Morava modules and the K(n)-local Picard group

Transcription

1 Morava modules and the K(n)-local Picard group Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy University of Melbourne Department of Mathematics and Statistics Drew Heard August 19, 2014

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3 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 different ways. The first is to closely study the category of Morava modules. It turns out that this category is equivalent to the category of complete E E-comodules. Using this, we develop a theory of (relative) homological algebra for Morava modules. We use this to give an explicit identification of the E 2 -term of the K(n)-local E n -based Adams spectral sequence. This turns out to be related to work of Goerss-Henn-Mahowald and Rezk [GHMR05], as constructed 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 groups the Tate spectrum Ep 1 tg always vanishes. Such a result was previously known to be true K(n)-locally, but we show it holds even before this. We use this to deduce some selfduality results for the K(n)-local Spanier-Whitehead dual of the homotopy fixed point spectra E hg n. In the final chapter we study the K(n)-local Picard group. In particular we show that, when p is an odd prime, the subgroup κ n of elements such that E X E as continuous modules over the Morava stabiliser group is always a p-group, and decomposes as a direct product of cyclic groups. Then, specialising to the case of n = p 1, we discuss the decomposition of the group of exotic elements, by studying the map from the Picard group of the K(n)-local category to the Picard group of En hg -modules, where G G n is a maximal finite subgroup of the Morava stabilizer group. We finish by explaining the connection to Gross-Hopkins duality, and outline an approach to constructing elements of κ n when n > 2. In fact this method already allows us to (independently) construct elements of κ 2 that are non-zero in I 2, the Gross-Hopkins dual of the sphere. 3

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5 Declaration This is to certify that: (i) the thesis comprises only my original work towards the PhD except where indicated in the Preface, (ii) due acknowledgement has been made in the text to all other material used, (iii) the thesis is fewer than words in length, exclusive of tables, maps, bibliographies and appendices. 5

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7 Acknowledgements I owe a great thanks to a number of people for this project. First and foremost I wish to thank my advisor Craig Westerland. Since he essentially taught me all the algebraic topology I know, it is no exaggeration to say that this thesis would not exist without his help. Beyond that, his encouragement and support was invaluable, and the numerous trips he helped fund shaped many of the results within this document. I d also like to thank Nora Ganter and Alex Ghitza for being members of my PhD committee, and to Nora for agreeing 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 my many s regarding these series of papers and related matters. I also had helpful conversations regarding this and related matters with Mark Behrens, Daniel Davis, Mark Hovey, Tyler Lawson and Vesna Stojansoka amongst others. Charles Rezk also provided a key lemma needed. It was great to be doing this project at the same time as TriThang and Jeff - I m sorry I 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 with the strange life of a PhD student. A special thanks to Sonja for the whiteboard and for encouraging me to do this in the first place. 7

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9 Table of Contents Introduction 11 1 Prerequisites Bousfield localisation and Morava K-theory Lubin-Tate theory and Morava E-theory The action of the Morava stabilizer group Homotopy fixed point spectra for subgroups of the Morava stabilzier group Finite subgroups at height n = p Morava modules and the K(n)-local category The K(n)-local category Morava modules and complete E E-comodules Homological algebra for Morava modules Relative homological algebra The K(n)-local E n -Adams spectral sequence Homotopy fixed point and Tate spectra The Tate spectrum A description of E n as a F -module The Tate cohomology of G Relation to the work of Mathew-Meier The Picard group of the K(n)-local category Background Some cohomological results

10 10 Table of Contents 4.3 A filtration on κ n The E(n)-local case The K(n)-local case The structure of κ n Remark on the finiteness of κ n The decomposition of the group of exotic elements Brown-Comenetz duality and G-exotic elements An approach to constructing G-exotic elements Bibliography 83 A Tate cohomology 89

11 Introduction From a computational standpoint, the chromatic approach to stable homotopy theory largely reduces the study of π X, for a finite p-local spectrum X, to the study of the localisations π L K(n) X. In theory, the homotopy of π X can then be recovered via the chromatic tower and the chromatic fracture square. This naturally leads to the study of the category of K(n)-local spectra, M S,K(n), itself. In this thesis we broadly study this category in a number of related ways. One of the most significant recent developments in chromatic homotopy theory was the 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 our understanding of the K(2)-local homotopy category at the prime 3; see the series of papers [GHM; GH12; GHMR12; Kar10]. The construction of this resolution starts with an algebraic resolution of the trivial Z p [[G n ]-module Z p by permutation modules of the form Z p [[G n /F ] for some finite subgroup F G n. This leads to an exact sequence of Morava modules, where a Morava module is a complete E -module such that the action of the Morvava stabilizer group, G n, is compatible with the E -module structure. We denote the category of such objects by EG n. An obstruction theory argument using the E n -Hurewicz map π F (E hh 1 n, E hh 2 n ) Hom EGn (E E hh 1 n, E E hh 2 n ) 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 homotopy fixed point spectral sequence. After identifying the category of Morava modules with the 11

12 12 Introduction category of complete E E-comodules of [Dev95], we show that the above map is in fact the edge homomorphism of the K(n)-local E n -Adams spectral sequence. Theorem convergent spectral sequence 1) If E X and E Y are Morava modules then there is a strongly E s,t 2 = Ext s,t EG n (E X, E Y ) π t s F (X, Y ), isomorphic to the K(n)-local E n -Adams spectral sequence. 2) If Y = E hf n for F a closed subgroup of G n then there is an isomorphism of E 2 -terms: Ext s,t EG n (E X, E E hf ) H s c (F ; E t X). Here the bifunctor Ext s EG n (, ) is a suitable relative version of Ext constructed in the (non-abelian) category EG n. The third chapter is computational in nature. Let G G n be a maximal finite subgroup (of order divisible by p) at height n = p 1. The spectrum En hg is meant to interpolate between the spectrum E n, whose homotopy groups are easy to compute and E hgn n L K(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 Tate spectrum E tg n is the cofiber of this map, i.e. there is a fiber sequence (E n ) hg E hg n E tg n. Our next result was previously known to be true after K(n)-localisation (for example, it follows from [Rog08] since E hg n E n is a K(n)-local G-Galois extension). Theorem The Tate spectrum E tg n is contractible. As an example of an application of this result we show the following, where D n En hg monodial duality in the K(n)-local category, i.e. D n X = F (X, L K(n) S 0 ). is the Corollary En hg is K(n)-locally self-dual up to suspension. In fact D n En hg Σ N En hg where N n 2 mod (2pn 2 ) and N is only uniquely defined modulo 2p 2 n 2. We then discuss a generalisation of this result due to Mathew and Meier [MM13].

13 Introduction 13 In the final chapter we investigate the Picard group, Pic n, of the K(n)-local category. In particular we are interested in the subgroup κ n of invertible spectra such that there is an isomorphism E X E of Morava modules. Our main result is the following, which partially answers a conjecture of Hopkins [Str92]. Theorem Let p > 2. Then κ n is a p-group. More specifically κ n is a direct product of cyclic p-groups. We then give an extended remark regarding the finiteness (or otherwise) of κ n. In particular, at height n = p 1, we define a group κ n (N) we believe should be closely related to κ n and prove that it is a finite p-group. Again, restricting to the case of n = p 1 we study the map from Pic n Pic(En hg ), the Picard group of En hg -modules. We show that under this map X Pic n always maps to a suspension of En hg. In the case that X κ n we explicitly identify this suspension. In particular we show that there is a natural decomposition (as observed at height 2 in [GHMR12]) into elements that are detected by a finite subgroup of G n and elements that are not detected by any finite subgroup. When n > 2 we are not able to produce any new elements of κ n. In the case n = 2, however we can partially (independently) recover results from [GH12; GHMR12]. Theorem There is an element P 2 κ 2 such that P 2 E hg 24 2 Σ 48 E hg 24 2 and there is an equivalence I 2 S 2 S 0 det P 2. We remark that this does not uniquely specify P 2. We finish by outlining a methodology for generalising these calculations to the case where n > 2.

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15 CHAPTER 1 Prerequisites The purpose of this introductory chapter is to provide an introduction to the cohomology theories K(n) and E n, and the Morava stabilizer group, G n. When n = p 1 we review the classification of the finite subgroups of the Morava stabilizer group, as we will need them throughout this document Bousfield localisation and Morava K-theory In commutative algebra one often studies a ring by studying its localisations. This is also true in the category of ring spectra; here classical localisation is replaced by Bousfield localisation. Let E be a cohomology theory. We say that a spectrum X is E-acyclic if E X is null and that a spectrum Y is E-local if [X, Y ] is null whenever X is E-acyclic. Then a map f : X Y of spectra is an E-equivalence if the fibre is E-acyclic (equivalently, if E (f) is an isomorphism). Bousfield [Bou79] has constructed a functor L E from X to a (unique) E-local spectrum L E X along with a natural transformation η : X L E X which is an E -equivalence. The map η E is terminal amongst E -equivalences out of X. If both E and X are connective, then Bousfield localisation behaves nicely. Example [Bou79] Let E = HQ be the rational Eilenberg-MacLane spectrum. Then if X is connective we have π (L HQ X) = π (X) Q. 15

16 16 Chapter 1: Prerequisites In general when E and X are connective spectra localisation is the same as localisation or completion with respect to some set of primes. Let M S denote a suitable category of spectra, such as the S-modules constructed in [EKMM97]. Then Bousfield localisation is a functor L E : M S M S,E, where M S,E M S is the full-subcategory of E-local S-modules. The homotopy category obtained by inverting the weak equivalences will be denoted by D S,E. We will be particularly interested in two cases. The first is localisation with respect to Morava K(n)-theory, denoted K(n). spectrum (at least when p > 2) with coefficient ring This is a complex oriented commutative ring K(n) F p [v ±1 n ], n > 0, with v n = 2(p n 1) (the dependence on the prime p is always suppressed from the notation). The associated formal group law Γ n (x, y) has p-series [p] Γn (x) = v n x pn. The Morava K-theories were first constructed by Morava in the seventies, although this work was never published. The first published reference was by Johnson and Wilson [JW75]. The construction proceeds using the Baas-Sullivan construction [Baa73] to create versions of complex cobordism for manifolds with singularities. The modern theories of ring spectra, for example [EKMM97], significantly ease the construction of ring spectra such as these by constructing them as suitable quotients of the Brown-Peterson spectrum, BP. We note that the complex orientation of K(n) arises as a morphism of ring spectra ψ : BP K(n), which induces a map given by ψ : BP K(n), v n if k = n, ψ (v k ) = 0 else. By convention we will take K(0) = HQ to be the rational Eilenberg-MacLane spectrum, and K( ) = HF p to be mod-p homology. The Morava K-theories have a number of useful properties: Proposition [JW75; Rav84] Let p be a prime. Then for all 0 n :

17 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 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 example 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 the stable homotopy category. This statement can be interpreted in a number of ways. For example 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 this definition 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 wedge sum K(0) K(n), which we denote by L n (by [Hov95, Corollary 1.12] this is also localisation 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 any spectrum X, L n X X L n S. 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 L n assemble into a tower of localisations, known as the chromatic tower: L n X L n 1 X L 1 X L 0 X. 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 chromatic tower. Hence we can hope to study π X by studying each of the π L n X. This description underlines the whole approach to chromatic homotopy theory.

18 18 Chapter 1: Prerequisites Slightly more is true in fact; for every X there is a homotopy pullback square (1.2) L n X L K(n) X L n 1 X L n 1 L K(n) X, and so in order to study L n X we can inductively study L K(n) X as well as some attaching maps. This underlines the importance of the category M S,K(n), of K(n)-local spectra. Remark Hopkins chromatic splitting conjecture [Hov95] includes conjectures relating to the fracture square Equation (1.2). In particular Hopkins conjectures that there is a map L K(n) X L n 1 X making the fracture square commute. This implies that there is a splitting L n 1 L K(n) X L n 1 X of the bottom horizontal map. The chromatic splitting conjecture is known to be true for all primes when n = 1 and for n = 2, p Lubin-Tate theory and Morava E-theory We noted above that the localisation L K(0) K(n) is also localisation with respect to a cohomology theory called Morava E-theory, constructed in [Mor85]. This is a periodic Landweber exact cohomology theory, with coefficient ring (E n ) W(F p n)[[u 1,..., u n 1 ][u ±1 ], with u i = 0 and u = 2. Here W(F p n) denotes the Witt vectors over F p n, which can be constructed by adjoining a primitive (p n 1)-st root of unity to Z p. 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 knowledge of formal group laws as covered in [Rav86, Appendix 2]. Let Γ n be a formal group law of height n over a perfect field k of characteristic p > 0. We will define a functor Def k,γn ( ) from the category of complete local rings to the category of groupoids. Fix a complete local ring B, with maximal ideal m and projection map π : B B/m. Then the objects of Def k,γn (B) consist of pairs (G, i) where G is a formal group law over B and i is the projection map i : k B/m B such that i Γ = π G. Such data is called a deformation. Morphisms (G 1, i 1 ) (G 2, i 2 ) are defined only when i 1 = i 2, and are given by isomorphisms of formal group laws f : i 1 i 2 such that π f = id. These are called -isomorphisms. To a morphism φ : B C of complete local rings we define Def k,γn (φ)

19 1.2. Lubin-Tate theory and Morava E-theory 19 by sending (G, i) to (φ G, φ B/m i). The groupoid Def k,γn (B) splits into a disjoint union of Def k,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] (see also the notes of Rezk [Rez98]). Following Rezk we use the notation π 0 of a groupoid to be the π 0 of the corresponding classifying space. In particular this is just the set of isomorphism classes of the groupoid. Theorem The functor π 0 Def k,γn ( ) from groupoids to sets sending B to its set of -isomorphism classes is corepresentable. In particular, there exists a ring E(k, Γ n ) W(k)[[u 1,..., u n 1 ] such that Hom c W(k) alg (E(k, Γ n), B) π 0 (Def k,γn (B)) with i : k E(k, Γ n ) and a formal group law F over E(k, Γ n ) such that the pair (F, id) is a universal deformation. In other words, for each -isomorphism class (G, i) of π 0 (Def k,γn (B)) there is a unique ring homomoprhism ψ : E(k, Γ n ) B and a unique -isomorphism f : ψ F G. We will be interested in the case where k = F p n and Γ n is the Honda formal group law with p-series [p](x) = x pn. Then the ring E(F p n, Γ n ) W(F p n)[[u 1,..., u n 1 ] carries a formal group law Γ, coming from the universal deformation of Γ n. Let u have degree 2. It is not hard to check that u Γ(u 1 x, u 1 y) still forms a formal group law, now over the ring (E n ) := W(F p n)[[u 1,..., u n 1 ][u ±1 ]. This is a p-typical formal group law, and there is a map BP (E n ), defined by u i u 1 pi 1 i n v i u 1 pn i = n 0 i > n. In other words we choose the universal deformation Γ to have p-series (1.3) (x) = px u [p] Γn + Γn 1 x p u + Γn + Γn n 1 x pn 1 x + Γn pn. The map BP (E n ) satisfies the criteria of the Landweber exact functor theorem, and so there is an associated homology theory E n, with coefficient ring (E n ). This explicit construction of Morava E-theory highlights some properties of it. Let S n = Aut(Γ n ) be the group of automorphisms of the formal group law Γ n. Such an automorphism f is defined by a power series f F p n[[x]. Given a deformation (G, i)

20 20 Chapter 1: Prerequisites lift i f B/m[[x] along the quotient map to B[[x] and apply this to G, obtaining a new deformation. It can be shown that this is independent of the choice of lift, up to -isomorphism, and so this gives an action of S n on Def Fp n,γ n ( ), which in turn induces an action on the corepresenting object E(k, Γ). Since the Honda formal group law is defined over F p n, a similar argument gives an action of Gal(F p n/f p ) by pre-composition with the map i : F p n B/m. Together these give an action of G n := Gal(F p n/f p ) S n on E(F p n, Γ n ), or even (E n ). Brown representability implies that there is an action of G n on E n in the stable homotopy category. In fact much more is true. Let FG be the category of pairs (k, Γ) where k is a perfect field of finite characteristic p and Γ is a formal group law of height n over k. Then Hopkins and Miller [Rez98] (in the A case, followed by Goerss and Hopkins [GH04] for the E -case) showed that there is a functor ψ : FG op E ring spectra (k, Γ) E(k, Γ) such that E(k, Γ) is a commutative ring spectrum whose associated formal group law is the universal deformation of Γ. The Goerss-Hopkins-Miller theorem also implies that the action of G n on E n can be taken to be one of E -maps The action of the Morava stabilizer group We can give a more explicit description of the group G n. Define the non-commutative polynomial ring O n = W(F p n) S /(S n = p, Sw = w σ S) where w W(F p n) := W and σ is a lift of the Frobenius to W. Explicitly by [Rav86, A2.2.15] we can uniquely write where w pn i w = w i p i W i=0 w i = 0, and then the Frobenius is given by w σ = w p i pi. i=0 O n is the endomorphism ring of Γ n and S n = O n [Die57; Lub64]. Although we will not give prove this statement we can at least describe how the endomorphisms arise.

21 1.3. The action of the Morava stabilizer group 21 Recall that the Honda formal group has p-series [p](x) = x pn. In order to simply our notation we write (1.4) f(t) = Γ n a k t pk, a k Z p k=0 for the sum taken with respect to the formal group law Γ n. An endomorphism of Γ n is a power 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 a pn k = a k, and so a k F p n [Rav86, Lemma A2.2.19]. A simple endomorphism is given by S(t) = t p : [p](s(t)) = (t p ) pn = t pn+1 = S([p](t)). Since [p](x) = x pn we get the relation S n = p. Then for any a F p n, 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] form End Fp n (Γ n ) = { θ(t) = } Γ n a k t pk : a k F p n. k=0 The Witt vectors appear in the description of O n as the subring of elements of the Γ n a k t pk, n k whilst w corresponds to the endomorphism t wt; this gives the relation Sw = w σ S. The algebra O n is a free rank n module over W with generators 1, S,, S n 1 and is the ring of integers in a division algebra D over Q p of dimension n 2 and Hasse invariant 1/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) min{ν(a), ν(b)}. The valuation is normalised such that ν(p) = 1. This ensures that the usual valuation on Q p extends uniquely to the valuation on D. The subring O n is given by O n = {x D n ν(x) 0} which implies that O n = {x D n ν(x) = 0}.

22 22 Chapter 1: Prerequisites We write an arbitrary element a O n a = a i S i 0 i n 1 where a i W. The Morava stabilizer group S n is then the set of elements with a 0 a unit in W. The Galois group is generated by the Frobenius σ and acts on the coefficients in the obvious way, namely σ(a 0 + a 1 S +... a n 1 S n 1 ) = σ(a 0 ) + σ(a 1 )S + + σ(a n 1 )S n 1. The group S n acts on O n by right multiplication through left W(F p n)-module homomorphisms and so there is a homomorphism S n GL n (W(F p n)) det W(F p n). The image of this map is exactly Z p, and since the whole map is Gal(F p n/f p )-invariant, there is an induced homomorphism G n Z p. We identify Z p with µ p 1 (1 + pz p ) µ p 1 Z p, and thus by composition with the quotient map to Z p we get the reduced determinant map G n Z p. As is customary we denote by G 1 n the kernel of this map, i.e. there is a short exact sequence 1 G 1 n G n det Z p 1. We also define SG n as the kernel of the determinant to Z p : 1 SG n G n Z p 1. The two are related by the short exact sequence 1 SG n G 1 n µ p 1 1. We will need to know the action on the central subgroup of G n, which can be seen to be Z p (for example the center of G n is the same as the center of S n, and one can consider the equation gω = ωg for ω a (p n 1)-st root of unity). The following lemma is well-known; we learnt it from [Hen07, Lemma 22]. Lemma Let g Z p G n. Then g u i = u i and g u = gu. Proof. We first describe how to calculate the action (see [DH95, Section 1]). Let g S n.

23 1.4. Homotopy fixed point spectra for subgroups of the Morava stabilzier group 23 Lift g to a power series g(x) (E n ) 0 [[x] and define a new formal group law by H(x, y) = g ( Γ ) 1 n ( g(x), g(y)) By Lubin-Tate theory there is a homomorphism g : (E n ) (E n ), which defines a - isomorphism h from (g ( Γ n ), g (u)) to (H, g (0)u). The action of g on u is then given by the formula g (u) = g (0)h (0)u. Now assume that g Z p G n is contained in the central subgroup. The morphism Z End Fp n (Γ n ) given by sending m to the homomorphism m : Γ n Γ n extends to a map Z p End Fp n (Γ n ). This lifts to the action of the universal deformation, which gives h(x) = x and g(x) = [g] Γn (x). Equation (1.3) then implies that g (u) = gu and g (u i ) = u i Homotopy fixed point spectra for subgroups of the Morava stabilzier group For the action of a discrete group G on a spectrum X we can form the homotopy fixed point spectrum X hg as the G fixed points of the function spectrum F (EG +, X), and there is an associated homotopy fixed point spectral sequence H (G; X Z) π F (Z, X hg ) arising from the skeletal filtration of EG. Note that the action is required to be on the point-set level, and not just up to homotopy. In the case we are interested in, namely X = E n and G = G n, both G n and E n are profinite, and the action is continuous, and so this construction does not apply. The work of Goerss-Hopkins-Miller described earlier implies that the action of G n can be rigidified to act on the spectrum level, and so we can define X hg for G G n finite in the usual way. Furthermore Devinatz and Hopkins [DH04] have given a construction of homotopy fixed point spectra En dhf for arbitrary closed subgroups F G n. This starts with the construction of En dhu for U G n an open subgroup of G n. Since G n is a profinite compact p-adic analytic Lie group there is a sequence of normal subgroups G n = U 0 U 1 U 2

24 24 Chapter 1: Prerequisites such that k U i = {e}. Then for closed subgroups F G n Devinatz and Hopkins define E dhf n := L K(n) colim i E hdu if n. These come with homotopy fixed points spectral sequences E s,t 2 = H s c (F ; π t (E n )) π t s (E hf n ), and in the case F = G n, an equivalence En dhgn L K(n) S 0. In this later case the spectral sequence sequence for En hdgn coincides with the Adams-Novikov spectral sequence for L K(n) S 0. Cohomology here is continuous group cohomology, to be described in more detail below. In the case that F is finite the spectrum En dhf constructed by Devinatz and Hopkins coincides with the usual homotopy fixed point spectrum En hf. In describing this construction we have used the notation En dhf because the spectrum is not constructed with respect to a continuous F -action. Davis [Dav06] has constructed such a continuous action, and has shown that En hf notation from now on. En dhf, and so we will omit this Remark In recent work Quick [Qui13a; Qui13b] has constructed a stable model category of profinite G-spectra, and defines X hg := F G (EG +, X), where the maps are now taken inside this category of profinite G-spectra. Quick shows that E n has a canonical model in the category of profinite G n -spectra and so can define En hg for G G n without using the Devinatz-Hopkins construction of En hu. Moreover these are shown to be equivalent to the construction of Devinatz-Hopkins and Davis. However this construction does not (directly) prove that E hg n is an E -ring spectrum. We now describe in careful detail the definition of continuous group cohomology that we use, starting with some definitions from [SW00]. Let G be a profinite group. Then M is a discrete G-module if M is a discrete abelian group and there is a continuous action ψ : G M M. by D p (G). The category of p-torsion discrete left G-modules will be denoted Let F p (G) denote the full subcategory of D p (G) whose objects are finite, discrete left Z p [[G]-modules. Finally C p (G) will denote the category of topological Z p [[G]- modules whose objects consist of inverse limits of objects in F p (G). Group cohomology with coefficients in D p (G) can be defined in the usual way [SW00] as the derived functor Ext Z p[[g] (Z p, M). A module M C p (G) will be called type F P if it has a projective resolution P in C p (G) where each P i can be chosen to be finitely-generated and free. A profinite group G is of type type p-f P if the trivial Z p [[G]-module Z p is of type F P. Compact p-

25 1.4. Homotopy fixed point spectra for subgroups of the Morava stabilzier group 25 adic analytic groups are always of type p F P [SW00, Proposition 5.1.2]. We have the following finiteness result. Lemma [SW00, Proposition 4.2.2] Let G be a profinite group of type p F P. Then for all F F p (G) H n (G; F ) < +. We now work with the case G = G n. Let I = (p i 0, i u i 1 1,..., i u i n 1 n 1 ) (E n) be an ideal. Such a system of ideals forms a cofiltered system such that (E n ) = lim I π t E n /Iπ t E n. Since each π t E n /Iπ t E n is a finite discrete Z p [[G n ]-module this displays (E n ) as an object of C p (G n ). In the sequence of ideals I there is a chain of ideals I 0 I 1 I k which can be realized as a sequence of generalised Moore spectra M Ik [HS99b, Section 4] such that π t (E n M Ik X) π t (E n X)/I k π t (E n X) whenever X is finite. For finite X we can then define (1.5) H s c (G n, π t (E n X)) = lim k H s c (G n, π t (E n X M Ik )). Remark From now we, we will usually omit the subscript c from our notation for continuous group cohomology Finite subgroups at height n = p 1 In this section we now assume that n = p 1; this is an important case due to the presence of p-torsion in G n, which implies that the cohomological dimension of G n is infinite. We will principally be interested in En hf when F G n is a maximal finite subgroup (up to conjugacy) of G n, whose order is divisible by p. These sometimes go by the name of the higher real K-theories, and are sometimes 1 denoted EO n, for reasons that the following simple example explains. 1 although we do not use this notation

26 26 Chapter 1: Prerequisites Example We consider the simplest case of n = 1 and p = 2. Note that for n = 1 the Honda formal group law over F 2 is isomorphic to the multiplicative formal group law: G m (x, y) = x + y xy. There is an isomorphism End F2 (G m ) Z 2, and so Aut F2 (G m ) Z 2 Z 2 C 2, where C 2 is generated by -1. Now E(F 2, G m ) = Z 2, and in fact the universal deformation is G m, now considered over Z 2. Finally then, by adjoining a degree 2 element u, we see that E 1 is nothing other than p-completed complex K-theory, and the action of G 1 is just the (p-adic) Adams operations. There is a unique maximal finite subgroup of Z 2, namely C 2 {±1}, and so we can form the homotopy fixed points spectrum E hc 2 1. By calculating the homotopy groups of this spectrum we can see that this is actually 2-complete real K-theory; E hc 2 1 KO2 (for example [Rog08, p ]). We now move to the general case. The maximal finite subgroups of S n have been classified by Hewett, in his work on subgroups of finite division algebras [Hew99], whilst the case of G n has been studied by Bujard [Buj12]. We will be interested in finite subgroups of order divisible by p. By [Buj12, Theorem 1.3.1] for n = p 1 there are exactly 2 conjugacy classes of maximal finite subgroups of S n represented by F 0 = C p n 1 and F = F 1 = C p C n 2 where the action of the right factor on the left is given by the mod n reduction map C n 2 Aut(C p ) C n We choose a presentation of the group F as the following F = ζ, τ ζ p = 1, τ n2 = 1, τ 1 ζτ = ζ e, where e (Z/p) is a generator. It is worthwhile to see how these finite subgroups arises. Again, we refer the reader to the standard references [Buj12; Hew99] for more detailed information, as well as [Hen07, Section 3.6].

27 1.4. Homotopy fixed point spectra for subgroups of the Morava stabilzier group 27 Let ω W S n be a (p n 1)-st root of unity. Then X := ω (p 1)/2 S D n is an element such that X n = p. It can be shown that the subfield Q p (X) D n is isomorphic to the cyclotomic extension Q p (ζ p ) generated by a primitive p-th root of unity ζ p [Hen07, Lemma 19]. A simple check shows that XωX 1 = ω p. Then, if we let η = ω (pn 1)/n 2 W(F p n) we have ηxη 1 = η n X; since η n is an n-th root of unity, conjugation by η induces an automorphism of Q p (X). We will write τ for the image of η in S n. The subgroup generated by ζ p and τ is precisely the maximal finite p-subgroup of F of S n. That this is unique up to conjugacy follows from an argument using the Skolem-Noether theorem, as in [Buj12, Example 1.33]. This relies on the fact that XηX 1 = η p, which implies that the subgroup F is normalised by X (note that X commutes with ζ.) Example Let n = 2, p = 3 and ω be a primitive 8-th root of unity. Then the element ζ = 1 2 (1 + ωs) is an element of order 3 and ω2 ζω 2 = ζ 2, so that F is generated by ζ and τ := ω 2. Here F usually goes by the name G 12 and is abstractly isomorphic to the non-trivial semi-direct product C 3 C 4. We wish to extend this to G n ; that is, to find a group G such that there is an extension of the form 1 F G Gal(F p n/f p ) 1 where F is a maximal finite subgroup in S n. This is in fact thoroughly analysed in [Buj12, Chapter 4], although we can analyse our case more simply [Sym04]. Let N Sn ( ζ ) be the normalizer of the subgroup of order p inside S n. Every conjugate of F inside G n is in fact in S n [Sym04]; from the discussion above it is therefore conjugate to F inside S n. This implies that N Gn (F )/N Sn (F ) Gal. We choose c to have image σ (the Frobenius), under this isomorphism. Then F and c generate a subgroup G of order pn 3. In fact, as in [Sym04] we can choose c such that it acts trivially on ζ for c to have no pro-p part. Example Again let n = 2, p = 3 Then let ψ = ωφ, where φ is the generator of the Galois group. The group generated by ψ, ζ and τ goes by the name G 24. The group generated by τ and ψ is the quaternion group Q 8 and there is an abstract isomorphism G 24 C 3 Q 8.

28

29 CHAPTER 2 Morava modules and the K(n)-local category The goal of this chapter is to investigate the category EG n of Morava modules, a good invariant of the K(n)-local category. As mentioned in the introduction this category plays an important role in the construction of the Goerss-Henn-Mahowald-Rezk resolution of the K(2)-local sphere [GHMR05]. In particular they construct, and make heavy use of, the commutative diagram (2.1) π E n [[G n /H 1 ] hh 2 ((E n ) [[G n /H 1 ]) H 2 π F (E hh 1 n, E hh 2 n ) Hom EGn (E E hh 1 n, E E hh 2 n ). Here the bottom horizontal map is the E n -Hurewicz map, and the top is the edge homomorphism in the homotopy fixed point spectral sequence. Note that if E is a spectrum and X = lim i X i is an inverse limit of a sequence of finite sets, then we define E[[X ] := holim i E (X i ) +, and (E n ) [[G n ] = lim i (E n ) [G n /U i ], 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 E n -Adams spectral sequence, and the two spectral sequences are in fact 29

30 30 Chapter 2: Morava modules and the K(n)-local category isomorphic. In fact, whilst the Goerss-Henn-Mahowald-Rezk construction requires that H 2 is finite, our result holds more generally when H 2 is a closed subgroup of G n (we note that Behrens and Davis [BD10] have constructed the rightmost vertical arrow in Equation (2.1) for an arbitrary closed subgroup of G n ) The K(n)-local category In this thesis we will primarily work in the category M S,K(n) of K(n)-local spectra, which we now briefly describe. For more details see [HS99b]. M S,K(n) is defined to be the full subcategory of the catgeory of spectra whose objects are K(n)-local. M S,K(n) is a closed symmetric monoidal category; since the smash product of two K(n)-local spectra need not be K(n)-local, the monoidal product is given by the K(n)-local smash product L K(n) (X Y ). To keep our notation compact we will usually write X Y for the K(n)-local smash product. The closed structure is given by the function space F (X, Y ); it is easy to check (using the universal property of Bousfield localisation) that this is K(n)-local whenever Y is. In a suitable category of spectra these form a adjoint pair; L K(n) (X ) is left adjoint to F (X, ). Definition Let X be a spectrum. Then we define E X := π L K(n) (E X). This has been seen to be a more natural covariant analogue of E X than E X, despite the fact that it is not a homology theory (as it does not preserve coproducts and filtered homotopy colimits). The functor E ( ) lands not just in the category of E -modules, but rather in the subcategory M of L-complete E -modules, which are discussed in detail in [HS99b, Appendix A]. We review the basics of this theory now. Definition Let L s M 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 L 0 M Mm 1. In fact the natural map M Mm factors as the composite M ηm L 0 M ɛm Mm. Definition M is L-complete if η M is an isomorphism. 1 It 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.

31 2.1. The K(n)-local category 31 The map ɛ M is surjective with kernel [HS99b, Theorem A.2] (2.2) lim 1 k Tor E 1 (E /m k, M). The category of L-complete E -modules is an abelian subcategory of E -modules, and is closed under extensions and inverse limits. One salient feature of this category is that Ext s M(M, N) Ext s E (M, N) whenever M and N are L-complete E -modules [Hov04, Theorem 1.11]. Remark In the language of Salch [Sal10], M is the best reflexive abelian approximation 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 then L K(n) X = holim I X M I where M I is a generalised Moore spectrum (as in Section 1.4.) This description, along with the fact that L n is smashing, then gives a Milnor exact sequence (2.3) 0 lim 1 E +1 (X M I ) E X lim E (X M I ) 0, which can be used to show that E X is L-complete. Definition An L-complete E -module is pro-free if it is isomorphic to the completion of a free E -module. Equivalently, these are the projective objects in the category M. These will play an important role for us; in fact all spectra we will consider will have this property by the following lemma. Lemma [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 follows since pro-freeness implies that E (X M I ) E X/(p i 0,..., v i n 1 n )) 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 If E X is pro-free then Hom E (E X, E Y ) π F (X, L K(n) (E Y )).

32 32 Chapter 2: Morava modules and the K(n)-local category Proof. Let M, N M E,K(n). conditionally convergent, spectral sequence of E -modules Hovey [Hov04] has constructed a natural, strongly and E s,t 2 = Ext s,t M (π M, π N) Ext s,t E (π M, π N) π s+t F E (M, N) Set M = L K(n) (E X) and N = L K(n) (E X). Note then that F E (L K(n) (E X), L K(n) (E Y ))) F E (E X, L K(n) (E Y )) F (X, L K(n) (E Y )), where the second isomorphism is [EKMM97, Corollary III.6.7], giving a spectral sequence E s,t 2 = Ext s,t M (E X, E Y ) Ext s,t E (E X, E Y ) π s+t F (X, L K(n) (E Y )) Since E X is pro-free it is projective in M and so the spectral sequence collapses, giving the desired isomorphism. One of the peculiarities of the m-adic topology is the fact that any homomorphism between complete 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 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(m n M) is a subset of m n N; in turn m n M is a subset of f 1 (m n N). Therefore f 1 (m n N) is a union of m n M-cosets. It follows from the fact that m n M is open that f 1 (m n N) is open in the m-adic topology. M is a symmetric monoidal category with monoidal product L 0 (M E N) for two L-complete E -modules. However if M and N are pro-free then L-completion of the tensor product has a slightly simpler description. We define the complete tensor product, M E N, of two complete E -modules M and N, to be the m-adic completion of M E N. We thank Mark Hovey for the proof of second part of the following lemma. Lemma Let M and N be in M S,K(n). If π M and π N are pro-free E -modules then so is π L K(n) (M E N) and there is an isomorphism (2.4) π L K(n) (M E N) L 0 (π M E π N) π M E π N

33 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 isomorphism π M E π N (π M E π N) m. Thus it will suffice to show that L 0 (π M E π N) (π M E π N) m By Equation (2.2) there is a short exact sequence 0 lim 1 k Tor E 1 (E /m k, π M E π N) L 0 (π M E π N) (π M E π N) m 0 and so it will further suffice to show that Tor E 1 (E /m k, π M E π N) = 0 for all k. Inductively using the short exact sequence 0 m k 1 /m k E /m k E /m k 1 0 we can reduce this further to the statement that Tor E 1 (E /m, π M E π N) = 0. Now by [HS99b, Theorem A.9] if π M is pro-free then Tor E s (π M, E /m) = 0 for s > 0. It follows that Tor E s (π M, K) = 0 whenever mk = 0, since such a module is a direct sum of copies of E /m. Let P 1 P 0 = E E /m 0 be a projective resolution of E /m and split this into a short exact sequence of E -modules 0 K 1 E E /m 0 and a surjection P 1 K 1 0. Since π M is pro-free Tor E s (π M, E /m) = 0 (again, for s > 0) and so there is a short exact sequence 0 π M E K 1 π M π M/m 0 and a surjection π M E P 1 π M E K 1 0. Similarly since π N is pro-free Tor E 1 (π N, π M/m) = 0 and we get a short exact sequence 0 (π N E π M) E K 1 (π N E π M) (π N E π M)/m 0

34 34 Chapter 2: Morava modules and the K(n)-local category and a surjection (π N E π M) E P 1 (π N E π M) E K 1 0. This is precisely the statement that Tor E 1 (E /m, π M E π N) = 0. Taking M = L K(n) (E X) and N = L K(n (E Y ) gives the following corollary, which should be compared with Equation (1.1). Corollary If E X and E Y are pro-free E -modules then there are isomorphisms E (X Y ) L 0 (E X E E Y ) E X E E Y Morava modules and complete E E-comodules From Chapter 1 we know that G n acts on E, and so in turn there is an action of G n on E X. This motivates the following definition, which we take from [GHMR05]. Definition A Morava module M is a complete E -module with a continuous action of G n such that g(ax) = g(a)g(x) for g G n, a (E n ), x M We will write EG n 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 -module M we give Hom c (G n, M) the diagonal G n -action, and the obvious E -module structure. Since M is complete, this gives Hom c (G n, M) the structure of a Morava module. Here we define Hom c (G n, M) lim k lim map(g i n /U i, M/m k M), where U i is, as previously, a system of open subgroups of G n with U i = {e}. Remark Suppose M is a complete E -module. Since G n is profinite and M is complete, we have an isomorphism (see, for example [Tor07, Lemma 4.9]) Hom c (G n, M) Hom c (G n, E ) E M.

35 2.2. Morava modules and complete E E-comodules 35 The functor M Hom c (G n, M) from complete E -modules to Morava modules is right adjoint to the forgetful functor [GHMR05, Remark 2.3]. We wish to relate the category of Morava modules to the category of comodules over a certain Hopf algebroid. The difficultly here is the we do not consider E E but rather E E, and this is not a Hopf algebroid as usually considered. m 2. In the following let R be a complete regular Noetherian local ring with maximal ideal We simply take what we need from Devinatz [Dev95], although the exposition is simplified based on our assumptions on R. Definition Let Mod c R be the category of complete R-modules M. Here complete means with respect to the m-adic filtration; that is, there is an isomorphism M lim k M/m k M. A morphism of complete R-modules is an R-module homomoprhism f : M N that is continuous with respect to the m-adic topology. Mod c R is an additive category with kernels and cokernels, however it is not an abelian category. The completed tensor product is defined, as above, to be the m-adic completion of the ordinary tensor product. The tensor product endows Mod c R with a symmetric monoidal structure (with unit R). A complete R-algebra A is then a commutative monoid object in this category. We denote the category of such objects by Alg c R. Finally we can give the definition we need: Definition A complete Hopf algebroid is a cogroupoid object in Alg c R. In particular it is a pair (A, Γ), where A, Γ Alg c R along with a collection of maps η R : A Γ (source) η L : A Γ (target) : Γ Γ ˆ A Γ (composition) ɛ : 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 Comod c Γ in the usual way. The following result of Devinatz will give us the example of complete Hopf algebroids 2 In this case regular means that if m = (r 1,..., r n), where n is as small as possible, then the Krull dimension of R is n.

36 36 Chapter 2: Morava modules and the K(n)-local category that we require. Theorem [Dev05] Let A be a complete Noetherian regular local ring, and let S be a profinite group acting on A via ring homomorphisms. Let Γ = Hom c (S, A). Then (A, Γ) has a canonical structure of a complete Hopf algebroid. Here Hom c (S, A) has m-adic filtration induced by that on A; i.e. F i Hom c (S, A) = Hom c (S, m i A). Devinatz shows that the maps are defined by η R (a)(s) = a, η L (a)(s) = s 1 a, ɛ(f) = f(e), c(f)(s) = s 1 (f(s 1 )). To define, there is an isomorphism σ : Hom c (S, A) A Hom c (S, A) Hom c (S S, A), defined by σ(f 1 f 2 )(s 1, s 2 ) = s 1 2 (f 1(s 1 )) f 2 (s 2 ). Then = σ 1 ˆ, where ˆ : Hom c (S, A) Hom c (S S, A) is induced by the multiplication S S S. This immediately implies that (E 0, (E 0 ) (E 0 )) is a complete Hopf algebroid. There is also an appropriate graded version of all these definitions and (E, E E) is a graded complete Hopf algebroid. See [Dev95, Remark 1.13] for details. Here the fact that everything is 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) the right adjoint takes the form M Hom c (G n, E ) E M for a complete E -module M. By Remark we have Hom c (G n, E ) E M Hom c (G n, M). Thus the right adjoint functors Hom c (G n, ) : complete E -modules Morava modules and Hom c (G n, E ) E ( ) : complete E -modules E E-comodules

37 2.3. Homological algebra for Morava modules 37 are, in some sense, isomorphic, although a priori they lie in different categories. We now make this precise by showing that the category of Morava modules is equivalent to the category of E E-comodules. Proposition There is an equivalence of symmetric monoidal categories between the 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 we need it is given in [Tor07] (where Morava modules are called complete twisted E G n - modules). E -module) the map Given a complete E E-comodule M (which is, of course, also a complete M ψ M E E E M Hom c (G n, E ) E M ev(g) id E E M M, where ev(g) is the evaluation map of g G n, defines a compatible G n -action on M, which Torii shows is continuous. Conversely given a Morava module M the adjoint of the G n -action map gives a map M Hom c (G n, 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 the symmetric monoidal structure Homological algebra for Morava modules We wish to consider homological algebra for Morava modules. Given that the category of Morava modules is equivalent to the category of complete E E-comodules, we will instead consider homological algebra for complete E E-comodules, as this simplifies the exposition somewhat. This has previously been considered in [Dev95] Relative homological algebra The category of complete E E-comodules is not abelian; it is an additive category with cokernels. Hence we need to use the methods of relative homological algebra, which we briefly review now. For a more thorough exposition see [EM65] (although in general one needs 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 An injective class I in a category C is a pair (D, S) where D is a class of objects and S is a class of morphisms such that: