morava modules and the k -local picard group · 12 introduction categoryofcomplete e∨...

Morava modules and the K(n)-localPicard group

Submitted in total fulfilment of the requirements of the degree ofDoctor of Philosophy

University of MelbourneDepartment of Mathematics and Statistics

Drew Heard

August 19, 2014

Abstract

The chromatic approach to homotopy theory naturally leads to the study of the K(n)-local stable homotopy category. In this thesis we study this category in three differentways.

The first is to closely study the category of Morava modules. It turns out that thiscategory is equivalent to the category of complete E∨∗ E-comodules. Using this, we developa theory of (relative) homological algebra for Morava modules. We use this to give anexplicit identification of the E2-term of the K(n)-local En-based Adams spectral sequence.This turns out to be related to work of Goerss-Henn-Mahowald and Rezk [GHMR05], asconstructed as part of their resolution of the K(2)-local sphere at the prime 3.

The second part is computational in nature; we show that for a large class of groupsthe Tate spectrum EtGp−1 always vanishes. Such a result was previously known to be trueK(n)-locally, but we show it holds even before this. We use this to deduce some self-duality results for the K(n)-local Spanier-Whitehead dual of the homotopy fixed pointspectra EhGn .

In the final chapter we study the K(n)-local Picard group. In particular we showthat, when p is an odd prime, the subgroup κn of elements such that E∨∗X ' E∗ ascontinuous modules over the Morava stabiliser group is always a p-group, and decomposesas a direct product of cyclic groups. Then, specialising to the case of n = p−1, we discussthe decomposition of the group of exotic elements, by studying the map from the Picardgroup of the K(n)-local category to the Picard group of EhGn -modules, where G ⊂ Gn isa maximal finite subgroup of the Morava stabilizer group. We finish by explaining theconnection to Gross-Hopkins duality, and outline an approach to constructing elementsof κn when n > 2. In fact this method already allows us to (independently) constructelements of κ2 that are non-zero in I2, the Gross-Hopkins dual of the sphere.

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Declaration

This is to certify that:

(i) the thesis comprises only my original work towards the PhD except where indicatedin the Preface,

(ii) due acknowledgement has been made in the text to all other material used,

(iii) the thesis is fewer than 100 000 words in length, exclusive of tables, maps, bibliogra-phies and appendices.

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Acknowledgements

I owe a great thanks to a number of people for this project. First and foremost I wishto thank my advisor Craig Westerland. Since he essentially taught me all the algebraictopology I know, it is no exaggeration to say that this thesis would not exist without hishelp. Beyond that, his encouragement and support was invaluable, and the numerous tripshe helped fund shaped many of the results within this document. I’d also like to thankNora Ganter and Alex Ghitza for being members of my PhD committee, and to Nora foragreeing to be a co-supervisor.

Many of the results in this paper were inspired by the series of papers [GHM; GHMR05;GH12; GHMR12] and I also thank Paul Goerss and Hans-Werner Henn for answering mymany emails regarding these series of papers and related matters. I also had helpfulconversations regarding this and related matters with Mark Behrens, Daniel Davis, MarkHovey, Tyler Lawson and Vesna Stojansoka amongst others. Charles Rezk also provideda key lemma needed.

It was great to be doing this project at the same time as TriThang and Jeff - I’m sorryI made you sit through so many chromatic homotopy theory talks.

Finally I’d like to thank my family and friends for their support, and putting up withthe strange life of a PhD student. A special thanks to Sonja for the whiteboard and forencouraging me to do this in the first place.

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Table of Contents

Introduction 11

1 Prerequisites 151.1 Bousfield localisation and Morava K-theory . . . . . . . . . . . . . . . . . . 151.2 Lubin-Tate theory and Morava E-theory . . . . . . . . . . . . . . . . . . . . 181.3 The action of the Morava stabilizer group . . . . . . . . . . . . . . . . . . . 201.4 Homotopy fixed point spectra for subgroups of the Morava stabilzier group 23

1.4.1 Finite subgroups at height n = p− 1 . . . . . . . . . . . . . . . . . . 25

2 Morava modules and the K(n)-local category 292.1 The K(n)-local category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Morava modules and complete E∨∗ E-comodules . . . . . . . . . . . . . . . . 342.3 Homological algebra for Morava modules . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Relative homological algebra . . . . . . . . . . . . . . . . . . . . . . 372.3.2 The K(n)-local En-Adams spectral sequence . . . . . . . . . . . . . 41

3 Homotopy fixed point and Tate spectra 433.1 The Tate spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 A description of En as a F -module . . . . . . . . . . . . . . . . . . . . . . . 453.3 The Tate cohomology of G . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Relation to the work of Mathew-Meier . . . . . . . . . . . . . . . . . . . . . 55

4 The Picard group of the K(n)-local category 594.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Some cohomological results . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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10 Table of Contents

4.3 A filtration on κn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 The E(n)-local case . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.2 The K(n)-local case . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 The structure of κn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4.1 Remark on the finiteness of κn . . . . . . . . . . . . . . . . . . . . . 68

4.5 The decomposition of the group of exotic elements . . . . . . . . . . . . . . 714.6 Brown-Comenetz duality and G-exotic elements . . . . . . . . . . . . . . . . 75

4.6.1 An approach to constructing G-exotic elements . . . . . . . . . . . . 81

Bibliography 83

A Tate cohomology 89

Introduction

From a computational standpoint, the chromatic approach to stable homotopy theorylargely reduces the study of π∗X, for a finite p-local spectrum X, to the study of thelocalisations π∗LK(n)X. In theory, the homotopy of π∗X can then be recovered via thechromatic tower and the chromatic fracture square. This naturally leads to the study ofthe category of K(n)-local spectra, MS,K(n), itself. In this thesis we broadly study thiscategory in a number of related ways.

One of the most significant recent developments in chromatic homotopy theory wasthe construction of a resolution of the K(2)-local sphere at the prime 3 by Goerss, Henn,Mahowald and Rezk [GHMR05], which has already lead to significant advances in ourunderstanding of the K(2)-local homotopy category at the prime 3; see the series ofpapers [GHM; GH12; GHMR12; Kar10]. The construction of this resolution starts withan algebraic resolution of the trivial Zp[[Gn]]-module Zp by permutation modules of theform Zp[[Gn/F ]] for some finite subgroup F ⊂ Gn. This leads to an exact sequence ofMorava modules, where a Morava module is a complete E∗-module such that the actionof the Morvava stabilizer group, Gn, is compatible with the E∗-module structure. Wedenote the category of such objects by EGn. An obstruction theory argument using theEn-Hurewicz map

π∗F (EhH1n , EhH2

n )→ HomEGn(E∨∗ EhH1n , E∨∗ E

hH2n )

then gives a resolution of spectra. In order to study the Hurewicz map, Goerss, Henn,Mahowald and Rezk show that it is isomorphic to the edge homomorphism in a homotopyfixed point spectral sequence. After identifying the category of Morava modules with the

11

12 Introduction

category of complete E∨∗ E-comodules of [Dev95], we show that the above map is in factthe edge homomorphism of the K(n)-local En-Adams spectral sequence.

Theorem 2.3.10. 1) If E∨∗X and E∨∗ Y are Morava modules then there is a stronglyconvergent spectral sequence

Es,t2 = Exts,tEGn(E∨∗X,E∨∗ Y )⇒ πt−sF (X,Y ),

isomorphic to the K(n)-local En-Adams spectral sequence.

2) If Y = EhFn for F a closed subgroup of Gn then there is an isomorphism of E2-terms:

Exts,tEGn(E∨∗X,E∨∗ EhF ) ' Hs

c (F ;E−tX).

Here the bifunctor ExtsEGn(−,−) is a suitable relative version of Ext constructed in the

(non-abelian) category EGn.The third chapter is computational in nature. Let G ⊂ Gn be a maximal finite sub-

group (of order divisible by p) at height n = p − 1. The spectrum EhGn is meant to“interpolate” between the spectrum En, whose homotopy groups are easy to compute andEhGnn ' LK(n)S

0 for which computing the homotopy groups seems an impossible task.There is a norm map between homotopy orbits and homotopy fixed points, and the Tatespectrum EtGn is the cofiber of this map, i.e. there is a fiber sequence

(En)hG → EhGn → EtGn .

Our next result was previously known to be true after K(n)-localisation (for example, itfollows from [Rog08] since EhGn → En is a K(n)-local G-Galois extension).

Theorem 3.3.6. The Tate spectrum EtGn is contractible.

As an example of an application of this result we show the following, where DnEhGn is the

monodial duality in the K(n)-local category, i.e.

DnX = F (X,LK(n)S0).

Corollary 3.3.11. EhGn is K(n)-locally self-dual up to suspension. In fact DnEhGn '

ΣNEhGn where N ≡ −n2 mod (2pn2) and N is only uniquely defined modulo 2p2n2.

We then discuss a generalisation of this result due to Mathew and Meier [MM13].

Introduction 13

In the final chapter we investigate the Picard group, Picn, of the K(n)-local category.In particular we are interested in the subgroup κn of invertible spectra such that there isan isomorphism E∨∗X ' E∗ of Morava modules. Our main result is the following, whichpartially answers a conjecture of Hopkins [Str92].

Theorem 4.4.1. Let p > 2. Then κn is a p-group. More specifically κn is a direct productof cyclic p-groups.

We then give an extended remark regarding the finiteness (or otherwise) of κn. In partic-ular, at height n = p− 1, we define a group κn(N) we believe should be closely related toκn and prove that it is a finite p-group.

Again, restricting to the case of n = p − 1 we study the map from Picn → Pic(EhGn ),the Picard group of EhGn -modules. We show that under this map X ∈ Picn always mapsto a suspension of EhGn . In the case that X ∈ κn we explicitly identify this suspension.In particular we show that there is a natural decomposition (as observed at height 2in [GHMR12]) into elements that are detected by a finite subgroup of Gn and elementsthat are not detected by any finite subgroup.

When n > 2 we are not able to produce any new elements of κn. In the case n = 2,however we can partially (independently) recover results from [GH12; GHMR12].

Theorem 4.6.8. There is an element P2 ∈ κ2 such that P2 ∧ EhG242 ' Σ48EhG24

2 and thereis an equivalence

I2 ' S2 ∧ S0〈det〉 ∧ P2.

We remark that this does not uniquely specify P2.We finish by outlining a methodology for generalising these calculations to the case

where n > 2.

CHAPTER 1

Prerequisites

The purpose of this introductory chapter is to provide an introduction to the coho-mology theories K(n) and En, and the Morava stabilizer group, Gn. When n = p− 1 wereview the classification of the finite subgroups of the Morava stabilizer group, as we willneed them throughout this document.

1.1. Bousfield localisation and Morava K-theory

In commutative algebra one often studies a ring by studying its localisations. This isalso true in the category of ring spectra; here classical localisation is replaced by Bousfieldlocalisation.

Let E be a cohomology theory. We say that a spectrum X is E-acyclic if E∧X is nulland that a spectrum Y is E-local if [X,Y ] is null whenever X is E-acyclic. Then a mapf : X → Y of spectra is an E-equivalence if the fibre is E-acyclic (equivalently, if E∗(f) isan isomorphism). Bousfield [Bou79] has constructed a functor LE from X to a (unique)E-local spectrum LEX along with a natural transformation η : X → LEX which is anE∗-equivalence. The map ηE is terminal amongst E∗-equivalences out of X.

If both E and X are connective, then Bousfield localisation behaves nicely.

Example 1.1.1. [Bou79] Let E = HQ be the rational Eilenberg-MacLane spectrum.Then if X is connective we have

π∗(LHQX) = π∗(X)⊗Q.

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16 Chapter 1: Prerequisites

In general when E and X are connective spectra localisation is the same as localisationor completion with respect to some set of primes. ♦

Let MS denote a suitable category of spectra, such as the S-modules constructedin [EKMM97]. Then Bousfield localisation is a functor LE :MS →MS,E , whereMS,E ⊂MS is the full-subcategory of E-local S-modules. The homotopy category obtained byinverting the weak equivalences will be denoted by DS,E .

We will be particularly interested in two cases. The first is localisation with respectto Morava K(n)-theory, denoted K(n). This is a complex oriented commutative ringspectrum (at least when p > 2) with coefficient ring

K(n)∗ ' Fp[v±1n ], n > 0,

with |vn| = 2(pn − 1) (the dependence on the prime p is always suppressed from thenotation). The associated formal group law Γn(x, y) has p-series

[p]Γn(x) = vnxpn.

The Morava K-theories were first constructed by Morava in the seventies, althoughthis work was never published. The first published reference was by Johnson and Wil-son [JW75]. The construction proceeds using the Baas-Sullivan construction [Baa73] tocreate versions of complex cobordism for manifolds with singularities. The modern the-ories of ring spectra, for example [EKMM97], significantly ease the construction of ringspectra such as these by constructing them as suitable quotients of the Brown-Petersonspectrum, BP . We note that the complex orientation of K(n) arises as a morphism ofring spectra

ψ : BP → K(n),

which induces a mapψ∗ : BP∗ → K(n)∗,

given by

ψ∗(vk) =

vn if k = n,

0 else.

By convention we will take K(0) = HQ to be the rational Eilenberg-MacLane spectrum,and K(∞) = HFp to be mod-p homology.

The Morava K-theories have a number of useful properties:

Proposition 1.1.2. [JW75; Rav84] Let p be a prime. Then for all 0 ≤ n ≤ ∞:

1.1. Bousfield localisation and Morava K-theory 17

(a) K(n)∗ is a graded field; every graded module over it is free.

(b) There is a Künneth formula; for all X and Y the natural map

(1.1) K(n)∗X ⊗K(n)∗ K(n)∗Y'−→ K(n)∗(X ∧ Y )

is an equivalence.

(c) If X is any spectrum then K(n) ∧X is a wedge of suspensions of K(n).

(d) If m 6= n then K(m) ∧K(n) is contractible.

(e) When n = 1,K(1) is a wedge summand of mod p complex K-theory.

These useful properties mean that K(n)∗X is reasonably computable - for exam-ple Ravenel and Wilson [RW80] have computed the Morava K-theory of the Eilenberg-MacLane spaces.

The Morava K-theories can, in some sense, be thought of as the (prime) fields of thestable homotopy category. This statement can be interpreted in a number of ways. Forexample one definition, due to Hopkins, calls a ring spectrum a prime field if, for all X,E ∧X is a wedge of suspension of E, and E is indecomposable as a spectrum. Given thisdefinition Hopkins and Smith [HS98] have shown that if E is a prime field then E ' K(n)for some n.

The second case we will be interested in is localisation with respect to the wedgesum K(0) ∨ · · · ∨ K(n), which we denote by Ln (by [Hov95, Corollary 1.12] this is alsolocalisation with respect to Morava E-theory, to be described in more detail in Section 1.2).These localisations have the important property that they are smashing; that is, for anyspectrum X,

LnX ' X ∧ LnS.

Such a statement is far from true in general.These localisation functors have been shown to be extremely useful in stable homotopy

theory. For example, the Ln assemble into a tower of localisations, known as the chromatictower:

· · · LnX Ln−1X · · · L1X L0X.

Then the chromatic convergence theorem of Hopkins and Ravenel [Rav92, Theorem 7.5.7]states that if X is a finite p-local spectrum then X is the homotopy limit of the chromatictower. Hence we can hope to study π∗X by studying each of the π∗LnX. This descriptionunderlines the whole approach to chromatic homotopy theory.


Slightly more is true in fact; for every X there is a homotopy pullback square

(1.2)LnX LK(n)X

Ln−1X Ln−1LK(n)X,

and so in order to study LnX we can inductively study LK(n)X as well as some attachingmaps. This underlines the importance of the categoryMS,K(n), of K(n)-local spectra.

Remark 1.1.3. Hopkins’ chromatic splitting conjecture [Hov95] includes conjectures relat-ing to the fracture square Equation (1.2). In particular Hopkins conjectures that there isa map LK(n)X → Ln−1X making the fracture square commute. This implies that there isa splitting Ln−1LK(n)X → Ln−1X of the bottom horizontal map. The chromatic splittingconjecture is known to be true for all primes when n = 1 and for n = 2, p ≥ 3.

1.2. Lubin-Tate theory and Morava E-theory

We noted above that the localisation LK(0)∨···∨K(n) is also localisation with respect toa cohomology theory called Morava E-theory, constructed in [Mor85]. This is a periodicLandweber exact cohomology theory, with coefficient ring

(En)∗ 'W(Fpn)[[u1, . . . , un−1]][u±1],

with |ui| = 0 and |u| = −2. Here W(Fpn) denotes the Witt vectors over Fpn , which canbe constructed by adjoining a primitive (pn − 1)-st root of unity to Zp.

To see why Morava E-theory is so useful it is necessary to briefly review its construction(for a detailed construction see [Rez98]). To do so we will need to assume basic knowledgeof formal group laws as covered in [Rav86, Appendix 2].

Let Γn be a formal group law of height n over a perfect field k of characteristic p > 0.We will define a functor Defk,Γn(−) from the category of complete local rings to thecategory of groupoids. Fix a complete local ring B, with maximal ideal m and projectionmap π : B → B/m. Then the objects of Defk,Γn(B) consist of pairs (G, i) where G is aformal group law over B and i is the projection map i : k → B/mB such that i∗Γ = π∗G.Such data is called a deformation.

Morphisms (G1, i1) → (G2, i2) are defined only when i1 = i2, and are given by iso-morphisms of formal group laws f : i1 → i2 such that π∗f = id. These are called?-isomorphisms. To a morphism φ : B → C of complete local rings we define Defk,Γn(φ)

1.2. Lubin-Tate theory and Morava E-theory 19

by sending (G, i) to (φ∗G,φ|B/mi).The groupoid Defk,Γn(B) splits into a disjoint union of Defk,Γn(B)i for a fixed i : k →

B/m. Then the following result is (part of) a theorem due to Lubin and Tate [LT66] (seealso the notes of Rezk [Rez98]). Following Rezk we use the notation π0 of a groupoidto be the π0 of the corresponding classifying space. In particular this is just the set ofisomorphism classes of the groupoid.

Theorem 1.2.1. The functor π0 Defk,Γn(−) from groupoids to sets sending B to its setof ?-isomorphism classes is corepresentable. In particular, there exists a ring E(k,Γn) 'W(k)[[u1, . . . , un−1]] such that

HomcW(k)−alg(E(k,Γn), B) ' π0(Defk,Γn(B))

with i : k '−→ E(k,Γn) and a formal group law F over E(k,Γn) such that the pair (F, id)is a universal deformation.

In other words, for each ?-isomorphism class (G, i) of π0(Defk,Γn(B)) there is a uniquering homomoprhism ψ : E(k,Γn)→ B and a unique ?-isomorphism f : ψ∗F → G.

We will be interested in the case where k = Fpn and Γn is the Honda formal grouplaw with p-series [p](x) = xp

n . Then the ring E(Fpn ,Γn) 'W(Fpn)[[u1, . . . , un−1]] carries aformal group law Γ, coming from the universal deformation of Γn. Let u have degree −2.It is not hard to check that uΓ(u−1x, u−1y) still forms a formal group law, now over thering (En)∗ := W(Fpn)[[u1, . . . , un−1]][u±1]. This is a p-typical formal group law, and thereis a map BP∗ → (En)∗, defined by

vi 7→

uiu

1−pi 1 ≤ i ≤ n

u1−pni = n

0 i > n.

In other words we choose the universal deformation Γ to have p-series

(1.3) [p]Γn(x) = px+Γn

u1xp +Γn

· · ·+Γnun−1x

pn−1 +Γnxp

n.

The map BP∗ → (En)∗ satisfies the criteria of the Landweber exact functor theorem,and so there is an associated homology theory En, with coefficient ring (En)∗.

This explicit construction of Morava E-theory highlights some properties of it. LetSn = Aut(Γn) be the group of automorphisms of the formal group law Γn. Such anautomorphism f is defined by a power series f ∈ Fpn [[x]]. Given a deformation (G, i)


lift i∗f ∈ B/m[[x]] along the quotient map to B[[x]] and apply this to G, obtaining anew deformation. It can be shown that this is independent of the choice of lift, up to?-isomorphism, and so this gives an action of Sn on DefFpn ,Γn(−), which in turn inducesan action on the corepresenting object E(k,Γ).

Since the Honda formal group law is defined over Fpn , a similar argument gives an ac-tion of Gal(Fpn/Fp) by pre-composition with the map i : Fpn → B/m. Together these givean action of Gn := Gal(Fpn/Fp)oSn on E(Fpn ,Γn), or even (En)∗. Brown representabilityimplies that there is an action of Gn on En in the stable homotopy category.

In fact much more is true. Let FG be the category of pairs (k,Γ) where k is a perfectfield of finite characteristic p and Γ is a formal group law of height n over k. ThenHopkins and Miller [Rez98] (in the A∞ case, followed by Goerss and Hopkins [GH04] forthe E∞-case) showed that there is a functor

ψ : FGop E∞ − ring spectra

(k,Γ) E(k,Γ)

such that E(k,Γ) is a commutative ring spectrum whose associated formal group law isthe universal deformation of Γ. The Goerss-Hopkins-Miller theorem also implies that theaction of Gn on En can be taken to be one of E∞-maps.

1.3. The action of the Morava stabilizer group

We can give a more explicit description of the group Gn. Define the non-commutativepolynomial ring

On = W(Fpn)〈S〉/(Sn = p, Sw = wσS)

where w ∈ W(Fpn) := W and σ is a lift of the Frobenius to W. Explicitly by [Rav86,A2.2.15] we can uniquely write

w =∞∑i=0

wipi ∈W

where wpn

i − wi = 0, and then the Frobenius is given by

wσ =∞∑i=0

wpi pi.

On is the endomorphism ring of Γn and Sn = O×n [Die57; Lub64]. Although we willnot give prove this statement we can at least describe how the endomorphisms arise.

1.3. The action of the Morava stabilizer group 21

Recall that the Honda formal group has p-series [p](x) = xpn . In order to simply our

notation we write

(1.4) f(t) =∞∑k=0

Γnaktpk, ak ∈ Zp

for the sum taken with respect to the formal group law Γn. An endomorphism of Γn is apower series of the form f(t) that commutes with the [p]-series; writing [p](f(t)) = f([p](t))and equating coefficients we find that ap

n

k = ak, and so ak ∈ Fpn [Rav86, Lemma A2.2.19].A simple endomorphism is given by S(t) = tp:

[p](S(t)) = (tp)pn = tpn+1 = S([p](t)).

Since [p](x) = xpn we get the relation Sn = p. Then for any a ∈ Fpn , at also commutes with

[p] and has the required form. We can then take all sums and limits to show that [Rav86,Lemma A2.2.20]

EndFpn (Γn) =θ(t) =

∞∑k=0

Γnaktpk : ak ∈ Fpn

.

The Witt vectors appear in the description of On as the subring of elements of theform ∑

n|k

Γnaktpk,

whilst w corresponds to the endomorphism t 7→ wt; this gives the relation Sw = wσS.The algebra On is a free rank n module over W with generators 1, S, · · · , Sn−1 and is

the ring of integers in a division algebra D over Qp of dimension n2 and Hasse invariant1/n. There is a discrete valuation ν on D such that

1) ν(ab) = ν(a) + ν(b);

2) ν(a) =∞ if and only if a = 0; and,

3) ν(a+ b) ≥ minν(a), ν(b).

The valuation is normalised such that ν(p) = 1. This ensures that the usual valuation onQp extends uniquely to the valuation on D. The subring On is given by

On = x ∈ Dn|ν(x) ≥ 0

which implies thatO×n = x ∈ Dn|ν(x) = 0.


We write an arbitrary element a ∈ On

a =∑

0≤i≤n−1aiS

i

where ai ∈W. The Morava stabilizer group Sn is then the set of elements with a0 a unitin W. The Galois group is generated by the Frobenius σ and acts on the coefficients inthe obvious way, namely

σ(a0 + a1S + . . . an−1Sn−1) = σ(a0) + σ(a1)S + · · ·+ σ(an−1)Sn−1.

The group Sn acts on On by right multiplication through left W(Fpn)-module homo-morphisms and so there is a homomorphism

Sn → GLn(W(Fpn)) det−−→W(Fpn)×.

The image of this map is exactly Z×p , and since the whole map is Gal(Fpn/Fp)-invariant,there is an induced homomorphism Gn → Z×p . We identify Z×p with µp−1 × (1 + pZp)× 'µp−1 × Zp, and thus by composition with the quotient map to Zp we get the reduceddeterminant map Gn → Zp. As is customary we denote by G1

n the kernel of this map, i.e.there is a short exact sequence

1→ G1n → Gn

det−−→ Zp → 1.

We also define SGn as the kernel of the determinant to Z×p :

1→ SGn → Gn → Z×p → 1.

The two are related by the short exact sequence

1→ SGn → G1n → µp−1 → 1.

We will need to know the action on the central subgroup of Gn, which can be seento be Z×p (for example the center of Gn is the same as the center of Sn, and one canconsider the equation gω = ωg for ω a (pn − 1)-st root of unity). The following lemma iswell-known; we learnt it from [Hen07, Lemma 22].

Lemma 1.3.1. Let g ∈ Z×p ⊂ Gn. Then g∗ui = ui and g∗u = gu.

Proof. We first describe how to calculate the action (see [DH95, Section 1]). Let g ∈ Sn.

1.4. Homotopy fixed point spectra for subgroups of the Morava stabilzier group 23

Lift g to a power series g(x) ∈ (En)0[[x]] and define a new formal group law by

H(x, y) = g−1(Γn(g(x), g(y))

)By Lubin-Tate theory there is a homomorphism g∗ : (En)∗ → (En)∗, which defines a ?-isomorphism h from (g∗(Γn), g∗(u)) to (H, g′(0)u). The action of g on u is then given bythe formula g∗(u) = g′(0)h′(0)u.

Now assume that g ∈ Z×p ⊂ Gn is contained in the central subgroup. The morphismZ → EndFpn (Γn) given by sending m to the homomorphism m : Γn → Γn extends toa map Zp → EndFpn (Γn). This lifts to the action of the universal deformation, whichgives h(x) = x and g(x) = [g]Γn

(x). Equation (1.3) then implies that g∗(u) = gu andg∗(ui) = ui.

1.4. Homotopy fixed point spectra for subgroups of the Morava stabilziergroup

For the action of a discrete group G on a spectrum X we can form the homotopy fixedpoint spectrum XhG as the G fixed points of the function spectrum F (EG+, X), and thereis an associated homotopy fixed point spectral sequence

H∗(G;X∗Z)⇒ π∗F (Z,XhG)

arising from the skeletal filtration of EG. Note that the action is required to be on thepoint-set level, and not just up to homotopy.

In the case we are interested in, namely X = En and G = Gn, both Gn and En areprofinite, and the action is continuous, and so this construction does not apply. The workof Goerss-Hopkins-Miller described earlier implies that the action of Gn can be rigidifiedto act on the spectrum level, and so we can define XhG for G ⊂ Gn finite in the usualway. Furthermore Devinatz and Hopkins [DH04] have given a construction of homotopyfixed point spectra EdhFn for arbitrary closed subgroups F ⊂ Gn. This starts with theconstruction of EdhUn for U ⊂ Gn an open subgroup of Gn. Since Gn is a profinitecompact p-adic analytic Lie group there is a sequence of normal subgroups

Gn = U0 ) U1 ) U2 ) · · ·


such that⋂k Ui = e. Then for closed subgroups F ⊂ Gn Devinatz and Hopkins define

EdhFn := LK(n) colimiEhdUiFn .

These come with homotopy fixed points spectral sequences

Es,t2 = Hsc (F ;πt(En))⇒ πt−s(EhFn ),

and in the case F = Gn, an equivalence EdhGnn ' LK(n)S

0. In this later case the spectralsequence sequence for EhdGn

n coincides with the Adams-Novikov spectral sequence forLK(n)S

0. Cohomology here is continuous group cohomology, to be described in moredetail below. In the case that F is finite the spectrum EdhFn constructed by Devinatz andHopkins coincides with the usual homotopy fixed point spectrum EhFn .

In describing this construction we have used the notation EdhFn because the spectrumis not constructed with respect to a continuous F -action. Davis [Dav06] has constructedsuch a continuous action, and has shown that EhFn ' EdhFn , and so we will omit thisnotation from now on.

Remark 1.4.1. In recent work Quick [Qui13a; Qui13b] has constructed a stable model cat-egory of profinite G-spectra, and defines XhG := FG(EG+, X), where the maps are nowtaken inside this category of profinite G-spectra. Quick shows that En has a canonicalmodel in the category of profinite Gn-spectra and so can define EhGn for G ⊂ Gn withoutusing the Devinatz-Hopkins construction of EhUn . Moreover these are shown to be equiv-alent to the construction of Devinatz-Hopkins and Davis. However this construction doesnot (directly) prove that EhGn is an E∞-ring spectrum.

We now describe in careful detail the definition of continuous group cohomology thatwe use, starting with some definitions from [SW00]. Let G be a profinite group. Then Mis a discrete G-module if M is a discrete abelian group and there is a continuous actionψ : G × M → M . The category of p-torsion discrete left G-modules will be denotedby Dp(G). Let Fp(G) denote the full subcategory of Dp(G) whose objects are finite,discrete left Zp[[G]]-modules. Finally Cp(G) will denote the category of topological Zp[[G]]-modules whose objects consist of inverse limits of objects in Fp(G). Group cohomologywith coefficients in Dp(G) can be defined in the usual way [SW00] as the derived functorExt•Zp[[G]](Zp,M).

A module M ∈ Cp(G) will be called type FP∞ if it has a projective resolution P• inCp(G) where each Pi can be chosen to be finitely-generated and free. A profinite groupG is of type type p-FP∞ if the trivial Zp[[G]]-module Zp is of type FP∞. Compact p-


adic analytic groups are always of type p− FP∞ [SW00, Proposition 5.1.2]. We have thefollowing finiteness result.

Lemma 1.4.2. [SW00, Proposition 4.2.2] Let G be a profinite group of type p − FP∞.Then for all F ∈ Fp(G)

|Hn(G;F )| < +∞.

We now work with the case G = Gn. Let I = (pi0 , iui11 , . . . , i

uin−1n−1 ) ⊂ (En)∗ be an ideal.

Such a system of ideals forms a cofiltered system such that

(En)∗ = lim←−I

πtEn/IπtEn.

Since each πtEn/IπtEn is a finite discrete Zp[[Gn]]-module this displays (En)∗ as an objectof Cp(Gn).

In the sequence of ideals I there is a chain of ideals

I0 ⊃ I1 ⊃ · · · ⊃ Ik ⊃ · · ·

which can be realized as a sequence of generalised Moore spectra MIk[HS99b, Section 4]

such that πt(En ∧MIk∧X) ' πt(En ∧X)/Ikπt(En ∧X) whenever X is finite.

For finite X we can then define

(1.5) Hsc (Gn, πt(En ∧X)) = lim←−

k

Hsc (Gn, πt(En ∧X ∧MIk

)).

Remark 1.4.3. From now we, we will usually omit the subscript c from our notation forcontinuous group cohomology.

1.4.1. Finite subgroups at height n = p− 1

In this section we now assume that n = p − 1; this is an important case due tothe presence of p-torsion in Gn, which implies that the cohomological dimension of Gn

is infinite. We will principally be interested in EhFn when F ⊂ Gn is a maximal finitesubgroup (up to conjugacy) of Gn, whose order is divisible by p. These sometimes go bythe name of the higher real K-theories, and are sometimes1 denoted EOn, for reasons thatthe following simple example explains.

1although we do not use this notation


Example 1.4.4. We consider the simplest case of n = 1 and p = 2. Note that for n = 1the Honda formal group law over F2 is isomorphic to the multiplicative formal group law:

Gm(x, y) = x+ y − xy.

There is an isomorphism EndF2(Gm) ' Z2, and so AutF2(Gm) ' Z×2 ' Z2×C2, whereC2 is generated by -1. Now E(F2,Gm) = Z2, and in fact the universal deformation isGm, now considered over Z2. Finally then, by adjoining a degree 2 element u, we see thatE1 is nothing other than p-completed complex K-theory, and the action of G1 is just the(p-adic) Adams operations.

There is a unique maximal finite subgroup of Z×2 , namely C2 ' ±1, and so we canform the homotopy fixed points spectrum EhC2

1 . By calculating the homotopy groups ofthis spectrum we can see that this is actually 2-complete real K-theory; EhC2

1 ' KO∧2 (forexample [Rog08, p. 5.5.2]). ♦

We now move to the general case. The maximal finite subgroups of Sn have been classifiedby Hewett, in his work on subgroups of finite division algebras [Hew99], whilst the case ofGn has been studied by Bujard [Buj12]. We will be interested in finite subgroups of orderdivisible by p.

By [Buj12, Theorem 1.3.1] for n = p − 1 there are exactly 2 conjugacy classes ofmaximal finite subgroups of Sn represented by

F0 = Cpn−1

andF = F1 = Cp o Cn2

where the action of the right factor on the left is given by the mod n reduction mapCn2 → Aut(Cp) ' Cn

We choose a presentation of the group F as the following

F = 〈ζ, τ |ζp = 1, τn2 = 1, τ−1ζτ = ζe〉,

where e ∈ (Z/p)× is a generator.

It is worthwhile to see how these finite subgroups arises. Again, we refer the reader tothe standard references [Buj12; Hew99] for more detailed information, as well as [Hen07,Section 3.6].


Let ω ∈ W× ⊂ Sn be a (pn − 1)-st root of unity. Then X := ω(p−1)/2S ∈ Dn is anelement such that Xn = −p. It can be shown that the subfield Qp(X) ∈ Dn is isomorphicto the cyclotomic extension Qp(ζp) generated by a primitive p-th root of unity ζp [Hen07,Lemma 19]. A simple check shows that XωX−1 = ωp. Then, if we let η = ω(pn−1)/n2 ∈W(Fpn) we have

ηXη−1 = η−nX;

since η−n is an n-th root of unity, conjugation by η induces an automorphism of Qp(X).We will write τ for the image of η in Sn. The subgroup generated by ζp and τ is

precisely the maximal finite p-subgroup of F of Sn. That this is unique up to conjugacyfollows from an argument using the Skolem-Noether theorem, as in [Buj12, Example 1.33].This relies on the fact that XηX−1 = ηp, which implies that the subgroup F is normalisedby X (note that X commutes with ζ.)

Example 1.4.5. Let n = 2, p = 3 and ω be a primitive 8-th root of unity. Then theelement ζ = −1

2(1 +ωS) is an element of order 3 and ω2ζω−2 = ζ2, so that F is generatedby ζ and τ := ω2. Here F usually goes by the name G12 and is abstractly isomorphic tothe non-trivial semi-direct product C3 o C4. ♦

We wish to extend this to Gn; that is, to find a group G such that there is an extensionof the form

1→ F → G→ Gal(Fpn/Fp)→ 1

where F is a maximal finite subgroup in Sn. This is in fact thoroughly analysed in [Buj12,Chapter 4], although we can analyse our case more simply [Sym04]. Let NSn(〈ζ〉) be thenormalizer of the subgroup of order p inside Sn. Every conjugate of F inside Gn is infact in Sn [Sym04]; from the discussion above it is therefore conjugate to F inside Sn.This implies that NGn(F )/NSn(F ) ' Gal. We choose c to have image σ (the Frobenius),under this isomorphism. Then F and c generate a subgroup G of order pn3. In fact, asin [Sym04] we can choose c such that it acts trivially on 〈ζ〉 for 〈c〉 to have no pro-p part.

Example 1.4.6. Again let n = 2, p = 3 Then let ψ = ωφ, where φ is the generator ofthe Galois group. The group generated by ψ, ζ and τ goes by the name G24. The groupgenerated by τ and ψ is the quaternion group Q8 and there is an abstract isomorphismG24 ' C3 nQ8.

CHAPTER 2

Morava modules and the K(n)-local category

The goal of this chapter is to investigate the category EGn of Morava modules, a goodinvariant of the K(n)-local category. As mentioned in the introduction this category playsan important role in the construction of the Goerss-Henn-Mahowald-Rezk resolution ofthe K(2)-local sphere [GHMR05]. In particular they construct, and make heavy use of,the commutative diagram

(2.1)π∗En[[Gn/H1]]hH2 ((En)∗[[Gn/H1]])H2

π∗F (EhH1n , EhH2

n ) HomEGn(E∨∗ EhH1n , E∨∗ E

hH2n ).

' '

Here the bottom horizontal map is the En-Hurewicz map, and the top is the edgehomomorphism in the homotopy fixed point spectral sequence. Note that if E is a spectrumand X = limiXi is an inverse limit of a sequence of finite sets, then we define

E[[X]] := holimiE ∧ (Xi)+,

and(En)∗[[Gn]] = lim

i(En)∗[Gn/Ui],

is the completed group ring.In this section we will show that the bottom map is also an edge homomorphism in

the K(n)∗-local En-Adams spectral sequence, and the two spectral sequences are in fact

29

30 Chapter 2: Morava modules and the K(n)-local category

isomorphic. In fact, whilst the Goerss-Henn-Mahowald-Rezk construction requires thatH2

is finite, our result holds more generally when H2 is a closed subgroup of Gn (we note thatBehrens and Davis [BD10] have constructed the rightmost vertical arrow in Equation (2.1)for an arbitrary closed subgroup of Gn).

2.1. The K(n)-local category

In this thesis we will primarily work in the category MS,K(n) of K(n)-local spectra,which we now briefly describe. For more details see [HS99b].MS,K(n) is defined to be the full subcategory of the catgeory of spectra whose objects

areK(n)-local. MS,K(n) is a closed symmetric monoidal category; since the smash productof two K(n)-local spectra need not be K(n)-local, the monoidal product is given by theK(n)-local smash product LK(n)(X ∧ Y ). To keep our notation compact we will usuallywrite X ∧ Y for the K(n)-local smash product. The closed structure is given by thefunction space F (X,Y ); it is easy to check (using the universal property of Bousfieldlocalisation) that this is K(n)-local whenever Y is. In a suitable category of spectra theseform a adjoint pair; LK(n)(X ∧ −) is left adjoint to F (X,−).

Definition 2.1.1. Let X be a spectrum. Then we define

E∨∗X := π∗LK(n)(E ∧X).

This has been seen to be a more natural covariant analogue of E∗X than E∗X, despitethe fact that it is not a homology theory (as it does not preserve coproducts and filteredhomotopy colimits).

The functor E∨∗ (−) lands not just in the category of E∗-modules, but rather in the sub-category M of L-complete E∗-modules, which are discussed in detail in [HS99b, AppendixA]. We review the basics of this theory now.

Definition 2.1.2. Let LsM be the s-th left derived functor of the completion functor(−)∧m.

Not that completion is not right (nor in fact left) exact; thus it is not true that L0M 'M∧m

1. In fact the natural map M →M∧m factors as the composite M ηM−−→ L0MεM−−→M∧m .

Definition 2.1.3. M is L-complete if ηM is an isomorphism.1It still makes sense to consider the left derived functor of an arbitrary additive functor; the fact that it

is not right exact simply implies that the zeroth derived functor fails to coincide with the original functor.

2.1. The K(n)-local category 31

The map εM is surjective with kernel [HS99b, Theorem A.2]

(2.2) lim←−1k TorE∗1 (E∗/mk,M).

The category of L-complete E∗-modules is an abelian subcategory of E∗-modules, andis closed under extensions and inverse limits. One salient feature of this category is thatExts

M(M,N) ' ExtsE∗(M,N) whenever M and N are L-complete E∗-modules [Hov04,

Theorem 1.11].

Remark 2.1.4. In the language of Salch [Sal10], M is the best reflexive abelian approx-imation to the category of complete E∗-modules, which is not itself an abelian category(for more on the category of complete E∗-modules see Section 2.2)

By [HS99b, Proposition 8.4] the functor E∨∗ (−) always takes value in L-complete E∗-modules. In fact we know (see [HS99b, Proposition 7.10]) that if X is E-local thenLK(n)X = holimI X ∧MI where MI is a generalised Moore spectrum (as in Section 1.4.)This description, along with the fact that Ln is smashing, then gives a Milnor exactsequence

(2.3) 0→ lim←−1E∗+1(X ∧MI)→ E∨∗X → lim←−E∗(X ∧MI)→ 0,

which can be used to show that E∨∗X is L-complete.

Definition 2.1.5. An L-complete E∗-module is pro-free if it is isomorphic to the comple-tion of a free E∗-module.

Equivalently, these are the projective objects in the category M. These will play animportant role for us; in fact all spectra we will consider will have this property by thefollowing lemma.

Lemma 2.1.6. [HS99b, Proposition 8.4] If K(n)∗X is concentrated in even dimensions,then E∨∗X is pro-free and concentrated in even dimensions.

Note that whenever E∨∗X is pro-free the lim←−1 term in Equation (2.3) vanishes (this fol-

lows since pro-freeness implies that E∗(X ∧MI) ' E∗X/(pi0 , . . . , vin−1n )) and there is an

isomorphism E∨∗X ' (E∗X)∧m.We will need the following version of the universal coefficient theorem (for Y = S this

is [Hov04, Corollary 4.2]).

Lemma 2.1.7. If E∨∗X is pro-free then

HomE∗(E∨∗X,E∨∗ Y ) ' π∗F (X,LK(n)(E ∧ Y )).


Proof. Let M,N ∈ ME,K(n). Hovey [Hov04] has constructed a natural, strongly andconditionally convergent, spectral sequence of E∗-modules

Es,t2 = Exts,tM

(π∗M,π∗N) ' Exts,tE∗(π∗M,π∗N)⇒ πs+tFE(M,N)

Set M = LK(n)(E ∧X) and N = LK(n)(E ∧X). Note then that

FE(LK(n)(E ∧X), LK(n)(E ∧ Y ))) ' FE(E ∧X,LK(n)(E ∧ Y )) ' F (X,LK(n)(E ∧ Y )),

where the second isomorphism is [EKMM97, Corollary III.6.7], giving a spectral sequence

Es,t2 = Exts,tM

(E∨∗X,E∨∗ Y ) ' Exts,tE∗(E∨∗X,E

∨∗ Y )⇒ πs+tF (X,LK(n)(E ∧ Y ))

Since E∨∗X is pro-free it is projective in M and so the spectral sequence collapses, givingthe desired isomorphism.

One of the peculiarities of the m-adic topology is the fact that any homomorphism betweencomplete E∗-modules (for example, if E∨∗ Y is pro-free, this applies to the above lemma)is, in fact, continuous with respect to this topology. We learnt this from Charles Rezk,who also provided the following proof.

Lemma 2.1.8. Let f : M → N be a E∗-module homomorphism between complete E∗-modules. Then f is continuous.

Proof. f is a E∗-module homomoprhism and so f(mnM) is a subset of mnN ; in turn mnM

is a subset of f−1(mnN). Therefore f−1(mnN) is a union of mnM -cosets. It follows fromthe fact that mnM is open that f−1(mnN) is open in the m-adic topology.

M is a symmetric monoidal category with monoidal product L0(M ⊗E∗ N) for twoL-complete E∗-modules. However if M and N are pro-free then L-completion of thetensor product has a slightly simpler description. We define the complete tensor product,M⊗E∗N , of two complete E∗-modulesM and N , to be the m-adic completion ofM⊗E∗N .

We thank Mark Hovey for the proof of second part of the following lemma.

Lemma 2.1.9. Let M and N be in MS,K(n). If π∗M and π∗N are pro-free E∗-modulesthen so is π∗LK(n)(M ∧E N) and there is an isomorphism

(2.4) π∗LK(n)(M ∧E N) ' L0(π∗M ⊗E∗ π∗N) ' π∗M⊗E∗π∗N

2.1. The K(n)-local category 33

Proof. The first part of the lemma, as well as the first equivalence of Equation (2.4)is just [Hov04, Corollary 5.6]. Since the ideal m is invariant there is an isomorphismπ∗M⊗E∗π∗N ' (π∗M ⊗E∗ π∗N)∧m. Thus it will suffice to show that L0(π∗M ⊗E∗ π∗N) '(π∗M ⊗E∗ π∗N)∧m

By Equation (2.2) there is a short exact sequence

0→ lim←−1k TorE∗1 (E∗/mk, π∗M ⊗E∗ π∗N)→ L0(π∗M ⊗E∗ π∗N)→ (π∗M ⊗E∗ π∗N)∧m → 0

and so it will further suffice to show that TorE∗1 (E∗/mk, π∗M ⊗E∗ π∗N) = 0 for all k.Inductively using the short exact sequence

0→ mk−1/mk → E∗/mk → E∗/m

k−1 → 0

we can reduce this further to the statement that TorE∗1 (E∗/m, π∗M ⊗E∗ π∗N) = 0.

Now by [HS99b, Theorem A.9] if π∗M is pro-free then TorE∗s (π∗M,E∗/m) = 0 fors > 0. It follows that TorE∗s (π∗M,K) = 0 whenever mK = 0, since such a module is adirect sum of copies of E∗/m.

Let· · · → P1 → P0 = E∗ → E∗/m→ 0

be a projective resolution of E∗/m and split this into a short exact sequence of E∗-modules

0→ K1 → E∗ → E∗/m→ 0

and a surjectionP1 K1 → 0.

Since π∗M is pro-free TorE∗s (π∗M,E∗/m) = 0 (again, for s > 0) and so there is a shortexact sequence

0→ π∗M ⊗E∗ K1 → π∗M → π∗M/m→ 0

and a surjectionπ∗M ⊗E∗ P1 π∗M ⊗E∗ K1 → 0.

Similarly since π∗N is pro-free TorE∗1 (π∗N, π∗M/m) = 0 and we get a short exact sequence

0→ (π∗N ⊗E∗ π∗M)⊗E∗ K1 → (π∗N ⊗E∗ π∗M)→ (π∗N ⊗E∗ π∗M)/m→ 0


and a surjection

(π∗N ⊗E∗ π∗M)⊗E∗ P1 (π∗N ⊗E∗ π∗M)⊗E∗ K1 → 0.

This is precisely the statement that TorE∗1 (E∗/m, π∗M ⊗E∗ π∗N) = 0.

Taking M = LK(n)(E ∧ X) and N = LK(n(E ∧ Y ) gives the following corollary, whichshould be compared with Equation (1.1).

Corollary 2.1.10. If E∨∗X and E∨∗ Y are pro-free E∗-modules then there are isomor-phisms

E∨∗ (X ∧ Y ) ' L0(E∨∗X ⊗E∗ E∨∗ Y ) ' E∨∗X⊗E∗E∨∗ Y.

2.2. Morava modules and complete E∨∗ E-comodules

From Chapter 1 we know that Gn acts on E∗, and so in turn there is an action of Gn

on E∨∗X. This motivates the following definition, which we take from [GHMR05].

Definition 2.2.1. AMorava moduleM is a complete E∗-module with a continuous actionof Gn such that

g(ax) = g(a)g(x) for g ∈ Gn, a ∈ (En)∗, x ∈M

We will write EGn for the category of Morava modules.The prototypical example of a Morava module is E∨∗X, whenever this is pro-free (as

then M is complete as an E∗-module).Given a complete E∗-moduleM we give Homc(Gn,M) the diagonal Gn-action, and the

obvious E∗-module structure. Since M is complete, this gives Homc(Gn,M) the structureof a Morava module. Here we define

Homc(Gn,M) ' lim←−k

lim−→i

map(Gn/Ui,M/mkM),

where Ui is, as previously, a system of open subgroups of Gn with⋂Ui = e.

Remark 2.2.2. Suppose M is a complete E∗-module. Since Gn is profinite and M iscomplete, we have an isomorphism (see, for example [Tor07, Lemma 4.9])

Homc(Gn,M) ' Homc(Gn, E∗)⊗E∗M.

2.2. Morava modules and complete E∨∗ E-comodules 35

The functor M 7→ Homc(Gn,M) from complete E∗-modules to Morava modules isright adjoint to the forgetful functor [GHMR05, Remark 2.3].

We wish to relate the category of Morava modules to the category of comodules overa certain Hopf algebroid. The difficultly here is the we do not consider E∗E but ratherE∨∗ E, and this is not a Hopf algebroid as usually considered.

In the following let R be a complete regular Noetherian local ring with maximal idealm2. We simply take what we need from Devinatz [Dev95], although the exposition issimplified based on our assumptions on R.

Definition 2.2.3. Let ModcR be the category of complete R-modules M . Here completemeans with respect to the m-adic filtration; that is, there is an isomorphism

M ' lim←−k

M/mkM.

A morphism of complete R-modules is an R-module homomoprhism f : M → N

that is continuous with respect to the m-adic topology. ModcR is an additive categorywith kernels and cokernels, however it is not an abelian category. The completed tensorproduct is defined, as above, to be the m-adic completion of the ordinary tensor product.

The tensor product endows ModcR with a symmetric monoidal structure (with unit R).A complete R-algebra A is then a commutative monoid object in this category. We denotethe category of such objects by AlgcR.

Finally we can give the definition we need:

Definition 2.2.4. A complete Hopf algebroid is a cogroupoid object in AlgcR.

In particular it is a pair (A,Γ), where A,Γ ∈ AlgcR along with a collection of maps

ηR : A→ Γ (source)

ηL : A→ Γ (target)

∆ : Γ→ Γ⊗AΓ (composition)

ε : Γ→ A (identity)

c : Γ→ Γ (inverse).

satisfying the usual identifies for a Hopf algebroid.We can then construct the category of complete left comodules ComodcΓ in the usual

way. The following result of Devinatz will give us the example of complete Hopf algebroids2In this case regular means that if m = (r1, . . . , rn), where n is as small as possible, then the Krull

dimension of R is n.


that we require.

Theorem 2.2.5. [Dev05] Let A be a complete Noetherian regular local ring, and let S bea profinite group acting on A via ring homomorphisms. Let Γ = Homc(S,A). Then (A,Γ)has a canonical structure of a complete Hopf algebroid.

Here Homc(S,A) has m-adic filtration induced by that on A; i.e. F i Homc(S,A) =Homc(S,miA). Devinatz shows that the maps are defined by

ηR(a)(s) = a,

ηL(a)(s) = s−1a,

ε(f) = f(e),

c(f)(s) = s−1(f(s−1)).

To define ∆, there is an isomorphism

σ : Homc(S,A)⊗A Homc(S,A) '−→ Homc(S × S,A),

defined byσ(f1 ⊗ f2)(s1, s2) = s−1

2 (f1(s1)) · f2(s2).

Then ∆ = σ−1 · ∆, where

∆ : Homc(S,A)→ Homc(S × S,A)

is induced by the multiplication S × S → S.This immediately implies that (E0, (E0)∨∗ (E0)) is a complete Hopf algebroid. There is

also an appropriate graded version of all these definitions and (E∗, E∨∗ E) is a graded com-plete Hopf algebroid. See [Dev95, Remark 1.13] for details. Here the fact that everythingis concentrated in even degrees only is important.

As usual [Rav86, Appendix A.1.2.1] the forgetful functor from Γ-comodules to A-modules has a right adjoint, given by Γ⊗A(−). Specializing to the case (E∗, E∨∗ E) theright adjoint takes the form M 7→ Homc(Gn, E∗)⊗E∗M for a complete E∗-module M .By Remark 2.2.2 we have Homc(Gn, E∗)⊗E∗M ' Homc(Gn,M). Thus the right adjointfunctors

Homc(Gn,−) : complete E∗-modules→ Morava modules

andHomc(Gn, E∗)⊗E∗(−) : complete E∗-modules→ E∨∗ E-comodules

2.3. Homological algebra for Morava modules 37

are, in some sense, isomorphic, although a priori they lie in different categories. We nowmake this precise by showing that the category of Morava modules is equivalent to thecategory of E∨∗ E-comodules.

Proposition 2.2.6. There is an equivalence of symmetric monoidal categories betweenthe category of Morava modules and the category of complete (E∗, E∨∗ E)-comodules.

Proof. The argument essentially goes back to Devinatz [Dev95], although the proof as weneed it is given in [Tor07] (where Morava modules are called complete twisted E∗ − Gn-modules). Given a complete E∨∗ E-comodule M (which is, of course, also a completeE∗-module) the map

MψM−−→ E∨∗ E⊗E∗M ' Homc(Gn, E∗)⊗E∗M

ev(g)⊗id−−−−−→ E∗⊗E∗M 'M,

where ev(g) is the evaluation map of g ∈ Gn, defines a compatible Gn-action on M , whichTorii shows is continuous.

Conversely given a Morava module M the adjoint of the Gn-action map gives a mapM → Homc(Gn,M) ' E∨∗ E⊗E∗M , which is continuous, and hence defines an E∨∗ E-comodule structure on M .

Both categories are symmetric monoidal and the equivalence of categories respects thesymmetric monoidal structure.

2.3. Homological algebra for Morava modules

We wish to consider homological algebra for Morava modules. Given that the categoryof Morava modules is equivalent to the category of complete E∨∗ E-comodules, we willinstead consider homological algebra for complete E∨∗ E-comodules, as this simplifies theexposition somewhat. This has previously been considered in [Dev95].

2.3.1. Relative homological algebra

The category of complete E∨∗ E-comodules is not abelian; it is an additive category withcokernels. Hence we need to use the methods of relative homological algebra, which webriefly review now. For a more thorough exposition see [EM65] (although in general oneneeds to dualise what they say, since they mainly work with relative projective objects).Our work is in fact similar to that of Miller and Ravenel [MR77].

Definition 2.3.1. An injective class I in a category C is a pair (D,S) where D is a classof objects and S is a class of morphisms such that:


1) I is in D if and only if for each f : A→ B in S

f∗ : HomC(B, I)→ HomC(A, I)

is an epimorphism. We call such objects relative injectives.

2) A morphism f : A→ B is in S if and only if for each I ∈ D the sequence

f∗ : HomC(B, I)→ HomC(A, I)

is an epimorphism. These are called the relative monomorphisms.

3) For any object A ∈ C there exists an object Q ∈ D and a morphism f : A→ Q in S.

Remark 2.3.2. Note that given either S and D, the other class is determined by the re-quirements above, and that the third conditions ensures the existence of enough relativeinjectives.

It is not hard to check that D is closed under retracts and that if the composite morphismA

f−→ B → C is in S then so is f : A→ B.

Example 2.3.3 (The split injective class). The split injective class Is has Ds equalto all objects of C and Ss all morphisms that satisfy Definition 2.3.1, i.e. HomC(f,−) issurjective for all objects. One can easily check that this is equivalent to the requirementthat f : A→ B is a split monomorphism. ♦

Example 2.3.4. If we let D be the set of all injective modules over some ring R, and Sis the set of all maps, then we are in the usual situation of homological algebra. ♦

One way to construct an injective class is via a method known as reflection of adjointfunctors. Suppose that C and F are additive categories with cokernels, and there is a pairof adjoint functors

T : C F : U.

If (D,S) is an injective class in C then we define an injective class (D′,S ′) in F where theclass of objects are given by the set of all retracts of T (D) and the class of morphisms isgiven by all morphisms whose image (under U) is in S. For a proof see [EM65, pp. 15]- here it is proved for relative projectives, but it is essentially formal to dualise the givenargument. Here is sketch of the argument. First note that, since relative injectives are


closed under retracts, to show that D′ is as claimed, it suffices to show that T (I) is relativeinjective, whenever I ∈ D. Let A→ B be in S ′ and I ∈ D; then the map

HomF (B, T (I))→ HomF (A, T (I))

is equivalent to the epimorphism

HomC(S(B), I)→ HomC(S(A), I).

A similar method shows that the relative monomorphisms are as claimed. Finally weobserve that for all A ∈ F there exists a Q ∈ D such that S(A) → Q ∈ S. Then theadjoint A → T (Q) satisfies Condition 3. To see this note that S(A) → Q factors asS(A) → S(T (Q)) → Q; since relative monomorphisms are closed under left factorisation(see above) S(A)→ S(T (Q)) ∈ S. Then A→ TQ ∈ S ′ as required.

Definition 2.3.5. An extended complete E∨∗ E-comodules is a comodule isomorphic toone of the form E∨∗ E⊗E∗M where M is a complete E∗-module. Here the comultiplicationis given by the map

E∨∗ E⊗E∗M∆⊗id−−−→ E∨∗ E⊗E∗E∨∗ E⊗E∗M.

Example 2.3.6. [Dev95] Let ModcE∗ be the category of complete E∗-modules, with thesplit injective class. Then the adjunction

HomModcE∗

(A,B) = HomE∨∗ E(A,E∨∗ E⊗E∗B)

defines an injective class in complete E∨∗ E-comodules. In particular we have

1) S is the class of all comodule morphisms f : A → B whose underlying map ofcomplete E∗-modules is a split monomorphism.

2) D is the class of complete E∨∗ E-comodules which are retracts of extended completeE∨∗ E-comodules.

Note that for any complete E∨∗ E-comodule M the coaction map Mψ−→ E∨∗ E⊗E∗M is a

relative monomorphism into a relative injective. ♦

We will say that a three term complex M f−→ Ng−→ P of comodules is relative short exact

if gf = 0 and f : M → N is a relative monomorphism. A relative injective resolution is aresolution of relative injectives formed by splicing together relative short exact sequences.Note that, essentially by definition, relative exact sequence are precisely those that give


exact sequences of abelian groups after applying HomE∨∗ E(−, I), whenever I is relativeinjective.

We have the usual comparison theorem for relative injective resolutions. The proofis nearly identical to the standard inductive homological algebra proof - in this contextsee [HM70, Theorem 2.2].

Proposition 2.3.7. Let M and M ′ be objects in an additive category C with relativeinjective resolutions P∗ and P ′∗. Suppose there is a map f : M → M ′. Then, there existsa chain map f∗ : P∗ → P ′∗ extending f that is unique up to chain homotopy.

Definition 2.3.8. (c.f [EM65, pp. 7].) Let M and N be complete E∨∗ E-modules. Let I•

be a relative injective resolution of N . Then we define

ExtsE∨∗ E(M,N) = Hs(HomcE∨∗ E

(M, I•))

Remark 2.3.9. It would possibly be more correct to write this as Exts(C,D)(M,N) to ex-plicitly identify the fact that this is relative to an injective class, however we choose tonot use this notation.

Note that Proposition 2.3.7 implies that the derived functor is independent of the choiceof relative injective resolution.

As in [MR77] we have the standard, or cobar resolution Ω∗(E∨∗ E,M), with

Ωn(E∨∗ E,M) = E∨∗ E⊗E∗ · · · ⊗E∗E∨∗ E︸︷︷︸n+1 times

⊗E∗M

and differential

d(e0 ⊗ · · · ⊗ es ⊗m) =n∑i=0

(−1)ie0 ⊗ · · · ei−1 ⊗∆(ei)⊗ ei ⊗ · · · ⊗m

+ (−1)s+1e0 ⊗ · · · es ⊗ ψ(m).

The usual contracting homotopy of [MR77] given by

s(e0 ⊗ · · · ⊗ es ⊗m) = ε(e0)e1 ⊗ · · · ⊗ en ⊗m

shows that this defines a relative injective resolution.


2.3.2. The K(n)-local En-Adams spectral sequence

The K(n)∗-local En-Adams spectral sequence (see [DH04, Appendix A]) is obtainedby mapping X into a K(n)∗-local En-Adams resolution of Y . If Y is a K(n)-local CW-spectrum this converges to π∗F (X,Y ) [DH04, Proposition A.3]. By [DH04, Remark A.9]there is a standard choice of En-Adams resolution for Y , namely the cosimplicial complexwith Cj(Y ) = LK(n)(Y ∧Ej+1). In the case of Y = EhFn we follow Devinatz-Hopkins andlabel the cosimplicial complex by CjGn/F

.Note that by taking s = 0 and X = EhH1

n in the second part of this theorem, and usingthe equivalence E∗EhH1

n ' E∗[[Gn/H1]] (see, for example, [GHMR05, Proposition 2.5]) werecover the rightmost vertical arrow of Equation (2.1). Note that we write ExtEGn forExtE∨∗ E using the equivalence of categories; this shows how this generalises [GHMR05].

Theorem 2.3.10. 1) If E∨∗X and E∨∗ Y are pro-free E∗-modules then there is a stronglyconvergent spectral sequence

Es,t2 = Exts,tEGn(E∨∗X,E∨∗ Y )⇒ πt−sF (X,Y ),

isomorphic to the K(n)-local En-Adams spectral sequence.

2) If Y = EhH2n for H2 a closed subgroup of Gn then there is an isomorphism of E2-

terms:Exts,tEGn

(E∨∗X,E∨∗ EhH2) ' Hsc (H2;E−tX).

Proof. The equivalence of categories between complete E∨∗ E-comodules and EGn meanswe can equivalently prove the statements for complete E∨∗ E-comodules.

Start with a relative injective resolution for E∨∗ Y , which we can take to be the cobarcomplex Ω∗(E∨∗ E,E∨∗ Y ) defined above.

Then we have

ExtsE∨∗ E(E∨∗X,E∨∗ Y ) = Hs(HomE∨∗ E(E∨∗X, (E∨∗ E)⊗•⊗E∨∗ Y ))

By the adjunction between extended complete E∨∗ E-comodules and E∗-modules wehave

HomE∨∗ E(E∨∗X, (E∨∗ E)⊗(j+1)⊗E∨∗ Y ) ' HomcE∗(E

∨∗X, (E∨∗ E)⊗j⊗E∨∗ Y )

Note that since E∨∗ E and E∨∗ Y are pro free, inductively using Corollary 2.1.10 givesan isomorphism

(E∨∗ E)⊗j⊗E∗E∨∗ Y ' E∨∗ (Ej ∧ Y )


The same corollary also implies this is pro-free. Then by Lemma 2.1.8 there is an isomor-phism

HomcE∗(E

∨∗X,E

∨∗ (Ej ∧ Y )) ' HomE∗(E∨∗X,E∨∗ (Ej ∧ Y ))

and so Lemma 2.1.7 applies to show

HomE∗(E∨∗X,E∨∗ (Ej ∧ Y )) ' π∗F (X,LK(n)(Ej+1 ∧ Y ))

In the terminology introduced above LK(n)(Ej+1 ∧ EhF ) = Cj(Y ). This identifies thecobar complex as the standard En-Adams resolution. This proves part (i).

Part (ii) is essentially [DH04, Theorem 2(ii)]. In particular [DH04, Equation 6.5]shows that πtF (X,C∗Gn/F

) ' D∗FE−tn X, where D∗FEtnX is a cochain complex computing

H∗c (F ; (En)∗X).

Remark 2.3.11. A version of the Morava change of rings theorem, as given in [Dev95] isthat (at least if E∨∗ Y is pro-free)

Ext∗,∗E∨∗ E(E∗, E∨∗ Y ) ' H∗c (Gn, E∨∗ Y ).

This is consistent with Theorem 2.3.10 in the sense that if Y = EhH2n Morava’s result gives

an isomorphism

Ext∗,∗E∨∗ E(E∗, E∨∗ EhH2n ) ' H∗c (Gn, E

∨∗ E

hH2n ) ' H∗c (H2, E∗),

where the last isomorphism is the Shapiro lemma, which agrees with Theorem 2.3.10.

CHAPTER 3

Homotopy fixed point and Tate spectra

In this chapter we will define the Tate spectrum EtGn and show that when G is amaximal finite subgroup of Gn for n = p − 1 and p > 2, then EtGn ' ∗ (this is also trueat n = 1, p = 2 - see [HS14], although it was certainly known earlier than this). Such atheorem is well known to hold K(n)-locally by a result of Greenlees and Sadofsky [GS96].Recently Meier and Mathew [MM13] have given a conceptual reason for this equivalence(see Section 3.4). Our method is by brute force; we simply calculate EtGn using the Tatespectral sequence and show that it vanishes. In fact the calculation of H∗(G; (En)∗) isessentially done in [Sym04], although Symonds does not use this to calculate EtGn .

The computational method also has the advantage of allowing an easy description of(most of) the homotopy orbit spectral sequence (see Equation (A.1)). This can be useful;for example with Stojanoska [HS14] we used a vanishing result for the Tate spectrumof KU tC2 to calculate the Anderson dual of KO. In particular the spectral sequence forcomputing the Anderson dual ofKO was shown to be the linear dual of the homotopy orbitspectral sequence for KO. A similar result is also true for tmf [1/2] [Sto12]. Behrens andOrmsby [BO12] also use the vanishing of the Tate spectrum TMF1(5)tF

×5 and the resulting

calculation of the homotopy orbit spectral sequence to compute hidden extensions in thehomotopy fixed point spectral sequence for TMF0(5).

Since we calculate it along the way, we also give a partial description of π∗(EhGn ), tobe used in Chapter 4. This calculation goes back to Nave [Nav99], who attributes it toHopkins, although Nave does not include the action of the Galois group. The detailed caseof n = 2, p = 3 is included in [GHMR05], and the general result is sketched in [Hen07].

43

44 Chapter 3: Homotopy fixed point and Tate spectra

As an application of this calculation we show that the K(n)-local Spanier-Whiteheaddual Dn(EhGn ) is self-dual up to suspension.

3.1. The Tate spectrum

We recall that when G is a finite group and X is a G-spectrum we define homotopyfixed point and homotopy orbit spectra by

XhG = F (EG+, X)G and XhG = EG+ ∧G X,

and that there are spectral sequences

Hs(G;πtX)⇒ πt−sXhG,

andHs(G;πtX)⇒ πt+sXhG.

We wish to recall briefly the construction of the Tate spectrum XtG. This continues thepattern above; namely there is a spectral sequence

(3.1) Hs(G;πtX)⇒ πt−sXtG.

Here Hs(G, πtX) is the Tate cohomology of G, whose definition is reviewed in Appendix A.If X is a ring spectrum then so are XhG and XtG, and the homotopy fixed point and Tatespectral sequence carry the structure of of a differential graded algebra. Additionally,there is a natural map between them which is compatible with the differential algebrastructure.

The spectrumXtG is part of the generalised Tate cohomology of Greenlees and May [GM95].Note that we will implicitly work with naive G-spectra, in the language of [LMSM86], andthat we also may omit the necessary change of universe functors. We start with the cofibersequence of G-spaces

EG+ → S0 → EG,

where the first map sends all points except the base point to the unbased part of S0. One

3.2. A description of En as a F -module 45

can then construct the following commutative diagram of cofiber sequences

EG+ ∧X X EG ∧X

EG+ ∧ F (EG+, X) F (EG+, X) EG ∧ F (EG+, X)

'

The left hand vertical arrow is always an equivalence. Greenlees and May call tG(X) :=EG ∧ F (EG+, X) the Tate spectrum; we reserve that for the G-fixed points, and defineXtG :=

(EG ∧ F (EG+, X)

)G. By the Adams isomorphism [LMSM86, p. II.7.2] (EG+ ∧

X)G ' EG+ ∧G X and so we have a cofiber sequence

(3.2) XhG → XhG → XtG.

The map XhG → XhG is called the norm map; it is clear that the norm map is anequivalence if and only if XtG vanishes. Our goal is this chapter is to use the spectralsequence of Equation (3.1) when X = En to show that EtGn ' ∗.

3.2. A description of En as a F -module

Hopkins and Devinatz [DH95] have calculated the action of the ring Sn on (En)∗, butthe formulas are difficult to compute with directly. The calculations of Nave and othersrely on the idea that there is a nicer presentation of (En)∗ for which the action of a finitesubgroup K ⊂ Gn is easier to describe.

The following conjecture of Hopkins was presented by Mike Hill at Oberwolfach.

Conjecture 3.2.1 (Hopkins, [Hil07]). If K ⊂ Sn is a finite subgroup then there is K-equivariant isomorphism

(En)∗ ' SW(Fpn )(ρ)[N−1]∧I ,

where ρ ∈ En is places in degree -2, S(ρ) denotes the graded symmetric algebra, N is atrivial representation corresponding to the multiplicative norm over the group on Mn andI is an ideal in degree 0.

In the case of n = p− 1 this has even been verified when K = G is the maximal finitesubgroup of Gn, although there does not appear to be a proof in the literature1.

We will in fact sketch this isomorphism when K = F is the maximal finite subgroup of1We ask for the readers patience when we fix n = p− 1 but continue to write expressions such as 2pn2


Sn, since the (collapsing) Lyndon-Hochschild-Serre spectral sequence gives an isomorphism

H∗(F ;E∗)Gal ' H∗(G;E∗).

We remind the reader that

F = 〈ζ, τ |ζp = 1, τn2 = 1, τ−1ζτ = ζe〉,

where e ∈ (Z/p)× is a generator, and we choose τ so that it is the image in Sn of η =ω(pn−1)/n2 ∈W(Fpn), where ω is a primitive (pn − 1)-st root of unity.

The calculation starts, as in [GHMR05; GHM04], by defining an action of F on W(Fpn)by

ζ(a) = a and τ(a) = ωpn−1

n2 a.

Let χ be the associated representation and denote by χ′ its restriction to Cn2 , the subgroupgenerated by τ .

We define an F -module ρ by the short exact sequence

(3.3) 0→ χ→W[F ]⊗W[Cn2 ] χ′ → ρ→ 0,

where the first map is given by m 7→ (1 + ζ + ζ2 + · · · + ζp−1) ⊗ m, for m a generatorof χ. The claim is that there is a morphism of F -modules ρ → E−2 such that there is aF -equivariant isomorphism

(3.4) (En)∗ ' S(ρ)[N−1]∧I ,

where N :=∏g∈F g∗(m) ∈ S(ρ) (see [GHMR05, Lemma 3.2]) and I is the preimage of

m = (p, u1, . . . , un−1) under the morphism S(ρ)[N−1]→ (En)∗.Again we simply sketch this result. Hopkins and Devinatz have shown that the divided

power envelope of E0 has canonical coordinates w,wi on which the action of Sn is easierto express. In fact by [DH95, Proposition 3.3] if we express g ∈ Sn by g =

∑n−1j=0 ajS

j then

g(w) = a0w +n−1∑j=1

aφj

n−jwwj ,

and

g(wwi) =paiw + paφn−1

i+1 wwn−1 + · · ·+ paφi+1

n−1wwi+1

+ aφi

0 wwi + · · ·+ aφi−1ww1,

3.2. A description of En as a F -module 47

where aφ denotes the Frobenius. The action on the w and wi is related to that of u andui by [DH95, Proposition 4.9], which gives (see also [Nav10])

(3.5) g(u) ≡ a0u+ aφn−1uu1 + · · ·+ aφn−1

1 uun−1 mod (p,m2)

and

(3.6) g(uui) ≡ aφi

0 uui + · · ·+ aφi−1uu1 mod (p,m2).

This gives the following action for τ , with 1 ≤ i ≤ n:

(3.7) τ(uui) ≡ ηpiuui mod (p,m2),

where un = 1 and u0 = p.For an element g, of order p, Nave shows that we can write g =

∑∞j=0 aiS

i with a0 ≡ 1mod (p) and a1 ∈W(Fpn)×. With Equations (3.5) and (3.6) the action modulo (p,m2) ofζ on the ordered basis u, uun−1, . . . , uu1, is given by the matrix

1 aφn−1

1 · · · aφ2

n−2 aφn−1

0 1 · · · aφ2

n−3 aφn−2...

......

...0 0 · · · 1 aφ10 0 · · · 0 1

.

Note that if we write this matrix as A = I +B where B is now a strictly upper triangularmatrix we see (note Bn = 0) that

u+ ζ∗(u) + · · ·+ ζp−1∗ (u) ≡ 0 mod (p,m2).

Together these calculations, along with the sequence of Equation (3.3), imply thatρ⊗W Fpn ' E−2⊗E0 E0/(p,m2) as F -modules and that we can choose the residue class ofu as a generator. The idea now is to find a class z ∈ E−2 with the same reduction as usuch that τ(z) = ηz and (1 + ζ + · · ·+ ζp−1)z = 0. We rely on the following result due toNave, who credits it to Hopkins.

Lemma 3.2.2. [Nav10; Nav99] There are z, z1, . . . , zn−1 ∈ (En)∗ such that

1) z ≡ cu mod (p,m2) for c ∈W(Fpn)×,

2) zi ≡ ciuui mod (p, u1, . . . , ui−1,m2) for ci ∈W(Fpn)×,


3) (1 + ζ + · · ·+ ζp−1)z = 0,

4) τ(z) = ηz,

5) (ζ − 1)z = zn−1 and (ζ − 1)zi+1 = zi for 1 ≤ i < n− 1,

6) ζ(zi) ≡ zi mod (p, z1, . . . , zi−1) for each i.

The morphism ρ → E−2 is then defined by sending the generator of ρ to z. Thisdefines a morphism of W(Fpn)-algebras S(ρ)→ E∗. Now S(ρ) is polynomial over W(Fpn)on n generators, and we can choose a generator such that its image in E∗ is the (invert-ible) element z from above. After inverting N this is an inclusion onto a dense subring;completing then gives the isomorphism described above.

3.3. The Tate cohomology of G

Let tr be the transfer map, defined as follows:

tr:A AG

x∑g∈G gx.

In what follows let A = S(ρ)[N−1].

Lemma 3.3.1. [GHMR05, Lemma 3.3] Let Cp ⊂ Sn be the subgroup generated by ζ. Thenthere is an exact sequence

Atr−→ H∗(Cp, A) ' Fpn [a, b, d±1]/(a2),

with |a| = (1,−2), |b| = (2, 0) and |d| = (0,−2p).

Proof. Let J be the F -module W[F ] ⊗W[Cn2 ] χ′. Choose W(Fpn)-generators of J named

x1, . . . , xp so that ζ(xi) = xi+1 for 1 ≤ i ≤ p − 1 and ζ(xp) = x1. Furthermore we can,and do, choose x1 so that it maps to the generator of m of ρ. Specifically

S(J) = W(Fpn)[x1, x2, . . . , xp].

The kernel of the map J → ρ is the principal ideal generated by σ1 = x1 +x2 + · · ·+xp,and there is a short exact sequence of F -modules

(3.8) 0→ S(J)⊗ χ→ S(J)→ S(ρ)→ 0,

3.3. The Tate cohomology of G 49

where we set the degree of χ to be -2 so that this is an exact sequence of graded modules.The orbit of a monomial under the Cp-action has p elements, unless it is a power ofσn := x1x2 . . . xp. Therefore as a Cp-module S(J) splits into a direct sum of free modulesand trivial modules generated by powers of σn, and there is a short exact sequence

S(J) tr−→ H∗(Cp, S(J))→ Fpn [b, σn]→ 0

with |b| = (2, 0) and |σn| = (0,−2p). Similar considerations show that the cohomology ofH∗(Cp, S(J)⊗ χ) is the same, however the internal degree is shifted by -2.

We then use the long exact sequence associated to Equation (3.8) and the fact thatH∗(Cp, S(J)) and H∗(Cp, S(J) ⊗ χ) are concentrated in even degrees to get short exactsequences

0→ H2k−1(Cp, S(ρ))→ H2k(Cp, S(J)⊗ χ)→ H2k(Cp, S(J))→ H2k(Cp, S(ρ))→ 0.

The middle map is zero when k > 0 (as it in induced by multiplication by σ1, which is inthe image of the transfer) and so this implies that there is an exact sequence

S(ρ) tr−→ H∗(Cp, S(ρ))→ Fpn [a, b, d]/(a2)→ 0,

where d is the image of σn, b is the image of b and a maps to b ∈ H2(Cp, S0(J) ⊗ χ)under the boundary map, and thus has bidegree (1,−2). Note that graded commutativityforces the element a to satisfy a2 = 0. The effect of inverting the element N is essentiallyinversion of the element σn ∈ S(ρ)Cp , since N is the image of a power of σn (c.f the proofof [GHMR05, Proposition 3.5]).

Lemma 3.3.2. Let Cp ⊂ G be the subgroup generated by ζ. Then there is an isomorphism

H∗(Cp, A) ' Fpn [a, b±1, d±1]/(a2).

Proof. The map from cohomology to Tate cohomology is an isomorphism in positive di-mensions (see Appendix A) and preserves the cup product structure. Note that the imageof the transfer map is in dimension 0, and hence must be killed by multiplication by b

since there is no corresponding group in positive dimensions. These observations, the cal-culations of Lemma 3.3.1 and Example A.8 imply that we can obtain the Tate cohomologyby simply inverting the degree 2 class b in the ordinary group cohomology. By definitionthe elements that are in the image of the transfer map are zero in Tate cohomology, and


so we have:H∗(Cp, A) ' H∗(Cp, A)[b−1] ' Fpn [a, b±1, d±1]/(a2).

In order to go from Cp to F we need to work out the quotient actions of τ on a, b andd.

Lemma 3.3.3. The action of τ on H∗(Cp;S(ρ)[N−1]) and H∗(Cp;S(ρ)[N−1]) is given by

τ(a) = eηa τ(b) = eb τ(d) = ηpd,

where e ∈ Z×p is the element such that τ−1ζτ = ζe.

Proof. The action on d is easily found as τ acts by multiplication by ωpn−1

n2 on each xi ofS(J).

The action on b is more complicated. This can be found as Proposition 6.2 of Nave’sthesis [Nav99] (see also [Wei94, Example 6.7.10]). Let M be an F -module. First notethat by naturality it suffices to find the action for the cohomology of H∗(Cp;S(J)). Asexplained in [Bro82, pp. 80], the action is induced by conjugation. Let α be the conjugationmap α(ζ) = τ−1ζτ and write M(α) for M with action induced by α. Define a mapf : M(α)→M by sending x to τ · x. Then the action is given as the composite

(3.9) (α, f)∗ : H∗(Cp,M) α∗−→ H∗(Cp,M(α)) f∗−→ H∗(Cp,M)

By taking two copies of the standard resolution, we can get an induced chain map betweenthem

· · · Z[Cp] Z[Cp] Z[Cp] Z 0

· · · Z[Cp] Z[Cp] Z[Cp] Z 0

φ2

N(p)

φ1

ζ−1 ε

φ0

N(p) ζ−1 ε

with the chain map satisfying φ∗(gx) = α(g)φ∗(x). Here N(k) = 1 + ζ + · · ·+ ζk−1. Navecalculates that the first few maps are φ0 = α and φ1(g) = φ2(g) = N(e)α(g).

To compute (α, f)∗ (see Equation (3.9)) we first apply HomZ[Cp](−;M) to the top row,HomZ[Cp](−;M(α)) to the bottom row, and then follow with HomZ[Cp](−; f) to get the


following (c.f [Nav99, pp. 22]):

M M M · · ·

M M M · · ·

ζ−1

τ

N(p)

τN(e) τN(e)ζ−1

N(p)

Taking M = S(J) = W[x1, . . . , xp] and tracing what happens to b we find that τ(b) = eb.The action of a then follows when we recall it is the pre-image of b with respect to the

isomorphismH1(Cp, S(ρ))→ H2(Cp;S0(J)⊗ χ)

and so the action is twisted by the representation χ.

Recall that e ∈ (Z/p)× is a generator satisfying τ−1ζτ = ζe. Before we can calculate theinvariants we need to obtain a relationship between ηp−1 (recall that η is the image of τ)and e.

Lemma 3.3.4. [Nav99, Page 24]

ηp−1 ≡ e mod (p)

Proof. Since ζ has order p we have ζ = 1 + a1S mod (S2) where a1 is a unit (recall ourdescription of elements of order p from above). Then we have (working modulo S2)

ζe = (1 + a1eS) = τ−1ζτ = 1 + τ−1a1Sτ = 1 + τp−1a1S.

It follows that a1τp−1 ≡ a1e mod (p) and the result follows.

Proposition 3.3.5. There are isomorphisms

H∗(G; (En)∗) ' Fp[α, β±1,∆±1]/(α2)

andH∗(G; (En)∗) ' Fp[α, β,∆±1]/(α2),

where the second isomorphism is modulo the image of the transfer.

Proof. We need to pass from H∗(Cp; (En)∗) to H∗(G; (En)∗). We do this by using the (col-lapsing) Lyndon-Hochschild-Serre spectra sequence twice. First note that H∗(F ; (En)∗) '


H∗(Cp, En)〈τ〉 since the composite of the restriction and transfer maps to the trivial groupis multiplication by n2, which is a unit in W(Fpn).

Now τ(dk) = ηpkdk which is equal to dk if and only if n2 divides k. Define elements

α := a

d∈ H1(C3, (S(ρ)[N−1])2p−2)

β := b

dn∈ H2(C3, (S(ρ)[N−1])2pn)

∆ := 1dn2 ∈ H0(C3, (S(ρ)[N−1])2pn2).

It is easy to check that these are invariant under the action of τ and an argument using,for example, [Sym04, Lemma 4.1], shows that β,∆, and their inverses, along with α, infact generate the invariants.

The effect of the Galois action is only to reduce coefficients to Fp. Indeed, as explainedin [Sym04], in bidegree (s, t) the cohmology group is either Fpn or 0, and so the invariantsare either Fp or 0. We can always replace the generators ∆, α and β by any non-zeroelement of the cohomology group they appear in, and so we may assume they are invariantunder the Galois action; it follows that

H∗(G,S(ρ)[N−1]) ' Fp[α, β±1,∆±1]/(α2)

Finally a completion argument, as given in [GHM04, Theorem 6] gives that

H∗(G, (En)∗) ' H∗(G,S(ρ)[N−1]∧I ) ' Fp[α, β±1,∆±1]/(α2).

The group cohomology calculation proceeds in a similar manner.

We are now in position to prove the main result of this chapter.

Theorem 3.3.6. The Tate spectrum EtGn is contractible.

Proof. We have calculated the E2-term of the spectral sequence above. The differentialsin the Tate spectral sequence can be inferred from the differentials in the homotopy fixedpoint spectral sequence, which are now well known. In particular α and β denote theimage, under the unit map S0 → EhGn , of α1 ∈ π2p−3S

0 and β1 ∈ π2(p2−p−1)S0.

Note that in the spectral sequence Es,t2 is concentrated in degrees t ≡ 0 mod (2(p−1));this implies that dr = 0 unless r ≡ 1 mod (2(p − 1)). For example the first possibledifferential is a d2n+1. Toda [Tod68] has shown that α1β

p1 = 0 ∈ π∗(S0); thus we must

have αβp = 0 in our spectral sequence, and the only possibility is that d2n+1(∆β) = αβp,or in other words, d2n+1(∆) = αβp−1 (at least up to multiplication by a unit in Fpn).


Likewise Toda’s relation βpn+11 = 0 ∈ π∗(S0) implies that βpn+1 must die; sparsity

again implies that the only possible differential is d2n2+1(∆nαβn) = βpn+1. This gives thedifferential d2n2+1(∆nα) = βn

2+1. The homotopy fixed point spectral sequence collapsesat this stage for degree reasons. In the Tate spectral sequence we have that β is invertible,and so the second differential implies that everything in the spectral sequence dies, andindeed EtGn ' ∗. This is shown in Figure 3.1 for n = 2.

−18 −14 −10 −6 −2 2 6 10 14 18 22−10

−6

−2

2

6

10

1α

β

∆

Figure 3.1: The Tate spectrum sequence to compute EtGn , shown for n = 2, p = 3. Herethe dashed lines representation multiplication by α and the vertical lines are differentials.

We have also computed the following which we we use in Chapter 4.

Lemma 3.3.7. In the spectral sequence H∗(G, (En)∗) ⇒ π∗(EhGn ) we have (in positivecohomological degrees)

Eeven,∗∞ ' Fp[∆±p][β]

βn2+1 and Eodd,∗∞ ' αFp[∆±p]1,∆, · · · ,∆p−2[β]

βn.

Remark 3.3.8. We can say slightly more about the image of the transfer, again referringto Nave for the proof. There is a natural map

Fp[∆±1]→ H0(G, (En)∗)/p

which is a split injection. If we let M denote the other summand determined by thesplitting then

H∗(G, (En)∗) = H0(G, (En)∗)[α, β](pα)(pβ)(α2)(Mα)(Mβ)) .


Remark 3.3.9. Note that EhGn is periodic of order 2p2n2 and that ∆p detects a periodicityclass.

We record some corollaries of the vanishing of the Tate spectrum. The first is obviousgiven the cofiber sequence Equation (3.2).

Corollary 3.3.10. There is an equivalence (En)hG ' EhGn .

For the next corollary let DnX = F (X,LK(n)S0) be theK(n)-local Spanier-Whitehead

dual of X. We learnt this technique from [Beh06, Proposition 2.5.1].

Corollary 3.3.11. EhGn is K(n)-locally self-dual up to suspension. In fact DnEhGn '

ΣNEhGn where N ≡ −n2 mod (2pn2) and N is only uniquely defined modulo 2p2n2.

Proof. By definition and the contractibility of the Tate spectrum we have

Dn(EhGn ) ' F (EhGn , LK(n)S0) ' F ((En)hG, LK(n)S

0) ' F (En, LK(n)S0)hG ' (DnEn)hG

and so there is an associated homotopy fixed point spectral sequence

Hs(G, (DnEn)t))⇒ πt−s(DnEn)hG.(3.10)

By [Str00] there is an isomorphism Dn(En) ' Σ−n2En and so the E2-term of the spectral

sequence (3.10) can be written as

H∗(G, (DnEn)∗) ' H∗(G,Σ−n2E∗).

This is a spectral sequence of modules over the spectral sequence of algebras

Hs(G, (En)t)⇒ πt−s(EhGn ),(3.11)

and so the spectral sequence of (3.10) is isomorphic to the spectral sequence (3.11) up toa shift congruent to −n2 modulo 2pn2 (the periodicity of the E2-term). In other wordsthere exists a map

SN → Dn(EhGn )

which is detected on the zero line of the spectral sequence (3.10) which extends, using thefact that Dn(EhGn ) is an EhGn -module, to an equivalence

ΣNEhGn'−→ Dn(EhGn ),

3.4. Relation to the work of Mathew-Meier 55

where N ≡ −n2 mod (2pn2), and N is only uniquely determined modulo 2p2n2, by Re-mark 3.3.9.

Example 3.3.12. By Hahn and Mitchell [HM07] or alternatively [HS14],D1EC21 ' Σ−1EC2

1 ,whilst Behrens [Beh06] has calculated that D2E

hG242 ' Σ44EhG24

2 .

Remark 3.3.13. This calculation is interesting to us because of its potential use in detectingexotic elements in the K(n)-local Picard group and their relation to Brown-Comenetzduality. See Section 4.6 for details.

3.4. Relation to the work of Mathew-Meier

Recent work of Mathew and Meier [MM13] gives a conceptual reason for this vanish-ing, which we wish to describe. Since we do not use the language of (derived) algebraicgeometry elsewhere in this thesis, we do not wish to spend to much time introducing thelanguage; rather we just refer the reader to [MM13] in this case. Throughout this sectionwe let K ⊂ Gn be a finite group. In the language of Rognes [Rog08], EhKn → En is aK(n)-local K-Galois extension. By [Rog08, Proposition 6.3.3] this gives another proofthat, K(n)-locally, EtKn vanishes. The work of [MM13] allows one to promote this to astatement before K(n)-localisation.

Since En is an even-periodic Landweber-exact E∞ ring spectrum then [MM13, Propo-sition 2.13], and the discussion beforehand, implies that there is a map of stacks

(3.12) (Specπ0En)//K →MFG,

where MFG is the moduli stack of formal groups, and that there is an even periodicrefinement of Equation (3.12). Here an even periodic refinement X, of a stack X, is asheaf Otop of even periodic E∞ ring spectra on the affine étale site of X. The map(Specπ0En)//K → MFG is in fact flat; this follows from the fact that Specπ0En →(Specπ0En)//K is faithfully flat, and Specπ0En →MFG is flat.

We thank the authors of [MM13] for allowing us to include the following argument.

Theorem 3.4.1 (Mathew-Meier). Let K ⊂ Gn be a finite subgroup. Then EtKn iscontractible.

Proof. By [MM13, Theorem 5.6], and the discussion above, we need only show that themap X = Spec(π0En)//K → MFG is affine. Let MFG(n) denote the moduli stack offormal groups with a coordinate modulo degree n; this is a smooth, affine cover ofMFG


and so it suffices to show that Spec(π0En)//K×MF GMFG(n) is an affine scheme for some

n.Let K = K ∩ Sn. Then any element of K corresponds to a power series in Fpn (given

by automorphisms of the chosen formal group law Γ). Let ` > 1 be some number suchthat the coefficients of all these power series have at least one non-trivial coefficient indegrees between 1 and `, and define the scheme

Y (`) = Specπ0En ×MF GMFG(`).

We claim this scheme is affine and Noetherian. Since Specπ0En is affine and Noetherianwe just need to check that the morphism MFG(`) → MFG is both affine and of finitetype. It is enough to check this after base change to an fpqc-cover; we chose the morphismSpecL → MFG, where L is the Lazard ring. Hence we must show that SpecL ×MF G

MFG(`)→ SpecL is affine and of finite type. But the functor that Spec(L)×MF GMFG(`)

represents is equivalent to the functor that sends a ring R to the set of all formal group lawsF on R with an additional coordinate up to level `. Coordinates on F are in one to onecorrespondence with power series over R, and hence a coordinate up to level ` correspondsto a truncated power series. Thus Spec(L)×MF G

MFG(`) ' SpecL[b0, . . . , b`−1], which isboth affine and of finite type over SpecL.

The group K acts on Y (`) and we say an element of K is inert if it has a closed fixedpoint on whose residue field it acts trivially. We claim that no such points exist. Indeedthe closed points are given by the choice of a coordinate up to level ` in Γ. If k ∈ K isa non-trivial element of Sn then k cannot stabilise such a closed point as it changes thecoordinate. If k has a non-trivial Galois action then it acts non-trivially on Fpn . It followsthat there are no inert elements, as claimed.

Since Y (`) = SpecR is Noetherian [Gro71, Expose V, Proposition 2.3,2.6] shows thatthe map Y (`)→ SpecRK is an étale K-torsor. The induced map SpecRK → Y (`)//K isan equivalence; this is precisely what we are required to show.

This in fact allows us to extend the result of Corollary 3.3.11. For example if K has ordercoprime to p then the homotopy fixed point spectral sequence collapses. We then havethe following.

Corollary 3.4.2. Suppose that K has order coprime to p. Then DnEhKn ' Σ−n2

EhKn .

Note that Behrens [Beh06, Proposition 2.61] has calculated that D2EhSD162 ' Σ44EhSD16

2and D2E

hD82 ' Σ44EhD8

2 , where these groups have order 16 and 8 respectively. There areequivalences Σ16EhSD16

2 ' EhSD162 and EhD8

2 ' Σ8EhD82 , and using this periodicity we see

that Behrens’ calculations are consistent with Corollary 3.4.2.

3.4. Relation to the work of Mathew-Meier 57

We suspect calculations such as these might be useful for Behrens’ conjecture [Beh06]regarding the existence of a fiber sequence involving the spectra Q(`) at the prime 2.

CHAPTER 4

The Picard group of the K(n)-local category

4.1. Background

Let C be a symmetric monoidal category with monoidal product ⊗ and unit I. Wesay that an object, X, of C, is invertible if there exists a Y such that X ⊗ Y ' I. Ifthe collection of equivalence classes of invertible objects forms a set, then it has a naturalgroup structure arising from ⊗. This group is called the Picard group of C, denoted byPic(C).

For example we can take C = Ho(S) to be the stable homotopy category. Thenthe group Pic(Ho(S) clearly contains Sk for all k ∈ Z, and in fact this is everything:Pic(Ho(S) ' Z. Hopkins, Mahowald and Sadofsky [HMS94] (see also [Str92]) introducedthe idea of studying the Picard group of the category of K(n)-local spectra, which wedenote Picn. As usual we will write X ∧ Y for LK(n)(X ∧ Y ).

Definition 4.1.1. A K(n)-local spectrum X is invertible (X ∈ Picn) if and only if thereexists a K(n) local spectrum-Y such that

X ∧ Y ' LK(n)S0.

Definition 4.1.2. A spectrum X is K(n)-locally dualisable if the natural map

F (X,LK(n)S0) ∧ Z → F (X,Z)

is a K(n)-local equivalence for all K(n)-local Z.

59

60 Chapter 4: The Picard group of the K(n)-local category

The following two (well-known) lemmas have obvious analogues in any closed symmetricmonoidal category.

Lemma 4.1.3. If X ∈ Picn then the inverse of X is

DnX = F (X,LK(n)S0).

Proof. Let Y be the inverse to X. Then − ∧ Y is an auto-equivalence ofMS,K(n) and sowe have equivalences

F (Z,DnX) ' F (Z ∧X,LK(n)S0) ' F (Z ∧X ∧ Y, Y ) ' F (Z, Y ).

The result follows from the Yoneda lemma.

Lemma 4.1.4. Any object X ∈ Picn is dualisable.

Proof. We have a series of K(n)-local equivalences

F (W,F (X,Z)) ' F (W ∧X,Z) ' F (W,Z ∧DnX) ' F (W,Z ∧ F (X,LK(n)S0)).

The result again follows from the Yoneda lemma.

The following is fundamental to the calculation of the groups Picn.

Theorem 4.1.5. [HMS94, Theorem 1.3] Given a K(n)-local spectrum Z, the followingare equivalent:

1) Z ∈ Picn;

2) dimK(n)∗ K(n)∗Z = 1; and,

3) E∨∗ Z is a free E∗ module of rank 1.

Let Picalgn be the Picard group of the category of Morava modules, with symmetric

monoidal product −⊗E∗− (see also Section 2.2). Theorem 4.1.5 implies that there is agroup homomoprhism:

εn : Picn Picalgn

X E∨∗X

Let Picalg,0n be the index 2 subgroup of modules concentrated in even degrees, and let

M ∈ Picalg,0n . Denote by eM a free generator of M in degree 0. Then for all g ∈ Gn there

4.1. Background 61

is an element ug ∈ (En)×0 such that g∗(eM ) = ugeM . The assignment g 7→ ug defines acrossed homomorphism fM : Gn → (En)×0 whose cohomology class is independent of thechoice of fM .

Theorem 4.1.6 ([HMS94]). The homomorphism

Picalg,0n

// H1(Gn, (En)×0 )

[M ] // [fM ]

is an isomorphism.

Remark 4.1.7. Note that Hopkins-Mahowald and Sadofsky define the algebraic Picardgroup in terms of isomorphism classes of rank one En−Sn-modules, i.e. En-modules witha Sn-action that commutes with the En-action. They then go on to prove in [HMS94,Chapter 8] that (relying on the fact that E0 is Noetherian and local) a module is in Picalg

n

if and only if the underlying E∗-module is free of rank 1.

As explained in [GHMR12] there are two basic elements ofH1(Gn, (En)×0 ) that we consider.These will in fact be generators of the cohomology group at n = 2, p > 2. The first one isgiven by the crossed homomorphism

ι : Gn// (En)×0

g // g∗(u)/u,

where u ∈ (En)−2 = (En)0(S2) is the generator.The second is given as the composite of the reduced norm and canonical inclusion

det : Gn → Z×p → (En)×0 .

The corresponding elements of H1(Gn, (En)×0 ) are also denoted by ι and det. Note thatthe element (En)∗(S2) ∈ Picalg,0

n is sent to ι. We will see in Section 4.6 that det can alsobe realised topologically; i.e. there is a spectrum S0〈det〉 whose image in H1(Gn, (En)×0 )(under the map εn and the isomorphism of Theorem 4.1.6) is det.

In the case n = 1 and p odd, G1 = Z×p and (E1)×0 = Z×p (note that (E1)×0 has trivialG1 action) and so

Picalg,01 ' H1(G1, (En)×0 ) ' Hom(Z×p ,Z×p ) ' Zp × Z/(p− 1)


and then Picalg1 ' Zp × Z/2(p− 1). In the case p = 2, Z×2 ' Z2 × Z/2 and

Picalg,01 ' H1(Z×2 ,Z

×2 ) ' Z×2 × Z/2.

This gives Picalg1 ' Z2 × Z/2× Z/2. The proofs of both of these statements can be found

in [HMS94].The case of n = 2 for odd primes has been considered by Hopkins for p > 3 (unpub-

lished, but see [Beh12] and the recent thesis of Lader [Lad13]) and Karamanov [Kar10]for p = 3.

Theorem 4.1.8. Let n = 2 and p > 2. Then

Picalg2 ' Z2

p × Z/2(p2 − 1).

In both cases ι and det generate the cohomology group.

Definition 4.1.9. Let κn be the kernel of εn : Picn → Picalgn . This will be called the

group of exotic elements of Picn.

By definition these will be the group of spectra X such that E∨∗X ' E∗ as Moravamodules, i.e. they cannot be detected by doing algebra over the Morava stabilizer algebra.

When n is not divisible by p−1 and n2 ≤ 2p−2 then an argument with the (collapsing)Adams-Novikov spectral sequence [HMS94, Section 7] shows that κn = 0. In addition wehave

κ1 =

Z/2 p = 2,

0 p > 2.

and

κ2 =

? p = 2,

Z/3⊕ Z/3 p = 3,

0 p > 5

These were all proved in [HMS94], with the exception of n = 2, p = 3 which is the workof [GHMR12]. κ2 also appears to be non-zero at the prime 2 [HMS94, Section 6].

4.2. Some cohomological results

Note that in the calculations above κn is always a finite p-group. Hopkins [Str92]conjectured that this was in fact always the case. The purpose of this first part of this

4.2. Some cohomological results 63

chapter will be to verify that κn is a p-group (when p is odd), and decomposes as a directproduct of cyclic groups. Unfortunately we can not yet prove that κn is finite. Note thatin this chapter, unless otherwise mentioned, we will always assume p is odd.

We start with two preliminary lemmas. The first is a simple generalisation of theargument for the case of n = 2, p = 3 given in [GHM, Proposition 4.2], and gives the usualsparseness result for the Adams spectral sequence. The second, which is also based on thecase of n = 2, p = 3 in [GHM, Proposition 4.2] (some modification is needed since n and pare not necessarily coprime in our case) implies that in positive cohomological degrees theE2-term of the Adams-Novikov spectral sequence for LK(n)S

0 is torsion when the internaldegree t 6= 0.

Proposition 4.2.1. Suppose t is not divisible by 2(p− 1). Then H∗(Gn, (En)t) = 0.

Proof. Since p is assumed to be odd, the central subgroup Z×p ⊂ Gn is isomorphic toµp−1 ⊕ (1 + pZp)×. There is an associated spectral sequence

Hp(Gn/µp−1, Hq(µp−1, (En)t))⇒ Hp+q(Gn, (En)t).

Let g ∈ Z×p . From Lemma 1.3.1 we know that g∗ui = ui and g∗u = gu. In particular ifζ is a generator of µp−1 then ζ∗uk = ζkuk, and we see that the group Hq(µp−1, (En)t)) iszero unless q = 0 (since p− 1 and p are coprime) and t is a multiple of 2(p− 1).

Proposition 4.2.2. Let t = 2(p− 1) · pkm with m 6≡ 0 mod (p). Then

pk+1H∗(Gn, (En)t) = 0.

Proof. Let the projective Morava stabilizer group [behrensrevisit ] be defined as thequotient of Gn by the central subgroup:

1→ Z×p → Gn → PGn → 1.

The Lyndon-Hochshild-Serre spectral sequence takes the form

(4.1) Hp(PGn, Hq(Z×p , (En)t))⇒ Hp+q(Gn, (En)t).

To calculate Hq(Z×p , (En)t) there is a further spectral sequence (arising from the decom-position of Z×p above, where we further identify (1 + pZp)× with Zp)

(4.2) Hr(µp−1, Hs(Zp, (En)t))⇒ Hr+s(Z×p , (En)t).


Let ψ be a topological generator of Zp. Then ψ ≡ 1 + p mod (p2). Note that since p isodd there is a short projective resolution for Zp:

(4.3) 0→ Zp[[Zp]]ψ−1−−−→ Zp[[Zp]]→ Zp → 0

By the binomial formula we then have

(ψ − 1)∗(ut/2) = (ψt/2 − 1)ut/2

= ((1 + p)(p−1)pkm − 1)ut/2

≡ (p− 1)pk+1mut/2 mod (pk+2)

We then see that Hs(Zp, (En)t) = 0 unless s = 1 and

pk+1H1(Zp, (En)t) = 0.

Since µp−1 has order coprime to p the spectral sequence of Equation (4.2) collapses, andso we find that Hq(Z×p , (En)t) = 0 when q 6= 1 and pk+1H1(Z×p , (En)t) = 0. The resultthen follows from the spectral sequence of Equation (4.1).

In what follows let Es,tr (S0) be the r-th page of the K(n)-local En-based Adams spectralsequence (e.g. Es,t2 = Hs(Gn, (En)t). We claim that this spectral sequence has a horizontalvanishing line at some finite page. This is in fact a stronger result than needed, howeverit is of interest in its own right.

Proposition 4.2.3. There exist r0, s0 such that Es,∗r (S0) = 0 whenever r ≥ r0 and s ≥ s0.

Proof. We recall from Equation (1.5) (with X = S0) that we have

Hs(Gn, (En)tX) = lim←−k

Hs(Gn, (En)tMIk).

A version of Morava’s change of rings theorem gives an isomorphism

Exts,tE(n)∗E(n)(E(n)∗, E(n)∗MIk) ' Hs(Gn, (En)tMIk

),

and there is an associated spectral sequence

Es,t2 = Exts,tE(n)∗E(n)(E(n)∗, E(n)∗MIk)⇒ πt−sLnMIk

Taking the inverse limit of these spectral sequences [HMS94, Section 7] gives the usual

4.3. A filtration on κn 65

Adams-Novikov spectral sequence

Es,t2 = Hs(Gn, (En)t)⇒ πt−sLK(n)S0,

since holimk LnMIk' LK(n)S

0 [HS99b, Theorem 7.10].By [HS99b, Proposition 6.5] there exists r0 and s0 (independent of MIk

) such thatEs,tr = 0 whenever r ≥ r0 and s ≥ s0 and hence the same is true for Es,tr .

Remark 4.2.4. This result is uninteresting when p is large compared to n (specifically whenn2 < 2p−1) since in this range Gn has finite cohomological dimension, and so the vanishingline exists already on the E2-page. When n = 1, p = 2 it is not hard to see that there is ahorizontal vanishing line on the E4-page, whilst at n = 2, p = 3 Goerss-Henn-Mahowaldand Rezk [GHMR12, Theorem 4.2] show that

Es,t10 = 0, s ≥ 13.

4.3. A filtration on κn

4.3.1. The E(n)-local case

Let Ln be the category of E(n)-local spectra. Hovey and Sadofsky [HS99a] showedthat there is a splitting Pic(Ln) = Z ⊕ Pic(Ln)0 where the factor of Z corresponds toLnS

m, and that if X is in Pic(Ln)0 then E(n)∗X ' E(n)∗ as an E(n)∗E(n)-comodule(the converse is also true by [KS04]). This implies that the E2-term in the E(n)-basedAdams spectral sequence is the same as that of the sphere. An important observation isthat if the identity class ιX ∈ Ext0,0

E(n)∗E(n)(E(n)∗, E(n)∗X) ' Z(p) survives the Adams-Novikov spectral sequence then the resulting map ιX : S0 → X gives rise to an equivalenceLnS

0 ' X.Kamiya and Shimomura [KS04] used this to define a finite filtration of Pic(Ln)0. Let

q = 2(p− 1). Note that a simpler sparsity result to Proposition 4.2.1 applies; this spectralsequence is only non-zero when the internal degree t ≡ 0 mod (q). In other words theonly differentials possible in the E(n)-based Adams-Novikov spectral sequence are of theform dkq+1. For k > 0 they define

Pic(Ln)0k = X ∈ Pic(Ln)0|ds(ιX) = 0 for s < kq + 1

There is natural inclusion Pic(Ln)0k+1 ⊂ Pic(Ln)0

k, and, since the spectral sequence


has a vanishing line, for large enough k we have Pic(Ln)0k = LnS0. Each Pic(Ln)0

k is asemigroup, and we define the associated graded group

GPic(Ln)0 =⊕k>0

(Pic(Ln)0k/Pic(Ln)0

k+1)

This is isomorphic to a subgroup of⊕

r>2E(n)r,r−1r (S0). Together these imply that

there is a monomorphism

φ : GPic(Ln)0 →⊕r>2

E(n)r,r−1r (S0).

Example 4.3.1. When n = 1, p = 2 this sum is Z/2, and is isomorphic to Pic(L1)0.Likewise at n = 2, p = 3 this sum is Z/3⊕ Z/3 and is isomorphic to Pic(L2)0. ♦

4.3.2. The K(n)-local case

We now repeat the above argument in the K(n)-local case (in fact this argumentwould work more generally for R-localisations satisfying analogues of Proposition 4.2.2and Proposition 4.2.3). We note that a similar filtration is used by Hopkins, Mahowaldand Sadofsky [HMS94, Proposition 7.6] (on the full Picard group Picn) to show that Picnis a set.

Definition 4.3.2. Let k ≥ 1. Define

Fkκn = X ∈ κn|ds(ιX) = 0 for s < k

Note that F1κn = κn and that Fnκn ⊃ Fn+1κn. Proposition 4.2.3 implies that for largeenough k we have Fkκn = LK(n)S

0. Arguing as in [KS04] we get the following.

Lemma 4.3.3. The filtration Fkκn satisfies the following properties:

1) If X,Y ∈ Fkκn then X ∧ Y ∈ Fkκn

2) For X ∈ Fkκn there exists X ′ ∈ Fkκn such that X ∧X ′ ∈ Fk+1κn

3) If X ∧ Y and X are in FkXm then so is Y .

Proof. Since this is essentially [KS04, Lemma 2.4] we will be brief. The first part followsfrom the Leibniz rule for the differentials.

4.4. The structure of κn 67

For the next two statements the key point is that if dk(ιX) = xιX and dk(ιY ) = yιY

then the Leibniz rule gives

dk(ιX∧Y ) = (x+ y)(ιX∧Y ).

Then for second statement we write dk(ιX) = xιX for some x ∈ Ek,k−1k . Since Ek,k−1

k

is torsion by Proposition 4.2.2, x has finite order, say m. Let X ′ = X∧(m−1) and one has(since X∧(m−1) ∈ Fkκn by Item 1)) that dk(ιX′) = (m − 1)xιX′ = −xιX′ . It follows thatdk(ιX∧X′) = (x− x)ιX∧X′ = 0.

Finally for the third statement assume we have ds(ιY ) = yιY for some s < k. Then0 = ds(ιX∧Y ) = (x+ y)ιX∧Y = yιX∧Y and so y = 0.

Define an equivalence relation on Fkκn by X ∼ Y if X ∧Y ′ ∈ Fk+1κn, where Y ′ is definedin Item 2) of Lemma 4.3.3.

Lemma 4.3.4. Let Fkκn/Fk+1κn = Fkκn/ ∼. Then Fkκn/Fk+1 is isomorphic to a sub-group of Ek,k−1

k (S0).

Proof. Again, we omit most of the details since this follows as in [KS04, Lemma 2.8]. Herethe map (which we denote f) is defined by taking the isomorphism class of X to dk(ιX) ∈Ek,k−1k (S0). This is well defined, and if f([X]) = f([Y ]) = yιY then f([Y ′]) = −yιY . It

follows that dk(ιX∧Y ′) = (y − y)ιX∧Y ′ = 0 and so X ∧ Y ′ ∈ Fk+1κn, and [X] = [Y ]. Thusf is injective.

The multiplication is defined by [X][Y ] = [X∧Y ]; this gives an abelian group structurewith unit [LK(n)S

0]. The inverse of [X] is [X ′] (which is actually the K(n)-local Spanier-Whitehead dual of X).

Let Gκn denote the associated group to this filtration.

Corollary 4.3.5. There is a monomoprhism

φ : Gκn →⊕r>1

Er,r−1r

Note that this sum is finite by Proposition 4.2.3.

4.4. The structure of κn

We are now in a position to prove the following result on the structure of κn.


Theorem 4.4.1. Let p > 2. Then κn is a p-group. More specifically κn is a direct productof cyclic p-groups.

Remark 4.4.2. This does not rule out the possibility that κn is infinite.

Proof. We recall that we have a monomoprhism φ : Gκn →⊕

r>1Er,r−1r . Note that

each Er,r−1r is a sub-quotient (i.e.a quotient of a subgroup) of Er,r−1

2 = Hr(Gn, (En)r−1).From Proposition 4.2.2 we see that Er,r−1

2 has bounded exponent; hence the same is truefor Er,r−1

r (as both the exponent of a quotient and of a subgroup must divide the exponentof the group). The same is then true for Gκn. By the Prüfer-Baer theorem [Fuc70] thereis a decomposition

Gκn =r⊕i=0

Z/(pi+1)(αi).

The claimed result then follows since Gκn is the associated graded of a finite filtration ofκn.

Remark 4.4.3. There is an isomorphism

H∗(Gn, (En)t) ' lim←−H∗(Gn, (En)∗MIk

) ' lim←−H∗(Gn, (En)∗/(pi0 , . . . , uin−1

n−1 ))

Since (En)∗/(pi0 , . . . , uin−1n−1 ) is finite and Gn is a p-adic compact analytic Lie group

Lemma 1.4.2 implies that each group is finite. A similar argument (alternatively aninductive approach) to previous shows that it has finite exponent; thus we can deducethat H∗(Gn, (En)∗MIk

) is a finite p-group and Er,r−12 is a pro p-group. This does not

conflict with the description of κn above. Note that this also implies that H∗(Gn, (En)t)is finite or uncountable [RZ10, Proposition 2.3.1].

4.4.1. Remark on the finiteness of κn

In this subsection (which is really an extended remark) we wish to give some heuristicsregarding the finiteness of κn.

(i) When n = p− 1 the high dimension cohomology of H∗(Gn, (En)∗) is known. Specif-ically Symonds’ calculation [Sym04] of the Tate-Farrell cohomology of the Moravastabilizer group implies that

Hs(Gn, (En)t) ' Fp[β±1,∆±1]⊗ Λ(α, x1, . . . , xn−1), for s > n2

where α, β,∆ are as previously defined, and |xi| = (2i− 1, 0). Here the n2 arises asthe virtual cohomologcial dimension of Gn. Clearly then Er,r−1

2 is finite whenever

4.4. The structure of κn 69

s > n2. In fact, in the case n = 2 since the first target of a differential for E0,02 is

a d5, this already implies that κ2 is a finite 3-group. When n = 4, p = 5 the onlypossibility of a non-finite group is H9(G4, (E4)8).

(ii) Recall that Sp−1 has a unique conjugacy class of subgroups of order p. Pick arepresentative and denote it by E and let N = NGn(E) be its normalizer inside ofGn. This is an infinite closed normal subgroup of Gn. We expect the structure ofHs(N,E∗) to be similar to Hs(Gn, E∗). In fact these groups are isomorphic whens > n2 (this is related to the calculation in Item (i)).

Let κ(N) denote the group of isomorphism classes of spectra such that there is anisomorphism E∨∗X ' E∨∗ E

hNn of Morava modules (the latter forms a group under

the K(n)-local smash product over EhNn ). Note that the Shapiro lemma gives anisomorphism

Es,t2 = Hs(Gn, (En)tX) ' Hs(Gn, (En)t(EhNn )) ' Hs(N, (En)t).

There is a map κn → κ(N) sending X to X ∧EhNn , and we hope that the two groupsare similar.

Arguing in a similar manner as to before there is a filtration of κ(N) such that theassociated graded gives a monomorphism

Gκ(N) →⊕r>2

Er,r−1r .

Note that sparsity means that the first possible non-zero entry of this sum is E2p−1,2p−22 .

We claim that the groups Er,r−1r are always finite p-groups. In fact we claim

Hs(N,Et) is finite p-torsion for s > n (the result then follows since 2p−1 = 2n+1 >n). To see this we use the fact that Henn has constructed a finite resolution of EhNn .In particular by [Hen07, Proposition 25] there is a resolution of length n:

EhNn → X0 → · · · → Xn,

withXr =

∨(i1,...,ir)

Σ2p2n(i1+···+ir)EhGn ,

with EhGn as in Section 1.4 and the wedge is taken over all sequences (i1, . . . , ir) withn− 1 ≥ i1 > i2 > . . . > ir ≥ 0.


As in [GHMR12, Section 4.2] we can construct a spectral sequence

Ep,q,∗1 = Hp(Gn, (E2)∗Xq)⇒ Hp+q(Gn, E∨∗ (EhNn )) ' Hp+q(N,E∗).

This spectral sequence has its E1 term made up of copies of H∗(G,E∗); for examplewe have Ep,0,∗1 = Hp(G,E∗). The important point is that 0 ≤ q ≤ n. We want tofind terms in Hp+q(N,E∗) when p + q > n; the restriction on q implies that theycan only come from Hp(G,E∗) when p > 0, and then Lemma 3.3.7 implies that thisis always finite p-torsion. Since Er,r−1

r is a subquotient of Er,r−12 it is also finite

p-torsion; since the sum is finite it follows that Gκ(N), and in turn κ(N) itself arealso finite p-groups.

(iii) The above shows that the finiteness of κn is equivalent to the finiteness of the kernelof the map κn → κ(N), i.e. the set

κn(N) = X ∈ κn|X ∧ EhNn ' EhNn .

Let X ∈ κn(N) and let ιX ∈ H0(Gn, EnX) ' Zp be a unit element. Then we havea spectral sequence

Hs(N, (En)tX)⇒ πt−s(En ∧X)hN ' πt−s(EhNn ∧X)

and the assumptions on X imply that ιZ survives to E∞ and defines a homotopyclass in π0E

hNn ∧X. There are obstructions to lifting this to a class in π0E

hGnn ∧X

however, as we now explain. LK(n)S0 and EhNn fit into a resolution [Hen07]

LK(n)S0 → Y0 = EhNn → Y1 → . . .→ Yd,

where each Yi is a summand in a finite wedge of En’s (this even implies that the Yiare En-injective). The resolution refines to a tower of fibrations

LK(n)S0 Cd−1 Cd−2 · · · EhNn = Y0

Σ−dYd Σ−d+1Yd−1 Σ−d+2Yd−2

We can show that Yk ∧ X ' Yk (this essentially follows because they are modulespectra over EhFq

n as described in [Hen07, Appendix A] where Fq is a group withorder coprime to p - see Remark 4.5.15) and, by assumption EhNn ∧ X ' EhNn .

4.5. The decomposition of the group of exotic elements 71

Hence, if we smash the tower with X, the spectral sequence associated to the towerof fibrations takes the form

πtYq ⇒ πt−qX.

The obstructions (i.e. differentials) to lifting the class in π0(EhNn ∧X) lie in πsYs+1.However these are non-zero since πsYs+1 will always be non-zero when s ≡ 0mod 2(p− 1). This shows that it may be difficult to get a handle on the set κn(N).

4.5. The decomposition of the group of exotic elements

Let G ⊂ Gn be a closed subgroup of Gn. By the work of Devinatz and Hopkins asdescribed in Chapter 1, EhGn is an E∞-ring spectrum and it makes sense to speak of thePicard group, Pic(EhGn ), of the category of EhGn -modules. Here the group structure is givenby the (K(n)-local) smash product over EhGn . There is an map Picn → Pic(EhGn ) sendingX → X ∧ EhGn . Part of this section is understanding the image of X under this map.In particular, when n = p − 1 and G is a maximal finite subgroup of Gn, Corollary 4.5.5shows that X ∧ EhGn is always just a suspension of EhGn itself.

We remind the reader that X ∈ Picn if and only if E∨∗X ' E∗(Sk) for some k.

Proposition 4.5.1. Let F be a closed subgroup of Gn. Then for X ∈ Picn we haveE∨∗ (X ∧ EhFn ) ' E∨∗ ΣkEhFn for some k.

Proof. Since X is in Picn it is K(n)-local dualisable by Lemma 4.1.4 . Since E∨∗X 'E∗(Sk), E∨∗X is always pro-free. Recall that E∨∗ EhFn = Hom(Gn/F, (En)∗) and that thisis pro-free. Then Lemma 2.1.9 implies that

E∨∗ (X ∧ EhFn ) ' E∨∗X⊗E∗E∨∗ EhFn .

The result follows since E∨∗X ' ΣkE∗.

Remark 4.5.2. Note that this does not imply that X ∧ EhFn ' ΣkEhFn even when F isfinite, as there could be spectra Y such that E∨∗ Y ' E∨∗ E

hFn . However this will be the

case when F has order coprime to p.

We now restrict to the case that n = p − 1 for p odd, since it is here we know thestructure of π∗EhGn . We recall from Chapter 3 that we had the following description ofthe E∞-page (in positive cohomological dimension) for the homotopy spectral sequence


for π∗EhGn :

Eeven,∗∞ = Fp[∆±p][β]

βn2+1 and Eodd,∗∞ = αFp[∆±p]1,∆, . . . ,∆p−2[β]

βn.

Note that this is periodic of degree 2p2n2.Let κ1

p−1 denote the group of EhGn modules such that E∨∗X ' E∨∗ EhGn as Morava

modules, with the product structure given by the smash product over EhGn (a priori it isnot clear that this is a subgroup of Pic(EhGn ), but this will be implied by Lemma 4.5.3).Then there is well defined group homomorphism τ1 : κ1

p−1 → H2p−1(G; (En)2p−2). Welearnt this construction from [GHMR12]. Choose an isomorphism f : E∨∗X

'−→ E∨∗ EhGn .

By Shapiro’s lemma this gives an isomorphism

f : H∗(Gn, E∨∗X) ' H∗(G,E∗).

We then construct the diagram

H0(G, (En)0)

f '

// H2p−1(G; (En)2p−2)

f'

H0(Gn; (En)0X)d2p−1

// H2p−1(Gn; (En)2p−2X).

Let ι ∈ H0(G, (En)0) ' Zp be the identity element, and define

τ1 : κ1p−1

// H2p−1(G; (En)2p−2)

[X] // [f−1d2p−1f(ι)]

This is a well-defined function, independent of the choice of f (as explained in [GHMR12]for other isomoprhism g there is a unit a ∈ Z×p such that g = af), and is a grouphomomorphism using the multiplicative nature of the Adams-Novikov spectral sequence.

Lemma 4.5.3. (c.f. [GHMR12, Proposition 5.4]) The homomorphism

τ1 : κp−11 → H2p−1(G, (Ep−1)2p−2) ' Z/p

is an isomorphism.

Proof. Note that our calculation of H∗(G, (En)∗) implies that

H2p−1(G; (En)2p−2) ' Z/pαβp−1∆−1,

4.5. The decomposition of the group of exotic elements 73

and that this is the only possible differential for ι. Thus if X is in the kernel of τ1 thenιX survives to the E∞ page of the homotopy fixed point spectral sequence. Using theEhGn -module structure the resulting class S0 → X extends to an equivalence X ' EhGn .Thus τ1 is injective.

To see that this is surjective recall once again that E∨∗ EhGn ' Homc(Gn/G, (En)∗)with the diagonal Gn-action. The description of the cohomology above shows that ∆ isan G-invariant element of (En)∗ of (internal) degree 2pn2, and post-multiplication by ∆defines an isomorphism E∨∗ E

hGn ' E∨∗ Σ2pn2

EhGn . However Σ2pn2EhGn is not equivalent to

EhGn due to the non-trivial differential on ∆. Hence Σ2pn2EhGn is an element of κp−1

1 thatmaps to a non-trivial element of Z/p under τ1 (else, the first paragraph would imply thatΣ2pn2

EhGn would be topologically equivalent to EhGn ). Hence τ1 is also surjective, and isan isomorphism.

Remark 4.5.4. It follows that if E∨∗X ' E∨∗ EhGn , and X is a EhGn -module, then X '

Σ2pn2`EhGn for ` ∈ Z/p.

When combined with Proposition 4.5.1 we have1 the following consequence, which can beconsidered as the natural higher chromatic version of (at least one direction of) [HMS94,Theorem 3.1], which is recovered when n = 1, p = 2 and G = C2. We remind the readerthat we are assuming that n = p− 1.

Corollary 4.5.5. If X ∈ Picn then X ∧ EhGn ' Σk+2pn2`EhGn for ` ∈ Z/p.

Remark 4.5.6. This gives a hint that perhaps Pic(EhGn ) ' Z/(2p2n2), given by suspensionsof EhGn . In fact it is true that Pic(KO) = Z/8. This is proven in forthcoming work ofLawson and Gepner, although it seems likely that it was known previously.

The following is then the analogue of the epimorphism Pic1 Z/8 (for p = 2) in [HMS94].

Corollary 4.5.7. There is an epimorphism ψ : Picn → Z/(2p2n2) given by taking X ∈Picn to k + 2pn2`.

Remark 4.5.8. There is also an epimorphism Picn → Z/(2(pn − 1)) - see [HS99b, Page68].

Example 4.5.9. Let S0〈det〉 be the determinantly twisted sphere spectrum, to be definedbelow in Definition 4.6.3. We will see that S0〈det〉 is in Picn and that S0〈det〉p−1 ∧

1In what follows the suspension is only unique modulo 2p2n2


EhGn ' EhGn . Hence S0〈det〉p−1 is in the kernel of ψ. When n = 2, p = 3 we will showin Lemma 4.6.5 that S0〈det〉 ∧EhG24

2 ' EhG242 , and hence when n = 2, S0〈det〉 itself is in

the kernel of ψ. ♦

Recall that if X is in κn then k = 0.

Corollary 4.5.10. If X ∈ κn then X ∧ EhGn ' Σ2pn2`EhGn for ` ∈ Z/p.

Remark 4.5.11. This also follows directly from Lemma 4.5.3.

Using terminology introduced in [GHMR12] we say that X is truly exotic whenever ` = 0,i.e. X ∧EhGn ' EhGn , and we say X is G−exotic if X is an element of κn such that ` 6= 0.The term truly exotic is appropriate considering the following lemma.

Lemma 4.5.12. Let X be a truly exotic element of κn. Then for all finite subgroupsK ⊂ Gn there is a weak equivalence X ∧ EhKn ' EhKn .

Proof. There is a spectral sequence

H∗(K,E∨∗X)⇒ π∗(E ∧X)hK ' π∗EhK ∧X,

where the last equivalence follows from Lemma 4.1.4.Let ιx ∈ H0(K,E∨0 X) be the unit element. If K has order coprime to p then this

spectral sequence always collapses, and the map resulting map S0 → EhK ∧X extends tothe desired equivalence.

If K has order divisible by p then the p-Sylow subgroup of K is conjugate to Cp, andin turn π∗EhKn is a retract of π∗E

hCpn , whose homotopy groups can be studied in a similar

way as of π∗EhGn in Chapter 3.More concretely the restriction map gives a commutative diagram in group cohomology

H∗(G,E∨∗X) H∗(Cp, E∨∗X)

H∗(K,E∨∗X),

as well as a corresponding diagram on the homotopy fixed points. Since Cp is an index n3

subgroup of G (and n3 is coprime to p) the map H∗(G,E∨∗X)→ H∗(Cp, E∨∗X) is injective,and hence the same is true for H∗(G,E∨∗X)→ H∗(K,E∨∗X). Running the homotopy fixedpoint spectral sequence we get an injection π∗(EhGn ∧X)→ π∗(EhKn ∧X). The end resultis that if the class ιX survives the spectral sequence for G, then it survives it for K, andwe can again extend the resulting map to the desired equivalence.

4.6. Brown-Comenetz duality and G-exotic elements 75

By Corollary 4.5.10 there is a well defined map φ : κn → Z/p given by sending X to `.The next result is clear.

Corollary 4.5.13. If there exists a (non-trivial) G−exotic element then φ : κn → Z/p issurjective.

Example 4.5.14. When n = 2, p = 3 there is a G24-exotic element denoted P2 and asplit short exact sequence

0→ Z/3→ κ2 → Z/3→ 0.

The surjection takes X ∈ κ2 to X ∧ EhG242 = Σ24kEhG24

2 . In [GH12] a canonical splittingfor this short exact sequence is constructed. ♦

We suspect that, in general, if there is a non-trivial G−exotic element, then it has orderp inside of κn, which would imply that φ : κn → Z/p is a split surjection.

Remark 4.5.15. Although we talk of exotic and truly exotic elements in this section, whenn > 2 we have not yet succeeded in proving that such elements actually exist. Nonetheless,we are confident that G−exotic elements do indeed exist when n = p − 1, as has beenobserved for n = 1 and n = 2.

In fact, Henn has constructed a resolution of K(n)-local sphere when n = p− 1 and pis odd, made up of wedges of suspensions of EhGn , and spectra which are module spectraover EhFq

n , where Fq = Cpn−1 oGal. This is constructed in the same way as the resolutionof [GHMR05]; namely an exact sequence of Morava modules is constructed, and then Hennshows that the obstructions to lifting this to a sequence of spectra all vanish. Henn thenshows that this refines to a tower of spectra with LK(n)S

0 at the top.Now it follows from Lemma 4.5.3 that there is an isomorphism of Morava modules

E∨∗ EhGn ' E∨∗ Σ2pn2kEhGn but topologically EhGn 6' Σ2pn2kEhGn , at least when k 6≡ 0

mod (p). Hence we can use this isomorphism to construct a new, twisted, version ofHenn’s resolution of Morava modules. The obstructions to lifting this to a sequence ofspectra are the same as in Henn’s case, and so they all vanish, however we have not yetsucceeded in showing that this sequence refines to a tower of spectra (unless n = 2), sowe cannot yet produce exotic elements of κn this way.

4.6. Brown-Comenetz duality and G-exotic elements

In this section we outline a computational approach to detecting G-exotic elements inthe K(n)-local Picard group, as well as an application to the Brown-Comenetz dual of the


K(n)-local sphere. We are in fact able to give partial calculations at height 2, recoveringsome of [GH12; GHMR12] in a matter that we believe can be generalised to higher heights.

We start with the definition of Brown-Comenetz duality [BC76]. The Pontryaginduality functor

X 7→ HomZ(π−∗X,Q/Z)

defines a cohomology theory, since Q/Z is an injective abelian group, and so by Brownrepresentability is represented by a spectrum, IQ/Z. This is characterised by the fact that

πk(IQ/Z) = HomZ(π−kS0,Q/Z)

The Brown-Comenetz dual of a spectrum X is then defined as

IQ/ZX = F (X, IQ/Z).

This need not be K(n)-local, even when X is. The details of the K(n)-local variant ofBrown-Comenetz duality have been worked out by Gross and Hopkins [HG94]. Let MnX

be the fiber of the natural map LnX → Ln−1X. By [Sto12, Proposition 2.2] for any Xand Y the natural map F (LnX,Y )→ F (MnX,Y ) is K(n)-localisation. This leads to thefollowing definition.

Definition 4.6.1. For K(n)-local X the Gross-Hopkins dual of X is

InX = F (MnX, IQ/Z) = IQ/ZMnX.

We let In denote IMnS0. This is characterised by natural isomorphisms

[X, In] = Hom(π−∗(MnX),Q/Z) = Hom(π−∗(MnS ∧X),Q/Z),

since Mn is a smashing localisation [Rav92]. There are natural maps In−1 → In and IQ/Zcan be recovered as the direct limit of the In.

Gross-Hopkins duality is arguably the most important type of duality in theK(n)-localsetting; it is, in some sense, the analogue of Serre duality [HG94; Str00]. For example,when n = 1, p > 2 and X is finite, there is an isomorphism

π∗Σ2L1DX ⊗Z(p) Zp'−→ HomZ(p)(π∗L1X,Q/Z(p));

this reflects the fact that I1 ' S2 in this case [Dev90].

Lemma 4.6.2 (Gross-Hopkins). In is an element of Picn.


Proof. See [HS99b, Theorem 10.2e].

Hopkins and Gross have worked out the image of In under the map In 7→ E∨∗ In. Todescribe their result we need to introduce the determinantly twisted sphere spectrumS0〈det〉 (see [GHMR12] although we more closely follow [Wes12]). Again we work at anodd prime. Recall that SGn is the kernel of the determinant map Gn → Z×p . We canform the homotopy fixed point spectrum EhSGn

n and this retains an action by Z×p . WriteZ×p = µp−1 × (1 + pZp)× and choose g = ζ(1 + p), where ζ is a primitive (p− 1)-st root ofunity, to be the topological generator of Z×p . Then define Fγ by the fiber sequence

Fγ EhSGnn EhSGn

n .ψg−γ

Fγ is an invertible spectrum; in fact the map γ → Fγ defines a homomorphism Z×p →Picn [Wes12, Proposition 3.16]. In the case γ = 1 this defines the action by Z×p and so

F1 = (EhSGnn )hZ

×p ' EhGn

n ' LK(n)S0.

Definition 4.6.3. The determinately twisted sphere S0〈det〉 is the homotopy fiber Fg.

In [GHMR12] it is shown, using the isomorphism E∨∗ ESGnn ' Homc(Gn/SGn, E∗) '

Homc(Z×p , E∗), that E∨∗ S0〈det〉 = E∗〈det〉 := E∗ ⊗Zp Zp〈det〉2. The result of Hopkins-Gross (for a full proof see [Str00]) states that there is an isomorphism of Morava modules

(4.4) E∨∗ In ' Σn2−nE∗〈det〉.

Now E∗〈det〉 is an element of Picn since E∨∗ S0〈det〉 is free of rank one over (En)∗ and sowhenever κn = 0 there is a K(n)-local equivalence

In ' Σn2−nS0〈det〉.

However when κn 6= 0 this need not be the case. Note that by Corollary 4.5.5 aftersmashing with EhGn both In and S0〈det〉 are suspensions of EhGn ; by explicitly calculatingthese we can potentially detect G-exotic elements (note that this method can not detecttruly exotic element, and that In ' Σn2−n may hold even when κn 6= 0).

Example 4.6.4. When n = 1, p = 2 (when S0〈det〉 = S2) we have a spectral sequence to

2Here Zp〈det〉 is the module defined via the map Gn → Z×p .


compute π∗(EhC21 ∧ I1) namely

H∗(C2, E∨∗ I1) ' H∗(C2,Σ2E∗)⇒ π∗(E1 ∧ I1)hC2 ' π∗(EhC2

1 ∧ I1).

An argument similar to Corollary 3.3.11 shows that I1 ∧ EhC21 ' Σ2EhC2

1 or Σ6EhC21 .

Smashing with the mod 2 Moore spectrum V (0), we have I1∧LK(1)V (0) ' Σ−2LK(1)V (0)(see, for example [GH12]); since EhC2

1 ∧ V (0) is 8 periodic (as can be checked via the longexact sequence associated to the defining cofiber sequence S ·2−→ S → V (0)), we see thatwe must have I1 ∧ EhC2

1 ' Σ6EhC21 .

Now clearly S0〈det〉∧EhC21 = Σ2EhC2

1 and so the Hopkins-Gross equation immediatelyimplies the existence of P1 ∈ κ1 such that P1 ∧ EhC2

1 ' Σ4EhC21 . Indeed this is the non-

trivial element of κ1, and I1 ' Σ2P1. ♦

We can do something similar when n = 2, p = 3. We start with a preliminary lemma.

Lemma 4.6.5. There is an equivalence

S0〈det〉 ∧ EhG242 ' EhG24

2 .

We actually prove something more general; Lemma 4.6.5 then follows from the fact thatG24 lies in the kernel of the determinant map.

Proposition 4.6.6. Suppose F ⊂ Gn is a finite subgroup contained in the kernel of thedeterminant map. Then

S0〈det〉 ∧ EhFn ' EhFn .

Proof. Since F is contained in the kernel of the determinant F is a subgroup of SGn.Westerland [Wes12] has constructed an invertible map S0〈det〉 → EhSGn

n . Compositionwith the restriction map EhSGn

n → EhFn gives an invertible map S0〈det〉 → EhFn thatextends to the desired equivalence.

Remark 4.6.7. We consider the case of a finite subgroup that is not in the kernel of thedeterminant below in Proposition 4.6.12.

Theorem 4.6.8. There is an element P2 ∈ κ2 such that P2 ∧ EhG242 ' Σ48EhG24

2 andthere is an equivalence

I2 ' S2 ∧ S0〈det〉 ∧ P2.

Remark 4.6.9. (i) Our definition of P2 is not unique, in that it may be the case thatP2 = P ′2∧Q, with P ′2∧E

hG242 ' Σ48EhG24

2 and Q a (non-trivial) truly exotic element.


(ii) It is in fact known [GH12] that κ2 = Z/3P ⊕ Z/3Q and

I2 ' S2 ∧ S0〈det〉 ∧ P.

However our method is independent of [GH12]. It has the advantage that it does notinvolve smashing with V (1), since the Smith-Toda complexes do not in general existat higher heights. However our method is unable to determine if P2 decomposes asabove.

Proof. We have [Beh06, Proposition 2.5.1] that

I2 ∧ EhG242 ' Σ50EhG24

2

Since Lemma 4.6.5 implies that Σ2S0〈det〉 ∧EhG242 ' Σ2EhG24

2 the group structure on κ2,along with the Hopkins-Gross equation Equation (4.4), implies that there is some elementP2 ∈ κ2 such that P2 ∧ EhG24

2 ' Σ48EhG242 .

Remark 4.6.10. One could argue that the claim that this calculation does not involvesmashing with V (1) is misleading since it relies on the calculation of I2 ∧ EhG24

2 due toBehrens, which does use rely on calculations involving V (1). We can offer the followingalternative. There is an isomorphism

I2(EhG242 ) ' D2(EhG24

2 ) ∧ I2,

and an equivalence I2(EhG242 ) ' Σ22EhG24

2 . We know of two proofs of this fact in theliterature, due to Behrens [Beh06], using Mahowald-Rezk duality, and Stojanoska [Sto12]using Anderson duality.

We have calculated that D2(EhG242 ) ' Σ−4+24kEhG24

2 in Corollary 3.3.11. We believethat the following map is an equivalence that would prove that k = 2. There is a mapδ : Σ44EhG24

2 → LK(2)S0 arising from the Goerss-Henn-Mahowald-Rezk tower of fibrations,

and we believe that the composite

Σ44EhG242 → F (EhG24

2 ,Σ44EhG242 ) δ∗−→ F (EhG24

2 , LK(2)S0)

is an equivalence, where the first map is the adjoint of the (suspension of) the multiplica-tion.

We remind the reader that κ2 = Z/3 × Z/3. We cannot fully recover this result,however we have the following, which follows from Corollary 4.5.13.


Corollary 4.6.11. There is a surjection κ2 Z/3.

When K ⊂ Gn is a finite subgroup that is not (necessarily) in the kernel of thedeterminant we have the following. This owes much to [Wes12], as well as discussionswith Westerland.

Proposition 4.6.12. If K ⊂ Gn is a finite subgroup then there is an equivalence

S0〈det〉∧(p−1) ∧ EhKn ' EhKn .

Proof. The point of the proof is that K is always a subgroup of G1n, and so we wish to

lift the invertible map S0〈det〉 → EhSGnn of Westerland to a map S0〈det〉 → E

hG1n

n '(EhSGn

n )hµp−1 . By [Wes12, Proposition 3.19] there is a K(n)-local equivalence

(4.5)p−2∨k=0

S0〈det〉∧k ∧ EhG1n

n → EhSGnn .

Consider the following composite:

S0〈det〉p−1 → EhSGnn → EhG

1n

n ,

where the first map is the (p − 1)-st power of the invertible map S0〈det〉 → EhSGnn and

the second map is projection onto one of the split summands given in Equation (4.5).We wish to calculate what happens to this map when applying E∗. Now E∨∗ SGn '

Homc(Z×p , E∗) and the image of the fundamental class under the map E∗〈det〉 → EhSGnn is

the natural inclusion map Z×p → E∗. Hence the image of the (p− 1)-st power map is themap φ : Z×p → E∗ given by x 7→ xp−1. Note that since x ∈ Z×p we have xp−1 ≡ 1 mod (p).This means that φ factors through Z×p /µp−1, i.e. we have a commutative diagram:

Z×p E∗

Z×p /µp−1 = (1 + pZp)×

φ

This implies that the image of the fundamental class of E∨∗ S0〈det〉p−1 in E∨∗ EhSGnn lies in

the image of the restriction map from E∨∗ (EhSGnn )hµp−1 ' E∨∗ E

G1n

n ' Homc((1+pZp)×, E∗).The function x 7→ xp−1 has inverse x 7→ x1−p; together these imply that the existence ofan invertible map

S0〈det〉p−1 → EhG1n

n .


The argument is now finished as in Proposition 4.6.6. Any finite subgroup K ⊂ Gn,is a subgroup of G1

n; composition with the above map then gives an invertible mapS0〈det〉p−1 → EhKn which extends to the equivalence S0〈det〉p−1 ∧ EhKn ' EhKn .

Example 4.6.13. Here is a non-trivial example. Let SD16 ⊂ G2 be the semi-dihedralgroup of order 16 generated by ω, an 8-th root of unity, and φ, the generator of theGalois group Gal(F9/F3). SD16 is not in the kernel of the determinant; rather there is anisomorphism

det ↓G2SD16' χ

where χ is the non-trivial character of Z/2 pulled back along the quotient map SD16 →SD16/D8 ' Z/2, where D8 = 〈ω2, φ〉 is the dihedral group of order 8. This gives anisomorphism of SD16-modules [Beh06]

E∗ ⊗ Z3(χ) ' Σ8E∗

and so there is an isomorphism

H∗(SD16, E∗〈det〉) ' H∗(SD16,Σ8E∗).

The (collapsing) homotopy fixed point spectral sequence implies that S0〈det〉 ∧EhSD162 '

Σ8EhSD162 . However S0〈det〉2 ∧ EhSD16

2 ' Σ16EhSD162 ' EhSD16

2 by periodicity. ♦

4.6.1. An approach to constructing G-exotic elements

We now explain our approach to constructing G-exotic elements.

Step 1: Calculate In ∧ EhGn . For example one approach is to start with the homotopy fixedpoint spectral sequence

Es,t2 = Hs(G;Et(In))⇒ πt−sIn ∧ EhGn

Using the Hopkins-Gross formula there is an isomorphism of E2-termsHs(G;EtIn) 'Hs(G,Σn2−nEt〈det〉). Now when n > 2 it is no longer true that G is in the kernel ofthe determinant homomorphism - rather det(G) = Z/(p− 1). We hope an analysisas in Example 4.6.13, along with an analysis of the spectral sequence will completethe calculation of In ∧ EhGn .


An alternative approach is to use the fact that there is a K(n)-local equivalence

(4.6) In(EhGn ) ' In ∧Dn(EhGn ).

In Corollary 3.3.11 we gave a partial description of Dn(EhGn ), whilst In(EhGn ) couldbe analysed via the relation between Brown-Comenetz duality and Mahowald-Rezkor Anderson duality (as in [HS14; Sto12]). Since Dn(EhGn ) is self-dual up to suspen-sion Equation (4.6) can be used to determine In ∧ EhGn .

Step 2: Calculate S0〈det〉 ∧ EhGn , again using a homotopy fixed point spectral sequenceargument, along with Proposition 4.6.12.

Step 3: Use the Gross-Hopkins formula Equation (4.4) to deduce the existence of G-exoticelements.

This is essentially the methodology used in Theorem 4.6.8, and expanded on in Re-mark 4.6.10 although it seems more complicated when n > 2 since G is no longer in thekernel of the determinant homomorphism. This also relies on the existence of G-exoticelements that are smash summands in In; as we have seen, such a result is true at n = 1and n = 2.

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APPENDIX A

Tate cohomology

In this appendix we will review the properties of Tate cohomology that we require. Agood reference for this material is [Bro82, Chapter VI]. To start, let G be a finite groupand choose a free resolution of Z as a G-module:

· · · P2 P1 P0 Z 0.ε

For example one can take Pi = Z[Gi+1] with differential

d(d0, . . . , gi) =i∑

j=0(−1)j(g0, . . . , gj−1, gj+1, . . . , gi)

and ε is the augmentation.

Define P ∗i := HomZ(Pi,Z) and form the exact sequence

· · · P2 P1 P0 Z 0

0 Z P ∗0 P ∗1 P ∗2 · · · .

Set P−n = P ∗n−1 so that there is an exact sequence

89

90 Chapter A: Tate cohomology

· · · P1 P0 P−1 · · ·

Z

This is called a complete resolution of G. In general a complete resolution is an exactcomplex F• of projective Z[G]-modules, together with a map ε : F0 → Z such that · · · →F1 → F0 → Z forms a resolution in the usual sense. As usual complete resolutions areunique up to chain homotopy.

Definition A.1. Let M be a G-module and P• a complete resolution of G. Then theTate cohomology groups are

Hs(G;M) := Hs(HomG(P•,M)).

Remark A.2. Let N be∑s∈g s ∈ Z[G]. Given a G-module M there is an endomorphism

of M given by multiplication by N and this gives a homomorphism

N∗ : H0(G;M)→ H0(G;M).

Then we have

(A.1) Hs(G;M) =

Hs(G;M) for s ≥ 1

coker(N∗) for s = 0

ker(N∗) for s = −1

H−s−1(G;N) for s ≤ −2.

Indeed for s ≥ 1 this is by definition and this is not hard to verify for s ≤ −2. Thecases s = 0 and s = −1 require slightly more work, for which we again refer the readerto [Bro82].

Since it is defined by a resolution many of the usual properties of ordinary group coho-mology have analogues in Tate cohomology, as the next few propositions show.

Proposition A.3. For an exact sequences of G-modules

1→M ′ →M →M ′′ → 1

91

there is an exact sequence

(A.2) · · · → Hq(G,M ′)→ Hq(G,M)→ Hq(G,C) δ−→ Hq+1(G,M ′)→ · · ·

Proposition A.4 (Dimension Shifting). There are G-modules K and C such that forall i ∈ Z

H i(G,M) ' H i+1(G,K) and H i(G,M) ' H i−1(G,C).

There is also a Lyndon-Hochshild-Serre spectral sequence associated to an extension ofthe form

1→ N → G→ G/N → 1,

where N E G is a normal subgroup.

Proposition A.5. There is a spectral sequence

Es,t2 = Hp(G/N, Hq(N,M))⇒ Hp+q(G,M).

We also have cup products in Tate cohomology

Proposition A.6. There is a unique cup product

Hp(G;A)⊗ Hq(G;B)→ Hp+q(G,A⊗B)

such that

(i) The cup product is natural with respect to maps f : A→ A′ and g : B → B′;

(ii) When p = q = 0 the cup product is induced by the map

AG ⊗BG → (A⊗B)G; and,

(iii) The cup product is compatible with the boundary morphism δ of Equation (A.2).

Lemma A.7. [Bro82, Exercise VI.6.1] The natural map H∗ → H∗ preserves products.

Proof (sketch). In the case that p = q = 0 this follows from Item (ii) of Proposition A.6. Ingeneral use dimension shifting (which is compatible with the cup product) and inductionon p, and then q, to reduce to the case of p = q = 0 .

For us the most important calculation will be the Tate cohomology of a finite cyclic group,which is well known.

92 Chapter A: Tate cohomology

Example A.8. (Tate cohomology of cyclic groups.) Let Cp be generated by g and definemaps Z[Cp]→ Z[Cp] by

NCp : a 7→p−1∑i=0

gia and g − 1 : a 7→ ga− a

Then there is a projective resolution of Z given by

· · · Z[Cp] Z[Cp] Z[Cp] Z 0g−1 NCp ε

By splicing this resolution together with its dual one gets the following complete resolution

· · · Z[Cp] Z[Cp] Z[Cp] Z[Cp] · · ·

Z

g−1 NCp g−1−1 NCp

After applying HomZ(−, A) we get the following complex

· · · → Ag−1−−→ A

NCp−−−→ Ag−1−−→ A→ · · · .

Then in fact

H i(Cp, A) =

H0(Cp, A) i even,

H−1(Cp, A) i odd.

This isomorphism is in fact given by a cup product - again we refer the reader to [Bro82,Chapter VI.9]. ♦

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