{∗}}(B_{∗}^{A}(C), B_{∗})

and

E_{2}^{s,t} = D_{B_{∗}T}^{s} (B_{∗}^{A}(C), Ω^{t}B_{∗})

for t > 0. Here Ω^{t}B_{∗} is the t-th desuspension of the module B_{∗}. As usual for Bousfield–Kan spectral sequences, this spectral sequence is concentrated in the wedge-shaped region 0 ≤ s ≤ t.

The subscript B_{∗}T refers to a (Reedy cofibrant, etc.) simplicial E_{∞} operad T that resolves the commutative algebra operad in the sense of [GH04, Thm. 2.1], and B_{∗}T is the associated simplicial E_{∞} operad in the category of B_{∗}-modules. The Goerss–Hopkins André–Quillen cohomology groups D^{s} are the right derived functors of derivations of B_{∗}T -algebras in B_{∗}-modules, in the sense of Quillen’s homotopical algebra. As surveyed by Basterra and Richter [BasR04, 2.6], these groups D^{s} do not depend on the choice of resolving simplicial E_{∞} operad T, and agree with the André–Quillen cohomology groups AQ_{sE_{∞}}^{s} defined by Mandell in [Man03, 1.1] for E_{∞} simplicial B_{∗}-algebras. These do in turn agree with the André–Quillen cohomology groups AQ_{dgE_{∞}}^{s} defined by Mandell for E_{∞} differential graded B_{∗}-algebras [Man03, 1.8], and with Basterra’s topological André–Quillen cohomology groups TAQ^{s} of the Eilenberg–Mac Lane spectra associated to these algebras and modules [Man03, §7]. By the comparison result of Basterra and McCarthy [BMc02, 4.2], these are finally isomorphic to the Γ-cohomology groups HΓ^{s} of Robinson and Sarah Whitehouse [RoW02], when B_{∗}^{A}(C) is projective over B_{∗}, or more generally, when B_{∗}^{A}(C) is flat over B_{∗} and the universal coefficient spectral sequence from homology to cohomology collapses. So in these cases the Goerss–Hopkins groups can be rewritten as

D_{B_{∗}T}^{s} (B_{∗}^{A}(C), Ω^{t}B_{∗}) = HΓ^{s,−t}(B_{∗}^{A}(C)|B_{∗}; B_{∗}) .

It is not quite obvious from the above references that this chain of identifications preserves the internal t-grading of these cohomology groups, since this grading could be lost by the passage through Eilenberg–Mac Lane spectra. However, Birgit Richter has checked that both gradings are indeed respected, up to the sign indicated above. In our applications all of these cohomology groups will in fact be zero, so the finer point about the internal grading is not so important.

If B_{∗}^{A}(C) is an étale commutative B_{∗}-algebra (thus flat over B_{∗}), then by [RoW02, 6.8(3)] all Γ-homology and Γ-cohomology groups of B_{∗}^{A}(C) over B_{∗} are zero, so by the sequence of comparison results above (and the universal coefficient spectral sequence for TAQ), all the Goerss–Hopkins André–Quillen cohomology groups D_{B_{∗}T}^{s} (B_{∗}^{A}(C), Ω^{t}B_{∗}) vanish. Therefore one can conclude:

Corollary 10.1.5. Let C → B be a map of commutative A-algebras. If B_{∗}^{A}(C) is étale over B_{∗}, then the Goerss–Hopkins spectral sequence for

π_{∗}C_{A}(C, B) ≅ π_{∗}C_{B}(B ∧_{A} C, B)

collapses to the origin at the E_{2}-term, so C_{A}(C, B) is homotopy discrete (each path component is weakly contractible) with

π_{0}C_{A}(C, B) ≅ Alg_{B_{∗}}(B_{∗}^{A}(C), B_{∗}) .

10.2. Idempotents and connected S-algebras

The converse part of the Galois correspondence, begun in Theorem 7.2.3, should intrinsically characterize the intermediate extensions A → C → B that occur as K-fixed S-algebras C = B^{hK} by allowable subgroups K ⊂ G. Already in the algebraic case of a G-Galois extension R → T of discrete rings there are additional complications (compared to the field case) when T admits non-trivial idempotents, i.e., when the spectrum of T is not connected in the sense of algebraic geometry. See [Mag74] for a general treatment of these complications. We do not expect that these issues are so central to the extension of the theory from discrete rings to S-algebras, so we prefer to focus on the analog of the situation when T is connected.

We can identify the idempotents of a commutative ring T with the non-unital T -algebra endomorphisms T → T, taking an idempotent e (with e^{2} = e) to the homomorphism t ↦ et. The forgetful functor from T -algebras to non-unital T - algebras has a left adjoint, taking a non-unital T -algebra N to T ⊕ N, with the multiplication (t_{1}, n_{1}) · (t_{2}, n_{2}) = (t_{1}t_{2}, t_{1}n_{2} + n_{1}t_{2} + n_{1}n_{2}) and unit (1, 0). In particular, we can identify the set of idempotents E(T) = {e ∈ T | e^{2} = e} with the set of T -algebra maps

E(T) ≅ Alg_{T} (T ⊕ T, T) .

Here T ⊕ T ≅ T [x]/(x^{2} − x) is finitely generated and free as a T -module. It is étale as a commutative T -algebra by [Mil80, I.3.4], since (x^{2} − x)′ = 2x − 1 is its own multiplicative inverse in T [x]/(x^{2} − x).

This leads us to the following definitions.

Definition 10.2.1. Let B be a (cofibrant) commutative S-algebra. Let the space of idempotents

E(B) = N_{B}(B, B)

be the mapping space of non-unital commutative B-algebra [Bas99, §1] endomorphisms B → B. We say that B is connected if the map {0, 1} → E(B) taking 0 and 1 to the constant map and the identity map B → B, respectively, is a weak equivalence.

We shall not have need to do so, but if we wanted to express that the spectrum B has the property that π_{∗}(B) = 0 for all ∗ ≤ 0, we would say that B

10.2. IDEMPOTENTS AND CONNECTED S-ALGEBRAS 81

is 0-connected, reserving the term “connected” for the algebro-geometric interpretation just introduced. A spectrum B with π_{∗}(B) = 0 for ∗ < 0 will be called (−1)-connected or connective.

There is a homeomorphism

E(B) ≅ C_{B}(B ∨ B, B) ,

where B ∨B is defined as the split commutative S-algebra extension of B with fiber the underlying non-unital commutative S-algebra of B. Its unit B → B ∨ B is the inclusion on the first wedge summand, and its multiplication is the composite

(B ∨ B) ∧_{B} (B ∨ B) ≅ B ∨ (B ∨ B ∨ B) ⟶^{1∨∇} B ∨ B

where ∇ folds the last three wedge summands together.

Proposition 10.2.2. Let B be any commutative S-algebra. The space of idempotents E(B) is homotopy discrete, with π_{0}E(B) ≅ E(π_{0}(B)). In particular, the commutative S-algebra B is connected if and only if the commutative ring π_{0}(B) is connected.

Proof. We compute the homotopy groups of E(B) ≅ C_{B}(B ∨ B, B) by means of the Goerss–Hopkins spectral sequence (10.1.4), in the almost degenerate case when A = B and C = B ∨ B. Here A → B is of course a G-Galois extension, in the trivial case G = 1, so our discussion in Section 10.1 justifies the use of this spectral sequence. It specializes to

E_{2}^{s,t} ⟹ π_{t−s}E(B)

with

E_{2}^{0,0} = Alg_{B_{∗}}(B_{∗} ⊕ B_{∗}, B_{∗}) = E(B_{∗})

and

E_{2}^{s,t} = D_{B_{∗}T}^{s} (B_{∗} ⊕ B_{∗}, Ω^{t}B_{∗})

for t > 0. Here B_{∗} ⊕ B_{∗} = B_{∗}[x]/(x^{2} − x) is étale over B_{∗}, so all the André–Quillen cohomology groups D^{s} = HΓ^{s} vanish [RoW02, 6.8(3)], and we deduce that E(B) is homotopy discrete, with π_{0}E(B) ≅ E(B_{∗}) equal to the set of idempotents in the graded ring B_{∗}, which of course are the same as the idempotents in the ring π_{0}(B). In short, we have applied Corollary 10.1.5. □

The following argument, explained by Neil Strickland, shows that the above definition of connectedness for structured ring spectra is equivalent to another definition originally proposed by the author. We say that an S-algebra B is trivial if it is weakly contractible, i.e., if π_{∗}(B) = B_{∗} = 0, and non-trivial otherwise.

Lemma 10.2.3. A non-trivial commutative S-algebra B is either connected, or weakly equivalent to a product B_{1} × B_{2} of non-trivial commutative B-algebras, but not both.

Proof. If B is non-trivial and not connected then there exists an idempotent e ∈ π_{0}(B) different from 0 and 1. Let f_{1} and f_{2} : B → B be B-module maps inducing multiplication by e and 1 − e on π_{∗}(B), respectively. (These could also be taken to be non-unital commutative B-algebra maps by the previous proposition.) For i = 1, 2 let B[f_{i}^{−1}] be the mapping telescope for the iterated self-map f_{i}, and let

B_{i} = L_{B[f_{i}^{−1}]}^{B}B

be the Bousfield B[f_{i}^{−1}]-localization of B in the category of B-modules. Then there are commutative B-algebra maps B → B_{1} and B → B_{2} inducing isomorphisms eπ_{∗}(B) ≅ π_{∗}(B_{1}) and (1 − e)π_{∗}(B) ≅ π_{∗}(B_{2}), of nontrivial groups, and their product B → B_{1} × B_{2} is the asserted weak equivalence.

Conversely, if B ≃ B_{1} × B_{2} as commutative B-algebras (or even just as ring spectra), with B_{1} and B_{2} non-trivial, then π_{0}(B) is not connected as a commutative ring, so B is not connected as a commutative S-algebra. □

10.3. Separable closure

The following terminology presumes, in some sense, that each finite separable extension can be embedded in a finite Galois extension, i.e., a kind of normal closure. We will not prove this in our context, but keep the terminology, nonetheless.

Definition 10.3.1. Let A be a connected commutative S-algebra. We say that A is separably closed if there are no G-Galois extensions A → B with G finite and non-trivial and B connected, i.e., if each finite G-Galois extension A → B has G = {e} or B not connected.

A separable closure of A is a pro-G_{A}-Galois extension A → Ā such that Ā is connected and separably closed. The pro-finite Galois group G_{A} of Ā over A is the absolute Galois group of A.

The existence of a separable closure follows from Zorn’s lemma. However, we have not yet proved that two separable closures of A are weakly equivalent, so talking of “the” absolute Galois group is also a bit presumptive.

By Minkowski’s theorem on the discriminant [Ne99, III.2.17], for every number field K different from ℚ the inclusion ℤ → O_{K} is ramified at one or more primes. In particular, there are no Galois extensions ℤ → O_{K} other than the identity. The following inference appears to be well-known.

Proposition 10.3.2. The only connected Galois extension of the integers is ℤ itself, so ℤ = ℤ̄ is separably closed.

Proof. Let ℤ → T be a G-Galois extension of commutative rings, so T is a finitely generated free ℤ-module. Then ℚ → ℚ ⊗ T is also a G-Galois extension, so ℚ⊗T ≅ ∏_{i} K_{i} is a product of number fields [KO74, III.4.1]. Then T is contained in the integral closure of ℤ in ℚ ⊗ T, which is a product ∏_{i} O_{K_{i}} of number rings. The condition T ⊗ T ≅ ∏_{G} T and an index count imply, in combination, that T = ∏_{i} O_{K_{i}} and that each O_{K_{i}} is unramified over ℤ. By Minkowski’s theorem, this only happens when each K_{i} = ℚ, so T = ∏_{G} ℤ. If T is connected, this implies that G is the trivial group and that T = ℤ. □

In other words, to have interesting Galois extensions of ℤ one must localize away from one or more primes. We have the following analog in the context of commutative S-algebras. The examples in Section 5.4 demonstrate that after localization there are indeed interesting examples of (local) Galois extensions of S.

Theorem 10.3.3. The only (global, finite) connected Galois extension of the sphere spectrum S is S itself, so S = S̄ is separably closed.

Proof. Let S → B be any finite G-Galois extension of global, i.e., unlocalized, commutative S-algebras (Definition 4.1.3). Then B is a dualizable S-module (Proposition 6.2.1), hence of the homotopy type of (a retract of) a finite CW spectrum (Proposition 3.3.3). Thus H_{∗}(B) = H_{∗}(B; ℤ) is finitely generated in each degree, and non-trivial only in finitely many degrees.

Let k be minimal such that H_{k}(B) ≠ 0 and let ℓ be maximal such that H_{ℓ}(B) ≠ 0. The condition B ∧ B ≃ ∏_{G} B implies that k = ℓ = 0. For if k < 0 then H_{k}(B) ⊗ H_{k}(B) is isomorphic to H_{2k}(B ∧ B) ≅ ∏_{G} H_{2k}(B) = 0, which contradicts H_{k}(B) ≠ 0 and finitely generated. If ℓ > 0 then H_{ℓ}(B) ⊗ H_{ℓ}(B) injects into H_{2ℓ}(B ∧ B) ≅ ∏_{G} H_{2ℓ}(B) = 0, which again contradicts H_{ℓ}(B) ≠ 0 and finitely generated. Thus H_{∗}(B) = T is concentrated in degree 0.

By the Hurewicz theorem, B is a connective spectrum with π_{0}(B) ≅ H_{0}(B) = T. The Künneth formula then implies that T ⊗ T ≅ ∏_{G} T and Tor_{1}^{ℤ}(T, T) = 0, so the unit map ℤ → T makes T a free abelian ℤ-module of rank equal to the order of G. In particular, T is a faithfully flat ℤ-module.

The result of inducing B up along the Hurewicz map S → Hℤ has homotopy π_{∗}(Hℤ ∧ B) = H_{∗}(B) = T concentrated in degree 0, so there is a pushout square

\xymatrix{B \ar[r] & HT \\ S \ar[u] \ar[r] & Hℤ \ar[u]}

of commutative S-algebras. By a variation on the proof of Lemma 7.1.1, we shall now show that Hℤ → HT is G-Galois.

The map HT ∧_{Hℤ} HT → ∏_{G} HT is induced up from the weak equivalence B ∧ B → ∏_{G} B, cf. diagram (7.1.2), and is therefore a weak equivalence. Next, S → B is dualizable, so Hℤ → HT is dualizable (Lemma 6.2.3). Finally, T is faithfully flat over ℤ and so HT is faithful over Hℤ by the proof of Lemma 4.3.5. Thus Hℤ → HT is a faithful G-Galois extension (Proposition 6.3.2).

From Proposition 4.2.1 we deduce that ℤ → T is a G-Galois extension of commutative rings. By the classical theorem of Minkowski, this is only possible if G = {e} is the trivial group or T is not connected. And π_{0}(B) ≅ T, so either G is trivial or B is not connected (Proposition 10.2.2). Thus S is separably closed. □

Note that we did not have to (possibly) restrict attention to faithful G-Galois extensions S → B in this proof.

Question 10.3.4. Can the absolute Galois group G_{A}, or its maximal abelian quotient G_{A}^{ab}, be expressed in terms of arithmetic invariants of A, such as its algebraic K-theory K(A)? This would constitute a form of class field theory for commutative S-algebras. The author expects that there is a better hope for a simple answer in the maximally localized category of K(n)-local commutative S-algebras, than for general commutative S-algebras.

Question 10.3.5. If an E-local commutative S-algebra A is an even periodic Landweber exact spectrum, and A → B is a finite E-local G-Galois extension, does it then follow that B is also an even periodic Landweber exact spectrum, and that π_{0}(A) → π_{0}(B) is a G-Galois extension of commutative rings?

In the case of E = K(n) and A = E_{n}^{nr}, for which π_{0}(A) = 𝕎(𝔽̄_{p})[[u_{1}, …, u_{n−1}]] is separably closed, there are no non-trivial such algebraic extensions to a connected ring π_{0}(B), so it would follow that E_{n}^{nr} is K(n)-locally separably closed. In particular, E_{n}^{nr} would be the K(n)-local separable closure of L_{K(n)}S, with absolute Galois group 𝔾_{n}^{nr} = 𝕊_{n} ⋊ ℤ̂. This amounts to Conjecture 1.4 in the introduction.

Baker and Richter [BR05b] have partial results in this direction, in the global category. They are able to show that E_{n}^{nr} does not admit any non-trivial connected faithful abelian G-Galois extensions. So E_{n}^{nr} = E_{n}^{ab} is the maximal global faithful abelian extension of E_{n}.

CHAPTER 11 Galois theory II

As before, we are implicitly working E-locally, for some spectrum E.

11.1. Recovering the Galois group

The space of commutative A-algebra endomorphisms of B in a G-Galois extension A → B can be rewritten as

C_{A}(B, B) ≅ C_{B}(B ∧_{A} B, B) ≃ C_{B}(F (G_{+}, B), B) ,

in view of the weak equivalence h: B ∧_{A} B → F (G_{+}, B). When G is finite and discrete, and B admits no non-trivial idempotents, we can compute the homotopy groups of this mapping space by the Goerss–Hopkins spectral sequence.

When G is not discrete, these spectral sequence computations appear to be much harder, and we will not attempt them. We are therefore principally working in the context of the separable/étale extensions from Chapter 9.

Theorem 11.1.1. Let A → B be a finite G-Galois extension of commutative S-algebras, with B connected. Then the natural map

G → C_{A}(B, B) ,

giving the action of G on B through commutative A-algebra maps, is a weak equivalence. In particular, C_{A}(B, B) is a homotopy discrete grouplike monoid, so each commutative A-algebra endomorphism of B is an automorphism, up to a contractible choice.

Proof. This time we compute the homotopy groups of

C_{A}(B, B) ≃ C_{B}(∏_{G} B, B)

by means of (10.1.4), once again in the almost degenerate case when A = B and C = F (G_{+}, B) = ∏_{G} B. The E_{2}-term has

E_{2}^{0,0} = Alg_{B_{∗}}(∏_{G} B_{∗}, B_{∗}) ≅ G

since B_{∗} = π_{∗}(B) is connected in the graded sense, or equivalently, π_{0}(B) has no non-trivial idempotents. The remainder of the E_{2}-term is

E_{2}^{s,t} = D_{B_{∗}T}^{s} (∏_{G} B_{∗}, Ω^{t}B_{∗}) = 0 ,

since ∏_{G} B_{∗} is étale over B_{∗}. We are therefore in the collapsing situation of Corollary 10.1.5, and C_{A}(B, B) ≃ G follows. □

The extension to profinite pro-Galois extensions is straightforward.

Proposition 11.1.2. Let A → B = colim_{α} B_{α} be a pro-G-Galois extension, with each A → B_{α} a finite G_{α}-Galois extension and G = lim_{α} G_{α}. Suppose that B is connected. Then C_{A}(B, B) is homotopy discrete, and the natural map G → π_{0}C_{A}(B, B) is a group isomorphism.

Proof. Using (8.1.2), we rewrite the commutative A-algebra mapping space as

C_{A}(B, B) ≅ C_{B}(B ∧_{A} B, B)

≃ C_{B}(colim_{α} F (G_{α+}, B), B) ≃ holim_{α} C_{B}(F (G_{α+}, B), B) .

By the finite case, each C_{B}(F (G_{α+}, B), B) is homotopy discrete with

π_{0}C_{B}(F (G_{α+}, B), B) ≅ Alg_{B_{∗}}(∏_{G_{α}} B_{∗}, B_{∗}) ≅ G_{α} ,

when B is connected. So C_{A}(B, B) ≃ holim_{α} G_{α} is homotopy discrete, with π_{0}C_{A}(B, B) ≅ lim_{α} G_{α} ≅ G. □

11.2. The brave new Galois correspondence

We now turn to the converse part of the Galois correspondence. The proper role of the separability condition in the following result was found in a conversation with Birgit Richter.

Proposition 11.2.1. Let A → B be a G-Galois extension, with B connected and G finite and discrete, and let

A → C → B

be a factorization of this map through a separable commutative A-algebra C. Then C_{C}(B, B) is homotopy discrete, and the natural map C_{C}(B, B) → C_{A}(B, B) identifies K = π_{0}C_{C}(B, B) with a subgroup of G = π_{0}C_{A}(B, B). Furthermore, the action of C_{C}(B, B) ≃ K on B induces a weak equivalence

h: B ∧_{C} B → ∏_{K} B .

Proof. By assumption A → C is separable, so there are maps

C′ ⟶^{σ} C ∧_{A} C ⟶^{μ} C

of C-bimodules relative to A such that μσ: C′ → C is a weak equivalence. Inducing these maps and modules up along C → B, both as left and right modules, we get maps

B ∧_{C} C′ ∧_{C} B ⟶^{σ̄} B ∧_{A} B ⟶^{μ̄} B ∧_{C} B

of B-bimodules relative to A, such that the composite is a weak equivalence. We consider C∧_{A}C as a commutative C-algebra via the left unit C ≅ C∧_{A}A → C∧_{A}C, and similarly for B ∧_{A} B over B. Then μ is a map of commutative C-algebras and μ̄ is a map of commutative B-algebras.

At the level of homotopy groups, we get a diagram

B_{∗}^{C}(B) ⟶^{σ̄_{∗}} B_{∗}^{A}(B) ⟶^{μ̄_{∗}} B_{∗}^{C}(B)

of B_{∗}^{A}(B)-module homomorphisms, whose composite is the identity. Furthermore, μ̄_{∗} is a B_{∗}-algebra homomorphism. It follows from the B_{∗}^{A}(B)-linearity of the homomorphism σ̄_{∗} that it is also a B_{∗}-algebra homomorphism. For if x, y ∈ B_{∗}^{C}(B) then σ̄_{∗}x ∈ B_{∗}^{A}(B) acts on y through multiplication by its image μ̄_{∗}σ̄_{∗}x = x and on σ̄_{∗}y ∈ B_{∗}^{A}(B) by multiplication by σ̄_{∗}x. The B_{∗}^{A}(B)-linearity of σ̄_{∗} now provides the left hand equality below:

σ̄_{∗}x · σ̄_{∗}y = σ̄_{∗}(μ̄_{∗}σ̄_{∗}x · y) = σ̄_{∗}(x · y) .

Thus σ̄_{∗} is a B_{∗}-algebra homomorphism, so that B_{∗}^{C}(B) is a retract of B_{∗}^{A}(B), both in the category of B_{∗}^{A}(B)-modules and, more importantly to us, in the category of commutative B_{∗}-algebras.

Recall that B_{∗}^{A}(B) ≅ ∏_{G} B_{∗}, since A → B is G-Galois. Here ∏_{G} B_{∗} ≅ ⨁_{G} B_{∗} is a finitely generated free B_{∗}-module, since G is finite, so the retraction above implies that B_{∗}^{C}(B) is a finitely generated projective B_{∗}-module. We may therefore once more consider the Goerss–Hopkins spectral sequence (10.1.4), now for the mapping space

C_{C}(B, B) ≅ C_{B}(B ∧_{C} B, B) .

The E_{2}-term has

E_{2}^{s,t} = D_{B_{∗}T}^{s} (B_{∗}^{C}(B), Ω^{t}B_{∗})

for t > 0. The commutative B_{∗}-algebra retraction μ̄_{∗}: B_{∗}^{A}(B) → B_{∗}^{C}(B) induces a split injection from each of these cohomology groups to

D_{B_{∗}T}^{s} (B_{∗}^{A}(B), Ω^{t}B_{∗}) ,

which we saw was zero in the proof of Theorem 11.1.1, since B_{∗}^{A}(B) = ∏_{G} B_{∗} is étale over B_{∗}. We therefore have E_{2}^{s,t} = 0 away from the origin, also in the Goerss–Hopkins spectral sequence for π_{t−s}C_{C}(B, B).

Thus C_{C}(B, B) is homotopy discrete, in the sense that each path component is weakly contractible, with set of path components

K = π_{0}C_{C}(B, B) ≅ Alg_{B_{∗}}(B_{∗}^{C}(B), B_{∗}) .

The B_{∗}-algebra retraction μ̄_{∗} induces a split injection from this set to

Alg_{B_{∗}}(B_{∗}^{A}(B), B_{∗}) ≅ G .

It is clear that the natural map C_{C}(B, B) → C_{A}(B, B), viewing a map B → B of commutative C-algebras as a map of commutative A-algebras, is a monoid map with respect to the composition of maps. Therefore the injection K → G identifies K as a sub-monoid of G. But G is a finite group, so K is in fact a subgroup of G. This completes the proof of the first claims of the proposition.

The tautological action by C_{C}(B, B) on B through commutative C-algebra maps can be converted to an action by K on a commutative C-algebra B′ weakly equivalent to B. We hereafter implicitly make this replacement, so as to have K acting directly on B over C, and turn to the proof of the final claim.

The composite σ̄_{∗}μ̄_{∗}: B_{∗}^{A}(B) → B_{∗}^{A}(B) is an idempotent B_{∗}-algebra map. Under the isomorphism B_{∗}^{A}(B) ≅ ∏_{G} B_{∗} it corresponds to an idempotent B_{∗}- algebra endomorphism of ∏_{G} B_{∗}. Since B_{∗} is connected, it must be the retraction of ∏_{G} B_{∗} onto the subalgebra ∏_{K′} B_{∗}, for some subset K′ ⊂ G (containing e ∈ G). Thus B_{∗}^{C}(B) ≅ ∏_{K′} B_{∗}, which implies

K = Alg_{B_{∗}}(B_{∗}^{C}(B), B_{∗}) ≅ Alg_{B_{∗}}(∏_{K′} B_{∗}, B_{∗}) ≅ K′ .

Thus K = K′ as subsets of G, and the weak equivalence B ∧_{A} B ≃ ∏_{G} B retracts to a weak equivalence

h: B ∧_{C} B → ∏_{K} B .

It is quite clearly given by the action of K on B through commutative C-algebra maps, as in (4.1.2). □

This leads us to the converse part of the Galois correspondence for E-local commutative S-algebras, in the case of finite, faithful Galois extensions.

Theorem 11.2.2. Let A → B be a G-Galois extension, with B connected and G finite and discrete. Furthermore, let

A → C → B

be a factorization of this map through a separable commutative A-algebra C such that C → B is faithful, and let K = π_{0}C_{C}(B, B) ⊂ G.

If A → B is faithful, or more generally, if B is dualizable over C, then C ≃ B^{hK} as commutative C-algebras, and C → B is a faithful K-Galois extension.

Proof. We first prove that A → B faithful implies that B is dualizable over C.

By hypothesis, A → C is separable, so the multiplication map μ and its weak section σ make C a retract up to homotopy of C ∧_{A} C, as a C ∧_{A} C-module. Therefore μ makes C a dualizable C ∧_{A} C-module, by Lemma 3.3.2(c). Similarly, for each g ∈ G the twisted multiplication map

μ(1 ∧ g): C ∧_{A} C → C

and its weak section (1 ∧ g^{−1})σ make C a dualizable C ∧_{A} C-module. Inducing up along C → B, Lemma 6.2.3 implies that each map B ∧_{A} C → B, given algebraically as b ∧ c ↦ b · g(c), makes B a dualizable B ∧_{A} C-module. By Lemma 3.3.2(c) again, it follows that the natural map B ∧_{A} C → B ∧_{A} B makes B ∧_{A} B ≃ ∏_{G} B ≃ ⋁_{G} B a dualizable B ∧_{A} C-module. By Proposition 6.2.1 and hypothesis, B is dualizable and faithful over A, so by Lemma 6.2.3 and Lemma 4.3.3 we know that B ∧_{A} C is faithful and dualizable over C. Thus by Lemma 6.2.4 it follows that the natural map C → B makes B a dualizable C-module.

By Proposition 11.2.1, K = π_{0}C_{C}(B, B) ⊂ G acts on (a weakly equivalent replacement for) B through C-algebra maps, so that h: B ∧_{C} B → ∏_{K} B is a weak equivalence. By hypothesis (and the argument above), C → B is faithful and dualizable. Then by Proposition 6.3.2 the natural map i: C → B^{hK} is a weak equivalence, and so C → B is a faithful K-Galois extension. □

CHAPTER 12 Hopf–Galois extensions in topology

In this final chapter we work globally, i.e., not implicitly localized at any spectrum (other than at E = S).

12.1. Hopf–Galois extensions of commutative S-algebras

Let A → B be a G-Galois extension of commutative S-algebras, with G stably dualizable, as usual. The right adjoint α̃: B → F (G_{+}, B) of the group action map α: G_{+} ∧ B → B can be lifted up to homotopy through the weak equivalence νγ: B ∧ DG_{+} → F (G_{+}, B), to a map β: B → B ∧ DG_{+}. The group multiplication G × G → G induces a functionally dual map DG_{+} → D(G × G)_{+}, which likewise can be lifted up to homotopy through the weak equivalence ∧: DG_{+} ∧ DG_{+} → D(G × G)_{+} to a coproduct ψ: DG_{+} → DG_{+} ∧ DG_{+}. We shall require rigid forms of these structure maps.

Definition 12.1.1. A commutative Hopf S-algebra is a cofibrant commutative S-algebra H equipped with a counit ϵ: H → S and a coassociative and counital coproduct ψ: H → H ∧_{S} H, in the category of commutative S-algebras.

Note that we are not assuming that the coproduct ψ is (strictly) cocommutative, nor that it admits a strict antipode/conjugation χ: H → H. This would severely limit the number of interesting examples.

Example 12.1.2. Let X be an infinite loop space. The E_{∞} structure on X makes S[X] = S ∧ X_{+} an E_{∞} ring spectrum. The diagonal map Δ: X → X × X and X → ∗ induce a coproduct ψ: S[X] → S[X × X] ≅ S[X] ∧ S[X] and counit ϵ: S[X] → S, which altogether can be rigidified to make H ≃ S[X] a commutative Hopf S-algebra. The rigidification takes the coassociative and counital coproduct and counit on S[X] to a corresponding co-A_{∞} structure on H, which in turn can be rigidified to strictly coassociative and counital operations, by working entirely within commutative S-algebras. It is in general not possible to make a similar rigidification of co-E_{∞} structures, within commutative S-algebras.

Definition 12.1.3. Let A be a cofibrant commutative S-algebra, let B be a cofibrant commutative A-algebra and let H be a commutative Hopf S-algebra. We say that H coacts on B over A if there is a coassociative and counital map

β: B → B ∧ H

of commutative A-algebras. In this situation, let

h: B ∧_{A} B → B ∧ H

be the composite map (μ ∧ 1)(1 ∧ β) of commutative B-algebras.

Definition 12.1.4. The (Hopf) cobar complex C^{•}(H; B), for H coacting on B over A, is the cosimplicial commutative A-algebra with

C^{q}(H; B) = B ∧ H ∧ ⋯ ∧ H

(q copies of H) in codegree q. The coface maps are d_{0} = β∧1^{∧q}, d_{i} = 1^{∧i}∧ψ∧1^{∧(q−i)} for 0 < i < q and d_{q} = 1^{∧q} ∧η, where η: S → H is the unit map. The codegeneracy maps involve the counit ϵ: H → S. Let C(H; B) = Tot C^{•}(H; B) be its totalization. The algebra unit A → B induces a coaugmentation A → C^{•}(H; B), and a map

i: A → C(H; B) .

Definition 12.1.5. A map A → B of commutative S-algebras is an H-Hopf– Galois extension if H is a commutative Hopf S-algebra that coacts on B over A, so that the maps i: A → C(H; B) and h: B∧_{A}B → B∧H are both weak equivalences.

Note that there is no finiteness/dualizability condition on H in this definition. See [Chi00] for a recent text on Hopf–Galois extensions in the algebraic setting.

Example 12.1.6. Let G be a stably dualizable topological group. The weak coproduct on DG_{+} = F (G_{+}, S), derived from the group multiplication, can be rigidified to give H ≃ DG_{+} the structure of a commutative Hopf S-algebra. If G acts on B over A, then the weak coaction of DG_{+} on B can be rigidified to a coaction of H on B over A. Then the (Hopf) cobar complex C^{•}(H; B) maps by a degreewise weak equivalence to the (group) cobar complex C^{•}(G; B) from Definition 8.2.5. In codegree q it is weakly equivalent to the composite natural map

B ∧ DG_{+} ∧ ⋯ ∧ DG_{+} ⟶_{≃}^{∧} B ∧ DG_{+}^{q} ⟶_{≅}^{γ} DG_{+}^{q} ∧ B ⟶_{≃}^{ν} F (G_{+}^{q}, B) .

On totalizations, we obtain a weak equivalence C(H; B) ≃ B^{hG}. In this case, the definition of an H-Hopf–Galois extension A → B generalizes that of a G-Galois extension A → B, since i: A → B^{hG} factors as

A ⟶^{i} C(H; B) ⟶^{≃} B^{hG} ,

and h: B ∧_{A} B → ∏_{G} B factors as

B ∧_{A} B ⟶^{h} B ∧ H ⟶^{≃} F (G_{+}, B) .

Recall the Amitsur complex C^{•}(B/A) from Definition 8.2.1.

Definition 12.1.7. There is a natural map of cosimplicial commutative Aalgebras h^{•}: C^{•}(B/A) → C^{•}(H; B) given in codegree q by the map

h^{q}: B ∧_{A} B ∧_{A} ⋯ ∧_{A} B → B ∧ H ∧ ⋯ ∧ H

that is the composite of the maps

B^{∧_{A}(i+1)} ∧ H^{∧j} ≅ B^{∧_{A}(i−1)} ∧_{A} (B ∧_{A} B) ∧ H^{∧j}

⟶^{1^{∧(i−1)}∧h∧1^{∧j}} B^{∧_{A}(i−1)} ∧_{A} (B ∧ H) ∧ H^{∧j} ≅ B^{∧_{A}i} ∧ H^{∧(j+1)}

for j = 0, …, q − 1 and i + j = q. Upon totalization, it induces a map h′: A_{B}^{∧} → C(H; B) of commutative A-algebras.

The diagram chase needed to verify that h^{•} indeed is cosimplicial uses the strict coassociativity and counitality of the Hopf S-algebra structure on H.

Proposition 12.1.8. Suppose that H coacts on B over A, as above, and that h: B ∧_{A} B → B ∧ H is a weak equivalence. Then h′: A_{B}^{∧} → C(H; B) is a weak equivalence. As a consequence, A → B is an H-Hopf–Galois extension if and only if A is complete along B.

Proof. The cosimplicial map h^{•} is a weak equivalence in each codegree, so the induced map of totalizations h′ is a weak equivalence. Therefore the composite i = h′ ∘ η, of the two maps

A ⟶^{η} A_{B}^{∧} ⟶^{h′} C(H; B) ,

is a weak equivalence if and only if η is one. □

12.2. Complex cobordism

Let A = S be the sphere spectrum, B = MU the complex cobordism spectrum and H = S[BU] = Σ^{∞}BU_{+} the unreduced suspension spectrum of BU. Bott’s infinite loop space structure on BU makes H a commutative S-algebra, and the diagonal map Δ: BU → BU × BU induces the Hopf coproduct ψ: S[BU] → S[BU] ∧ S[BU]. The Thom diagonal

β: MU → MU ∧ BU_{+}

defines a coaction by S[BU] on MU over S. The induced map

h: MU ∧ MU → MU ∧ BU_{+}

is the weak equivalence known as the Thom isomorphism. The Bousfield–Kan spectral sequence associated to the cosimplicial commutative S-algebra C^{•}(MU/S) is the Adams–Novikov spectral sequence

E_{2}^{s,t} = Ext_{MU_{∗}MU}^{s,t}(MU_{∗}, MU_{∗}) ⟹ π_{t−s}(S) .

The convergence of this spectral sequence is the assertion that the coaugmentation

i: S → S_{MU}^{∧} = Tot C^{•}(MU/S)

is a weak equivalence. In view of Proposition 12.1.8 we can summarize these facts as follows:

Proposition 12.2.1. The unit map S → MU is an S[BU]-Hopf–Galois extension of commutative S-algebras. □

Remark 12.2.2. There is no topological group G such that S → MU is a G-Galois extension, but S[BU] is taking on the role of its functional dual DG_{+}, as in Example 12.1.6. So the commutative Hopf S-algebra S[BU] is trying to be the ring of functions on the non-existent Galois group of MU over S. Note that there is no bimodule section to the multiplication map μ: MU ∧ MU → MU, since the left and right units η_{L}, η_{R}: MU_{∗} → MU_{∗}MU are really different, so S → MU is not separable in the sense of Section 9.1.

Remark 12.2.3. There are similar S[X]-Hopf–Galois extensions S → T h(γ) to the Thom spectrum induced by any infinite loop map γ: X → BGL_{1}(S). For example, there is such an extension S → MUP to the even periodic version MUP of MU, which is the Thom spectrum of the tautological virtual bundle over X = ℤ × BU = Ω^{∞}ku. More generally, for any commutative S-algebra R and infinite loop map γ: X → BGL_{1}(R) there is an R-based Thom spectrum T h^{R}(γ), i.e., a γtwisted form of R[X] = R ∧ X_{+}, and an R[X]-Hopf–Galois extension R → T h^{R}(γ).

Remark 12.2.4. The extension S → MU is known not to be faithful, since by [Ra84, §3] or [Ra92, 7.4.2] MU_{∗}(cY) = 0 for every finite complex Y with trivial rational cohomology. Here cY denotes the Brown–Comenetz dual of Y. This faithlessness leaves the telescope conjecture [Ra84, 10.5] or [Ra92, 7.5.5] a significant chance to be false. Recall that if F (n) is a finite complex of type n (with a v_{n}-self map), and T (n) = v_{n}^{−1}F (n) its mapping telescope, the conjecture is that the natural map λ: T (n) → L_{n}F (n) is a weak equivalence. After inducing up to MU, 1 ∧ λ: MU ∧ T (n) → MU ∧ L_{n}F (n) is an equivalence, by the localization theorem v_{n}^{−1}MU ∧ F (n) ≃ L_{n}MU ∧ F (n) [Ra92, 7.5.2]. Positive information about the faithfulness of Galois- or Hopf–Galois extensions (Question 4.3.6) might conceivably reflect back on this conjecture.

To conclude this paper, we wish to discuss how the Hopf–Galois extension S → MU provides a global, integral object whose p-primary K(n)-localization and nilpotent completion L_{K(n)}S → L̂_{K(n)}^{MU}MU governs the pro-Galois extensions L_{K(n)}S → E_{n}, for each rational prime p and integer n ≥ 0.

(12.2.5)  \xymatrix{MU & L̂_{K(n)}^{MU}MU \ar[r]^{t} & E_{n} \\ S \ar[u]^{S[BU]} & L_{K(n)}S \ar[u] \ar[ur]_{𝔾_{n}}}

This suggests that S → MU is a kind of near-maximal ramified Galois extension, and that its weak Galois group (“weak” in the analytic sense that it is only realized through its functional dual DG_{+} = S[BU] that coacts on MU) is a kind of nearabsolute ramified Galois group of the sphere. More precisely, the maximal extension may be the one obtained from the even periodic theory MUP by tensoring with the ring Oℚ̄ of algebraic integers.

Even if S → MU does not admit many Galois automorphisms, the Hopf coaction β: MU → MU ∧ BU_{+} still determines the Galois action of each element g ∈ 𝔾_{n} on E_{n}. By the Hopkins–Miller theory, each commutative S-algebra map g: E_{n} → E_{n} is uniquely determined by the underlying map of (commutative) ring spectra, so it is the description of the latter that we shall review.

Recall from 5.4.2 that Γ_{n} is the Honda formal group law over 𝔽_{p^{n}} and Γ̃_{n} its universal deformation, defined over π_{0}(E_{n}). By the Lubin–Tate theorem [LT66, 3.1], each automorphism g ∈ 𝕊_{n} ⊂ 𝔾_{n} of Γ_{n} determines a unique pair (ϕ, g̃), where ϕ: π_{0}(E_{n}) → π_{0}(E_{n}) is a ring automorphism and g̃: Γ̃_{n} → ϕ_{∗}Γ̃_{n} is an isomorphism of formal group laws over π_{0}(E_{n}), whose expansion g̃(x) ∈ π_{0}(E_{n})[[x]] ≅ E_{n}^{0}(ℂP^{∞}) reduces modulo (p, u_{1}, …, u_{n−1}) to the expansion g(x) ∈ 𝔽_{p^{n}}[[x]] of g. Then ϕ = π_{0}(g), when g is considered as a self-map of E_{n}. Furthermore,

g̃(x) ∈ E_{n}^{0}(ℂP^{∞}) ≅ Hom_{E_{n∗}}(E_{n∗}(ℂP^{∞}), E_{n∗})
≅ Alg_{E_{n∗}}(E_{n∗}(BU), E_{n∗}) ⊂ E_{n}^{0}(BU)

corresponds to a unique map of ring spectra g̃: S[BU] → E_{n} . Let t: MU → E_{n} be the usual complex orientation, corresponding to the graded version of Γ̃_{n}. Then the following diagram commutes up to homotopy:

(12.2.6)  \xymatrix{MU \ar[r]^{β} \ar[d]_{t} & MU ∧ BU_{+} \ar[d]^{μ(t∧g̃)} \\ E_{n} \ar[r]^{g} & E_{n}}

The composite g ∘ t = μ(t ∧ g̃)β determines g, in view of t^{∗}: E_{n}^{∗}(E_{n}) → E_{n}^{∗}(MU) being nearly injective. Only the Galois automorphisms in Gal ⊂ 𝔾_{n} are missing, but these may be ignored if we are focusing on { \atop E(n)}, or can be detected by passing to MUP and adjoining some roots of unity.

By analogy, for number fields K ⊂ L and primes 𝔭 ∈ O_{K}, a factorization 𝔭O_{L} = 𝔓_{1}^{e_{1}} ·. . .·𝔓_{r}^{e_{r}} leads to a splitting of completions K_{𝔭} → L⊗_{K} K_{𝔭} ≅ ∏_{i} L_{𝔓_{i}}. If the field extension K → L is G-Galois, then each local extension K_{𝔭} → L_{𝔓_{i}} is G_{𝔓_{i}}- Galois, where G_{𝔓_{i}} ⊂ G is the decomposition group of 𝔓_{i}, and G acts transitively on the finite set of primes over 𝔭. Thus when the global extension K → L is localized (i.e., completed), it splits as a product of smaller local extensions, in a way that depends on the place of localization.

(12.2.7)  \xymatrix{L & L ⊗_{K} K_{𝔭} \ar[r]^{pr_{i}} & L_{𝔓_{i}} \\ K \ar[u]^{G} & K_{𝔭} \ar[u] \ar[ur]_{G_{𝔓_{i}}}}

In the algebraic case of a pro-Galois extension K → K̄ there is a profinite set of places over each prime 𝔭, still forming a single orbit for the action by the absolute Galois group G_{K}.

In the topological setting of S → MU there is likewise a single orbit of chromatic primes of MU over the one of S that corresponds to the localization functor L_{K(n)} on M_{S}, namely those corresponding to the nilpotent completion functors L̂_{K(n)}^{MU} on M_{MU}, for all the various possible complex orientations MU → K(n). In the absence of a real Galois group of automorphisms of MU these do not form a geometric orbit of places, but the next-best thing is available, namely the S-algebraic coaction by S[BU] via the Thom diagonal, a sub-coaction of which indeed links the various complex orientations of K(n) into one “weak” orbit.

Jack Morava [Mo05] has developed this Galois theoretic perspective on the stable homotopy category further.

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Stably Dualizable Groups

John Rognes

Author address: Department of Mathematics, University of Oslo, Norway

E-mail address: rognes@math.uio.no

Abstract

We extend the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the p-complete study for p-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the E-local stable homotopy category, for any spectrum E. The principal new examples occur in the K(n)-local category, where the Eilenberg– Mac Lane spaces G = K(ℤ/p, q) are stably dualizable and nontrivial for 0 ≤ q ≤ n.

We show how to associate to each E-locally stably dualizable group G a stably defined representation sphere S^{adG}, called the dualizing spectrum, which is dualizable and invertible in the E-local category. Each stably dualizable group is Atiyah–Poincaré self-dual in the E-local category, up to a shift by S^{adG}. There are dimension-shifting norm- and transfer maps for spectra with G-action, again with a shift given by S^{adG}. The stably dualizable group G also admits a kind of framed bordism class [G] ∈ π_{∗}(L_{E}S), in degree dim_{E}(G) = [S^{adG}] of the P ic_{E}-graded homotopy groups of the E-localized sphere spectrum.

Received by the editor February 8, 2005.

2000 Mathematics Subject Classification. 55M05, 55P35, 57T05.

Key words and phrases. K(n)-compact group, adjoint representation, Poincaré duality, norm map, framed bordism class.

CHAPTER 1 Introduction

1.1. The symmetry groups of stable homotopy theory

Compact Lie groups occur naturally as the symmetry groups of geometric objects, e.g. as the isometry groups of Riemannian manifolds [MS39]. Such geometric objects can usefully be viewed as equivariant objects, i.e., as a spaces with an action by a Lie group. The homotopy theory of such equivariant spaces is quite well approximated by the corresponding stable equivariant homotopy theory, which in its strong “genuine” form relies, already in its construction, on the good representation theory for actions by Lie groups on finite-dimensional vector spaces.

As a first example of a useful stable result, consider the Adams equivalence Y/G ≃ (Σ^{−adG}Y)^{G} of [LMS86, II.7]. Here Y is any free G-spectrum, adG denotes the adjoint representation of G on its Lie algebra and Σ^{−adG}Y is the stably defined desuspension of Y with respect to this G-representation.

As a second example, Atiyah duality [At61] asserts that if M is a smooth closed manifold with stable normal bundle ν, the functional (Spanier–Whitehead) dual DM_{+} = F (M_{+}, S) of M_{+} is equivalent to the Thom spectrum T h(ν ↓ M). When M = G is a compact Lie group, and thus parallelizable, we can write this as a stable Poincaré duality equivalence DG_{+} ≃ T h(ϵ^{−n} ↓ G) = Σ^{−n}Σ^{∞}(G_{+}). But G acts on itself both from the left and the right, and the bi-equivariant form of this equivalence takes the more precise form

DG_{+} ∧ S^{adG} ≃ Σ^{∞}(G_{+})

where G acts by conjugation from the left on the one-point compactification S^{adG} of the adjoint representation and trivially from the right. See Theorem 3.1.4 below.

As a third example, the left-invariant framing of an n-dimensional compact Lie group G gives it an associated stably framed cobordism class [G] in Ω_{n}^{fr} ≅ π_{n}(S), the n-th stable stem. For example [S^{1}] = η ∈ π_{1}(S) realizes the stable class of the Hopf fibration η: S^{3} → S^{2}. It is of interest to see which stable homotopy classes actually occur in this way [Os82].

The formulation of these three results may appear to require that G admits a geometric representation theory, with tangent spaces, adjoint representations, etc., but in fact much less is required, and that is the main thrust of the present article.

1.2. Algebraic localizations and completions

Homotopy-theoretically, the main properties of compact Lie groups are (i) that they are compact manifolds, hence admit the structure of a finite CW complex, and (ii) that they are topological groups, hence are homotopy equivalent to loop spaces. Browder [Brd61, 7.9] showed that all finite H-spaces are Poincaré complexes, and recently Bauer, Kitchloo, Notbohm and Pedersen [BKNP04] showed that all finite loop spaces indeed are homotopy equivalent to manifolds. However, there are examples of finite loop spaces that are not even rationally equivalent to Lie groups [ABGP04].

A standard method in homotopy theory, and a key ingredient in [BKNP04], is the possibility to study homotopy types locally, say by Bousfield localization with respect to a homology theory [Bo75], [Bo79], or completion in the sense of Bousfield and Kan [BK72]. For later generalization, we recall that the Bousfield– Kan p-completion functor is derived from a monad (𝔽_{p}(−), μ, η), where X ↦ 𝔽_{p}(X) is a particular endofunctor on spaces with π_{∗}𝔽_{p}(X) = H̃_{∗}(X; 𝔽_{p}), equipped with suitable associative and unital natural transformations μ_{X}: (𝔽_{p} ∘ 𝔽_{p})(X) → 𝔽_{p}(X) and η_{X}: X → 𝔽_{p}(X). For each space X the monad generates a cosimplicial space

[q] ⟼ (𝔽_{p} ∘ ⋯ ∘ 𝔽_{p})(X)

(the (q+1)-fold iterate of the endofunctor), whose totalization defines the Bousfield– Kan p-completion of X, see [BK72, §I.4], which we denote by X_{p}^{∧}. We say that X is p-complete when the natural map X → X_{p}^{∧} is a weak equivalence. Each space 𝔽_{p}(X) is a product of mod p Eilenberg–Mac Lane spaces, i.e., the spaces in the Ωspectrum of the mod p Eilenberg–Mac Lane spectrum H𝔽_{p}, and X_{p}^{∧} is constructed as a limit of such spaces.

In the Bousfield–Kan p-complete category, the local incarnations of finite loop spaces are the p-compact groups of Dwyer and Wilkerson [DW94]. These are the topological groups G with finite mod p homology H_{∗}(G; 𝔽_{p}) and p-adically complete classifying space BG ≃ BG_{p}^{∧}. Dwyer and Wilkerson consider loop spaces instead of topological groups, but any loop space is equivalent to a topological group, and conversely, so there is no real distinction. We shall prefer to work with topological groups G and their classifying spaces BG, rather than with loop spaces ΩY and their deloopings Y, since we shall make extensive use of group actions by G, which would involve operad actions or coherence theory to formulate for loop spaces.

We think of a connected compact Lie group G as a geometric, integrally defined object, which can be analyzed one rational prime p at a time by way of its homotopytheoretic, locally defined p-compact pieces, namely the p-compact groups Ω(BG)_{p}^{∧} obtained by p-completing the classifying space BG at p and looping. In this case, the fact that the loop space of (BG)_{p}^{∧} still has finite mod p homology follows from the convergence of the mod p Eilenberg–Moore spectral sequence [Dw74]. In addition to these geometric examples, there are also other “exotic” p-compact groups that only exist locally, without the global origin of a compact Lie group, such as the Dwyer–Wilkerson 2-compact group DI(4), for which H^{∗}(BDI(4); 𝔽_{2}) realizes the ring of rank 4 Dickson invariants [DW93]. There is no compact Lie group with this cohomology algebra.

In his Ph.D. thesis, T. Bauer [Ba04] showed that for each p-compact group G one can produce a p-complete stable replacement for the adjoint representation sphere S^{adG}, for the purposes of p-complete stable homotopy theory. It suffices to work G-equivariantly in the “naive” sense, where the objects are spectra equipped with a G-action, and the (weak) equivalences are G-equivariant maps that are stable equivalences in the underlying non-equivariant category. Bauer showed that for a p-compact group G, analogous results to the Adams equivalence and the Atiyah–Poincaré duality equivalences above hold, with S^{adG} reinterpreted as the dualizing spectrum (Σ^{∞}G_{+})^{hG} = F (EG_{+}, Σ^{∞}G_{+})^{G} of W. Dwyer (unpublished) and J. R. Klein [Kl01], but formed in the p-complete category. Bauer also showed that a p-compact group G has the analogue of a framed bordism class [G] in π_{∗}(S_{p}^{∧}). For example, the Sullivan spheres (see Example 2.4.4) are examples of p-compact groups for p odd, and their framed bordism classes represent the generators α_{1} ∈ π_{2p−3}(S_{p}^{∧}). We shall generalize the constructions and results of Bauer to other, topologically defined, local categories.

1.3. Chromatic localizations and completions

In stable homotopy theory it is well-known, following Ravenel [Ra84], that for each prime p it is possible to interpolate in infinitely many “chromatic” stages between the algebraically p-local and rational situations, through a tower of Bousfield localizations with respect to the homology theories represented by the Johnson– Wilson spectra E(n), for n ≥ 0. Furthermore, one can isolate the individual strata of this filtration by way of the Bousfield localization with respect to the Morava Ktheory spectra K(n), for n ≥ 1. In a precise sense, these K(n)-local strata are the finest non-trivial localizations possible [HSt99, 7.5]. We therefore have the possibility of analyzing the stable images of compact Lie groups, or p-compact groups, in even finer detail than that offered by the algebraic localizations, by working in the p-primary K(n)-local category for one prime p and one natural number n, thereby focusing only on the “p-primary v_{n}-periodic” parts of their homotopy theory.

The topological groups G that have the finiteness property that K(n)_{∗}(G) is finite in each degree will be called K(n)-locally stably dualizable groups. Among these we can single out the K(n)-compact groups as those whose classifying space BG is a K(n)-complete space, in the sense of Bendersky, Curtis and Miller [BCM78] and [BT00, §2], which will be recalled in Section 2.4 below. These are the precise analogues, to the eyes of Morava K-theory, of the p-compact groups, to the eyes of mod p homology. However, the real work in this paper applies to all stably dualizable groups. There are examples of even more “exotic” K(n)-locally stably dualizable groups than the p-compact ones, that only exist K(n)-locally for some n. The simplest, abelian, instances of this are provided by the Eilenberg– Mac Lane spaces G = K(π, q), e.g. for π = ℤ/p, 0 ≤ q ≤ n [RaW80, 9.2], which for q ≥ 1 have infinite mod p homology and thus are never p-compact.

In this paper we generalize the duality theory of Lie groups and of p-compact groups by Klein and Bauer, to show that also for a K(n)-locally stably dualizable group G, the dualizing spectrum

S^{adG} = L_{K(n)}(Σ^{∞}G_{+})^{hG}

formed in the K(n)-local stable category has the properties that make it a stable substitute for the adjoint representation sphere of a compact Lie group. Here the G-homotopy fixed points are formed with respect to the standard right G-action on Σ^{∞}G_{+}. The dualizing spectrum S^{adG} is a dualizable and invertible spectrum in the K(n)-local category, cf. Theorem 3.3.4, which means that it has an equivalence class

[S^{adG}] ∈ Pic_{K(n)}

in the K(n)-local Picard group [HMS94]. In particular, suspending (smashing) by S^{adG} is an invertible self-equivalence of the K(n)-local category. The K(n)-local smash inverse of S^{adG} is its functional dual S^{−adG} = DS^{adG} = F (S^{adG}, L_{K(n)}S). See Propositions 2.5.7 and 3.2.3.

We show that there is a natural norm map

N: (X ∧ S^{adG})_{hG} → X^{hG}

for any spectrum X with G-action, which is a K(n)-local equivalence under slightly different conditions on X than those of the Adams equivalence. Up to rewriting, it is the canonical colim/lim exchange map for the G-homotopy orbit and G-homotopy fixed point constructions on X ∧ Σ^{∞}G_{+}. See Theorem 5.2.5. We also show that there is an (implicitly K(n)-local) natural Atiyah–Poincaré duality equivalence

DG_{+} ∧ S^{adG} ≃ Σ^{∞}G_{+} ,

which is G-equivariant from both the left and the right. See Theorem 3.1.4. Finally, we combine the norm map N: BG^{adG} = (S^{adG})_{hG} → S^{hG} = D(BG_{+}) for X = S with a bottom cell inclusion i: S^{adG} → BG^{adG} and the projection p: S^{hG} → S to obtain a natural map

pNi: S^{adG} → S ,

representing a homotopy class

[G] ∈ π_{∗}(L_{K(n)}S)

in the Pic_{K(n)}-graded homotopy groups of the K(n)-local sphere spectrum. See Definition 5.4.1. We informally think of this as the K(n)-locally framed bordism class of G.

The results discussed up to now hold in a uniform manner in the E-local stable category, for each fixed spectrum E and suitably defined E-locally stably dualizable groups. The terminology is chosen so that G is E-locally stably dualizable precisely when its suspension spectrum Σ^{∞}G_{+} is dualizable in the E-local stable category. See Definition 2.3.1. This is the generality in which the main body of the paper is written.

In Chapter 4 we develop calculational tools to study E-locally stably dualizable groups, mostly particular to the most local case E = K(n). The group structure on G makes H = K(n)_{∗}(G) a graded Frobenius algebra over R = K(n)_{∗} (Proposition 4.2.4), for the R-dual H^{∗} = K(n)^{∗}(G) is a free graded H-module of rank 1. There is a strongly convergent homological spectral sequence of Eilenberg–Moore type

E_{s,t}^{2} = Tor_{s,t}^{H}(R, H^{∗}) ⟹ K(n)^{−(s+t)}(S^{adG})

(Proposition 4.1.1). It collapses at the E^{2}-term to the line s = 0, and its dual identifies K(n)_{∗}(S^{adG}) with the H^{∗}-comodule primitives P_{H^{∗}}(H) ≅ Hom_{H}(H^{∗}, R) in Hom_{R}(H^{∗}, R) ≅ H = K(n)_{∗}(G) (Theorem 4.2.6). For example, when G = K(ℤ/p, n) is viewed as a K(n)-locally stably dualizable group, it follows that [G]: S^{adG} → S is an equivalence in the K(n)-local category (Example 5.4.6), so that the Atiyah–Poincaré duality equivalence takes the untwisted form

F (K(ℤ/p, n)_{+}, L_{K(n)}S) ≃ L_{K(n)}Σ^{∞}K(ℤ/p, n)_{+} .

1.4. Applications

It is conceivable that more invertible spectra in the K(n)-local category can be constructed in the form S^{adG} for K(n)-locally stably dualizable groups G, than just the localized integer sphere spectra L_{K(n)}Σ^{d}S for d ∈ ℤ. There are no such examples in the p-complete setting, but the K(n)-local Picard group is more subtle.

Likewise, it is conceivable that the associated homotopy classes [G] ∈ π_{∗}(L_{K(n)}S) can realize more homotopy classes than those that appear from Lie groups and p-compact groups. However, so far we have mostly studied the abelian examples of K(n)-locally stably dualizable groups given by Eilenberg–Mac Lane spaces, where this added potential is not realized. We think of these abelian groups as playing the role analogous to tori in the theory of compact Lie groups, and hope to develop a richer supply of non-abelian examples in joint work with Bauer, cf. Remark 2.4.8.

This work was simultaneously motivated by the author’s formulation [Rog08] of Galois theory of E-local commutative S-algebras. If A → B is an E-local G-Galois extension there is a useful norm equivalence N: (B ∧ S^{adG})_{hG} → B^{hG}, with A ≃ B^{hG}. For finite groups G this follows as in [Kl01], but the natural generality for the theory appears to be to allow topological Galois groups G that are E-locally stably dualizable, as considered here. The constructions in Chapters 3 and 5 of the present paper then find applications in the cited Galois theory.

Acknowledgments

The author wishes to thank Tilman Bauer for discussions starting with [Ba04] and leading to the present paper, and the referee for very constructive comments. Part of this work was done while the author was a member of the Isaac Newton Institute for Mathematical Sciences, Cambridge, in the fall of 2002, and he wishes to thank the INI for its hospitality and support.

CHAPTER 2 The dualizing spectrum

2.1. The E-local stable category

As our basic model for spectra we shall take the bicomplete, bitensored closed symmetric monoidal category ℳ_{S} of S-modules from [EKMM97]. The symmetric monoidal pairing is the smash product X ∧ Y, the unit object is the sphere spectrum S, and the internal function object is the mapping spectrum F (X, Y). We write DX = F (X, S) for the functional dual. For a based topological space T we write X ∧ T = X ∧ Σ^{∞}T and F (T, X) = F (Σ^{∞}T, X) for the resulting bitensors.

Let E be any S-module. It induces the (generalized, reduced) homology theory E_{∗} that takes an S-module X to the graded abelian group E_{∗}(X) = π_{∗}(E ∧ X). A map f: X → Y of S-modules is said to be an E-equivalence if the induced homomorphism f_{∗}: E_{∗}(X) → E_{∗}(Y) is an isomorphism, and an S-module Z is E-local if for each E-equivalence f: X → Y the induced homomorphism f^{#} : [Y, Z]_{∗} → [X, Z]_{∗} is an isomorphism.

Let ℳ_{S,E} be the full subcategory of ℳ_{S} of E-local S-modules. There is a Bousfield localization functor L_{E}: ℳ_{S} → ℳ_{S,E} [Bo79], [EKMM97, Ch. VIII] that comes equipped with a natural E-equivalence X → L_{E}X for each S-module X (with L_{E}X E-local). Let 𝒟_{S} = h̄ℳ_{S} be the homotopy category of ℳ_{S}, i.e., the stable category, and let 𝒟_{S,E} = h̄ℳ_{S,E} be the homotopy category of ℳ_{S,E}, i.e., the E-local stable category. It is a stable homotopy category in the sense of [HPS97, 1.2.2]. The induced E-localization functor L_{E}: 𝒟_{S} → 𝒟_{S,E} is left adjoint to the forgetful functor 𝒟_{S,E} → 𝒟_{S}.

The E-local category ℳ_{S,E} inherits the structure of a bicomplete, bitensored closed symmetric monoidal category from ℳ_{S} by applying L_{E} to each construction formed in ℳ_{S}. The symmetric monoidal pairing takes X and Y to L_{E}(X ∧ Y), and the unit object is the E-local sphere spectrum L_{E}S. The internal function object F (X, Y) is already E-local when Y is E-local, hence does not change when E-localized. In a similar fashion the (limits and) colimits in ℳ_{S,E} are obtained from those formed in ℳ_{S} by applying the E-localization functor, and likewise for tensors (and cotensors).

Example 2.1.1. We may take E = S, in which case every spectrum is S-local, ℳ_{S,S} = ℳ_{S} and the S-local stable category is the whole stable category.

Example 2.1.2. For a fixed rational prime p and number 0 ≤ n < ∞ we may take E = E(n), the n-th p-primary Johnson–Wilson spectrum, with

E(n)_{∗} = ℤ_{(p)}[v_{1}, …, v_{n}, v_{n}^{−1}] .

When n = 0, E(0) = Hℚ is the rational Eilenberg–Mac Lane spectrum and Eequivalence means rational equivalence. In each case L_{n} = L_{E(n)} is a smashing localization, L_{n}S is a commutative S-algebra and the E(n)-local category ℒ_{n} = ℳ_{S,E(n)}, as studied in [HSt99], is equivalent to the category ℳ_{L_{n}S} of L_{n}S-modules. In this case the forgetful functor ℳ_{S,E(n)} → ℳ_{S} preserves the symmetric monoidal pairing, but not the unit object.

Example 2.1.3. For each prime p and number 0 ≤ n ≤ ∞ we may alternatively take E = K(n), the n-th p-primary Morava K-theory spectrum. When n = 0, K(0) = E(0) = Hℚ, as discussed above. When 0 < n < ∞,

K(n)_{∗} = 𝔽_{p}[v_{n}, v_{n}^{−1}]

is a graded field, and 𝒦_{n} = 𝒟_{S,K(n)} is the K(n)-local stable category, again studied in [HSt99]. When n = ∞, K(∞) = H𝔽_{p} and ℳ_{S,H𝔽_{p}} is the category of H𝔽_{p}-local S-modules. For a connective spectrum X, H𝔽_{p}-localization amounts to algebraic p-completion. For 0 < n ≤ ∞ the forgetful functor to ℳ_{S} neither preserves the symmetric monoidal pairing nor the unit object.

Convention 2.1.4. Hereafter we shall work entirely within the E-local category ℳ_{S,E}. We refer to the objects of ℳ_{S,E} as E-local S-modules, or simply as spectra. For brevity we shall write X ∧ Y for the smash product L_{E}(X ∧ Y), S for the sphere spectrum L_{E}S and F (X, Y) for the function spectrum L_{E}F (X, Y) within this category. The same applies to the functional dual DX, limits, colimits, tensors and cotensors, all of which then take values in ℳ_{S,E}.

2.2. Dualizable spectra

Following Dold and Puppe [DP80], Lewis, May and Steinberger [LMS86, III.1] observe that in any closed symmetric monoidal category there are natural canonical maps ρ: X → DDX, ν: F (X, Y) ∧ Z → F (X, Y ∧ Z) and ∧: F (X, Y) ∧ F (Z, W) → F (X ∧ Z, Y ∧ W). We follow Hovey and Strickland [HSt99, 1.5] and say that a spectrum X is E-locally dualizable if the canonical map

ν: DX ∧ X → F (X, X)

(in the special case X = Z, Y = S) is an equivalence in ℳ_{S,E}. When the spectrum E is clear from the context, we simply say that X is dualizable. Lewis et al. then show [LMS86, III.1.2, 1.3]:

Lemma 2.2.1.

(1) The canonical map ρ: X → DDX is an equivalence if X is dualizable.

(2) The canonical map ν: F (X, Y) ∧ Z → F (X, Y ∧ Z) is an equivalence if X or Z is dualizable.

(3) The smash product map ∧: F (X, Y) ∧ F (Z, W) → F (X ∧ Z, Y ∧ W) is an equivalence if X and Z are dualizable, or if X is dualizable and Y = S.

It follows that the function spectrum F (X, Y) and smash product X ∧ Y are dualizable when X and Y are dualizable. In particular, DX is dualizable when X is dualizable.

Example 2.2.2. For E = S, a spectrum X is dualizable if and only if it is stably equivalent to a finite CW spectrum, i.e., if and only if X ≃ Σ^{∞}Σ^{d}K for some finite CW complex K and integer d ∈ ℤ. See e.g. [May96, XVI.7.4] for a proof, although the non-equivariant result must be older.

Example 2.2.3. For E = K(n) with 0 ≤ n ≤ ∞, Hovey and Strickland [HSt99, 8.6] show that a K(n)-local S-module X is dualizable if and only if K(n)_{∗}(X) is a finitely generated K(n)_{∗}-module. Note that this includes the cases n = 0 with K(0) = Hℚ and n = ∞ with K(∞) = H𝔽_{p}. In each case K(n)_{∗} is a graded field, so K(n)_{∗}(X) will automatically be free.

Lemma 2.2.4. If a spectrum X is H𝔽_{p}-locally dualizable then L_{K(n)}X is K(n)- locally dualizable for each 0 < n < ∞.

Proof. The Atiyah–Hirzebruch spectral sequence

E_{s,t}^{2} = H_{s}(X; π_{t}K(n)) ⟹ K(n)_{s+t}(X)

shows that if H_{∗}(X; 𝔽_{p}) is a (totally) finite 𝔽_{p}-module, then K(n)_{∗}(X) is a finitely generated K(n)_{∗}-module for each 0 < n < ∞. □

Remark 2.2.5. More generally, for natural numbers n < m it is not true that a K(m)-locally dualizable spectrum X must be K(n)-locally dualizable. The lemma above would correspond to the case m = ∞. The case X = K(n) provides an easy counterexample, with K(m)_{∗}(K(n)) = 0 and K(n)_{∗}(K(n)) infinitely generated over K(n)_{∗}. However, we are principally interested in unreduced suspension spectra X = Σ^{∞}T_{+}, in which case the issue is: Does K(m)_{∗}(T) being finite in each degree imply that K(n)_{∗}(T) is finite in each degree, for a topological space T? Replacing “finite in each degree” by “trivial” in this statement, it becomes a theorem of Bousfield [Bo99, 1.1], with a different proof under a finite type hypothesis by Wilson [Wi99, 1.1]. It is not clear to the author whether either of these proofs can be adapted to resolve the stronger question.

2.3. Stably dualizable groups

Let G be a topological group. We write S[G] = S ∧ G_{+} = L_{E}Σ^{∞}G_{+} for the Elocalization of the unreduced suspension spectrum on G, and DG_{+} = F (S[G], S) = F (G_{+}, L_{E}Σ^{∞}S^{0}) for its functional dual. We may always suppose that G is cofibrantly based and of the homotopy type of a based CW-complex.

Definition 2.3.1. A topological group G is E-locally stably dualizable if

S[G] = L_{E}Σ^{∞}G_{+}

is dualizable in ℳ_{S,E}.

Lemma 2.3.2. The product G = G_{1} × G_{2} of two E-locally stably dualizable groups is again E-locally stably dualizable.

Proof. If S[G_{1}] and S[G_{2}] are dualizable, then so is S[G] ≅ S[G_{1}] ∧ S[G_{2}], as remarked after Lemma 2.2.1. □

The examples of Section 2.2 carry over as follows. When E is clear from the context, we omit to say “E-locally”.

Example 2.3.3. If E = S, then G is a stably dualizable group if and only if G_{+} is stably equivalent to a finite CW complex, up to an integer suspension. So each compact Lie group G is stably dualizable, since G itself is then a finite CW complex.

2.4. E-COMPACT GROUPS 109

Example 2.3.4. For E = H𝔽_{p}, a topological group G is stably dualizable if and only if H_{∗}(G; 𝔽_{p}) is a (totally) finite 𝔽_{p}-module.

Example 2.3.5. For E = K(n), a topological group G is stably dualizable if and only if K(n)_{∗}(G) is a finitely generated K(n)_{∗}-module. By the calculations of Ravenel and Wilson [RaW80, 11.1] for p odd, and [JW85, Appendix] for p = 2, each Eilenberg–Mac Lane space G = K(π, q) = B^{q}π for π a finite abelian group is a stably dualizable group. More generally, by [HRW98, 1.1] any topological group G with only finitely many nonzero homotopy groups, each of which is a finite abelian p-group, has finite K(n)-homology, hence is stably dualizable.

Remark 2.3.6. By Lemma 2.2.4, compact Lie groups or p-compact groups provide examples of K(n)-locally stably dualizable groups, since they have finite mod p homology, and therefore have finite Morava K-theory. The Eilenberg–Mac Lane space examples above, for q ≥ 1, do not arise in this fashion, since they have infinite mod p homology. By Example 2.3.4 they do not extend to stably dualizable groups in the p-complete or integral category.

Building on Remark 2.2.5, if it turns out that K(m)_{∗}(G) being finite over K(m)_{∗} implies that K(n)_{∗}(G) is finite over K(n)_{∗}, for topological groups G and natural numbers n < m, then each K(m)-locally stably dualizable group will also be a K(n)-locally stably dualizable group. By [Bo99, 1.1], each K(m)-equivalence G_{1} → G_{2} is also a K(n)-equivalence, so there will then be a “chromatic” tower of K(n)-equivalence classes of K(n)-locally stably dualizable groups, for 1 ≤ n ≤ ∞, with maps from the set at height m to the set at height n, for all n < m.

2.4. E-compact groups

The material in this section is included to enable a more precise comparison with the Dwyer–Wilkerson theory of p-compact groups, but is not needed elsewhere in the paper.

Suppose that the spectrum E is in fact an S-algebra [EKMM97]. This includes all the examples E = HR for rings R, E = S, E = E(n) and E = K(n) considered above, although the S-algebra structure on e.g. K(n) is not at all unique. We now consider a version of Bousfield–Kan p-completion for the homology theory represented by E, following [BCM78] and [BT00, §2]. Let Ω^{∞}E denote the underlying infinite loop space of E, so that Σ^{∞} is left adjoint to Ω^{∞}. The following terminology extends that of [BK72, §I.5].

Definition 2.4.1. (a) Let E(X) = Ω^{∞}(E ∧ Σ^{∞}X) define an endofunctor of based topological spaces, with π_{∗}E(X) = Ẽ_{∗}(X) for all ∗ ≥ 0. The S-algebra multiplication μ: E ∧ E → E and the adjunction counit Σ^{∞}Ω^{∞}E → E induce a natural transformation μ_{X}: (E ∘ E)(X) = E(E(X)) → E(X). The S-algebra unit η: S → E and the adjunction unit X → Q(X) = Ω^{∞}Σ^{∞}X induce a natural transformation η_{X}: X → E(X). These make (E(−), μ, η) a monad.

(b) For each space X, the E-completion X_{E}^{∧} = Tot E(X)^{•} is defined as the totalization of the cosimplicial space

[q] ↦ E(X)^{q} = (E ∘ ⋯ ∘ E)(X)

(the (q + 1)-fold iterate of the endofunctor E(−)), with coface and codegeneracy maps induced by μ and η, respectively. There is a natural map X → X_{E}^{∧}. We say that X is E-complete if this map is a weak equivalence.

(c) Each space E(X)^{q} is E-local, so X_{E}^{∧} is E-local and there is a canonical factorization X → L_{E}X → X_{E}^{∧} through the Bousfield localization to the completion. We say that X is E-good if X → X_{E}^{∧} is an E-equivalence, or equivalently, if L_{E}X ≃ X_{E}^{∧}.

Definition 2.4.2. An E-compact group is an E-locally stably dualizable group G whose classifying space BG ≃ (BG)_{E}^{∧} is E-complete.

Example 2.4.3. If E = S, then S-completion equals Hℤ-completion [Ca91, II.3], so nilpotent spaces are S-complete. A connected compact Lie group G has simply-connected, thus nilpotent, classifying space BG, so such a group G is also an S-compact group.

Example 2.4.4. If E = H𝔽_{p}, H𝔽_{p}-completion equals p-completion, so a topological group G is H𝔽_{p}-compact if and only if G ≃ ΩBG is a p-compact group in the sense of Dwyer and Wilkerson [DW94].

The loop space Ω(BG)_{p}^{∧} of the p-completed classifying space of a connected compact Lie group provides some examples of a p-compact group, but there are also exotic examples, such as (i) the p-compact Sullivan sphere

(S^{2p−3})_{p}^{∧} = Ω(EW ×_{W} BA)_{p}^{∧}

for p odd, with A = Bℤ_{p}, BA = K(ℤ_{p}, 2) and W = (ℤ/p)^{∗} acting on A through multiplication by the p-adic roots of unity, and (ii) the 2-compact Dwyer–Wilkerson group DI(4) [DW93]. With the exception of (S^{3})_{3}^{∧} these only exist locally, in the sense that they do not extend to integrally defined stably dualizable groups.

Example 2.4.5. For q ≥ n and π a finite abelian group the Eilenberg–Mac-Lane spaces G = K(π, q) have K(n)_{∗}(BG) = K(n)_{∗} by [RaW80, 11.1], hence these can never be nontrivial K(n)-compact groups. When 0 ≤ q < n and π is a finite abelian p-group it is known that BG = K(π, q + 1) is K(n)-local by [Bo82, 7.4], so if these classifying spaces are also K(n)-good, then the Eilenberg–Mac Lane spaces K(π, q) are K(n)-compact groups.

Remark 2.4.6. It is to be expected that connected compact Lie groups, pcompact groups or the more exotic K(n)-locally stably dualizable groups of Example 2.3.5 provide examples of K(n)-compact groups through K(n)-completion at the level of classifying spaces. However, it is not clear in what generality the natural map

G → Ω(BG)_{K(n)}^{∧}

is a K(n)-equivalence. There is a K(n)-based Eilenberg–Moore spectral sequence [JO99]

(2.4.7)  E_{2}^{s,t} = T̂or_{K(n)^{∗}(Y)}^{s,t}(K(n)^{∗}, K(n)^{∗}) ⟹ K(n)^{s+t}(ΩY) ,

where T̂or^{s} = T̂or_{−s} are the left derived functors of the completed tensor product, but too little is known about its convergence. Certainly the space Y should be K(n)-local for this to have a chance, but by analogy with the mod p case, it is more plausible that the correct condition is that Y should be K(n)-complete. Since a K(n)-complete space is a limit of spaces of the form Ω^{∞}(K(n) ∧ T), it may suffice to verify convergence for such spaces, or for the individual spaces Y = \underline{K(n)}_{q} of the Ω-spectrum of K(n), for q ≥ 0.

Bauer has made some progress in this direction. So if (i) G is a K(n)-locally stably dualizable group, (ii) BG is K(n)-good, (iii) K(n)^{∗}(BG) is a finitely generated power series ring over K(n)^{∗}, or more generally, T̂or_{K(n)^{∗}(BG)}^{∗∗}(K(n)^{∗}, K(n)^{∗}) is finite over K(n)^{∗}, and (iv) the K(n)-based Eilenberg–Moore spectral sequence for Y = (BG)_{K(n)}^{∧} converges, then it follows that ΩY = Ω(BG)_{K(n)}^{∧} is a K(n)-compact group.

Remark 2.4.8. The examples that are abelian topological groups can be expected to play a similar role to that of tori in the theory of compact Lie groups. For non-abelian examples it is natural to look to finite Postnikov systems, as in [HRW98], or to looped completed Borel constructions of the form

G = Ω(EW ×_{W} BA)_{K(n)}^{∧}

where A is an abelian topological group, such as A = K(π, q), the Weyl group W is a finite group acting on A from the left, EW ×_{W} BA = B(W ⋉ A) is the classifying space of the semi-direct product W ⋉ A and (−)_{K(n)}^{∧} denotes the K(n)-completion of spaces. To analyze the K(n)-homology of G it is again necessary to study the convergence properties of the K(n)-based Eilenberg–Moore spectral sequence in the path–loop fibration of Y = B(W ⋉ A)_{K(n)}^{∧}.

Remark 2.4.9. In his Master’s thesis, Håkon Schad Bergsaker [Be05, 6.6] has shown that for A = K(ℤ_{p}, n) and W = (ℤ/p)^{∗} acting through multiplication by the p-adic roots of unity, the construction above produces a loop space (or topological group) G with the Morava K-theory of a (2m − 1)-sphere, for m = (p^{n} − 1)/ gcd(p − 1, n), under the assumption that the appropriate K(n)-based Eilenberg–Moore spectral sequence (2.4.7) converges. Conversely, he shows [Be05, 4.15] that for each prime p and height n there are only finitely many m for which G = L_{K(n)}S^{2m−1} can be a K(n)-locally stably dualizable group, assuming the existence of a map t: Bℤ/p → BG with nontrivial K(n)^{∗}(t).

2.5. The dualizing and inverse dualizing spectra

Let EG = B(∗, G, G) be the usual free, contractible right G-space. Let X be a spectrum with right G-action, and let Y be a spectrum with left G-action. We define the G-homotopy fixed points of X to be

X^{hG} = F (EG_{+}, X)^{G}

and the G-homotopy orbits of Y to be

Y_{hG} = EG_{+} ∧_{G} Y .

In all cases G acts on EG from the right. These constructions only involve naive G-equivariant spectra, or spectra with G-action, in the sense that no deloopings with respect to non-trivial G-representations are involved. Each G-equivariant map X_{1} → X_{2} that is an equivalence induces an equivalence X_{1}^{hG} → X_{2}^{hG} of homotopy fixed points, and similarly for homotopy orbits.

Definition 2.5.1. Let G be an E-locally stably dualizable group. The group multiplication provides the suspension spectrum S[G] = L_{E}Σ^{∞}G_{+} with mutually commuting standard left and right G-actions. We define the dualizing spectrum S^{adG} of G to be the G-homotopy fixed point spectrum

S^{adG} = S[G]^{hG} = F (EG_{+}, S[G])^{G}

of S[G], formed with respect to the standard right G-action. The standard left action on S[G] induces a left G-action on S^{adG}.

Remark 2.5.2. A discrete group G of type F P (e.g. the classifying space BG is finitely dominated) is called a duality group if H^{∗}(G; ℤ[G]) is concentrated in a single degree n and torsion free. The G-module D = H^{n}(G; ℤ[G]) is then called the dualizing module of G, cf. [Brn82, VIII.10]. The spectrum level construction above is analogous to this algebraic notion, and was previously considered for topological groups by Dwyer and by Klein [Kl01, §1], and for p-compact groups by Bauer [Ba04, 4.1]. However, in algebra the focus is on G with a finiteness condition on BG, whereas in the topological cases G itself satisfies a finiteness condition. Klein writes D_{G} and Bauer writes S_{G} for the dualizing spectrum of G. We use D for the functional dual and S for the sphere spectrum, so we prefer to write S^{adG} instead, in view of the compact Lie group example recalled immediately below. Our construction differs a tiny bit from that of Dwyer and Klein, due to our implicit E-localization.

Examples 2.5.3. (a) When G is a finite group, there is a canonical equivalence S[G] = S ∧ G_{+} ≃ F (G_{+}, S), so S[G]^{hG} ≃ F (G_{+}, S)^{hG} ≅ F (EG_{+}, S) ≃ S is naturally equivalent to the sphere spectrum.

(b) More generally, when G is a compact Lie group Klein [Kl01, 10.1] shows that the dualizing spectrum S^{adG} is equivalent as a spectrum with left G-action to the suspension spectrum of the representation sphere associated to the adjoint representation adG of G, i.e., the left conjugation action of G on its tangent space T_{e}G at the identity.

(c) In the case of a p-compact group G, Bauer [Ba04] shows that S^{adG} ≃ (S^{d})_{p}^{∧} for some integer d = dim_{p} G called the p-dimension of G, and that S^{adG} takes over the role of the representation sphere in the duality theory in that context. The present paper extends some of Bauer’s work to the E-local stable category.

Lemma 2.5.4. When G is abelian, the left G-action on S^{adG} is homotopically trivial, in the sense that it extends over the inclusion G ⊂ EG to an action by the contractible topological group EG.

Proof. When G is abelian, the left and right G-actions on S[G] agree. In S^{adG} = F (EG_{+}, S[G])^{G} the right action on S[G] is equal to the right action on EG_{+}, which in the commutative case factors as

EG_{+} ∧ G_{+} ⊂ EG_{+} ∧ EG_{+} ⟶ EG_{+} .□

Remark 2.5.5. It can be inconvenient to study the E-homology of S^{adG} directly from its definition as a homotopy fixed point spectrum. We shall soon see that this dualizing spectrum is the functional dual of another spectrum S^{−adG}, which we call the inverse dualizing spectrum, and which admits a computationally more convenient construction as a homotopy orbit spectrum. Once we know that these two spectra are indeed dualizable, and mutually dual, this provides a convenient route to E-homological calculations.

Definition 2.5.6. Let G be a stably dualizable group. The left and right G-actions on S[G] induce standard right and left G-actions on its functional dual DG_{+} = F (S[G], S), respectively, by acting in the source of the mapping spectrum. We define the inverse dualizing spectrum S^{−adG} of G to be the G-homotopy orbit spectrum

S^{−adG} = (DG_{+})_{hG} = EG_{+} ∧_{G} DG_{+}

of DG_{+}, formed with respect to the standard left G-action. These left and right actions commute, so the standard right action on DG_{+} induces a right G-action on S^{−adG}.

Proposition 2.5.7. There is a natural equivalence

S^{adG} ≃ DS^{−adG}

between the dualizing spectrum and the functional dual of the inverse dualizing spectrum, as spectra with left G-action.

Proof. The canonical equivalence ρ: S[G] → DDG_{+} = F (DG_{+}, S) induces an equivalence ρ^{hG} of G-homotopy fixed points, from S^{adG} to

F (DG_{+}, S)^{hG} = F (EG_{+}, F (DG_{+}, S))^{G} ≅ F (EG_{+} ∧_{G} DG_{+}, S) = DS^{−adG} .□

CHAPTER 3 Duality theory

3.1. Poincaré duality

Let G be a stably dualizable group. The topological group structure on G makes S[G] a cocommutative Hopf S–algebra, with product φ: S[G] ∧ S[G] → S[G], unit η: S → S[G], coproduct ψ: S[G] → S[G] ∧ S[G], counit ϵ: S[G] → S and conjugation (antipode) χ: S[G] → S[G], induced by the group multiplication m: G × G → G, unit inclusion {e} → G, diagonal map Δ: G → G × G, collapse map G → {e} and group inverse i: G → G, respectively.

The product φ and unit η makes S[G] an E-local S-algebra in ℳ_{S,E}, while the coproduct, counit and conjugation need only be defined in the E-local stable category 𝒟_{S,E}.

The standard right G-action on DG_{+} makes DG_{+} a right S[G]-module. The module action is given by the map

α: DG_{+} ∧ S[G] ⟶ DG_{+}

that in symbols takes ξ∧x to ξ∗x: y ↦ ξ(xy). Inspired by [Ba04, §4.3], we consider the following shearing equivalence. Its definition is simpler than that considered by Bauer, but the key idea is the same.

Definition 3.1.1. Let the shear map sh: DG_{+} ∧ S[G] → DG_{+} ∧ S[G] be the composite map

sh: DG_{+} ∧ S[G] ⟶^{1∧ψ} DG_{+} ∧ S[G] ∧ S[G] ⟶^{α∧1} DG_{+} ∧ S[G] .

Algebraically, sh: ξ ∧ x ↦ ∑(ξ ∗ x′) ∧ x′′ where ψ(x) = ∑ x′ ∧ x′′.

The standard left and right G-actions on S[G] (and DG_{+}) can be converted into right and left G-actions on S[G] (and DG_{+}), respectively, by way of the group inverse i: G → G. We refer to these non-standard actions as actions through inverses. For example, the left G-action through inverses on DG_{+} is given by the composite map

S[G] ∧ DG_{+} ⟶_{≅}^{γ} DG_{+} ∧ S[G] ⟶^{1∧χ} DG_{+} ∧ S[G] ⟶^{α} DG_{+} ,

where γ: X ∧ Y → Y ∧ X denotes the canonical twist map. Algebraically, this action takes (x, ξ) to ξ ∗ χ(x): y ↦ ξ(χ(x)y).

Lemma 3.1.2. The shear map sh is equivariant with respect to each of the following three mutually commuting G-actions:

(1) The first, left G-action given by the action through inverses on DG_{+} and the standard action on S[G] in the source, and the standard action on S[G] in the target; (2) The second, right G-action given by the action through inverses on DG_{+} in the source, and the action through inverses on DG_{+} in the target;

(3) The third, right G-action given by the standard action on S[G] in the source

and by the standard actions on DG_{+} and S[G] in the target. Each action is trivial on the remaining smash factors.

Proof. In each case this is clear by inspection. □

Lemma 3.1.3. The shear map sh is an equivalence, with homotopy inverse given by the composite map

DG_{+} ∧S[G] ⟶^{1∧ψ} DG_{+} ∧S[G]∧S[G] ⟶^{1∧χ∧1} DG_{+} ∧S[G]∧S[G] ⟶^{α∧1} DG_{+} ∧S[G] .

Proof. This is an easy diagram chase, using coassociativity of ψ, the fact that α is a right S[G]-module action with respect to the product φ on S[G], the Hopf conjugation identities φ(χ ∧ 1)ψ ≃ ηϵ ≃ φ(1 ∧ χ)ψ, counitality for ψ and unitality for α. □

Theorem 3.1.4. Let G be a stably dualizable group. There is a natural equivalence

DG_{+} ∧ S^{adG} ⟶^{≃} S[G] .

It is equivariant with respect to the first, left G-action through inverses on DG_{+}, the standard left action on S^{adG} and the standard left action on S[G]. It is also equivariant with respect to the second, right G-action through inverses on DG_{+}, the trivial action on S^{adG} and the standard right action on S[G].

Proof. The shear equivalence sh: DG_{+} ∧ S[G] → DG_{+} ∧ S[G] induces a natural equivalence

(sh)^{hG}: (DG_{+} ∧ S[G])^{hG} ⟶^{≃} (DG_{+} ∧ S[G])^{hG}

of G-homotopy fixed points with respect to the third, right G-action. Note that this action is different in the source and in the target of sh.

There is a natural equivalence to the source of (sh)^{hG}:

DG_{+} ∧ S^{adG} = DG_{+} ∧ S[G]^{hG} ⟶^{≃} (DG_{+} ∧ S[G])^{hG} .

To see that this map is an equivalence, consider the commutative square

\xymatrix{DG_{+} ∧ S[G]^{hG} \ar[r] \ar[d]^{≃} & (DG_{+} ∧ S[G])^{hG} \ar[d]^{≃} \\ F (G_{+}, S[G]^{hG}) \ar[r]^{≅} & F (G_{+}, S[G])^{hG} .}

The vertical maps are equivalences, because S[G] is dualizable and passage to homotopy fixed points respects equivalences. Hence the upper horizontal map is also an equivalence.

There is also a (composite) natural equivalence from the target of (sh)^{hG}:

(DG_{+} ∧ S[G])^{hG} ⟶^{≃} F (G_{+}, S[G])^{hG} ⟶^{≃} S[G] .

The left hand map is an equivalence because S[G] is dualizable, by the same argument as above. The right hand map is the composite equivalence

F (G_{+}, S[G])^{hG} ≅ F (EG_{+} ∧ G_{+}, S[G])^{G} ≅ F (EG_{+}, S[G]) ⟶^{≃} S[G] .

Here the middle isomorphism uses that G acts freely on G_{+} in the source.

The composite of these three natural equivalences is the desired natural equivalence DG_{+} ∧ S^{adG} → S[G]. The equivariance statements follow by inspection. □

Remark 3.1.5. We call DG_{+} ∧ S^{adG} ≃ S[G] the Poincaré duality equivalence. It shows how S[G] is functionally self-dual, up to a shift by the dualizing spectrum. See also Remark 3.3.5. The equivariance statements in the theorem express the standard left and trivial right G-actions on S^{adG} in terms of the more familiar G-actions on DG_{+} and S[G].

Lemma 3.1.6. Let G_{1} and G_{2} be stably dualizable groups. There is a natural equivalence

S^{adG_{1}} ∧ S^{adG_{2}} ≃ S^{ad(G_{1}×G_{2})}

of spectra with standard left (and trivial right) (G_{1} × G_{2})-actions.

Proof. The Poincaré duality equivalences for G_{1}, G_{2} and (G_{1} × G_{2}) compose to an equivalence

DG_{1+} ∧ S^{adG_{1}} ∧ DG_{2+} ∧ S^{adG_{2}} ≃ S[G_{1}] ∧ S[G_{2}]

≃ S[G_{1} × G_{2}] ≃ D(G_{1} × G_{2})_{+} ∧ S^{ad(G_{1}×G_{2})} .

It is equivariant with respect to the first, left (G_{1} × G_{2})-action that involves the standard left action on S^{adG_{1}}, S^{adG_{2}} and S^{ad(G_{1}×G_{2})}, as well as with respect to the second, right (G_{1}×G_{2})-action through inverses on DG_{1+}∧DG_{2+} and D(G_{1}×G_{2})_{+}. Taking homotopy fixed points with respect to the second, right action we obtain the desired equivalence, which is equivariant with respect to the first, left action.□

Remark 3.1.7. A similar relation S^{adG} ≃ S^{adH} ∧ S^{adQ} is likely to hold for an extension 1 → H → G → Q → 1 of stably dualizable groups, cf. [Kl01, Thm. C], but for simplicity we omit the then necessary discussion of how to promote S^{adH} to a spectrum with G-action, etc.

3.2. Inverse Poincaré duality

The aim of this section is to establish an inverse Poincaré equivalence

S[G] ∧ S^{−adG} ≃ DG_{+} .

The initial idea is to functionally dualize the construction of the shear map in Section 3.1, and to apply homotopy orbits in place of homotopy fixed points. Following Milnor and Moore [MM65, §3], we identify the functional dual of a smash product X ∧ Y of dualizable spectra with the smash product DX ∧ DY, in that order, via the canonical equivalence

DX ∧ DY = F (X, S) ∧ F (Y, S) ⟶_{≃}^{∧} F (X ∧ Y, S ∧ S) = D(X ∧ Y) .

However, to form homotopy orbits we need genuine G-equivariant maps, and it is generally not the case that a G-equivariant inverse can be found for the (weak) equivalence displayed above. Thus some care will be in order.

Working for a moment in the E-local stable category 𝒟_{S,E} = h̄ℳ_{S,E}, let

β: S[G] ⟶ S[G] ∧ DG_{+}

be dual to the module action map α: DG_{+} ∧ S[G] → DG_{+}. It makes S[G] a right DG_{+}-comodule spectrum, up to homotopy, where DG_{+} has the weakly defined coproduct ψ′: DG_{+} → DG_{+} ∧ DG_{+} that is dual to φ. Furthermore, let

φ′: DG_{+} ∧ DG_{+} ⟶ DG_{+}

be the (strictly defined) product on DG_{+} that is dual to ψ. The functional dual sh^{#} of the shear map is then the composite

sh^{#} : S[G] ∧ DG_{+} ⟶^{β∧1} S[G] ∧ DG_{+} ∧ DG_{+} ⟶^{1∧φ′} S[G] ∧ DG_{+} ,

which is an equivalence by Lemma 3.1.3 and duality.

Returning to the category ℳ_{S,E}, we shall now obtain G-equivariant representatives for these maps.

Definition 3.2.1. Let φ̃: S[G] → F (S[G], S[G]) be right adjoint to the opposite product map φγ: S[G] ∧ S[G] → S[G]. Algebraically, φ̃: x ↦ (y ↦ yx). Let ψ^{#} : F (S[G] ∧ S[G], S[G] ∧ S) → F (S[G], S[G]) be given by precomposition by ψ: S[G] → S[G] ∧ S[G] and postcomposition with S[G] ∧ S ≅ S[G].

The dual shear map sh′: S[G] ∧ DG_{+} → F (S[G], S[G]) is defined to be the composite map:

sh′: S[G] ∧ DG_{+} ⟶^{φ̃∧1} F (S[G], S[G]) ∧ DG_{+}

⟶_{≃}^{∧} F (S[G] ∧ S[G], S[G] ∧ S) ⟶^{ψ^{#}} F (S[G], S[G]) .

It is equivariant with respect to the left G-action given by the standard left actions on S[G] and DG_{+} on the left hand side, and the left action through the standard right action on the S[G] in the source of the mapping spectrum.

Theorem 3.2.2. The dual shear map sh′ is homotopic to the composite map

S[G] ∧ DG_{+} ⟶_{≃}^{sh^{#}} S[G] ∧ DG_{+} ⟶_{≃}^{νγ} F (S[G], S[G]) .

In particular, sh′ is an equivalence. On G-homotopy orbit spectra it induces an equivalence

DG_{+} ≃ S[G] ∧ S^{−adG} .

Proof. The right action map α factors up to homotopy as the composite

DG_{+} ∧ S[G] ⟶^{ψ′∧1} DG_{+} ∧ DG_{+} ∧ S[G]

⟶^{1∧γ} DG_{+} ∧ S[G] ∧ DG_{+} ⟶^{ϵ∧1} S ∧ DG_{+} = DG_{+} .

Here ϵ: DG_{+} ∧ S[G] → S is the pairing that evaluates a function on an element in its source. Let η: S[G] ∧ DG_{+} → S be its functional dual, in the homotopy category. Then the dual map β factors up to homotopy as

S[G] ≅ S ∧ S[G] ⟶^{η∧1} S[G] ∧ DG_{+} ∧ S[G]

⟶^{1∧γ} S[G] ∧ S[G] ∧ DG_{+} ⟶^{φ∧1} S[G] ∧ DG_{+} .

A diagram chase then verifies that φ̃ is homotopic to the composite

S[G] ⟶^{β} S[G] ∧ DG_{+} ⟶_{≅}^{γ} DG_{+} ∧ S[G] ⟶_{≃}^{ν} F (S[G], S[G]) .

A similar chase shows that the diagram

\xymatrix{S[G] ∧ DG_{+} ∧ DG_{+} \ar[rr]^{1∧φ′} \ar[d]^{νγ∧1}_{≃} &  & S[G] ∧ DG_{+} \ar[d]^{νγ}_{≃} \\ F (S[G], S[G]) ∧ DG_{+} \ar[r]^{∧}_{≃} & F (S[G] ∧ S[G], S[G] ∧ S) \ar[r]^{ψ^{#}} & F (S[G], S[G])}

homotopy commutes.

Taken together, these diagrams show that νγ ∘ sh^{#} ≃ sh′. Applying Ghomotopy orbits to the chain of equivalences

S[G] ∧ DG_{+} ⟶_{≃}^{sh′} F (S[G], S[G]) ⟵_{≃}^{νγ} S[G] ∧ DG_{+}

we obtain the desired chain of equivalences

DG_{+} ≃ (S[G] ∧ DG_{+})_{hG} ⟶_{≃}^{(sh′)_{hG}} F (S[G], S[G])_{hG}

⟵_{≃}^{(νγ)_{hG}} (S[G] ∧ DG_{+})_{hG} ≃ S[G] ∧ S^{−adG} . □

Proposition 3.2.3. Let G be a stably dualizable group. The dualizing spectrum S^{adG} and the inverse dualizing spectrum S^{−adG} are both dualizable spectra. Hence

S^{−adG} ≃ DS^{adG}

as spectra with right G-action. The inverse Poincaré equivalence

S[G] ∧ S^{−adG} ≃ DG_{+}

is equivariant with respect to the dual G-actions to those of Theorem 3.1.4: The first of these is the right G-action through inverses on S[G], the standard right action on S^{−adG} and the standard right action on DG_{+}. The second is the left G-action through inverses on S[G], the trivial action on S^{−adG} and the standard left action on DG_{+}.

Proof. It suffices to prove that S^{−adG} is dualizable, in view of Proposition 2.5.7 and Theorem 3.1.4. We must show that the canonical map

ν: DS^{−adG} ∧ S^{−adG} ⟶ F (S^{−adG}, S^{−adG})

is an equivalence. We first check that ν smashed with the identity map of S[G] is an equivalence. This map factors as the composite

DS^{−adG} ∧ S^{−adG} ∧ S[G] ≃ DS^{−adG} ∧ DG_{+} ⟶_{≃}^{ν} F (S^{−adG}, DG_{+})

≃ F (S^{−adG}, S^{−adG} ∧ S[G]) ⟵_{≃}^{ν} F (S^{−adG}, S^{−adG}) ∧ S[G] .

Here the first and third equivalences follow from the inverse Poincaré equivalence, while the second and fourth equivalences are consequences of the dualizability of DG_{+} and S[G], respectively. Thus ν ∧ 1_{S[G]} is an equivalence. Since S is a retract of S[G], it follows that also ν itself is an equivalence. Hence S^{−adG} is dualizable.□

3.3. The Picard group

The Picard group of the category of E-local S-modules was introduced by M. Hopkins; see [HMS94].

Definition 3.3.1. An E-local S-module X is invertible if there exists a spectrum Y with X ∧ Y ≃ S in ℳ_{S,E}. Then Y is also invertible. The smash product X ∧ X′ of two invertible spectra X and X′ is again invertible.

The E-local Picard group Pic_{E} = Pic(ℳ_{S,E}) is the set of equivalence classes of invertible E-local S-modules. We write [X] ∈ Pic_{E} for the equivalence class of X. The abelian group structure on Pic_{E} is defined by [X] + [X′] = [X ∧ X′] and −[X] = [Y], with X, Y and X′ as above.

Example 3.3.2. The only invertible spectra in ℳ_{S} are the sphere spectra S^{d} = Σ^{d}S for integers d ∈ ℤ, so Pic_{S} ≅ ℤ. Similarly, in the p-complete category ℳ_{S,H𝔽_{p}} the invertible spectra are precisely the p-completed sphere spectra (S^{d})_{p}^{∧} for d ∈ ℤ, so Pic_{H𝔽_{p}} ≅ ℤ too.

Example 3.3.3. Hopkins, Mahowald and Sadofsky [HMS94, 1.3] show that a K(n)-local spectrum X is invertible if and only if K(n)_{∗}(X) is free of rank one over K(n)_{∗}. These authors also prove [HMS94, 2.1, 2.7, 3.3] that for n = 1 and p ≠ 2 there is a non-split extension

0 → ℤ_{p}^{×} ⟶ Pic_{K(1)} ⟶ ℤ/2 → 0

while for n = 1 and p = 2 there is a non-split extension

0 → ℤ_{2}^{×} ⟶ Pic_{K(1)} ⟶ ℤ/8 → 0 .

Furthermore, they show [HMS94, 7.5] that when n^{2} ≤ 2p−2 and p > 2 there is an injection α: Pic_{K(n)} → H^{1}(𝕊_{n}; π_{0}(E_{n})^{×}), where E_{n} is the Hopkins–Miller commutative S-algebra and 𝕊_{n} is (one of the variants of) the n-th Morava stabilizer group. This permits an algebraic identification of Pic_{K(2)} for p odd. The homomorphism α seems to have a non-trivial kernel for n = 2 and p = 2, cf. [HMS94, §6].

Theorem 3.3.4. Let G be a stably dualizable group. Then

S^{adG} ∧ S^{−adG} ≃ S

so S^{adG} and S^{−adG} are mutually inverse invertible spectra in the E-local stable category. Hence the equivalence classes [S^{adG}] and [S^{−adG}] represent inverse elements in the E-local Picard group Pic_{E}.

Proof. The Poincaré duality equivalence and the inverse Poincaré equivalence provide a chain of equivalences

S[G] ∧ S^{−adG} ∧ S^{adG} ≃ DG_{+} ∧ S^{−adG} ≃ S[G] ,

which is equivariant with respect to the standard left action on both copies of S[G], the trivial action on S^{−adG} and the standard left action on S^{adG}. Taking G-homotopy orbits of both sides yields the required equivalence

S^{−adG} ∧ S^{adG} ≃ S[G]_{hG} ≃ S .□

Remark 3.3.5. These results show that the shift given by smashing with S^{adG}, as in the Poincaré duality equivalence, is really an invertible self-equivalence of the stable homotopy category of spectra with G-action, in that it can be undone by smashing with S^{−adG} ≃ DS^{adG}.

Definition 3.3.6. Let the E-dimension of G be the equivalence class

dim_{E}(G) = [S^{adG}] ∈ Pic_{E}

of the dualizing spectrum S^{adG} in the E-local Picard group.

Example 3.3.7. For E = S the S-dimension of a compact Lie group G equals its manifold dimension in Pic_{S} ≅ ℤ. Similarly, for E = H𝔽_{p} the H𝔽_{p}-dimension of a p-compact group G is the same as its p-dimension.

CHAPTER 4 Computations

4.1. A spectral sequence for E-homology

Suppose that the S-module E is an S-algebra. The standard left G-action α′ on DG_{+} makes E_{∗}(DG_{+}) = E^{−∗}(G) a left E_{∗}(G)-module via the composite action map

E_{∗}(G) ⊗ E_{∗}(DG_{+}) → E_{∗}(S[G] ∧ DG_{+}) ⟶^{α′_{∗}} E_{∗}(DG_{+}) .

Proposition 4.1.1. Let E be an S-algebra and let G be a stably dualizable group. There is a spectral sequence

E_{s,t}^{2} = Tor_{s,t}^{E_{∗}(G)}(E_{∗}, E^{−∗}(G)) ⟹ E_{s+t}(S^{−adG})

converging strongly to E_{∗}(S^{−adG}) ≅ E^{−∗}(S^{adG}).

Proof. This is the E-homology homotopy orbit spectral sequence, which is a special case of the Eilenberg–Moore type spectral sequence [EKMM97, IV.6.4] for the E-homology of

S^{−adG} = EG_{+} ∧_{G} DG_{+} ≅ S[EG] ∧_{S[G]} DG_{+} .

(Other names in use are the bar spectral sequence and the Steenrod–Rothenberg spectral sequence.) Here E_{∗}(S[EG]) ≅ E_{∗}, E_{∗}(S[G]) ≅ E_{∗}(G) and E_{∗}(DG_{+}) ≅ E^{−∗}(G). The duality S^{−adG} ≃ DS^{adG} from Proposition 3.2.3 relates the abutment to the E-cohomology of S^{adG}. □

4.2. Morava K-theories

In this and the following section (4.3) we specialize to the case when E = K(n), for some fixed prime p and number 0 ≤ n ≤ ∞. Hence stably dualizable means K(n)-locally stably dualizable, etc.

Lemma 4.2.1. Let G be a stably dualizable group, so that H = K(n)_{∗}(G) is a finitely generated (free) module over R = K(n)_{∗}. Then H is a graded cocommutative Hopf algebra over R, and its R-dual H^{∗} = K(n)^{∗}(G) = Hom_{R}(H, R) is a graded commutative Hopf algebra over R.

Proof. By [HSt99, 8.6], a topological group G is stably dualizable if and only if H = K(n)_{∗}(G) is finitely generated over R = K(n)_{∗}. The group multiplication and diagonal map on G induce the Hopf algebra structure on H, in view of the Künneth isomorphism

K(n)_{∗}(X) ⊗_{K(n)_{∗}} K(n)_{∗}(Y) ⟶^{≅} K(n)_{∗}(X ∧ Y)

in the case X = Y = S[G]. The identity K(n)^{∗}(G) ≅ Hom_{R}(H, R) is a case of the universal coefficient theorem

K(n)^{∗}(X) ⟶^{≅} Hom_{K(n)_{∗}}(K(n)_{∗}(X), K(n)_{∗}) .

This also leads to the Hopf algebra structure on H^{∗}. □

Proposition 4.2.2. Let G be a stably dualizable group. Then K(n)_{∗}(S^{adG}) ≅ Σ^{d}R for some integer d, and K(n)_{∗}(S^{−adG}) ≅ Σ^{−d}R.

Proof. By Theorem 3.3.4, S^{adG} is an invertible K(n)-local spectrum with inverse S^{−adG}, so by the Künneth theorem

K(n)_{∗}(S^{adG}) ⊗_{R} K(n)_{∗}(S^{−adG}) ≅ K(n)_{∗}(S) = R .

This implies that K(n)_{∗}(S^{adG}) and K(n)_{∗}(S^{−adG}) both have rank one over R. (Alternatively, use Theorem 3.1.4 and the Künneth theorem to obtain the isomorphism

H^{∗} ⊗_{R} K(n)_{∗}(S^{adG}) ≅ H .

The total ranks of H^{∗} and H as R-modules are equal, and finite, so K(n)_{∗}(S^{adG}) must have rank one. In view of [HMS94, 1.3] or [HSt99, 14.2], this also provides an alternative proof that S^{adG} is invertible in the K(n)-local category.) □

Definition 4.2.3. Let the integer d = deg_{K(n)}(G) such that K(n)_{∗}(S^{adG}) ≅ Σ^{d}R be the K(n)-degree of G. When 0 < n < ∞ this number is only well-defined modulo |v_{n}| = 2(p^{n} − 1).

Remark 4.2.4. The evident homomorphism deg: Pic_{K(n)} → ℤ/|v_{n}| takes the K(n)-dimension of G to its K(n)-degree. By [HMS94, 1.3] or [HSt99, 14.2] we also have { \atop E(n)}^{∗}(S^{adG}) ≅ Σ^{d}{ \atop E(n)}^{∗}, where { \atop E(n)} = L_{K(n)}E(n). Similarly E_{n}^{∗}(S^{adG}) ≅ Σ^{d}E_{n}^{∗}, where E_{n} is the Hopkins–Miller commutative S-algebra. Taking into account the action of the n-th Morava stabilizer group on E_{n}^{∗}(S^{adG}) it is in principle possible to recover much more information about the K(n)-dimension of G than just the K(n)-degree.

For any graded commutative ring R and R-algebra H, we may consider both H and its R-dual H^{∗} = Hom_{R}(H, R) as left H-modules in the standard way. Recall from e.g. [Pa71, §4] that H is called a (graded) Frobenius algebra over R if

(1) H is finitely generated and projective as an R-module, and

(2) H and some suspension Σ^{d}H^{∗} are isomorphic as left H-modules. It follows that H is also isomorphic to Σ^{d}H^{∗} as right H-modules, and conversely. A (left or right) module M over a Frobenius algebra H is projective if and only if it is injective.

Proposition 4.2.4. Let G be a stably dualizable group. Then H = K(n)_{∗}(G) is a Frobenius algebra over R = K(n)_{∗}. In particular, H^{∗} = K(n)^{∗}(G) is an injective and projective (left) H-module. In fact, it is free of rank one.

4.2. MORAVA K-THEORIES 123

Proof. Applying K(n)-homology to the equivalence of Theorem 3.1.4 gives an isomorphism

H^{∗} ⊗_{R} Σ^{d}R = K(n)_{∗}(DG_{+}) ⊗_{K(n)_{∗}} K(n)_{∗}(S^{adG}) ≅ K(n)_{∗}(S[G]) = H .

Here H acts from the left via the inverse of the second G-action, i.e., by the standard left action on H^{∗}, the trivial action on K(n)_{∗}(S^{adG}) = Σ^{d}R, and the left action through inverses on H. We continue with the isomorphism

χ_{∗}: H = K(n)_{∗}(G) ⟶^{≅} K(n)_{∗}(G) = H

induced by the conjugation χ on S[G], which takes the left action through inverses to the standard left action. Then the composite of these two isomorphisms exhibits H as a Frobenius algebra over R.

It is a formality that H^{∗} is injective as a left H-module, so the general theory implies that it is also projective. But we can also see this directly in our case, since H^{∗} ≅ Σ^{−d}H is an isomorphism of left H-modules, and the right hand side is free of rank one and thus obviously projective. □

Theorem 4.2.5. Let G be a K(n)-locally stably dualizable group. The spectral sequence

E_{s,t}^{2} = Tor_{s,t}^{H}(R, H^{∗}) ⟹ K(n)_{s+t}(S^{−adG})

collapses to the line s = 0 at the E^{2}-term. The natural map i: DG_{+} → S^{−adG} identifies

Σ^{−d}R = K(n)_{∗}(S^{−adG}) ≅ R ⊗_{H} H^{∗}

with the left H = K(n)_{∗}(G)-module indecomposables of H^{∗} = K(n)_{∗}(DG_{+}) = K(n)^{−∗}(G). Dually, the natural map p: S^{adG} → S[G] identifies

Σ^{d}R = K(n)_{∗}(S^{adG}) ≅ Hom_{H}(H^{∗}, R)

with the left H^{∗}-comodule primitives in H.

Proof. The spectral sequence is that of Proposition 4.1.1 in the special case E = K(n). By Proposition 4.2.4, H^{∗} is a free left H-module of rank one, hence flat. Thus Tor_{s,t}^{H}(R, H^{∗}) = 0 for s > 0, while for s = 0, Tor_{0,∗}^{H}(R, H^{∗}) = R ⊗_{H} H^{∗}. Hence the spectral sequence collapses to the line s = 0, and the edge homomorphism corresponding to the inclusion i: DG_{+} → EG_{+} ∧_{G} DG_{+} = S^{−adG} is the surjection H^{∗} = K(n)_{∗}(DG_{+}) → K(n)_{∗}(S^{−adG}) = R ⊗_{H} H^{∗}. Thinking of H^{∗} as a left H-module, these are the H-module coinvariants, or indecomposables, of H^{∗}.

Passing to duals, the projection p: S^{adG} = F (EG_{+}, S[G]) → S[G] is functionally dual to the inclusion above, hence induces the R-dual injection Hom_{R}(R ⊗_{H} H^{∗}, R) → Hom_{R}(H^{∗}, R) in K(n)-homology. Thus K(n)_{∗}(S^{adG}) is identified with Hom_{R}(R ⊗_{H} H^{∗}, R) ≅ Hom_{H}(H^{∗}, R), sitting inside Hom_{R}(H^{∗}, R) ≅ H. The left H-module structure on H^{∗} dualizes to a left H^{∗}-comodule structure on H. The inclusion Hom_{H}(H^{∗}, R) → Hom_{R}(H^{∗}, R) ≅ H then identifies Hom_{H}(H^{∗}, R) with the H^{∗}-comodule primitives in H. □

Remark 4.2.6. We sometimes write Q_{H}(H^{∗}) = R ⊗_{H} H^{∗} for the left Hmodule indecomposables of H^{∗}, and dually P_{H^{∗}}(H) = Hom_{H}(H^{∗}, R) for the left H^{∗}-comodule primitives in H. Then

K(n)_{∗}(S^{−adG}) ≅ Q_{H}(H^{∗})  and  K(n)_{∗}(S^{adG}) ≅ P_{H^{∗}}(H) .

To be explicit, an element x ∈ H ≅ Hom_{R}(H^{∗}, R) lies in Hom_{H}(H^{∗}, R) if and only if (y ∗ ξ)(x) = ξ(xy) equals ϵ(y)ξ(x) = ξ(xϵ(y)) for each y ∈ H and ξ ∈ H^{∗}. Here ϵ: H → R is the augmentation. This condition is equivalent to asking that xy = 0 for each y ∈ ker(ϵ), i.e., x ∈ H multiplies to zero with each element in the augmentation ideal of H. So P_{H^{∗}}(H) is the left annihilator ideal of the augmentation ideal of H.

4.3. Eilenberg–Mac Lane spaces

We can make the identifications in Theorem 4.2.5 explicit in the cases when G = K(ℤ/p, q) is an Eilenberg–Mac Lane space. For p an odd prime the K(n)- homology H = K(n)_{∗}K(ℤ/p, q) was computed by Ravenel and Wilson in [RaW80, 9.2], as we now recall.

Writing K(n)_{∗}K(ℤ, 2) ≅ K(n)_{∗}{β_{m} | m ≥ 0} with |β_{m}| = 2m there are classes a_{m} ∈ K(n)_{∗}K(ℤ/p, 1) in degree |a_{m}| = 2m for 0 ≤ m < p^{n} such that the Bockstein map K(ℤ/p, 1) → K(ℤ, 2) takes each a_{m} to β_{m}. Let a_{(i)} = a_{p^{i}} in degree |a_{(i)}| = 2p^{i} for 0 ≤ i < n. The q-fold cup product K(ℤ/p, 1) ∧ ⋯ ∧ K(ℤ/p, 1) → K(ℤ/p, q) takes a_{(i_{1})} ⊗ ⋯ ⊗ a_{(i_{q})} to a class a_{I} ∈ K(n)_{∗}K(ℤ/p, q), where I = (i_{1}, …, i_{q}) and |a_{I}| = 2(p^{i_{1}} + ⋯ + p^{i_{q}}).

For q = 0, G = K(ℤ/p, 0) = ℤ/p is a finite group and not very special to the K(n)-local situation. For each q > n, K(ℤ/p, q) has the K(n)-homology of a point. The intermediate cases 0 < q ≤ n are more interesting.

For 0 < q < n there is an algebra isomorphism

K(n)_{∗}K(ℤ/p, q) ≅ ⨂_{I} K(n)_{∗}[a_{I}]/(a_{I}^{p^{ρ(I)}}) ,

where I = (i_{1}, …, i_{q}) ranges over all integer sequences with 0 < i_{1} < ⋯ < i_{q} < n, and ρ(I) = s + 1 where s ∈ {0, 1, …, q} is maximal such that the final s-term subsequence has the form

(i_{q−s+1}, …, i_{q}) = (n − s, …, n − 1) .

Equivalently, s is minimal such that i_{q−s} < n − s − 1.

For q = n there is an algebra isomorphism

K(n)_{∗}K(ℤ/p, n) ≅ K(n)_{∗}[a_{I}]/(a_{I}^{p} + (−1)^{n}v_{n}a_{I}) ,

where I = (0, 1, …, n − 1). Here |a_{I}| = 2(1 + p + ⋯ + p^{n−1}) = 2(p^{n} − 1)/(p − 1).

Proposition 4.3.1. For G = K(ℤ/p, q) with 0 < q < n, K(n)_{∗}(S^{adG}) is generated over K(n)_{∗} by the product π = ∏_{I} a_{I}^{p^{ρ(I)}−1}. Its K(n)-degree is 0 modulo 2(p^{n} − 1).

Proof. By Theorem 4.2.5 we identify K(n)_{∗}(S^{adG}) with the left H^{∗}-comodule primitives in H, which by Remark 4.2.6 consists of the elements of H that multiply to zero with every element in the augmentation ideal of H. These are generated by the product π above. Its degree deg_{K(n)}(G) ≡ |π| can be computed by grouping together the integer sequences with the same value of ρ(I) = s + 1:

|π| = ∑_{I} 2(p^{i_{1}} + … p^{i_{q}})(p^{ρ(I)} − 1)

= ∑_{{0≤s≤q \atop 1≤i_{1}<···<i_{q−s}≤n−s−2}} 2(p^{i_{1}} + ⋯ + p^{i_{q−s}} + p^{n−s} + ⋯ + p^{n−1})(p^{s+1} − 1)

≡ ∑_{{0≤s≤q \atop s+2≤j_{s+1}<···<jq ≤n−1}} 2(p^{j_{s+1}} + ⋯ + p^{j_{q}} + p^{1} + ⋯ + p^{s})

− ∑_{{0≤s≤q \atop 1≤i_{1}<···<i_{q−s}≤n−s−2}} 2(p^{i_{1}} + ⋯ + p^{i_{q−s}} + p^{n−s} + ⋯ + p^{n−1})

= ∑_{1≤j_{1}<···<j_{q}≤n−1} 2(p^{j_{1}} + ⋯ + p^{j_{q}}) − ∑_{1≤i_{1}<···<i_{q}≤n−1} 2(p^{i_{1}} + ⋯ + p^{i_{q}}) = 0

modulo 2(p^{n} − 1). □

Proposition 4.3.2. For G = K(ℤ/p, n), K(n)_{∗}(S^{adG}) is generated by
π = a_{I}^{p−1} + (−1)^{n}v_{n}2

as a K(n)_{∗}-module. Its K(n)-degree is 0 modulo 2(p^{n} − 1).

Proof. In this case the primitives in K(n)_{∗}(S^{adG}) are generated by π = a_{I}^{p−1}+ (−1)^{n}v_{n} in degree |v_{n}| = 2(p^{n} − 1). So also in this case deg_{K(n)}(G) ≡ 0. □

Remark 4.3.3. It would be interesting to produce non-integer elements in the K(n)-local Picard group Pic_{K(n)} as the class [S^{adG}] of the dualizing spectrum of a K(n)-locally stably dualizable group G. The Eilenberg–Mac Lane examples above do not decisively produce any such non-integer elements. Together with Lemmas 2.3.2 and 3.1.6, and [HRW98], this indicates that the required stably dualizable group G should not even be homotopy commutative. This adds interest to the construction suggested in Remark 2.4.8.

CHAPTER 5 Norm and transfer maps

5.1. Thom spectra.

The Thom space of a G-representation V is the reduced Borel construction, or homotopy orbit space, BG^{V} = EG_{+} ∧_{G} S^{V}, where S^{V} is the representation sphere of V. Generalizing the compact Lie group case, when S^{adG} is the (suspension spectrum of) the representation sphere of the adjoint representation adG, we make the following definition:

Definition 5.1.1. Let G be a stably dualizable group. The Thom spectrum BG^{adG} of its dualizing spectrum is the homotopy orbit spectrum

BG^{adG} = (S^{adG})_{hG} = EG_{+} ∧_{G} S^{adG} .

The inclusion G ⊂ EG induces the bottom cell inclusion i: S^{adG} → BG^{adG}.

Note that for G abelian, S^{adG} is a spectrum with EG-action by Lemma 2.5.4, so in these cases

BG^{adG} ≃ BG_{+} ∧ S^{adG} .

As in Proposition 4.1.1, when E is an S-algebra there is a strongly convergent spectral sequence

E_{s,t}^{2} = Tor_{s,t}^{E_{∗}(G)}(E_{∗}, E_{∗}(S^{adG}))

⟹ E_{s+t}(BG^{adG}) . (5.1.2)

When E = K(n) we have K(n)_{∗}(S^{adG}) ≅ Σ^{d}K(n)_{∗} by Proposition 4.2.2, with d = deg_{K(n)}(G). Thus the spectral sequence takes the form

E_{s,t}^{2} = Tor_{s,t}^{K(n)_{∗}(G)}(K(n)_{∗}, Σ^{d}K(n)_{∗})

⟹ K(n)_{s+t}(BG^{adG}) . (5.1.3)

When S^{adG} is K(n)-orientable, so that the bottom cell inclusion i: S^{adG} → BG^{adG} induces a nonzero homomorphism i_{∗}: Σ^{d}K(n) ≅ K(n)_{∗}(S^{adG}) → K(n)_{∗}(BG^{adG}), then this spectral sequence 5.1.3 is a free comodule over the corresponding bar spectral sequence for K(n)_{∗}(BG), on a single generator in degree d.

5.2. The norm map and Tate cohomology.

Let X be a spectrum with left G-action. We give it the trivial right G-action. The smash product X ∧S[G] then has the diagonal left G-action, as well as the right G-action that only affects S[G]. Consider forming homotopy orbits with respect to the left action, and forming homotopy fixed points with respect to the right action.

We shall construct the norm map in three steps. First, there is a canonical colimit/limit exchange map

(5.2.1)  κ: ((X ∧ S[G])^{hG})_{hG} ⟶ ((X ∧ S[G])_{hG})^{hG}

induced by the familiar map

EG_{+} ∧ F (EG_{+}, Y) ⟶ F (EG_{+}, EG_{+} ∧ Y) ,

in the case Y = X ∧ S[G].

Second, there is a natural map ν: X ∧ S^{adG} = X ∧ S[G]^{hG} → (X ∧ S[G])^{hG}, since G acts trivially on X from the right. It can be identified with the chain of weak equivalences

X ∧ S^{adG} ⟶^{≃} F (G_{+}, X ∧ S^{adG})^{hG} ⟵_{≃}^{ν^{hG}} (X ∧ DG_{+} ∧ S^{adG})^{hG} ⟶^{≃} (X ∧ S[G])^{hG}

where the middle map uses that G is stably dualizable and the right hand map uses the Poincaré duality equivalence of Theorem 3.1.4. In particular ν: X ∧ S^{adG} → (X ∧ S[G])^{hG} is a weak equivalence, and it induces a weak equivalence

ν_{hG}: (X ∧ S^{adG})_{hG} ⟶^{≃} ((X ∧ S[G])^{hG})_{hG}

on homotopy orbits with respect to the left actions. Note that ν_{hG} maps to the left hand side of (5.2.1). In the special case X = S, the maps ν and ν_{hG} are isomorphisms.

Third, there is an untwisting equivalence ζ: (X ∧S[G])_{hG} → X ∧S[G]_{hG} ≃ X ∧ S ≅ X, cf. [LMS86, p. 76], that takes the remaining right action on (X ∧ S[G])_{hG} to the right action on X through the inverse of the left action. Hence there is an equivalence

ζ^{hG}: ((X ∧ S[G])_{hG})^{hG} ⟶^{≃} X^{hG}

of homotopy fixed points, formed with respect to these right actions. Note that ζ^{hG} maps from the right hand side of (5.2.1).

Definition 5.2.2. Let X be a spectrum with left G-action. The (homotopy) norm map

N: (X ∧ S^{adG})_{hG} ⟶ X^{hG}

is the composite of the natural maps:

(X ∧ S^{adG})_{hG} ⟶_{≃}^{ν_{hG}} ((X ∧ S[G])^{hG})_{hG} ⟶^{κ} ((X ∧ S[G])_{hG})^{hG} ⟶_{≃}^{ζ^{hG}} X^{hG} ,

where ν_{hG} and ζ^{hG} are weak equivalences. The G-Tate cohomology spectrum X^{tG} of X is the cofiber of the norm map:

(X ∧ S^{adG})_{hG} ⟶^{N} X^{hG} → X^{tG} .

Remark 5.2.3. (a) Note that X^{tG} ≃ ∗ if and only if the norm map N is a weak equivalence, which in turn is equivalent to the canonical colimit/limit exchange map κ in (5.2.1) being a weak equivalence. So G-Tate cohomology measures the failure of G-homotopy orbits and G-homotopy fixed points to commute.

(b) In view of [GM95, 3.5], it is reasonable to expect that if X is an S-algebra with G-action, then X^{tG} is an S-algebra and X^{hG} → X^{tG} is a map of S-algebras. We do not know how to give a direct model for X^{tG}, say as the “G-fixed points” of the spectrum ẼG ∧ F (EG_{+}, X) with G-action, so it is not so easy to verify our expectation. Here, as usual in this context, ẼG is the mapping cone of the collapse map c: EG_{+} → S^{0}.

In the special case when X = S with trivial G-action, the norm map simplifies to the canonical colimit/limit exchange map

BG^{adG} = (S^{adG})_{hG} = (S[G]^{hG})_{hG} ⟶^{κ} (S[G]_{hG})^{hG} ≃ S^{hG} ≅ D(BG_{+}) .

Here we use that S^{hG} = F (EG_{+}, S)^{G} ≅ F (BG_{+}, S) = D(BG_{+}) is the functional dual of BG_{+}, since S has trivial G-action. Hence there is a cofiber sequence

BG^{adG} ⟶^{N} D(BG_{+}) ⟶ S^{tG} .

In the case of a compact Lie group G, the G-Tate cohomology X^{tG} is the same as that denoted t_{G}(X)^{G} by Greenlees and May [GM95] and ℍ̂(G, X) by Bökstedt and Madsen [BM94].

Definition 5.2.4. A spectrum with G-action X is in the thick subcategory generated by spectra of the form G_{+} ∧ W, if X can be built from ∗ in finitely many steps by (1) attaching cones on induced G-spectra of the form G_{+} ∧ W, with W any spectrum, (2) passage to (weakly) equivalent spectra with G-action and (3) passage to retracts. For instance, any finite G-cell spectrum has this form.

Theorem 5.2.5. Let G be a stably dualizable group. If a spectrum with Gaction X is in the thick subcategory generated by spectra of the form G_{+} ∧ W, then:

(1) The norm map N: (X ∧ S^{adG})_{hG} → X^{hG} for X is an equivalence.

(2) The G-Tate cohomology X^{tG} ≃ ∗ is contractible.

Proof. If X = G_{+}∧W is induced up from a spectrum W with trivial G-action, the source of the norm map can be identified with

(G_{+} ∧ W ∧ S^{adG})_{hG} ≃ W ∧ S^{adG}

while the target of the norm map can be identified with

(G_{+} ∧ W)^{hG} ≃ (DG_{+} ∧ S^{adG} ∧ W)^{hG} ≃ F (G_{+}, W ∧ S^{adG})^{hG} ≃ W ∧ S^{adG} .

These identifications are compatible, as can be checked by starting with the case W = S, hence in this case the norm map is itself an equivalence. The general case follows by induction on the number of attachments made. □

Remark 5.2.6. This result generalizes the third case of [Kl01, Thm. D], from compact Lie groups to stably dualizable groups. For compact Lie groups G this norm equivalence can be compared with the genuinely G-equivariant Adams equivalence Y/G ≃ (Y ∧S^{−adG})^{G} for Y a free G-spectrum [LMS86, II.7]. Any such Y is a filtered colimit of finite, free G-spectra, which are in the thick subcategory generated by S[G] ≅ G_{+}∧S. But, while genuine G-fixed points (Y ↦ Y^{G}) commute with filtered colimits, this is not generally the case for G-homotopy fixed points (Y ↦ Y^{hG}). Therefore we cannot extend Theorem 5.2.5 to all spectra X with free G-action.

There is also a dual construction X_{tG} that is to Tate homology as the Tate construction X^{tG} is to Tate cohomology. To define it, we suppose that X is a spectrum with right G-action, and give it the trivial left G-action. The smash product DG_{+} ∧ X has the left G-action that only affects DG_{+}, and the diagonal right G-action. There is a canonical colimit/limit exchange map

κ: ((DG_{+} ∧ X)^{hG})_{hG} ⟶ ((DG_{+} ∧ X)_{hG})^{hG} .

The source of κ receives an equivalence from X_{hG} obtained by applying homotopy orbits to the equivalence X → F (G_{+}, X)^{hG}. The target of κ admits a weak equivalence to (S^{−adG} ∧ X)^{hG} obtained by taking homotopy fixed points of the isomorphism

(DG_{+} ∧ X)_{hG} ⟶^{≅} (DG_{+})_{hG} ∧ X = S^{−adG} ∧ X ,

which exists because G acts trivially on X from the left.

Definition 5.2.7. Taken together, these maps yield the alternate norm map N′, defined as the composite:

N′: X_{hG} ⟶^{≃} ((DG_{+} ∧ X)^{hG})_{hG} ⟶^{κ} ((DG_{+} ∧ X)_{hG})^{hG} ⟶^{≃} (S^{−adG} ∧ X)^{hG} .

Its homotopy fiber X_{tG} is the G-Tate homology spectrum, and sits in a cofiber sequence

X_{tG} ⟶ X_{hG} ⟶^{N′} (S^{−adG} ∧ X)^{hG} .

If X is an S-coalgebra with G-action, it again appears likely that X_{tG} is such a coalgebra and that X_{tG} → X_{hG} is a map of S-coalgebras.

5.3. The G-transfer map.

Definition 5.3.1. Let X be a spectrum with left G-action. The G-transfer map

trf_{G}: (X ∧ S^{adG})_{hG} ⟶ X

is the composite of the norm map N: (X ∧ S^{adG})_{hG} → X^{hG} and the forgetful map p: X^{hG} → X.

When X = S[Y] is the unreduced suspension spectrum of a G-space Y, this is the dimension-shifting G-transfer map associated to the principal G-bundle Y ≃ EG × Y → EG ×_{G} Y.

5.4. E-local homotopy classes.

Let G be a stably dualizable group, with dualizing spectrum S^{adG}. The composite of the bottom cell inclusion i: S^{adG} → BG^{adG} and the dimension-shifting G-transfer trf_{G}: BG^{adG} → S is the composite map

S^{adG} ⟶^{i} (S^{adG})_{hG} = (S[G]^{hG})_{hG} ⟶^{κ} (S[G]_{hG})^{hG} ≃ S^{hG} ⟶^{p} S .

Noting that the projection p amounts to forgetting G-homotopy invariance, this map can also be expressed as the composite

S^{adG} = S[G]^{hG} ⟶^{p} S[G] ⟶^{i} S[G]_{hG} ≃ S .

Definition 5.4.1. The composite map pκi: S^{adG} → S represents a class denoted [G] ∈ π_{∗}(L_{E}S) in the Pic_{E}-graded homotopy groups of the E-local sphere spectrum, in degree ∗ = dim_{E}(G) = [S^{adG}] ∈ Pic_{E}. We might call [G] the E-local stably framed bordism class of G.

Example 5.4.2. For the circle group G = S^{1} and E = S we have [G] = η ∈ π_{1}(S). For the p-complete Sullivan sphere G = (S^{2p−3})_{p}^{∧} and E = H𝔽_{p} we have [G] = α_{1} ∈ π_{2p−3}(S_{p}^{∧}), when p is odd. These examples are also detected K(1)-locally, i.e., for G considered as a K(1)-locally stably dualizable group.

Lemma 5.4.3. In the case E = K(n), the induced homomorphism

[G]_{∗}: Σ^{d}K(n)_{∗} ≅ K(n)_{∗}(S^{adG}) → K(n)_{∗}(S) = K(n)_{∗}

takes a generator of the K(n)^{∗}(G)-comodule primitives in K(n)_{∗}(G) to its image under the augmentation ϵ: K(n)_{∗}(G) → K(n)_{∗}.

Proof. Recall from Theorem 4.2.5 that K(n)_{∗}(S^{adG}) is identified with the H^{∗} = K(n)^{∗}(G)-comodule primitives Hom_{H}(H^{∗}, R), and the projection p: S^{adG} → S[G] induces the forgetful inclusion into Hom_{R}(H^{∗}, R) ≅ H = K(n)_{∗}(G). The inclusion i: S[G] → S[G]_{hG} ≃ S induces the augmentation ϵ, which establishes the claim □

Example 5.4.4. When E = K(n) and G is a finite discrete group we get H = R[G] and P_{H^{∗}}(H) ≅ R{N}, where N = ∑_{g∈G} g is the norm element in H. Then ϵ(N) = |G| equals the order of G, so [G]_{∗} multiplies by |G| in R = K(n)_{∗}.

Example 5.4.5. When E = K(n) and G = K(ℤ/p, q) for 0 < q < n, the H^{∗}- comodule primitives were found in Proposition 4.3.1 to be generated by an element π that lies in the augmentation ideal ker(ϵ), so the induced homomorphism [G]_{∗} is zero and [G]: S^{adG} → S has positive K(n)-based Adams filtration.

Example 5.4.6. When E = K(n) and G = K(ℤ/p, n), with q = n, Proposition 4.3.2 exhibited a generating element π = a_{I}^{p−1} + (−1)^{n}v_{n} for P_{H^{∗}}(H), which augments to the unit (−1)^{n}v_{n} ∈ K(n)_{∗}. Hence in this case [G]: S^{adG} → S induces an isomorphism on K(n)-homology, and so S^{adG} ≃ S in the K(n)-local category. By Lemma 2.5.4, the G-action on S^{adG} is homotopy trivial in this case. Hence the Poincaré duality equivalence 3.1.4 amounts to a K(n)-local self-duality equivalence

(5.4.7)  F (G_{+}, L_{K(n)}S) = DG_{+} ≃ S[G] = L_{K(n)}Σ^{∞}G_{+}

for G = K(ℤ/p, n), which is left and right G-equivariant up to homotopy, and which may be compared with [HSt99, 8.7].

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Index

A-algebra, 14 A-module, 14 𝒜_{A}, A-algebras, 14 absolute Galois group, 82, 84 action through inverse, 114 Adams conditions, 77, 78 Adams e-invariant, 26 Adams operation, 34, 43 Adams resolution, 57 Adams spectral sequence, 30, 58 Adams summand, 35, 63 Adams, John Frank, 65, 77 Adams–Novikov spectral sequence, 91 adelic, 6 𝒜Der , 64 adjoining roots of unity, 32 adjoint representation, 19 Â_{4}, binary tetrahedral group, 30 algebra, 10 algebraic geometry, 16, 69, 81 algebraic K-theory, 57 algebraic tensor product, 11 allowable normal subgroup, 52, 53 allowable subgroup, 52, 53, 62, 80 Amitsur cohomology, 58 Amitsur complex, 8, 29, 57, 90 André–Quillen cohomology, 79 André–Quillen homology, 67 Angeltveit, Vigleik, 63, 72 anti-equivalence, 11 𝒜_{S}, S-algebras, 14 associative derivation, 64, 65 associatively étale, 65 Atiyah duality, 104 Atiyah–Hirzebruch spectral sequence, 43 augmented commutative algebra, 67 Auslander, Maurice, 11, 45 Ausoni, Christian, 9, 63

B⟨⟨G⟩⟩, completed twisted group ring, 56 Baker, Andrew, 9, 27, 32, 49, 72, 73, 78, 84 bar construction, 37 base change, 7, 24, 43, 51, 71 Basterra, Maria, 66–68, 71, 79 Bauer, Tilman, 19, 105 Bendersky, Martin, 57 β, Bott equivalence, 26, 43 β, coaction, 89 BG^{adG}, Thom spectrum, 126 binary tetrahedral group, 30 Boardman, Michael E., 14 Bökstedt, Marcel, 63 B^{op}, opposite algebra, 61 Bott periodicity, 26 Bott, Raoul H., 91 bottom cell inclusion, 126 Bousfield localization, 15, 25, 65, 72, 76, 82 Bousfield nilpotent completion, 58 Bousfield, A. K., 15, 57, 58 Bousfield–Kan spectral sequence, 58, 79, 91 brave new rings, 3, 14 Brown–Comenetz dual, 92

c, complexification, 26, 42

𝒞_{A}, commutative A-algebras, 14

Carlsson, Gunnar, 57

𝒞_{B}/B, 67 𝒞Der, 67 characteristic map, 62 Chase, Stephen U., 11 Chern character, 43 Childs, Lindsay N., 90 chromatic filtration, 15, 73, 75 chromatic localization, 6 class field theory, 32, 83 closed symmetric monoidal category, 14 coaction, 89 cobar complex, 8, 37, 59 cobar construction, 37 cochain algebra, 4, 7, 37, 67 commutative A-algebra, 14 commutative derivation, 67 commutative S-algebra, 14 commutatively étale, 8 compact Hausdorff spaces, 11 compact Lie group, 18 completed tensor product, 11 completed twisted group ring, 56, 57 completion along a map, 8, 57–59, 77, 91 complex K-theory, 34, 48 complex cobordism, 6, 73 complex conjugation, 26 complex image-of-J spectrum, 35 complex K-theory, 4, 25 complexification, 26 connected, 4, 80–82, 85, 86, 88 connective, 81 connective K-theory, 25, 27 continuous homotopy fixed points, 29 continuous mapping spectrum, 56, 57 continuous section, 33, 52 convergence, 38, 91 converse part, 86, 88 cosimplicial resolution, 57 cover, 23 covering space, 61 𝒞_{S}, commutative S-algebras, 14

d, Kronecker delta, 62 𝒟_{A}, 18 𝒟_{A,E}, 15, 46 D_{A}M, functional dual, 17 Davis, Daniel G., 9, 28, 29 𝔡_{B/A}, discriminant map, 48 deck transformation, 10 decomposition group, 35 degree map, 32 descent problem, 57 Devinatz, Ethan S., 27, 29–31 diagonal map, 25 dimension-shifting transfer map, 129 discrete, 53, 61, 62, 64, 67, 78, 86, 88 discriminant, 12 discriminant map, 48 d_{K}, characteristic map, 62 double of A(1), 31 𝒟_{S}, 14 𝒟_{S,E}, 15, 21 dual shear map, 117 dual Steenrod algebra, 39 duality group, 112 dualizable, 17–19, 41, 42, 61, 66, 82, 88, 103,

107 dualizable base change, 51 dualizing module, 112 dualizing spectrum, 19, 48–50, 103, 112 Dwyer, William G., 19, 37, 38 DX, functional dual, 17 Dyer–Lashof operation, 39

E, 68 ℰ(B), space of idempotents, 80 E(n), Johnson–Wilson spectrum, 15 E(T), set of idempotents, 80 E_{∗}-equivalence, 15, 64, 106 E-acyclic, 15, 67 E-compact group, 110 E-completion, 109 E-dimension, 120 E-good, 110 E-local, 15, 64, 67, 106 E_{2} model structure, 78 EG, 19, 58 eigenspace, 43 Eilenberg–Moore spectral sequence, 38 Eilenberg–Mac Lane space, 18, 67 Eilenberg–Mac Lane spectrum, 3, 22, 79 E_{∞} mapping space, 7, 14, 77, 79, 80, 85, 87 Elmendorf, A. D., 14, 63 E_{n}, Lubin–Tate spectrum, 27 E_{n}^{nr}, maximal unramified extension, 28 EO_{n}, higher real K-theory, 30 ϵ, adjunction counit, 17, 49 equivalent allowable subgroup, 52 η, Hopf map, 26 étale, 5, 8, 61, 66, 67, 80, 81, 85 étale topology, 17 examples, 25 exchange map, 20, 47, 54 extended equivalence, 40 extended Morava stabilizer group, 28

F ((G_{+}, B)), continuous mapping spectrum,

56 faithful, 4, 23, 25, 26, 30, 32, 45–47, 50, 53,

83, 88 faithful abelian extension, 49, 84 faithful and dualizable, 43, 45, 51, 53, 58, 88 faithful base change, 7, 24, 33, 62 faithful, projective descent, 58 faithfully flat, 12 Fausk, Halvard, 9 f_{#}, f^{#}, 40 finite, 17, 83 finite cell module, 18 finite CW module, 18 finite Galois extension, 22, 78, 85, 88 fixed ring, 11 fixed subfield, 10 flat, 7 formal thickening, 6, 8 formally associatively étale, 65 formally étale, 5, 66–68 formally symmetrically étale, 63, 66, 68 formally taq-étale, 66 formally thh-étale, 63 forward part, 53 framed bordism class, 104 Frobenius algebra, 122 function object, 14 functional dual, 17

Galois cohomology, 58

Galois correspondence, 4, 53, 80

Galois descent, 73

Galois extension, 3, 22, 53, 59, 67 Galois extension of commutative S-algebras,

22 Galois extension of commutative rings, 12 Galois extension of fields, 10 Γ-cohomology, 79, 80 Γ-homology, 80 global, 16 global Galois extension, 3, 22, 82 global model, 6, 73, 75, 92 𝔾_{n}, extended Morava stabilizer group, 28 Goerss, Paul G., 77–79 Goerss–Hopkins obstruction theory, 14, 28,

29, 77 Goerss–Hopkins spectral sequence, 77, 79–

81, 85, 87 Goldman, Oscar, 11, 45 graded Frobenius algebra, 122 Greenlees, John, 37 Greither, Cornelius, 12, 82 Gross, Benedict H., 57 Gross–Hopkins duality, 57 Grothendieck model topology, 58 Grothendieck, Alexander, 67, 70 group cobar complex, 59, 90

h, 10, 11, 21, 40, 45, 89, 91 Harrison, David K., 11 h^{•}, h′, 59, 90 Henselian map, 6, 8, 68, 71, 75 Hess, Kathryn, 9 Hewett, Thomas James, 29 higher real K-theory, 29, 30 h_{M}, 40 Hochschild homology, 67 homotopically meaningful, 21, 22 homotopy discrete, 80, 81, 85, 86 homotopy fixed point spectral sequence, 23,

26, 30 homotopy fixed points, 8, 20, 59, 111 homotopy norm map, 20, 127 homotopy orbits, 20, 111 Honda formal group, 28, 92 Hopf cobar complex, 8, 90 Hopf S-algebra, 6, 89 Hopf–Galois extension, 6, 90, 91 Hopkins, Michael J., 27–31, 49, 57, 77–79 Hopkins–Miller theory, 7, 28, 77, 92 Hovey, Mark A., 14, 15, 17, 32, 48, 56, 57,

65, 73, 74, 76 Hurewicz map, 83 Hurewicz theorem, 83

i, 11, 21, 45, 46, 90, 91 I-adic tower, 8, 72 idempotent, 4, 7, 80, 81, 87 image-of-J spectrum, 34, 63 infinitesimal thickening, 8, 69 integers, 82 inverse dualizing spectrum, 113 invertible, 12 Iyengar, Srikanth, 37

j, 10, 11, 40, 45, 56 J, image-of-J spectrum, 34 j_{M}, 41 Johnson, David Copeland, 75 Johnson–Wilson spectrum, 15, 32 JU, complex image-of-J spectrum, 35

K(n), Morava K-theory spectrum, 15 K(n)-compact group, 103 K(n)-degree, 122 K(n)-local category, 18, 24, 73, 75 Kähler differentials, 5, 8 Kan, Daniel M., 58 κ, exchange map, 20, 54 Klein, John R., 19 Knus, Max–Albert, 61, 82 KO, real K-theory, 25 Křiž, Igor, 14, 63 Kronecker delta, 62 Krull topology, 29, 56 KU, complex K-theory, 25 Künneth formula, 24, 38, 83 Künneth spectral sequence, 23, 24, 37, 38

ℓ, connective Adams summand, 63 L, periodic Adams summand, 35 Landweber exact, 28 Lazarev, Andrey, 32, 61, 64, 65, 70, 72, 73 L_{E}, E-localization, 15 left lifting property, 69 Lewis, L. Gaunce, Jr., 17 L̂, nilpotent completion, 16, 58 L_{K(n)}, K(n)-localization, 15 L_{n}, E(n)-localization, 15 local Galois extension, 3, 15, 22 local S-module, 15 local stable homotopy category, 15 locally stably dualizable, 18 Loday, Jean–Louis, 63 Lubin, Jonathan, 92 Lubin–Tate deformation theory, 8, 28, 92 Lubin–Tate spectrum, 4, 27, 28, 30 Lydakis, Manos G., 14

ℳ_{A}, A-modules, 14 ℳ_{A,E}, E-local A-modules, 15, 77 Magid, Andy R., 80 Mahowald, Mark E., 30, 34, 49 Main Theorem of Galois Theory, 4 Mandell, Michael A., 14, 63, 67, 68, 79 maximal abelian extension, 35 maximal ideal, 10 maximal unramified extension, 7, 28, 31, 73 May, J. Peter, 14, 17, 63 McCarthy, Randy, 64, 67, 79 McClure, James E., 63 Miller, Haynes R., 28, 29, 34 Milne, James Stuart, 66, 70, 80 Minasian, Vahagn, 64, 67 Minkowski’s discriminant theorem, 4, 82 model category, 21 model site, 58 model structure, 21, 78 monochromatic category, 73 Morava change-of-rings, 27, 33, 57 Morava E-theory, 32 Morava K-theory, 15, 27 Morava module, 43 Morava stabilizer group, 4, 28, 30 Morava, Jack, 3, 9, 27, 93 ℳ_{S}, S-modules, 14 ℳ_{S,E}, E-local S-modules, 15, 21 MU, complex cobordism, 73, 91 MU-chromatic tower, 76 μ_{∞,p}, roots of order prime to p, 31 μ_{m}, m-th roots of unity, 31 MUP, even periodic MU, 92

N, norm map, 20, 127 Neukirch, Jürgen, 82 nilpotence theorem, 31 nilpotent, 27, 31 nilpotent action, 38 nilpotent completion, 8, 16, 57, 58, 72, 75,

76 non-commutative Galois extension, 38 non-commutative T HH, 63 norm map, 20, 45, 104, 127 normal closure, 82 not discrete, 78 not étale, 63 not faithful, 92 not Galois, 27, 36, 91 not semi-finite, 42 not separable, 91 not smashing, 74 not surjective, 48 ν′, canonical map, 41, 44 ν, canonical map, 17 number field, 12

obstruction to commutation, 20 Ojanguren, Manuel, 61, 82 Ω^{t}B_{∗}, t-th desuspension, 79 open problem, 4, 24, 27, 51, 63, 66, 72, 75,

76, 82, 83, 85 opposite multiplication, 61 orthogonal spectrum, 14

p-compact group, 19, 102 p-dimension, 112 Palmieri, John H., 56 Picard graded homotopy group, 104 Picard group, 48–50, 119 Pic_{E}, Picard group, 49 Poincaré duality, 19, 46, 47, 50, 104, 116 pointwise product, 11 principal bundle, 7, 11, 37, 38 pro-Galois extension, 27, 56, 75, 86 Prüfer ring, 28 pull-back, 69 pushout, 71

quadratic Galois extension, 26, 35, 42 quaternion algebra, 30 Quillen spectral sequence, 67 Quillen suspension, 68 Quillen, Daniel, 3

r, realification, 26 ramified cover, 10 ramified primes, 12 Ravenel, Douglas C., 15, 18, 74, 76, 92 real K-theory, 4, 25, 35, 48 realification, 26, 48 realization, 35, 36 regular covering space, 4, 10 regular sequence, 72–75 resolution model structure, 78 ρ, canonical map, 18 Richter, Birgit, 9, 32, 49, 78, 79, 84, 86 Riemann surface, 10 right lifting property, 69 ring of integers, 12 Robinson obstruction theory, 77 Robinson, Christopher Alan, 77, 79–81 roots of unity, 31 Rosenberg, Alex, 11 Rudyak, Yuli B., 27

S^{−adG}, inverse dualizing spectrum, 113 S-algebra, 14 S-algebraic scheme, 5 S-algebraic stack, 5, 73 S-module, 14, 106 S[BU], 6 S^{adG}, dualizing spectrum, 112 Sadofsky, Hal, 49, 74 Schwänzl, Roland, 31, 63 Schwede, Stefan, 14 Segal, Graeme B., 14 self-dual, 19, 49, 78 self-induced extension, 25 self-reference, 63, 136 semi-finite, 18, 42 separable, 4, 18, 61, 62, 64, 66, 86, 88 separable closure, 82, 84 separably closed, 4, 82 sequential suspension spectrum, 68 sh′, dual shear map, 117 sh, shear map, 114 shear map, 19, 48, 114 Shipley, Brooke E., 14, 21, 38 simplicial E_{∞} operad, 79 simplicial functor, 14 singular extension, 8, 70–72 small, 18 smash invertible, 19, 48–50, 103, 119 smash product, 14 smashing, 8, 65, 66, 68 smashing localization, 16, 65, 67 Smith, Jeffrey Henderson, 14, 21, 31, 32 𝕊_{n}, Morava stabilizer group, 28 sphere spectrum, 82 square-zero extension, 8, 67, 69–72 stable homotopy category, 14, 106 stably dualizable, 6, 38, 44, 108 stably dualizable group, 103 stably framed bordism class, 129 Staffeldt, Ross E., 63 standard action, 112, 113 Steenrod algebra, 39 Steinberger, Mark, 17 Strickland, Neil, 9, 15, 17, 32, 48, 56, 57, 73,

74, 76, 81 strong convergence, 38 strongly dualizable, 17 sub-monoid, 87 suspension spectrum, 14 symmetric bimodule, 64 symmetric spectrum, 14 symmetrically étale, 8, 61, 63, 64, 67 symmetrically Henselian map, 71, 73–75 synthetic, 29

t, complex conjugation, 26 TAQ, topological André–Quillen homology,

66 taq-étale, 61, 66 Tate cohomology spectrum, 127 Tate construction, 45, 127 Tate homology spectrum, 129 Tate, John T., 92 telescope conjecture, 92 tensored structure, 57, 67 T HH, topological Hochschild homology, 63 thh-étale, 8, 61, 63, 64 thh-Henselian map, 71, 73–75 thick subcategory, 17, 128 thickening, 6 Thom diagonal, 6, 91 Thom isomorphism, 91 Thom spectrum, 92, 126 Thompson, Robert D., 57 T h^{R}(γ), R-based Thom spectrum, 92 tmf, topological modular forms, 30 Toën, Bertrand, 17, 58, 69 topological K-theory, 4, 25, 34 topological André–Quillen cohomology, 79 topological André–Quillen homology, 5, 8,

66, 68, 71 topological closed category, 14, 16 topological cotensor, 14 topological Hochschild cohomology, 61 topological Hochschild homology, 5, 8 topological modular forms, 30 topological tensor, 14, 63 topology, 21 trace map, 12, 47, 48 trace pairing, 48 transfer map, 129 trf_{G}, transfer map, 129 triple resolution, 58 trivial Galois extension, 25 twisted group ring, 10, 40 twisted product, 12

unique lifting, 61, 66, 68, 71

universal coefficient spectral sequence, 79 unreduced suspension spectrum, 14, 18 unresolved, 9

Vezzosi, Gabriele, 17, 58, 69 Vogt, Rainer M., 32, 63

Waldhausen, Friedhelm, 3, 32

weak equivalence, 14 weak Galois group, 92 weak orbit, 93 wedge sum, 74 Whitehouse, Sarah, 79–81 Wilkerson, Clarence W., Jr., 19 Wilson, W. Stephen, 18 Witt ring, 27 Würgler, Urs, 27

X_{E}^{∧}, E-completion, 109

X^{hG}, homotopy fixed points, 20

X_{hG}, homotopy orbits, 20

X^{tG}, Tate construction, 20, 127 X_{tG}, Tate homology spectrum, 129

Yosimura, Zen-ichi, 75

Zariski topology, 16, 65, 73 ζ, canonical map, 63, 64, 67, 68