THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS MARK BEHRENS1 AND DANIEL G. DAVIS2 Abstract.Let E be a k-local profinite G-Galois extension of an E1 -ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithf* *ul k-local profinite extension which satisfies certain extra conditions, th* *en the forward direction of Rognes's Galois correspondence extends to the profi* *nite setting. We show the function spectrum FA((EhH )k, (EhK )k) is equivalent to the homotopy fixed point spectrum ((E[[G=H]])hK)k where H and K are closed subgroups of G. Applications to Morava E-theory are given, includ* *ing showing that the homotopy fixed points defined by Devinatz and Hopkins f* *or closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived * *functor of fixed points. Contents 1. Introduction 1 2. Discrete symmetric G-spectra 5 3. Homotopy fixed points of discrete G-spectra 12 4. Continuous G-spectra 18 5. Modules and commutative algebras of discrete G-spectra 21 6. Profinite Galois extensions 29 7. Closed homotopy fixed points of profinite Galois extensions 36 8. Applications to Morava E-theory 45 References 52 1.Introduction In [36], John Rognes develops a Galois theory of commutative S-algebras which mimics Galois theory for commutative rings. Let k be an S-module, and let (-)k denote Bousfield localization with respect to k. Given a k-local cofibrant comm* *uta- tive S-algebra A, and a cofibrant commutative A-algebra E that is k-local, Rogn* *es gives the following definition of a finite k-local Galois extension. Definition 1.0.1 (Finite Galois extension). The spectrum E is a k-local G-Galois extension of A, for a finite discrete group G, if it satisfies the following co* *nditions: ____________ 1The first author was supported by NSF grant DMS-0605100, the Sloan Foundati* *on, and DARPA. 2Part of the second author's work on this paper was supported by an NSF VIGR* *E grant at Purdue University, a visit to the Mittag-Leffler Institute, and a grant from th* *e Louisiana Board of Regents Support Fund. 1 2 MARK BEHRENS AND DANIEL G. DAVIS (1) G acts on E through commutative A-algebra maps. (2) The canonical map A ! EhG is an equivalence. (3) The canonical map (E ^A E)k ! Map (G, E) is an equivalence. E is said to be k-locally faithful over A if (M ^A E)k ' * implies that Mk ' * for every A-module M. In the context of k-local Galois extensions, we shall sim* *ply refer to such extensions as faithful. Remark 1.0.2. Rognes ([36, Prop. 6.3.3]; see also [1]) shows that a k-local G- Galois extension is faithful if and only if the additive form of Hilbert's Theo* *rem 90 holds: (EtG)k ' *. We will mostly consider faithful Galois extensions, because these are the Galois extensions for which the fundamental theorem of Galois theory holds. It is not known whether there exist non-faithful Galois extensions. Let G be a profinite group. Following (and slightly modifying) Rognes's defin* *i- tion [36, Def. 8.1.1] of a k-local pro-G-Galois extension, we define a (profait* *hful) k-local profinite G-Galois extension E of A to be a colimit (in the category of* * com- mutative A-algebras) of (faithful) k-local G=Uff-Galois extensions Effof A, for* * a cofinal system of open normal subgroups Uffof G (see Definition 6.2.1). Since a colimit of k-local spectra need not be k-local, the spectrum E is not necessari* *ly k-local. In [5], the second author developed a category of discrete G-spectra and defi* *ned their homotopy fixed points (see also [42], [25], [33], [12], [27]). In this pa* *per, we examine k-local profinite G-Galois extensions E of A as objects in the category* * of discrete G-spectra, and we study the spectra of A-module maps between the vario* *us homotopy fixed point spectra of E. Unfortunately, to say meaningful things it s* *eems that we must impose more hypotheses on our profinite Galois extensions. Assumption 1.0.3. In this paper, we shall only concern ourselves with localiza- tions (-)k which are given as a composite of two localization functors ((-)T)M , where (-)T is a smashing localization and (-)M is a localization with respect * *to a finite spectrum M. The spectra S, HFp, E(n), and K(n) are all examples of such localizations k (see [3], [20]). For a cofibrant commutative S-algebra B and a cofibrant commutative B-algebra C, the k-local Amitsur derived completion B^k,Cis the homotopy limit of the cos* *im- plicial spectrum Ck ) (C ^B C)k V (C ^B C ^B C)k . . . (see, for example, [36, Def. 8.2.1]). Definition 1.0.4. Let E be a k-local profinite G-Galois extension of A. (1) The extension E is consistent if the coaugmentation of the k-local Amits* *ur derived completion A ! A^k,E is an equivalence. (2) The extension E is of finite virtual cohomological dimension (finite vcd) if the profinite group G has finite vcd (that is, G has an open subgroup U of finite cohomological dimension, so that there exists a d such that Hsc(U; M) = 0 for each s > d and each discrete U-module M). THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 3 Assumption 1.0.3 ensures that if E has finite vcd, then the condition of E be* *ing consistent is equivalent to requiring that the map A ! (EhG )k is an equivalence. This is proven as Corollary 6.3.2. It then follows that the maps Eff! (EhUff)k are equivalences (Lemma 6.3.6). The consistency hypothesis may be unnecessary, since we do not know of any profinite Galois extensions which are not consisten* *t. The main concern of this paper is the study of the intermediate homotopy fixed point spectra EhH with respect to closed subgroups H of G. We prove the "forwar* *d" direction of the Galois correspondence. Theorem (7.2.1). Suppose that E is a consistent profaithful k-local profinite G- Galois extension of A of finite vcd, and that H is a closed subgroup of G. (1) The spectrum E is k-locally H-equivariantly equivalent to a consistent p* *ro- faithful k-local H-Galois extension of (EhH )k of finite vcd. (2) If H is a normal subgroup of G, then the spectrum EhH is k-locally equiv- alent to a profaithful k-local G=H-Galois extension of A. If the quotient G=H has finite vcd, then this extension is consistent (and of finite vcd* *) over A. We also identify the function spectrum of A-module maps between any two such homotopy fixed point spectra. Theorem (7.3.1). Let E be a consistent profaithful k-local profinite G-Galois e* *x- tension of finite vcd, and let H and K be closed subgroups of G. Then there is * *an equivalence (1.1) FA ((EhH )k, (EhK )k) ' ((E[[G=H]])hK )k. The spectrum E[[G=H]] that appears on the right-hand side of (1.1) is the con- tinuous G-spectrum with the diagonal action. The case where K = H = {e} was handled by Rognes [36, (8.1.3)]. In the context of Morava E-theory, (1.1) was proven in [13] under the additio* *nal assumption that K is finite, and it was suggested by the authors of [13] that (* *1.1) should be true with this extra assumption removed. Another source of motivation for this work arises from the fact that a special case of (1.1) (Corollary 7.3.* *2) was needed in an essential way by the first author in [2] (see [2, Thm. 2.3.2, Cor.* * 2.3.3]). One important example of a profinite Galois extension is given by Morava E- theory. Let k = K(n) be the nth Morava K-theory spectrum and let A = SK(n)be the K(n)-local sphere spectrum. Let G = Gn be the nth extended Morava stabilizer group Sn o Gal(Fpn=Fp). Let En be the nth Morava E-theory spectrum, where (En)* = W (Fpn)[[u1, ..., un-1]][u 1]. Goerss and Hopkins [14], building on wor* *k of Hopkins and Miller [35], have shown that Gn acts on En by maps of commutative S-algebras. Devinatz and Hopkins [8] have given constructions of homotopy fixed point spectra EdhHnfor closed subgroups H of Gn. In particular, they show that there is an equivalence EdhGnn' SK(n). Thus, the homotopy fixed point spectra of En are intimately related to the nth chromatic layer of the sphere spectrum. 4 MARK BEHRENS AND DANIEL G. DAVIS Rognes [36, Thm. 5.4.4, Prop. 5.4.9] proved for U an open normal subgroup of Gn, that the work of Devinatz and Hopkins [7, 8] shows that EdhUnis a faithful K(n)-local Gn=U-Galois extension of SK(n). Therefore, the discrete Gn-spectrum Fn = colimUEEdhUn oGn is a profaithful K(n)-local profinite Gn-Galois extension of SK(n). Additionall* *y, the profinite extension Fn of SK(n) is consistent and has finite vcd (Proposition 8* *.1.2). The spectrum En is recovered by the equivalence [8] En ' (Fn)K(n). As mentioned above, for any closed subgroup H of Gn, Devinatz and Hopkins [8] constructed the commutative S-algebra EdhHn. Further, they showed that EdhHn behaves like a homotopy fixed point spectrum with respect to a continuous action of H. In more detail, [8] showed that EdhHnhas the following properties: (a) th* *ere is a K(n)-local En-Adams spectral sequence Hsc(H; sst(En)) ) sst-s(EdhHn), where the E2-term is the continuous cohomology of H, with coefficients in the profinite H-module sst(En), and this spectral sequence has the form of a descent spectral sequence; (b) when H is finite, there is a weak equivalence EdhHn! EhH* *n, and the descent spectral sequence for EhHnis isomorphic to the spectral sequence in (a); and (c) EdhHnis an (N(H)=H)-spectrum, where N(H) is the normalizer of H in Gn. On the other hand, when H is not finite, EdhHnis not known to actually be the H-homotopy fixed point spectrum of En, because (a) it is not constructed with respect to a continuous H-action, and (b) it is not obtained by taking the total right derived functor of fixed points (and homotopy fixed points are, by defini* *tion, the total right derived functor of fixed points, in some sense - see [5, Remark* * 8.4] for the precise definition in the case of a continuous H-spectrum that arises f* *rom a tower of discrete H-spectra). To address this situation, in [5], the second a* *uthor showed that H does act continuously on En and there is an actual H-homotopy fixed point spectrum EhHn, with a descent spectral sequence Hsc(H; sst(En)) ) sst-s(EhHn). From the above discussion, we see from the properties of EdhHnand EhHnthat they should be equivalent to each other, and, by a result in the second author's thesis [4], they are. However, since this part of [4] was never published, we * *use the machinery of this paper to prove the equivalence of these two spectra. In m* *ore detail, we give proofs of the following two results (which originally appeared * *in [4]). Theorem (8.2.1). For every closed subgroup H of Gn, there is an equivalence EdhHn' EhHn between the Devinatz-Hopkins construction and the homotopy fixed points that are defined with respect to the continuous action of H. The above theorem shows that EdhHncan be referred to as a homotopy fixed point spectrum, whereas, previously, EdhHnwas only known to behave like a homotopy fixed point spectrum. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 5 Theorem (8.2.3, 8.2.4, 8.2.5). Let H be a closed subgroup of Gn and let X be a finite spectrum. Then there is an equivalence EdhHn^ X ' (En ^ X)hH and the K(n)-local En-Adams spectral sequence for ss*(EdhHn^ X) is isomorphic to the descent spectral sequence for ss*((En ^ X)hH ) from the E2-terms onward.* * In particular, (X)K(n)' (En ^ X)hGn. The paper is organized as follows. Our notion of homotopy fixed point spectra uses the framework of equivariant spectra (with respect to a profinite group) a* *s de- veloped by the second author [5]. The foundations in [5] use Bousfield-Friedlan* *der spectra. Since we need to work with structured ring spectra to do Galois theory, it is essential for this paper that we reformulate portions of [5] in the conte* *xt of symmetric spectra. A concise summary of these foundations appears in Section 2. In Section 3, we describe properties of the homotopy fixed point functor. In Se* *c- tion 4, we describe continuous G-spectra, generalizing somewhat the setting of * *[5]. In Section 5, we explain how to extend our constructions to categories of modul* *es and commutative algebras of spectra. In Section 6, we explain how profinite Gal* *ois extensions give rise to discrete G-spectra, and we show that the homotopy fixed points with respect to open subgroups of the Galois group give rise to intermed* *iate finite Galois extensions. In Section 7, we prove our results concerning the hom* *o- topy fixed point spectra with respect to closed subgroups of the Galois group. * *In Section 8, we show that the hypotheses on profinite Galois extensions which we require are satisfied by Morava E-theory. We then apply our machinery to show that the Devinatz-Hopkins homotopy fixed points agree with the second author's homotopy fixed points, and deduce some corollaries. Acknowledgments. The first author benefited from the input of Halvard Fausk, Paul Goerss, and Daniel Isaksen. The second author thanks Paul Goerss for helpf* *ul discussions, when he was a Ph.D. student, regarding the results in Sections 8.1* * and 8.2. Also, the second author is grateful to Paul for later helpful conversatio* *ns, to Mark Hovey for providing some intuition related to Theorem 2.3.2, and John Rognes for a helpful discussion regarding group actions. 2. Discrete symmetric G-spectra Let G be a profinite group. We begin this section by describing the basic cat* *e- gories of discrete G-objects that will be used in this paper. We then describe * *and compare the model structures on the categories of discrete G-objects in Bousfie* *ld- Friedlander and symmetric spectra. We end this section with descriptions of some basic constructions in the category of discrete G-spectra. More detailed accoun* *ts of some of these model categories and constructions can be found in [5]. 2.1. Simplicial discrete G-sets. A G-set Z is said to be discrete if, for every element z 2 Z, the stabilizer StabG(z) is open in G. We may express this condit* *ion by saying that Z is the colimit of its fixed points: Z = colimUZU , oG where the colimit is taken over all open subgroups. These conditions are equiva* *lent to the condition that the action map GxZ ! Z is continuous, when Z is given the 6 MARK BEHRENS AND DANIEL G. DAVIS discrete topology. A simplicial discrete G-set is a simplicial object in the ca* *tegory of discrete G-sets. Goerss showed that the category sSetG of simplicial discrete G-sets admits a model category structure [12]. Theorem 2.1.1 (Goerss). The category sSetG admits a model category structure where o the cofibrations are the monomorphisms, o the weak equivalences are those morphisms which are weak equivalences on underlying simplicial sets, o the fibrations are determined. Lemma 2.1.2. The model structure on sSetG is left proper and cellular. Proof.The model structure is left proper because the cofibrations and weak equi* *v- alences are precisely the cofibrations and weak equivalences on the underlying * *sim- plicial sets, and the model category structure on simplicial sets is left prope* *r. The model structure in [12] is cofibrantly generated, with generating cofibrations * *I and generating trivial cofibrations J, where: I = {G=U x @ n ,! G=U x n : U o G, n 0}, ae oe J = A j-!B : j#isBa trivialfcofibration,f.. Here, ff is a fixed infinite cardinal greater than the cardinality of G and #B * *de- notes the cardinality of the set of non-degenerate simplices of B (see the proo* *f of Lemma 1.13 of [12]). The axioms of being cellular are immediately verified from this description of the generating cofibrations. The category (sSetG)* of pointed simplicial discrete G-sets, being an under- category, inherits a model structure from sSetG. The cofibrations, weak equiv- alences, and fibrations are detected on the level of underlying simplicial disc* *rete G-sets. If K and L are pointed simplicial discrete G-sets, then their smash pro* *duct K ^ L is easily seen to be a simplicial discrete G-set. The smash product gives the c* *ategory (sSetG)* a symmetric monoidal structure. (It does not extend to a closed symmet* *ric monoidal structure.) Lemma 2.1.3. The model category structure on (sSetG)* is left proper and cellu- lar. With respect to the symmetric monoidal structure given by the smash produc* *t, the model category (sSetG)* is a symmetric monoidal model category. Proof.Left properness follows from the fact that sSetG is left proper. The model structure on (sSetG)* is cofibrantly generated with generating cofibrations (re* *spec- tively generating trivial cofibrations) I+ (respectively J+ ). Here, I+ and J+* * are the sets of maps obtained from I and J by adding a disjoint basepoint on which G acts trivially. The axioms of being a symmetric monoidal model category are eas* *ily verified. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 7 2.2. Discrete G-spectra. Define the category of discrete G-spectra SpG to be the category of Bousfield-Friedlander spectra of simplicial discrete G-sets. An obj* *ect X 2 SpG consists of a sequence {Xi}i 0, where each Xi is a pointed simplicial discrete G-set, together with G-equivariant maps oei: S1 ^ Xi! Xi+1. Here, S1 is given the trivial G-action. A map f : X ! Y of discrete G-spectra is a sequence of G-equivariant maps of pointed simplicial sets fi : Xi ! Yi which are compatible with the spectrum structure maps. In [5], the second author studied the following model structure. Theorem 2.2.1. The category SpG admits a model structure where o the cofibrations are the cofibrations of underlying Bousfield-Friedlander spectra, o the weak equivalences are the stable weak equivalences of the underlying Bousfield-Friedlander spectra, o the fibrations are determined. The method used in [5] was to transport a Jardine model structure on presheav* *es of spectra on an appropriate site. However, an alternative approach is given be* *low using the machinery of M. Hovey [21]. Proof.Observe that (sSetG)* satisfies the conditions of Definition 3.3 of [21].* * There- fore, the category SpG of S1-spectra of simplicial discrete G-sets admits a sta* *ble model category structure where o the cofibrations are those morphisms A ! B where the induced maps A0 ! B0 Ai[S1^Ai-1S1 ^ Bi-1 ! Bi n 1 are cofibrations. o the fibrant objects X are those spectra for which (1) the spaces Xi are fibrant as simplicial discrete G-sets, (2) the maps Xi! Xi+1are weak equivalences. o the weak equivalences f : X ! Y between fibrant objects are those f for which the maps fi: Xi! Yi are all weak equivalences. Clearly the cofibrations of SpG are the maps which are cofibrations of underlyi* *ng Bousfield-Friedlander spectra. We are left with verifying that the weak equival* *ences in SpG are precisely the stable equivalences of underlying Bousfield-Friedland* *er spectra. The forgetful functor U : sSetG ! sSet from simplicial discrete G-sets to simplicial sets is a left Quillen functor (i* *t preserves cofibrations and trivial cofibrations, and is left adjoint to the functor CoInd* *G1of Section 3.4). By Proposition 5.5 of [21], the induced forgetful functor U : SpG ! Sp is a left Quillen functor. Let (-)fG denote the functorial fibrant replacement* * in SpG, so that there are natural trivial cofibrations ffG,X : X ! XfG. 8 MARK BEHRENS AND DANIEL G. DAVIS Suppose that OE : X ! Y is a morphism in SpG , and consider the following diagram: OE X _______//Y ffG,X|| |ffG,Y| fflffl| fflffl| XfG _OE__//YfG fG We claim that OE is a stable equivalence in SpG if and only if the induced morp* *hism UOE is a stable equivalence of underlying Bousfield-Friedlander spectra. Becau* *se U is a left Quillen functor, and the morphisms ffG,- are trivial cofibrations, * *we may conclude that the morphisms UffG,- induce stable equivalences of underlying Bousfield-Friedlander spectra. Suppose that OE is a stable equivalence in SpG . Then the morphism OEfG is a stable equivalence between fibrant objects in SpG, and therefore is a levelwise* * weak equivalence. Therefore, UOEfG is a levelwise weak equivalence in Sp, and so is* * a stable equivalence of underlying spectra. Since the morphisms UffG,- are stable equivalences of underlying spectra, we deduce that UOE is a stable equivalence * *of underlying spectra. Suppose that OE is an equivalence of underlying spectra. Since the morphisms UffG,- are stable equivalences of underlying spectra, we deduce that UOEfG is a stable equivalence of underlying spectra. Since the underlying spectrum of a fi* *brant object in SpG is a fibrant object in Sp, we may conclude that UOEfG is a levelw* *ise weak equivalence of underlying spectra. Therefore, it is a stable equivalence i* *n SpG. Since the morphisms ffG,- are stable equivalences in SpG, we conclude that OE i* *s a stable equivalence in SpG. 2.3. Discrete symmetric G-spectra. Let Sp denote the category of symmetric spectra (see [22], [31] for accounts of symmetric spectra). Define the category* * of discrete symmetric G-spectra SpG to be the category of symmetric spectra of simplicial discrete G-sets. Let i denote the ith symmetric group. An object X 2 SpG consists of a sequence {Xi}i 0, where each Xi is a pointed simplicial discrete Gx i-set, together with suitably compatible Gx ix j-equivariant maps oei,j: Si^ Xj ! Xi+j. Here, Si = (S1)^i is given the trivial G-action, and i permutes the factors of the smash product (S1)^i. When G is finite, a discrete symmetric G-spectrum is simply a na"ive symmetric G-spectrum, and not a genuine equivariant symmetric G-spectrum in the sense of [30]. Maps f : X ! Y of discrete symmetric G-spectra are sequences of G x i- equivariant maps of pointed simplicial sets fi: Xi! Yi which are compatible with the spectrum structure maps. For a cofibrantly generated model category C, let C j denote the diagram cat- egory of j-equivariant objects in C with the projective model structure ([17, Thm. 11.6.1]). Lemma 2.3.1. In the projective model category structure on (sSetG)*j: o the cofibrations are those maps that are projective cofibrations in the * *un- derlying category sSet*j, THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 9 o the weak equivalences are those maps that are weak equivalences in the underlying category sSet*j, o the fibrations are determined. Proof.The statement concerning weak equivalences follows immediately from the definition of the weak equivalences in the projective model structure. The proj* *ective cofibrations in (sSetG)*j are generated by the set I+j = {( jx G=U x @ n)+ ,! ( jx G=U x n)+ : U o G, n 0}. Using the relative skeletal filtration, it is easy to see that the class of cof* *ibra- tions generated by the set I+j are the monomorphisms which are relative free j- complexes. However, these are precisely the projective cofibrations in sSet*j. Theorem 2.3.2. The category SpG admits a left proper cellular model structure where o the cofibrations are the cofibrations of underlying symmetric spectra, o the weak equivalences are the stable weak equivalences of underlying sym- metric spectra, o the fibrations are determined. Proof.Observe that (sSetG)* satisfies the conditions of Definition 8.7 of [21].* * There- fore, the category SpG of symmetric S1-spectra of simplicial discrete G-sets a* *dmits a stable model category structure where o the cofibrations are those morphisms A ! B where the induced maps A0 ! B0 Ai[LiA LiB ! Bi i 1 are projective cofibrations in (sSetG)*j, where Ln is the latching objec* *t of [21, Def. 8.4]. o the fibrant objects X are those spectra for which (1) the spaces Xi are fibrant as simplicial discrete G-sets, (2) the maps Xi! Xi+1are weak equivalences. o the weak equivalences f : X ! Y between fibrant objects are those f for which the maps fi : Xi ! Yi are all weak equivalences of underlying simplicial sets. The cofibrations are immediately seen to be the cofibrations of underlying symm* *et- ric spectra, using Lemma 2.3.1. The verification that the weak equivalences are precisely the stable equivalences of underlying symmetric spectra is identical * *to the argument given in the proof of Theorem 2.2.1. We have the following proposition, which helps to translate results in the ca* *tegory SpG, to the category SpG . Proposition 2.3.3. There is a Quillen equivalence VG : SpG o SpG : UG where UG is the forgetful functor. Proof.The functor VG is the left adjoint of UG : it is explicitly given by (see* * [22, Sec. 4.3]) VG (X) = S T(G1S1)GX 10 MARK BEHRENS AND DANIEL G. DAVIS where GX is the symmetric sequence given by (GX)i= ( i)+ ^ Xi (where G acts through its action on Xi), G1S1 is the symmetric sequence (*, S1, *, *, . .). (with trivial G action), T (G1S1) is the free monoid on G1S1 with respect to (which gives the symmetric monoidal structure on symmetric sequences), and S is the usual symmetric sequence (S0, S1, S2, . .).. We have the following commutat* *ive diagram of functors VG // Sp G_____ooSpG_ UG U || |U| fflffl|V fflffl| Sp ______//oSpo_ U where the bottom row is the Quillen equivalence of [22, Sec. 4]. The functor VG preserves cofibrations and trivial cofibrations because the fu* *nc- tors U and U reflect and detect cofibrations and trivial cofibrations, and the* * functor V is a left Quillen functor. Therefore (VG , UG ) forms a Quillen pair. To show that (VG , UG ) is a Quillen equivalence, we must show that for all c* *ofi- brant X in SpG and all fibrant Y in SpG , a morphism f : VG X ! Y is a weak equivalence if and only if its adjoint ef: X ! UG Y is a weak equivalence. However, since the functors U and U reflect and detect weak equivalences, it suffices to show that: ae oe ae oe U f : VUX ! U Y Ufe: UX ! UU Y is a weak equivalence if and only if is a weak equivalence. This follows from the fact that U preserves cofibrations (Theorem 2.2.1), U pr* *e- serves fibrant objects (proof of Theorem 2.3.2), and (V, U) form a Quillen equi* *va- lence [22, Thm. 4.2.5]. For the rest of this paper, we shall be working in the world of symmetric spe* *ctra, and shall refer to a symmetric spectrum as simply a spectrum. 2.4. Mapping spectra. Let K and L be discrete G-sets. Then the set of (non- equivariant) functions Map (K, L) is a G-set with G acting by conjugation. Thus, for g 2 G and f 2 Map (K, L), g . f is the map (g . f)(z) = gf(g-1z). Observe that Map (K, L) is not in general a discrete G-set, but it is if K is f* *inite. For a finite set K and a spectrum X, we define the mapping spectrum Map(K, X) to be the spectrum whose mth space is given by Map (K, X)m = Map (K, Xm ), where the n-simplices of Map (K, Xm ) is the set Map (K, (Xm )n). If X is a dis* *crete G-spectrum, and K is a finite discrete G-set, then the above definitions combin* *e to give that Map (K, X) is a discrete G-spectrum. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 11 If K = limffKffis a profinite set and X is a spectrum, then the spectrum of continuous maps is the spectrum Map c(K, X) = colimffMap(Kff, X). If K is a continuous G-space, with each Kffa discrete G-set, and X is a discrete G-spectrum, then Map c(K, X) is a discrete G-spectrum. Lemma 2.4.1. Let K = limffKffbe a profinite set, where each of the Kffare finit* *e, and each of the maps Kff! Kfiin the pro-system are surjections. The functor Mapc(K, -) : Sp ! Sp preserves all stable equivalences. Proof.In [37], it is shown that stable equivalences are preserved under finite * *prod- ucts. The argument goes as follows: the canonical map from a finite wedge to a finite product is a ss*-isomorphism, hence a stable equivalence, and stable equ* *iva- lences are preserved under finite wedges (this is easily checked from the defin* *ition of stable equivalences, by mapping into injective -spectra). Therefore, for ea* *ch ff, the functor Y X 7! Map (Kff, X) ~= X Kff preserves stable equivalences. Since we have assumed that the morphisms Kff! Kfi are surjections, the induced morphisms Map (Kfi, X) ! Map (Kff, X) are levelwise monomorphisms for every X. The category of symmetric spectra possesses an injective stable model structu* *re, where the injective cofibrations are the levelwise monomorphisms and the weak equivalences are the stable equivalences (see [22, pg. 199]). The directed syst* *em {Map (Kff, X)} is a directed system of injective cofibrations between injectively cofibrant ob* *jects (every object is cofibrant in the injective model structure). The colimit Map c(K, X) = colimffMap(Kff, X) may be computed on a cofinal ~-sequential subcategory of the indexing category * *of the system {Kff}, for some ordinal ~. We deduce, by [17, Prop. 17.9.1], that the functor X 7! Map c(K, X) = colimffMap(Kff, X) preserves stable equivalences. 2.5. Permutation spectra. Let K be a discrete G-set. Then for X a discrete G-spectrum, we may define the permutation spectrum X[K] to be the spectrum whose nth space is given by X[K]n = Xn ^ K+ . We let G act on the spectrum X[K] through the diagonal action. Lemma 2.5.1. The spectrum X[K] is a discrete G-spectrum. 12 MARK BEHRENS AND DANIEL G. DAVIS Proof.Note that 0 Xn ^ K+ ~=(colimNE(Xn)N ) ^ (( colimKN )+ ) oG N0EoG ~=(colim(Xn)N ) ^ ( colim(KN0)+ ) NEoG N0EoG ~= colim ((Xn)N ^ (KN0)+ ) N,N0EoG is a simplicial0discrete G-set, with G acting diagonally, since the simplicial * *set (Xn)N ^ (KN )+ has a diagonal G=(N \ N0)-action and the group G=(N \ N0) is finite. Thus, the spectrum X[K] is a discrete G-spectrum. 2.6. Smash products. Given discrete G-spectra X and Y , we define their smash product X ^ Y to be the smash product of the underlying symmetric spectra with G acting diago- nally. Since the smash product commutes with colimits, it follows, as in the pr* *oof of Lemma 2.5.1, that X ^Y is a discrete G-spectrum. Also, if K is a discrete G-* *set, then there is a G-equivariant isomorphism X ^ S[K] ~=X[K], where S, the sphere spectrum, has trivial G-action. 3.Homotopy fixed points of discrete G-spectra Much of the material in this section is assembled from [42], [26], [12], [27]* *, [33], [29], and [5]. Let G be a profinite group. We begin this section with an accoun* *t of the model category theoretic definition of G-homotopy fixed points. We then de- scribe the comparison with hypercohomology spectra. Finite index restriction and induction functors, as well as iterated homotopy fixed points for finite index * *sub- groups are then discussed. We explain how continuous homomorphisms of groups induce various "change of group functors," of which induction, coinduction, fix* *ed points, and restriction functors are all special cases. We then describe the va* *rious technical difficulties related to the homotopy fixed point construction for clo* *sed subgroups of G. The technical difficulties are observed to vanish if G has fin* *ite cohomological dimension. As alluded to above, Sections 3.3, 3.5, and 3.6 discuss the construction of i* *terated homotopy fixed points. Much of this material overlaps with portions of [6]: it * *was necessary to repeat some of the material from [6], so that certain issues are c* *lear and to give a context for the results of Section 7.1. We note that, as explained in Section 2.3, "spectrum" means "symmetric spec- trum," so that, for example, a "discrete G-spectrum" is a "discrete G-symmetric spectrum." 3.1. The homotopy fixed point spectrum. For a discrete G-spectrum X, we define the fixed point spectrum by taking the fixed points levelwise: (XG )i= (Xi)G . The G-fixed points functor is right adjoint to the functor triv, which associat* *es to a spectrum X the discrete G-spectrum X, where X now has the trivial G-action: triv: Sp o SpG : (-)G . THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 13 Lemma 3.1.1. The adjoint functors (triv, (-)G ) form a Quillen pair. Proof.The functor trivpreserves cofibrations and weak equivalences. Let ffG,X : X ! XfG denote a functorial fibrant replacement functor for the model category SpG , where ffG,X is a trivial cofibration of discrete G-spectr* *a. The homotopy fixed point functor (-)hG is the Quillen right derived functor of (-)G , and is thus given by XhG = (XfG)G . 3.2. Hypercohomology spectra. The functor G = Map c(G, -) is a coaug- mented comonad on the category of spectra, with coproduct _ : G = Map c(G, -) ! Map c(G x G, -) = G O G induced from the product on G, counit G = Map c(G, -) ! Map c(pt, -) = Id induced from the unit on G, and coaugmentation Id! Map c(G, -) given by the inclusion of the constant maps. Discrete G-spectra are coalgebras over the comonad G (this follows from con- sidering the map of spectra X ! G (X), x 7! (g 7! g . x), for any discrete G-spectrum X). Let C and D be categories, and suppose that is a comonad in C. Dualizing Definition 9.4 of [32], there is a notion of a -functor F : C ! D. Let Y be a -coalgebra. Dualizing Construction 9.6 of [32], one may associate to (F, , Y ) a cosimplicial object Co(F, , Y ) in D (the comonadic cobar constru* *ction), given by Cs(F, , Y ) = F sY. If is a coaugmented comonad, then the identity functor IdCis a -functor. We will let oY denote the cosimplicial object oY = Co(IdC, , Y ) in C. In [5], the homotopy fixed point spectrum was shown to have the following alternate description, provided G is sufficiently nice (see also [33], [12], [2* *5]). Theorem 3.2.1. Suppose that G has finite vcd, and that X is a discrete G- spectrum. Then there is an equivalence XhG ' holim oGX = Hc(G; X), where Hc(G; X) is the hypercohomology spectrum. 14 MARK BEHRENS AND DANIEL G. DAVIS Proof.In [5, Thm. 7.4] it is proven that there is an equivalence XhG ' holim oGXfG. (The cosimplicial object defining the hypercohomology spectrum is different, but isomorphic to that appearing in [5].) The result follows once we establish that* * the map induced from fibrant replacement holim oGX ! holim oGXfG is an equivalence. This map is deduced to be an equivalence from the following * *facts: (a) the fibrant replacement map X ! XfG is an equivalence; (b) the functor G preserves equivalences, by Lemma 2.4.1; (c) the homotopy limit construction sen* *ds levelwise equivalences to equivalences, since it is a Quillen derived functor. 3.3. Iterated homotopy fixed points. Let U be an open normal subgroup of G, so that G=U is finite. Proposition 3.3.1. Let X be a discrete G-spectrum. (1) The U-fixed point spectrum (XfG)U is fibrant as a discrete G=U-spectrum. (2) The fibrant discrete G-spectrum XfG is fibrant as a discrete U-spectrum. (3) The homotopy fixed point spectrum XhU is a G=U-spectrum. (4) There is an equivalence XhG ' (XhU )hG=U. Proof.To prove (1), observe that since U is normal, for any discrete G-spectrum* * Y , the U-fixed point spectrum Y U is naturally a G=U-spectrum. There is an adjoint pair of functors (ResGG=U, (-)U ) ResGG=U: SpG=U o SpG : (-)U , where ResGG=Uis defined by restriction along the quotient homomorphism G ! G=U. Since ResGG=Upreserves cofibrations and weak equivalences, the functor (-)U preserves fibrant objects. We verify (2) in a similar way (compare with [27, Rmk. 6.26]). Define the induction functor on a discrete U-spectrum Y to be the Borel construction IndGUY = G+ ^U Y. Here, the Borel construction is formed by regarding G and U as discrete groups,* * but this is easily seen to produce a discrete G-spectrum, since U is a normal subgr* *oup of finite index. The induction functor is the left adjoint of an adjunction IndGU: SpU o SpG : ResUG, where ResUGis restriction along the inclusion U ,! G. Since non-equivariantly t* *here is an isomorphism IndGUY ~=G=U+ ^ Y, we see that IndGUpreserves cofibrations and weak equivalences, from which it fo* *llows that ResUGpreserves fibrant objects. By (2), XfG is a fibrant discrete U-spectrum. Also, X ! XfG is a trivial cofibration of spectra and it is U-equivariant, so it is a trivial cofibration * *in SpU . Thus, XhU = (XfG)U , which is a G=U-spectrum. This proves (3). THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 15 (4) is proven using our fibrancy results. There are equivalences: XhG ' XGfG= (XUfG)G=U ' (XhU )hG=U. 3.4. Homomorphisms of groups. If f : H ! G is a continuous homomorphism of profinite groups, we may regard discrete G-sets as discrete H-sets. For a di* *screte H-set Z, we define the coinduced discrete G-set by f*Z = CoIndGHZ = Map cH(G, Z) = colimUEMapH(G=U, Z), oG where the G-action is defined by the formula (g . ff)(g0U) = ff(g0gU), for g 2 G and ff 2 Map H(G=U, Z). This construction extends to simplicial discr* *ete G-sets and discrete G-spectra in the obvious manner to give a functor f* : SpH ! SpG . The functor f* is the right adjoint of an adjoint pair (f*, f*), where f* = ResHG: SpG ! SpH is the restriction functor along the homomorphism f. Since f* clearly preserves cofibrations and weak equivalences, we have the following lemma. Lemma 3.4.1. The adjoint functors (f*, f*) form a Quillen pair. In particular, * *f* preserves fibrations and weak equivalences between fibrant objects. We make the following observations. (1) The Quillen pair (f*, f*) gives rise to a derived adjoint pair (Lf*, Rf** *). (2) Since the functor f* preserves all weak equivalences, there are equivale* *nces Lf*X ' f*X for all discrete G-spectra X. (3) If j : H ,! G is the inclusion of a closed subgroup, then for a discrete H-spectrum X, we have a non-equivariant isomorphism j*X = Map cH(G, X) ~=Map c(G=H, X). By Lemma 2.4.1, we see that j* preserves weak equivalences, and therefore there is an equivalence j*X ' Rj*X. (4) The adjoint pair (triv, (-)G ) of Section 3.1 agrees with the adjoint pa* *ir (r*, r*) when r : G ! {e} is the homomorphism to the trivial group. Therefore, the homotopy fixed point functor is given by (-)hG = Rr*. (5) Given continuous homomorphisms H f-!G g-!K, there are natural isomor- phisms (g O f)* ~=g* O f* and (g O f)* ~=f* O g*. We get similar formulas on the level of derived functors. (6) If i : U ,! G is the inclusion of an open subgroup, then the induction functor i!= IndGU(Proposition 3.3.1) is the left adjoint of the Quillen * *pair (i!, i*). We use these derived functors to prove a version of Shapiro's Lemma. Lemma 3.4.2. Let X be a discrete G-spectrum, and suppose that H is a closed subgroup of G. Then there is an equivalence Map c(G=H, X)hG '-!XhH . 16 MARK BEHRENS AND DANIEL G. DAVIS Proof.Consider the following diagram of groups. j H _____________//_BBG" BBB """" s BB__B"""r"" {e} If Z is a discrete G-set, there is a G-equivariant bijection ~= c ffi : j*j*Z = Map cH(G, Z) -! Map (G=H, Z). The map ffi sends a map ff in Map cH(G, Z) to the map ffi(ff) : gH 7! gff(g-1). The inverse ffi-1 sends a map fi in Map c(G=H, Z) to the map ffi-1(fi) : g 7! gfi(g-1H). The isomorphism ffi induces for a discrete G-spectrum Y an isomorphism ~= c ffi : j*j*Y -! Map (G=H, Y ), in SpG . By Lemma 3.4.1, the functor j* sends H-fibrant objects to G-fibrant objects. Therefore we have equivalences: Map c(G=H, X)hG~=Rr*j*j*X ' Rr*Rj*j*X ' Rs*j*X = XhH . 3.5. Iterated fixed points for closed subgroups. Let j : H ,! G be the in- clusion of a closed subgroup. We wish to extend the results of Section 3.3 to t* *he closed subgroup H. The following proposition may be compared to [27, Lem. 6.35]. Proposition 3.5.1. Let N be a closed normal subgroup of G, and let X be a discrete G-spectrum. Then there is an equivalence ((XfG)N )hG=N ' XhG . Proof.Consider the following diagram. q G _____________//_@@G=N z @@@ zzz r @@OO@__zszzz {e} There is an equivalence ((XfG)N )hG=N = Rs*Rq*X ' Rr*X = XhG . THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 17 The reader might wonder if XfG is fibrant as a discrete H-spectrum, but this does not appear to hold for nontrivial closed subgroups H that are not open. We discuss these difficulties in Section 3.6. Though (XfG)H is not known to always equal XhH , the following result identifies (XfG)H with a canonical colimit th* *at always maps to XhH . Corollary 3.5.2. There is an equivalence (XfG)H ' Hcolim UXhU . oG Proof.Since H acts discretely on XfG, we have an equality (XfG)H = Hcolim U(XfG)U . oG By Proposition 3.3.1, the spectrum XfG is fibrant as a discrete U-spectrum, so there are equivalences (XfG)U ' XhU . By Corollary 3.5.2, given a discrete G-spectrum X and a closed subgroup H, there is a natural map (XfG)H ' Hcolim UXhU ! XhH . oG If we restrict ourselves to the case where G has finite cohomological dimension, then, as shown below, the iterated homotopy fixed point spectrum behaves in a more satisfactory way. Proposition 3.5.3. Suppose that G has finite cohomological dimension, and sup- pose that X is a discrete G-spectrum. Then the natural map H colimUXhU ! XhH oG is an equivalence. Proof.For K a profinite group of finite vcd, and Y a discrete K-spectrum, let Er(K; Y ) denote the conditionally convergent descent spectral sequence E2(K; Y ) = H*c(K; ss*(Y )) ) ss*(Y hK). There is a map of spectral sequences E0r(H; X) := Hcolim UEr(U; X) ! Er(H; X), oG which is an isomorphism on the level of E2-terms by [44, Thm. 9.7.2]. The propo- sition now follows from [33, Prop. 3.3]. Corollary 3.5.4. Let G be of finite cohomological dimension, and let X be a discrete G-spectrum. (1) There is an equivalence (XfG)H ' XhH . (2) Suppose H is normal in G. Then, by using the model Hcolim UXhU oG for the H-homotopy fixed point spectrum, there is an equivalence (XhH )hG=H ' XhG . In Section 7.1, we will see that we may extend Proposition 3.5.3 to groups of finite virtual cohomological dimension provided that we are taking homotopy fix* *ed points of a consistent profaithful k-local profinite Galois extension. 18 MARK BEHRENS AND DANIEL G. DAVIS 3.6. The difficulties concerning arbitrary closed fixed points. Let H be a closed subgroup of an arbitrary profinite group G. We would be able to remove the finite cohomological dimension hypothesis in Section 3.5, if we knew that t* *he restriction functor ResHG: SpG ! SpH sends G-fibrant objects to H-fibrant objects. While we know of no counterexampl* *es to this assertion, we also doubt that it this is true in general. We saw in Proposition 3.3.1 that for U an open subgroup of G, the presence of an induction functor IndGUwhich was a left Quillen adjoint to ResUGallowed us to prove that ResUGpreserves fibrant objects. However, as pointed out to the second author by Jeff Smith, ResHGcannot posse* *ss a left adjoint, in general, since it does not preserve limits. This can be see* *n as follows. For a profinite group K and a diagram {Xff} in the category SpK , let limffKXff denote the limit computed in the category SpK . This limit is given by the fol* *lowing formula: limffKXff= colimUE(limSpXff)U . oKff Here, the limit limSpis the limit computed in the underlying category of symmet* *ric spectra. Thus, given a diagram {Xff} in SpG , the restriction of the limit is given by ResHGlimffGXff= colimUE(limSpXff)U . oGff However, the limit of the restriction is computed to be limffHResHGXff= colimV(ElimSpXff)V . oHff When H is not open in G, the lack of cofinality implies that these subspectra of limSpffXffin general do not agree. One might suspect that one could still prove that the map H colimUXhU ! XhH oG is an equivalence if G has finite virtual cohomological dimension, by a compari* *son of descent spectral sequences. This approach, however, also presents difficult* *ies. As in the proof of Proposition 3.5.3, there is a map of spectral sequences E0r(H; X) = Hcolim UEr(U; X) ! Er(H; X) oG which is an isomorphism on the level of E2-terms (see [44, Thm. 9.7.2]). The problem is that the colimit of the spectral sequences does not converge to the colimit of the abutments in general. 4. Continuous G-spectra In this paper, a continuous G-spectrum is a pro-object in the category of dis* *crete G-spectra. In this section, we extend some of our constructions for SpG to the category of continuous G-spectra. For continuous G-spectra that are indexed over {0 1 2 . .}., part of this material appears in more detail in [5]. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 19 4.1. Pro-objects in discrete G-spectra. Following standard usage, a pro-object in a category C is a cofiltered diagram in C. We define the category of continu* *ous G-spectra SpcGto be the category of pro-objects in SpG . Thus, a continuous G-spectrum is a cofiltered diagram X = {Xi}i2I of discrete G-spectra. Maps in the category of continuous G-spectra are given by SpcG(X, Y) = limjcolimi SpG (Xi, Yj). Any pro-spectrum X = {Xi} gives rise to a spectrum X via the homotopy limit functor: X = holimiXi. We shall always denote our pro-spectra by boldface type and their homotopy limi* *ts with non-boldface type. Remark 4.1.1. A more general theory of pro-spectra, including a model category structure, has been developed by Isaksen (see [24] and [11, Section 1.1]). Fau* *sk has developed a category of continuous genuine G-spectra where G is a compact Hausdorff topological group [10]. The notion of continuous G-spectrum in this paper (that is, a pro-discrete G-spectrum) corresponds roughly, in [10], to a p* *ro- G-spectrum_that is in the full subcategory of cofibrant objects in the Postnikov Lie(G)-model structure on pro-MS. For more detail, we refer the reader to [10, Section 11.3], especially the discussion centered around [op. cit., (11-15)]. 4.2. Continuous mapping spectra. Let K = limiKibe a profinite G-set. Given a continuous G-spectrum X, the continuous mapping spectrum Map c(K, X) is defined to be the continuous G-spectrum {Map c(K, Xj)}j. We denote the homotopy limit of Map c(K, X) by Map c(K, X). If the derived functors limsjMapc(K, sst(Xj)) = 0, for all s > 0 and all t 2 Z, then the Bousf* *ield- Kan spectral sequence limsjMapc(K, sst(Xj)) ) ss*(Map c(K, X)) collapses, and thus, ss*(Map c(K, X)) ~=Map c(K, limjss*(Xj)). 4.3. Continuous permutation spectra. Let K be as above, and let each finite set Kj, for each j in the indexing set for K, be a discrete G-set. Also, let X * *= {Xi}i be a continuous G-spectrum. Define the permutation spectrum X[[K]] to be the continuous G-spectrum given by {Xi[Kj]}i,j. We denote the homotopy limit of X[[K]] by X[[K]] = holimi,jXi[Kj]. Note that if limsi,jsst(Xi)[Kj] = 0, for all s > 0 and all t 2 Z (where sst(X* *i)[Kj] is an abelian group), then ss*(X[[K]]) ~=limi,jss*(Xi)[Kj]. 20 MARK BEHRENS AND DANIEL G. DAVIS If E is a discrete G-spectrum, we use E[[K]] to denote the continuous G-spectrum {E[Kj]}j. 4.4. Continuous homotopy fixed points. For a continuous G-spectrum X, we define the homotopy fixed point pro-spectrum XhG to be {XhGi}i. We denote the homotopy limit of XhG by XhG , and we refer to XhG as the homotopy fixed point spectrum. 4.5. Continuous hypercohomology spectra. We define G : (pro- Sp) ! (pro- Sp) to be the coaugmented comonad given by G (X) = Map c(G, X). Let G (X) be the homotopy limit of G (X). If X is a continuous G-spectrum, then it is a coalgebra over G . Let Hc(G; X) denote the pro-spectrum obtained by taking hypercohomology levelwise: Hc(G; X) = {Hc(G; Xi)}i. Let Hc(G; X) denote the homotopy limit of the pro-spectrum Hc(G; X). The fol- lowing result follows immediately from Theorem 3.2.1. Theorem 4.5.1. Suppose that X = {Xi} is a continuous G-spectrum. If G has finite vcd, then there is an equivalence XhG ' Hc(G; X). 4.6. Homotopy fixed point spectral sequence. Let G have finite vcd. Then Theorem 4.5.1 implies that XhG ' holimholimiMapc(Go, Xi), and, hence, the associated Bousfield-Kan spectral sequence has the form Es,t2(G; X) = ssssst(holimiMapc(Go, Xi)) ) sst-s(XhG ), giving the conditionally convergent homotopy fixed point spectral sequence for XhG . If limsiMapc(Gk, sst(Xi)) = 0, for all s > 0, all k 0, and all t 2 Z, then,* * for each k 0, the Bousfield-Kan spectral sequence limsiMapc(Gk, sst(Xi)) ) ss*(holimiMap c(Gk, Xi)) collapses, and thus, Es,t2(G; X)~=sss(limiMapc(Go, sst(Xi))) ~=Hs(Map c(Go, limsst(Xi))) i ~=Hs(Map c(Go, sst(X))) ~=Hsc(G; sst(X)). Here, Hsc(G; sst(X)) denotes the continuous cohomology of continuous cochains, with coefficients in the topological G-module sst(X) ~=limisst(Xi). THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 21 4.7. Completed smash product. If X and Y are continuous G-spectra, we define the completed smash product X ^cY to be the continuous G-spectrum {Xi^ Yj}i,j. The completed smash product gives SpcGa symmetric monoidal product, where the unit is {S0} (the sphere spectrum regarded as a diagram indexed by a single element). 5. Modules and commutative algebras of discrete G-spectra Let A be a commutative symmetric ring spectrum and let G be a profinite group. In this section, we describe the model categories of discrete G-A-modules and discrete commutative G-A-algebras. We show that the homotopy fixed points of a discrete G-A-module is an A-module and the homotopy fixed points of a discrete commutative G-A-algebra is a commutative A-algebra. These structured homotopy fixed point constructions are shown to agree, in the stable homotopy category, with the usual homotopy fixed points of the underlying discrete G-spectrum. We then make comparisons between filtered homotopy colimits and filtered colimits * *of modules and commutative algebras, and conclude that, when properly interpreted, they all coincide in the stable homotopy category. We conclude this section by describing how to make the hypercohomology spectra of discrete commutative G- A-algebras take values in the category of commutative A-algebras. 5.1. Modules of discrete G-spectra. Let A be a commutative symmetric ring spectrum. By a discrete G-A-module, we shall mean a discrete G-spectrum X that also possesses the structure of an A-module. We require these structures to be compatible in the following sense: for every element g 2 G, the following diagr* *am must commute. , A ^ X _____//X A^g || |g| fflffl|, fflffl| A ^ X _____//X Here, , is the A-module structure map. Let Mod G,Adenote the category of discre* *te G-A-modules, with morphisms being the G-equivariant maps that are also maps of A-modules. Note that, given discrete G-A-modules X and Y , their smash product X ^A Y is easily seen to be a discrete G-A-module with the diagonal action. The following simplified variant of D.M. Kan's "lifting theorem" will be used repeatedly to provide the desired model structures on structured categories like Mod G,A. Lemma 5.1.1. Suppose that M is a cofibrantly generated model category with generating cofibrations I and generating trivial cofibrations J. Furthermore, * *as- sume that the domains of I and J are ff-small for some cardinal ff. Suppose that we are given a complete and cocomplete category N and an adjoint pair (F, G) F : M o N : G, where: (1) G commutes with filtered colimits; and (2) G takes relative F J-cell complexes to weak equivalences. 22 MARK BEHRENS AND DANIEL G. DAVIS Then N admits an induced model category structure where the fibrations and weak equivalences are those morphisms which get sent to fibrations and weak equiva- lences by G, and the cofibrations are determined. This model category structure* * is cofibrantly generated with generating cofibrations F I and generating trivial c* *ofi- brations F J. The domains of F I and F J are ff-small in N . Proof.This theorem is a special case of Theorem 11.3.2 of [17]. To see that the hypotheses of this theorem are met in our situation, we must verify that the do* *mains of F I and F J are ff-small with respect to relative F I- and F J-cell complexe* *s, respectively. However, our hypotheses imply that F preserves all ff-small objec* *ts: given an ff-small object X 2 M, and a ~-sequence (~ ff) Y1 ! Y2 ! Y3 ! . .!.Yfi! . . .(fi < ~) in N , we have: colimiHomN (F X, Yi)~=colimiHomM(X, GYi) ~=Hom M (X, colimGYi) i ~=Hom M (X, G colimYi) i ~=Hom N (F X, colimYi). i In [31], a model category structure is defined on the category of A-modules. * *The fibrations and weak equivalences of this model structure are the fibrations and* * weak equivalences of underlying symmetric spectra, and the cofibrations are determin* *ed. Proposition 5.1.2. The category Mod G,Ais a model category, where the fibrations and weak equivalences are the fibrations and weak equivalences of the underlying discrete G-spectra, and the cofibrations are the cofibrations of the underlying* * A- modules. Proof.We apply Lemma 5.1.1 to the adjoint pair A ^ - : SpG o Mod G,A: U, where U is the forgetful functor. The category SpG is cofibrantly generated with generating cofibrations I = {Fn(G=U x @ n)+ ,! Fn(G=U x n)+ : U o G, n 0}. (Here, Fn is the left adjoint to the functor which returns the nth space of a s* *ym- metric spectrum.) Thus, the domains of the maps in I are !-small. Since the mod* *el structure on SpG is left proper and cellular, the Bousfield-Smith cardinality * *argu- ment [17, Prop. 4.5.1] implies that there exists a cardinal ff, such that the c* *ollection J of inclusions of I-cell complexes of size at most ff, which are weak equivale* *nces, generates the trivial cofibrations of SpG . In particular, since the I-cells a* *re finite, the domains of J are ff-small. By Lemma 5.3.4, the functor U preserves filtered colimits. We need to verify that U takes relative (A ^ J)-cell complexes to weak equivalences. However, the domains and codomains of the maps in J are cofibrant objects in Sp, and A ^ - preserves stable equivalences between cofibrant symmetric spectra. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 23 The model structure on Mod G,Agiven by Lemma 5.1.1 has the desired fibrations and weak equivalences. We need to check that the cofibrations of this model str* *uc- ture are precisely the cofibrations of underlying non-equivariant A-modules. The cofibrations are generated by A^I. Since all of the morphisms in I are cofibrat* *ions of underlying non-equivariant symmetric spectra, we deduce that all of the cofi* *bra- tions of G-A-modules are cofibrations of underlying A-modules. The converse is * *an argument similar to Corollary 1.10 of [12]. Given a discrete G-A-module X, let XfGA denote a functorial fibrant replace- ment in Mod G,A, so that the fibrant replacement map X ! XfGA is a trivial cofibration of discrete G-A-modules. We define the homotopy fixed * *point A-module to be the fixed point spectrum XhAG = (XfGA )G . The properties of our model category structures immediately give the following lemma. Lemma 5.1.3. The spectrum XfGA is fibrant as a discrete G-spectrum, and there exists a weak equivalence XfG '-!XfGA in the category SpG . Since (-)G preserves weak equivalences between fibrant spectra in SpG , we have the following corollary. Corollary 5.1.4. There is an equivalence XhG -'!XhAG . Since colimits of discrete G-A-modules are formed on the level of underlying symmetric spectra, the comonad G (-) = Map c(G, -) restricts to the category of A-modules, where, given an A-module X, the A-module structure is given by the composition A ^ Mapc(G, X) ! Map c(G, A ^ X) ,*-!Mapc(G, X). In this composition, the first map is given by the composite A ^ Mapc(G, X)= A ^ colimNEMap(G=N, X) oG ~= -! colimNEA ^ Map(G=N, X) oG ! colimNEMap(G=N, A ^ X) oG = Map c(G, A ^ X). Since fibrant discrete G-A-modules are fibrant in SpG , the following propos* *ition is an immediate consequence of Theorem 3.2.1. Proposition 5.1.5. Suppose G has finite vcd, and let X be a discrete G-A-module. Then there is an equivalence of A-modules XhAG ' holim oGX = Hc(G; X). 24 MARK BEHRENS AND DANIEL G. DAVIS Define the category Mod cG,Aof continuous G-A-modules to be the category of pro-objects in Mod G,A. Given a continuous G-A-module X = {Xi}i, we define XhAG := holimiXhAGi, Hc(G; X):= holimiHc(G; Xi). Proposition 5.1.5 has the following corollary. Corollary 5.1.6. Suppose G has finite vcd, and let X be a continuous G-A-module. Then there is an equivalence of A-modules XhAG ' Hc(G; X). We shall henceforth drop the distinction between (-)hAG and (-)hG: all homo- topy fixed points of discrete G-A-modules will implicitly be taken in the categ* *ory of discrete G-A-modules. 5.2. Commutative algebras of discrete G-spectra. By a discrete commutative G-A-algebra, we shall mean a discrete G-A-module E together with a commutative A-algebra multiplication ~ : E ^A E ! E, such that G acts on E through maps of commutative A-algebras. Let AlgA,Gdenote the category of discrete commutative G-A-algebras, with morphisms being those morphisms in Mod G,A that are also maps of commutative A-algebras. Following [31], to place a model structure on AlgA,G, we need to replace the model struct* *ure on SpG with a Quillen equivalent "positive" model structure. Lemma 5.2.1. The category of discrete G-spectra admits a positive stable model structure, which we denote by Sp+G, where the cofibrations are the positive co* *fi- brations of underlying symmetric spectra, the weak equivalences are the stable equivalences of underlying symmetric spectra, and the positive fibrations are d* *eter- mined. Proof.We follow [31, Sec. 14], in its construction of the positive stable model structure on symmetric spectra, from start to finish, with some mild alteration* *s. The category SpG admits a positive level model structure that is defined as fo* *llows: o the positive level fibrations are those maps f : X ! Y where the maps fi: Xi! Yi are fibrations of simplicial discrete G-sets for i 1; o the positive level weak equivalences are those maps f : X ! Y where the maps fi : Xi ! Yi are weak equivalences of underlying simplicial sets, f* *or all i 1; and o the positive cofibrations are those morphisms X ! Y where the induced map X0 ! Y0 is an isomorphism and each of the induced maps Xi! Yi, i = 1, Xi[LiX LiY ! Yi, i 2, is a projective cofibration in (sSetG)*i, where Li is the latching objec* *t of [21, Def. 8.4] (that is to say, they are precisely the positive cofibrat* *ions of underlying symmetric spectra). THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 25 The positive level model structure on SpG is left proper and cellular, and hen* *ce, it admits a localization with respect to the set of maps FnS1 ^ C ! Fn+1C, n 1, where, as usual, Fn is the left adjoint to the nth space evaluation functor and* * C runs through the cofibrant domains and codomains of the generating cofibrations of sSetG. This localized model structure is the positive stable model structure: o the cofibrations are those morphisms which are positive cofibrations on * *the underlying symmetric spectra; o the fibrant objects are the discrete G-spectra X for which (1) the spaces Xi are fibrant as simplicial discrete G-sets for i 1; (2) the spectrum structure maps Xi ! Xi+1 are weak equivalences for i 1, o the weak equivalences between fibrant objects are those morphisms that are positive level weak equivalences of discrete G-spectra. We are left with verifying that the weak equivalences of the positive stable mo* *del structure are precisely the stable equivalences of underlying symmetric spectra* *. It suffices to check this for morphisms between positive stable fibrant objects. * *Let OE : X ! Y be a morphism between positive stable fibrant objects in SpG , and consider the functorial fibrant replacement (in the non-positive stable model s* *truc- ture) of OE in SpG . ffG,X X ____//_XfG OE|| OEfG|| fflffl| fflffl| Y _ffG,Y//_//_YfG Note that the stably fibrant objects of SpG are positive stable fibrant. The t* *wo morphisms ffG,- are therefore positive stable equivalences between positive sta* *ble fibrant discrete G-spectra, and therefore are positive level weak equivalences.* * It follows that they induce isomorphisms on stable homotopy groups, and hence are stable equivalences [22, Thm. 3.1.11]. The same argument shows that if OE is a positive stable equivalence, then OE is a stable equivalence. Conversely, supp* *ose that OE is a stable equivalence. Then, by the two out of three axiom, the morph* *ism OEfG is a stable equivalence. Since OEfG is a stable equivalence between stable* * fibrant discrete G-spectra, we deduce that it is a levelwise equivalence. In particular* *, it is a positive level weak equivalence. This allows us to deduce that OE is a positi* *ve level weak equivalence, and thus, it is a positive stable equivalence. Lemma 5.2.2. The identity functor from discrete G-spectra with the positive stable model structure to discrete G-spectra with the stable model structure is* * the left adjoint of a Quillen equivalence. Proof.This follows from the fact that the weak equivalences in both model struc- tures are the same, and every positive cofibration is a cofibration. The category AlgS of commutative symmetric ring spectra admits a model cat- egory structure where the weak equivalences and fibrations are detected on the level of underlying symmetric spectra (with the positive model structure) [31].* * The category AlgA may be regarded as the category of commutative symmetric ring 26 MARK BEHRENS AND DANIEL G. DAVIS spectra under A, and hence inherits a model structure with the same cofibration* *s, fibrations, and weak equivalences. Theorem 5.2.3. The category of discrete commutative G-A-algebras admits a positive model structure where the cofibrations are the cofibrations of underly* *ing commutative symmetric ring spectra, the weak equivalences are the stable equiva- lences of underlying symmetric spectra, and the fibrations are the positive fib* *rations of discrete G-spectra. Proof.The proof follows [31, Thm 15.2(i)]. The category of discrete commutative G-A-algebras is the undercategory (A # AlgG), where A is regarded as a discrete* * G- spectrum with the trivial action and AlgG denotes the category AlgS,Gof discrete commutative G-algebras, and, thus, AlgA,Ginherits a model structure from AlgG, by [17, Thm 7.6.5(1)]. Therefore, it suffices to prove the theorem when A = S. * *We apply Lemma 5.1.1 to the adjoint pair P : Sp+Go AlgG : U, where U is the forgetful functor, and P is the free commutative algebra functor: ` P(X) = (X^i) i. i 0 The category Sp+Gis cofibrantly generated with generating cofibrations I+ = {Fn(G=U x @ n)+ ,! Fn(G=U x n)+ : U o G, n 1}. The domains of the maps in I+ are thus !-small. The same Bousfield-Smith car- dinality argument used in the proof of Proposition 5.1.2 shows that there exist* *s a cardinal ff and a set J+ of generating trivial cofibrations such that the domai* *ns of J+ are ff-small. By Lemma 5.3.4, the functor U preserves filtered colimits. We just need to ve* *rify that for any stable equivalence OE : X ! Y, the induced morphism P(OE) : P(X) ! P(Y ) is a stable equivalence. It suffices to verify that the map on coinvariants OEi: (X^i) i ! (Y ^i) i is a stable equivalence for every i. Consider the following diagram of canonic* *al maps. qX,i (X^i)h i____//_(X^i) i (OEi)h|| |OEi| fflffl| fflffl| (Y ^i)h iqY,i//_(Y ^i) i Since homotopy colimits preserve weak equivalences, we deduce that the maps (OEi)h are stable equivalences. The morphisms q-,iinduce isomorphisms of stable homotopy groups: this is seen by applying the geometric realization functor lev- elwise and by using Lemma 15.5 of [31] (the geometric realization of a positive cofibrant symmetric spectrum of simplicial sets is easily seen to be a positive* * cofi- brant symmetric spectrum of topological spaces). Therefore, by [22, Thm 3.1.11], THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 27 the morphisms q-,iare stable equivalences. We deduce that each OEi is a stable equivalence, as desired. We have the following corollary. Corollary 5.2.4. The following pairs of adjoint functors are Quillen adjoints: triv: Sp+ o Sp+G: (-)G , triv: AlgA o AlgA,G: (-)G . For a discrete G-spectrum X, we use XfG+ to denote the functorial fibrant replacement in the positive model structure, and we denote the corresponding ho- motopy fixed point spectrum by +G G Xh = (Xf+G ) . We have the following result. Lemma 5.2.5. If X is a discrete G-spectrum, then there is an equivalence +G ' hG Xh -! X . Proof.Since fibrant discrete G-spectra are positive fibrant discrete G-spectra,* * there is a stable equivalence ff : Xf+G ! XfG in the category SpG . Since (-)G preserves stable equivalences between positive fibrant discrete G-spectra, ff induces an equivalence +G G ff* G hG Xh = (Xf+G ) --! (XfG) = X . Given a discrete commutative G-A-algebra E, we shall denote the functorial fibrant replacement by EfGA-Alg. We define the homotopy fixed point commutative A-algebra to be the fixed point spectrum EhAlgG= (EfGA-Alg)G . Since EfGA-Alg is a positive fibrant discrete G-spectrum, a slight modification* * of the proof of Lemma 5.2.5 implies the following lemma. Lemma 5.2.6. If E is a discrete commutative G-A-algebra, then there is an equiv- alence +G ' h G Eh -! E Alg . If E is a discrete commutative G-A-algebra, then Corollary 5.1.4 and Lem- mas 5.2.5 and 5.2.6 imply that there is the following zig-zag of equivalences: +G ' h G EhAG ' EhG ' Eh ! E Alg . Therefore, we shall henceforth not distinguish between the four equivalent types of homotopy fixed points that appear in the above zig-zag. Also, homotopy fixed points of discrete commutative G-A-algebras will always implicitly be taken in * *the category of discrete commutative G-A-algebras. 28 MARK BEHRENS AND DANIEL G. DAVIS 5.3. Filtered colimits. We will make frequent use of filtered colimits. In this section, we show that filtered colimits of spectra are rather well-behaved. We * *begin with the following lemma, whose proof follows the proofs of [42, Lemma 5.5] and [33, Proposition 3.2] (written in the context of Bousfield-Friedlander spectra)* *. We remind the reader that a fibrant spectrum is positive fibrant. Lemma 5.3.1. In the category of symmetric spectra, filtered colimits preserve cofibrations, fibrations, and positive fibrations. Filtered colimits preserve * *weak equivalences between fibrant spectra, and weak equivalences between positive fi- brant spectra. Corollary 5.3.2. Given a filtered diagram {Xff}ff2Iof (positive) fibrant spectr* *a, there is a stable equivalence hocolimffXff'-!colimffXff. Proof.Let {Xeff} OE-!{Xff} be a cofibrant replacement in the projective model c* *at- egory of I-shaped diagrams of spectra, so that OE is a levelwise acyclic fibrat* *ion. Then each spectrum eXffis (positive) fibrant, and we have hocolimffXff= colimffeXffOE-!'colimffXff. Corollary 5.3.3. (Positive) fibrant discrete G-spectra are (positive) fibrant as non-equivariant spectra. Proof.Let X be a (positive) fibrant discrete G-spectrum. Let U be an open nor- mal subgroup of G. By Proposition 3.3.1(2) (and the obvious analog in the posi- tive fibrant case), X is (positive) fibrant as a discrete U-spectrum. Therefore* *, by Lemma 3.1.1 (Corollary 5.2.4), the U-fixed points XU is (positive) fibrant as a non-equivariant spectrum. The formula X = colimUEXU oG shows that X is (positive) fibrant as a non-equivariant spectrum. Lemma 5.3.4. Filtered colimits in both the category of A-modules and in the category of commutative A-algebras are formed in the category of spectra. Proof.We treat the case of commutative A-algebras _ the case of A-modules is similar. Let {Eff} be a filtered diagram of commutative A-algebras. Then the co* *l- imit in the category of spectra is easily seen to have the structure of a commu* *tative A-algebra with multiplication given by (colimffEff) ^A (colimfiEfi)~=colimffcolimfiEff^A Efi ~=colimEff^A Eff ff ! colimffEff. This filtered colimit is easily seen to satisfy the universal property. We shall henceforth always form filtered colimits of spectra, with the under- standing that we implicitly take functorial (positive) fibrant replacements bef* *ore computing the filtered colimit, if the terms in the colimit are not already (po* *sitive) THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 29 fibrant. Therefore, Corollary 5.3.2 implies that our filtered colimits will alw* *ays have the desired homotopy invariance, so we will never need to take a filtered homot* *opy colimit of spectra. If {Xff} is a filtered diagram of discrete commutative G-A-algebras, then whe* *n- ever we are taking the filtered colimit, we shall implicitly be taking the filt* *ered colimit colimff(Xff)fGA-Alg of the functorial fibrant replacements. Note that t* *he underlying spectrum of each (Xff)fGA-Alg is positive fibrant. Since we often take filtered colimits of commutative A-algebras, in the next * *result (whose proof is similar to that of Corollary 5.3.2), we point out a nice relati* *onship between this colimit and the homotopy colimit in AlgA. This result is also usef* *ul in relating the filtered colimits of Section 8 to the homotopy colimits of commuta* *tive S-algebras that appear in [8, Def. 1.5, Sec. 6]. Lemma 5.3.5. Suppose that {Eff} is a filtered diagram of fibrant commutative A-algebras. Then there is an equivalence colimffEff' hocolimffAlgEff, where the homotopy colimit hocolimAlgis taken in the category of commutative A-algebras. 5.4. Commutative hypercohomology algebras. Let E be a discrete commu- tative G-A-algebra. For any finite set K, the mapping spectrum Map (K, E) is naturally a commutative A-algebra, by using the diagonal on K. Therefore, by Lemma 5.3.4, the continuous mapping spectrum Map c(G, E) = colimUEMap(G=U, E) oG is a commutative A-algebra. Since the category of spectra with the positive model structure is Quillen eq* *uiv- alent to the category of spectra with the stable model structure [31, Prop. 14.* *6], there is an equivalence H+c(G; E) := holim+Map c(Go, E) ' holimMap c(Go, E) = Hc(G; E) between the homotopy limits computed in the positive and stable model structure* *s. Since the homotopy limit of commutative A-algebras is computed in the underlying category of spectra with the positive model structure, we have the following le* *mma. Lemma 5.4.1. The hypercohomology spectrum Hc(G; E) is equivalent to a com- mutative A-algebra H+c(G; E). When we take hypercohomology spectra of discrete commutative G-A-algebras, we shall always implicitly be taking the homotopy limit with respect to the pos* *itive model structure. 6. Profinite Galois extensions Although the homotopy limit of k-local objects is k-local, it is not true in * *general that k-localization commutes with homotopy limits. We begin this section by ex- plaining how, under a certain hypothesis, Assumption 1.0.3 allows us to commute these two functors. We then explain how a profinite Galois extension of a commu- tative symmetric ring spectrum A naturally gives rise to a discrete commutative G-A algebra, and we show that our consistency hypothesis allows us to recover t* *he intermediate finite Galois extensions using the homotopy fixed point constructi* *on. 30 MARK BEHRENS AND DANIEL G. DAVIS 6.1. Properties of k-localization. Recall that we assume that the k-localization functor is given by ((-)T)M , where localization with respect to T is smashing * *and M is a finite spectrum (Assumption 1.0.3). These localizations are the functori* *al fibrant replacements in appropriately localized model categories. In this subse* *ction we shall establish some lemmas concerning such k-localizations. Lemma 6.1.1. If X is a k-local spectrum, then it is a T -local spectrum. Proof.Let f : A ! B be a T -local equivalence. Then it induces an equivalence on T -local spectra fT : AT '-!BT, and hence on the M-localization fk : Ak = (AT)M -'!(BT)M = Bk. Therefore, f is a k-local equivalence, and since X is k-local, the induced map ~= f* : [B, X] -! [A, X] is an isomorphism. Since this is true for every T -local equivalence f, we ded* *uce that X is T -local. Lemma 6.1.2. Let {Xi} be a diagram of spectra. Then there is an equivalence (holimiXi)M ' holimi(Xi)M . Proof.The homotopy limit holimi(Xi)M is M-local, so there is a map f : (holimiXi)M ! holimi(Xi)M . Smashing with M, and using the fact that M is a finite complex, we have the following commutative diagram of equivalences M ^ holimiXi ____'___//holimi(M ^ Xi) '|| |'| fflffl| fflffl| M ^ (holimiXi)M M^f_//_holimi(M ^ (Xi)M ) from which we deduce that f is an M-local equivalence. Since f is a map between M-local spectra, the map f is an equivalence. Arbitrary localizations do not commute with homotopy limits. Our reason for making Assumption 1.0.3 on k-localization is that it allows us to deduce the fo* *l- lowing corollary. Corollary 6.1.3. Let {Xi} be a diagram of T -local spectra. Then there is an equivalence (holimiXi)k ' holimi(Xi)k. Proof.Since the spectra Xi are T -local, the homotopy limit holimiXi is T -loca* *l. Using Lemma 6.1.2, we have the following equivalences: (holimiXi)k' (holimiXi)M ' holimi(Xi)M ' holimi(Xi)k. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 31 Since T -localization is smashing, it possesses the following pleasant proper* *ties. Lemma 6.1.4. (1) Colimits of T -local spectra are T -local. (2) If X is a T -local spectrum and Y is any spectrum, then X ^ Y is T -loca* *l. (3) If X is T -local, then Map c(G, X) is T -local. We end this section with the following lemma and a corollary. Lemma 6.1.5. Suppose that f : X ! Y is a k-local equivalence of T -local discre* *te G-spectra. Then the induced map f* : Hc(G; X)k ! Hc(G; Y )k is an equivalence. Proof.Using Lemma 6.1.4, we see that the hypercohomology functor Hc(G; -) = holimMap c(Go, -) sends T -local spectra to T -local spectra. Therefore, we just need to check th* *at the map f* : Hc(G; X) ! Hc(G; Y ) is an M-local equivalence. Since M is finite and f is an M-local equivalence, we have M ^ Hc(G; X) ' Hc(G; M ^ X) (M^f)*-----! ' Hc(G; M ^ Y ) ' M ^ Hc(G; Y ), which implies that M ^ f* is an equivalence. Theorem 3.2.1 implies the following corollary. Corollary 6.1.6. Suppose that G has finite vcd and that f : X ! Y is a k-local equivalence of T -local discrete G-spectra. Then the induced map f* : XhG ! Y hG is a k-local equivalence. 6.2. Profinite Galois extensions as discrete G-spectra. We first give the def- inition of a profinite Galois extension, which is a slight modification of the * *notion of a pro-G-Galois extension, due to John Rognes (see [36, Section 8.1]). Let A * *be a k-local cofibrant commutative symmetric ring spectrum, let E be a commutative A-algebra, and let G be a profinite group. Definition 6.2.1 (Profinite Galois extension). The spectrum E is a (profaithful) k-local G-Galois extension of A if (1) there is a directed system of (faithful) finite k-local G=Uff-Galois ext* *ensions Effof A, for {Uff} a cofinal system of open normal subgroups of G; (2) all of the maps Eff! Efiare G-equivariant and are cofibrations of under- lying commutative A-algebras; 32 MARK BEHRENS AND DANIEL G. DAVIS (3) for ff fi, letting Kff,fidenote the quotient Uff=Ufi, the natural maps Eff! EhKff,fifiare equivalences; and (4) the spectrum E is the filtered colimit colimffEff. Remark 6.2.2. The spectra Effare k-local, but the spectrum E need not be k-local. However, Assumption 1.0.3 does imply that E is T -local. Proposition 6.2.3. The spectrum E in Definition 6.2.1 is a discrete commutative G-A-algebra. Proof.Clearly, Effis a discrete commutative G-A-algebra. Discrete commutative G-A-algebras are closed under filtered colimits taken in the category of commut* *ative A-algebras. Proposition 6.2.4 (Rognes [36, Sec. 8.1]). If E is a k-local G-Galois extension* * of A, then there are natural equivalences: (E ^A E)k '-!(Map c(G, E))k, (E[[G]])k ' FA (Ek, Ek). 6.3. The consistent hypothesis. In this subsection, we assume that E is a k- local profinite G-Galois extension of A. We recall from the Introduction some terminology: o E is consistent over A if the map A ! A^k,Eis an equivalence; and o E has finite vcd if the profinite group G has finite virtual cohomologic* *al dimension. Proposition 6.3.1. Let E be a k-local profinite G-Galois extension of A of fini* *te vcd. Then there is a natural equivalence A^k,E' (EhG )k, between the k-local Amitsur derived completion and the k-localization of the ho- motopy fixed point spectrum. Proof.By iterating Proposition 6.2.4, the natural map (E_^A_E_^A_.-.^.AEz_______")k ! (Map c(Gn, E))k n+1 is an equivalence. Totalizing the associated cosimplicial spectra and using Cor* *ol- lary 6.1.3 and Theorem 3.2.1, we have: A^k,E = holim(E^Ao+1)k -'! holim(Map c(Go, E)) k ' (holimMap c(Go, E))k ' (EhG )k. Corollary 6.3.2. Let E be a k-local profinite G-Galois extension of A of finite vcd. Then the extension is consistent if and only if the A-algebra unit map A ! (EhG )k is an equivalence. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 33 We shall say that a k-local A-module X is k-locally dualizable if the map (DA (X) ^A X)k ! FA (X, X) is an equivalence. Here, DA (-) = FA (-, A) is the Spanier-Whitehead dual in the category of A-modules. The following standard properties of k-local dualizabili* *ty are contained in [36, Lem. 3.3.2(a),(b)]. Lemma 6.3.3. (1) For k-local A-modules X, Y , and Z, the natural map (FA (X, Y ) ^A Z)k ! FA (X, (Y ^A Z)k) is an equivalence if either X or Z is k-locally dualizable. (2) If the k-local A-module X is k-locally dualizable, then DA (X) is also k- locally dualizable, and the natural map X ! DA (DA (X)) is an equivalence. We note the following useful consequence of k-local dualizability which makes use of Assumption 1.0.3. Lemma 6.3.4. Suppose that X is a k-local A-module which is k-locally dualizable, and that {Yi} is a diagram of T -local A-modules. Then the natural map (X ^A holimiYi)k ! (holimiX ^A Yi)k is an equivalence. Proof.The result follows from the following chain of equivalences: (X ^A holimiYi)k' (DA (DA (X)) ^A holimiYi)k ' (DA (DA (X)) ^A (holimiYi)k)k ' FA (DA (X), (holimiYi)k) ' FA (DA (X), holimi(Yi)k) ' holimiFA (DA (X), (Yi)k) ' holimi(DA (DA (X)) ^A (Yi)k)k ' holimi(DA (DA (X)) ^A Yi)k ' (holimiX ^A Yi)k, where the fourth and last equivalences follow from Corollary 6.1.3. We shall repeatedly use the following dualizability result [36, Props. 6.2.1,* * 6.4.7]. Proposition 6.3.5. If E is a finite k-local Galois extension of A (not required* * to be faithful), then E is a k-locally dualizable A-module. Also, there is a natu* *ral discriminant map (in the stable homotopy category) E ! DA (E), which is an equivalence. 34 MARK BEHRENS AND DANIEL G. DAVIS Given a k-local profinite G-Galois extension E = colimffEffof A, each of the spectra Effcarries a G-action, where the subgroup Uffacts trivially on Eff. Sin* *ce the maps Eff! Efi are G-equivariant, each of the maps Eff! colimfiEfi= E is G-equivariant. Since the subgroup Uffacts trivially on Eff, we get an induced G=Uff-equivariant map Eff! EUff! (EfGA-Alg)Uff' EhUff, where the last equivalence follows from Proposition 3.3.1. Being consistent implies the following consistency result. Lemma 6.3.6. Suppose that E = colimffEffis a consistent k-local profinite G- Galois extension of finite vcd. Then for each ff, the natural G=Uff-equivariant* * map Eff! ((EfGA-Alg)Uff)k ' (EhUff)k is an equivalence. Proof.Since Effis a k-local G=Uff-Galois extension of A, we have a chain of equ* *iv- alences: (Eff^A E)k' (Eff^A colimfiEfffi)k ' (colimfi(ff(Eff^A Eff) ^EffEfi))k ' (colimfi(ffMap(G=Uff, Eff) ^EffEfi))k ' (colimfiMffap(G=Uff, Efi))k ' (Map (G=Uff, E))k, where the G-action on the factor E in (Eff^A E)k corresponds to the conjugation action on (Map (G=Uff, E))k. By Corollary 6.3.2, the natural map (6.1) A ! (EhG )k is an equivalence. Smashing (6.1) over A with Eff, using Theorem 3.2.1, employ- ing the fact that Effis a k-locally dualizable A-module (Proposition 6.3.5 and Lemma 6.3.4), and applying Corollary 6.1.3 and Shapiro's Lemma (3.4.2), we have THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 35 the following equivalences: Eff' (Eff^A A)k ' (Eff^A EhG )k ' (Eff^A holimMap c(Go, E))k ' (holimMap c(Go, Eff^A E))k ' holimMap c(Go, Eff^A E)k ' holimMap c(Go, Map(G=Uff, E))k ' (holimMap c(Go, Map(G=Uff, E)))k ' (Map (G=Uff, E)hG)k ' (EhUff)k. Adding the profaithful hypothesis allows us to expand the particular system {Uff} of open normal subgroups of G to the collection of all open normal subgro* *ups of G. Proposition 6.3.7. Let E be a consistent profaithful k-local profinite G-Galois extension of A of finite vcd. (1) For each open normal subgroup U of G, (EhU )k is a faithful k-local G=U- Galois extension of A. (2) If U V are a pair of open subgroups of G with U normal in V , then (EhU )k is a faithful k-local V=U-Galois extension of (EhV )k. Proof.For both parts, we repeatedly use the fundamental theorem of Galois theory [36, Thm. 1.2]. (1) Choose ff so that Uffis contained in U. Then by Lemma 6.3.6, there is an equivalence Eff' (EhUff)k, so (EhUff)k is a faithful k-local G=Uff-Galois exten* *sion of A. Proposition 3.3.1 implies that there are equivalences (EhU )k ' ((EhUff)hU=Uff)k ' ((EhUff)k)hU=Uff. Therefore, the fundamental theorem of Galois theory implies that (EhU )k is a faithful k-local G=U-Galois extension of A. (2) Let N be an open normal subgroup of G contained in U. By (1), we know that (EhN )k is a faithful k-local G=N-Galois extension of A. By Proposition 3.* *3.1, we have (EhV )k ' ((EhN )hV=N)k ' ((EhN )k)hV=N. Thus, Galois theory implies that (EhN )k is a faithful k-local V=N-Galois exten* *sion of (EhV )k. As before, we have (EhU )k ' ((EhN )k)hU=N. Since U=N is normal in V=N, with quotient V=U, Galois theory implies that (EhU * *)k is a faithful k-local V=U-Galois extension of (EhV )k. 36 MARK BEHRENS AND DANIEL G. DAVIS 7.Closed homotopy fixed points of profinite Galois extensions Let E be a consistent profaithful k-local profinite G-Galois extension of A of finite vcd. We begin by showing that under these hypotheses, the H-homotopy fixed points functor is well-behaved whenever H is an arbitrary closed subgroup of G. We then prove the forward part of the profinite Galois correspondence, and we compute the homotopy type of the function spectrum between arbitrary k-local closed homotopy fixed point spectra of E. 7.1. Iterated Galois homotopy fixed points. In this subsection we will extend the results of Section 3.5 to all closed subgroups H of G. Let j : H ,! G be the inclusion of the closed subgroup H. Also, recall that, by hypothesis, G has fin* *ite vcd. Then we prove the following theorem and derive consequences from it. Theorem 7.1.1. The natural map (Hcolim UEhU )k ! (EhH )k oG is an equivalence. In order to prove Theorem 7.1.1, we need to introduce a spectrum E0 which is equivalent to E, but which has better point-set level properties. Observe that * *by Proposition 6.3.7, the collection of homotopy fixed point spectra {(EhU )k}UEoG gives rise to a k-local profinite G-Galois extension E0= colimUE(EhU )k. oG of A. Strictly speaking, given an open subgroup V of G, the spectrum E is not an (EhV )k-algebra. Since the spectrum (EhU )k is an (EhV )k-algebra for every open normal subgroup U of G contained in V , the spectrum E0 is an (EhV )k-algebra. Furthermore, by Lemma 6.3.6, the map E = colimffEff! colimff(EhUff)k ~=E0 is an equivalence of discrete commutative G-A-algebras. We shall need the following fundamental lemma. Lemma 7.1.2. Let V be an open normal subgroup of G. Then the natural map ((EhV )k ^(EhHV)k(E0)hH )k ! ((E0)h(H\V ))k, induced from the commutative diagram (EhHV )k______//(colimUEoG(EhU )k)H_____//(E0)hH | | | | fflffl| fflffl| (EhV )k_____//_(colimUEoG(EhU )k)H\V___//(E0)hH\V of commutative symmetric ring spectra, is an equivalence. Proof.The lemma will be proven by showing that there exists a zig-zag of k-local equivalences between (EhV )k ^(EhHV)k(E0)hH and (E0)hH\V THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 37 which are maps both of commutative (EhV )k-algebras and commutative (E0)hH - algebras. Let Q be the finite group HV=V ~=H=(H \ V ). Note that, by Proposi- tion 6.3.7, the map (EhHV )k ! (EhV )k is a k-local Q-Galois extension. Observe that there is a zig-zag of maps: (EhV )k ^(EhHV)k(E0)hH = (EhV )k ^(EhHV)k(E0fH)H ~= ((EhV )k ^(EhHV)kE0fH)H w-! (((EhV ) 0 H k ^(EhHV)kEfH )fH ) = ((EhV )k ^(EhHV)kE0fH)hH -' ((EhV ) 0 hH k ^(EhHV)kE ) . Each of the maps above is a map of commutative (EhV )k-algebras and commutative (E0)hH -algebras. Furthermore, the map w is a k-local equivalence, since we hav* *e a commutative diagram (EhV )k ^(EhHV)k(E0fH)H___u____//_(EhV )k ^(EhHV)kHc(H; E0fH) w || |w0| fflffl| fflffl| (((EhV )k ^(EhHV)kE0fH)fH )Hu0_//Hc(H; ((EhV )k ^(EhHV)kE0fH)fH ) where the maps u and u0are equivalences by Theorem 3.2.1, and the map w0is seen to be a k-local equivalence by using the fact that (EhV )k is a k-locally duali* *zable (EhHV )k-module (Proposition 6.3.5 and Lemma 6.3.4). The composite (7.1) (EhV )k ^(EhHV)kE0~=((EhV )k ^(EhHV)k(EhV )k) ^(EhV)kE0 (7.2) ! Map (Q, (EhV )k) ^(EhV)kE0 (7.3) -'!Map (Q, E0) is a k-local equivalence. Here, the smash product ^(EhV)kon the right-hand side of (7.1) uses the left (EhV )k-module structure on (EhV )k ^(EhHV)k(EhV )k. Und* *er the isomorphism E0~=(EhV )k ^(EhV)kE0 the G-action on E0 is transformed to the diagonal action on (EhV )k ^(EhV)kE0. Therefore, under (7.1)-(7.3), the H-action on E0is transformed to the conjugati* *on action on Map (Q, E0). Furthermore, under (7.1)-(7.3): (1) the E0-algebra structure on (EhV )k^(EhHV)kE0is sent to that given by the inclusion E0! Map (Q, E0) of the constant maps, and (2) the (EhV )k-module structure on (EhV )k ^(EhHV)kE0 is sent to that given by the composite (EhV )k ,-!Map(Q, (EhV )k) ! Map (Q, E0), where , is the adjoint of the Q-action map. 38 MARK BEHRENS AND DANIEL G. DAVIS Taking H-homotopy fixed points of (7.1)-(7.3) gives, by Corollary 6.1.6, a k-lo* *cal equivalence ((EhV )k ^(EhHV)kE0)hH ! Map (Q, E0)hH which is a map of commutative (EhV )k-algebras and of commutative (E0)hH - algebras. The proof of the lemma is completed by observing that the equivalence given by Shapiro's Lemma (Lemma 3.4.2) (E0)hH\V -'!Map (H=(H \ V ), E0)hH ~=Map (Q, E0)hH is a map of commutative (EhV )k-algebras and of commutative (E0)hH -algebras. Proof of Theorem 7.1.1.Choose an open normal subgroup V of G of finite coho- mological dimension. By Proposition 3.5.3, we see that the map (7.4) H colimU Eh(U\V )! Eh(H\V ) oHV is an equivalence. Let Q = HV=V ~=H=(H \ V ) be the finite quotient group. For each open subgroup U of HV containing H, we have UV = HV . Therefore, there is an isomorphism Q = UV=V ~=U=(U \ V ). By Proposition 6.3.7, the extensions (EhU )k ! (Eh(U\V ))k, (EhUV )k ! (EhV )k are faithful k-local Q-Galois extensions. Therefore, by Remark 1.0.2, the norm maps ((Eh(U\V ))hQ)k ! ((Eh(U\V ))hQ)k, ((EhV )hQ)k ! ((EhV )hQ)k THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 39 are equivalences. Therefore, by using Proposition 3.3.1 and Lemma 7.1.2, we have the following sequence of equivalences: (Hcolim UEhU )k' ( colim EhU )k oG H U oHV ' (H colimU (Eh(U\V ))hQ)k oHV ' (H colimU (Eh(U\V ))hQ)k oHV ' ((H colimU Eh(U\V ))hQ)k oHV ' ((EhH\V )hQ)k ' (((E0)hH\V )hQ)k ' (((EhV )k ^(EhHV)k(E0)hH )hQ)k ' (((EhV )hQ)k ^(EhHV)k(E0)hH )k ' (((EhV )hQ)k ^(EhHV)k(E0)hH )k ~=(((EhV )h(HV=V ))k ^(EhHV)k(E0)hH )k ' ((EhHV )k ^(EhHV)k(E0)hH )k ' ((E0)hH )k. ' (EhH )k. Using the methods of Section 3.5, Theorem 7.1.1 has the following corollary. Corollary 7.1.3. (1) There is an equivalence ((EfG)H )k ' (EhH )k. (2) If H is a normal subgroup of G, then, using the model for H-homotopy fixed points given by part (1), there is an equivalence ((EhH )hG=H )k ' (EhG )k. 7.2. Intermediate Galois extensions. In this subsection we will prove the for- ward direction of the profinite Galois correspondence. Theorem 7.2.1. Suppose that H is a closed subgroup of G. (1) The spectrum E is k-locally H-equivariantly equivalent to a consistent p* *ro- faithful k-local H-Galois extension of (EhH )k of finite vcd. (2) If H is a normal subgroup of G, then the spectrum EhH is k-locally G=H- equivariantly equivalent to a profaithful k-local G=H-Galois extension of A. If the quotient G=H has finite vcd, then this extension is consistent (and of finite vcd) over A. Remark 7.2.2. It is useful to note that if G is a compact p-adic analytic group, then for any closed normal subgroup H of G, the quotient group G=H is a compact p-adic analytic group, and therefore must also have finite vcd [9, Thm. 9.6], [* *41]. Remark 7.2.3. Theorem 7.2.1, when applied to the K(n)-local profinite Galois extension Fn of SK(n), provides extensions of [36, Thm. 5.4.4, Prop. 5.4.9]. 40 MARK BEHRENS AND DANIEL G. DAVIS Proof of part (1).We shall prove that E is k-locally H-equivariantly equivalent, as a discrete commutative H-algebra, to a spectrum L which is an extension of (EhH )k. We will prove that the extension (EhH )k ! L is a consistent profaithful k-local H-Galois extension by proving that there is* * a commutative diagram of commutative symmetric ring spectra (7.5) LO____'_____//_L0OOO | | | | | | (EhH )k__'__//((E0)hH )k where E0 is the discrete commutative G-A-algebra, equivalent to E, introduced before Lemma 7.1.2, and L0is a profaithful k-local H-Galois extension of ((E0)h* *H )k. The proof concludes by showing directly that L is consistent over (EhH )k. Since H is a closed subgroup of G, a group of finite vcd, we may conclude that H has finite vcd. The system {H \ V }V EoGis cofinal in the system of open norm* *al subgroups of H. Let U = H \ V be one of these open normal subgroups. ((E0)hU)k is k-locally H=U-Galois over ((E0)hH )k. We must check that the last * *two conditions of Definition 1.0.1 are satisfied. By Proposition 3.3.1 and Corollar* *y 6.1.3, we have (((E0)hU)k)hH=U' (((E0)hU)hH=U )k ' ((E0)hH )k, which verifies the second condition. The third condition is verified through t* *he use of Lemma 7.1.2 and the fact that (EhV )k is a faithful k-local HV=V -Galois extension of (EHV )k (Proposition 6.3.7): (((E0)hU)k ^((E0)hH)k((E0)hU)k)k ' ((((E0)hH )k ^(EhHV)k(EhV )k)k ^((E0)hH)k(((E0)hH )k ^(EhHV)k(EhV )k)k)k ' (((E0)hH )k ^(EhHV)k(EhV )k ^(EhHV)k(EhV )k)k ' (((E0)hH )k ^(EhHV)kMap (HV=V, (EhV )k))k ~=(((E0)hH )k ^(EhHV)kMap (H=U, (EhV )k))k ' Map (H=U, (((E0)hH )k ^(EhHV)k(EhV )k)k) ' Map (H=U, ((E0)hU)k). ((E0)hU)k is k-locally faithful over ((E0)hH )k. Suppose M is an ((E0)hH )k-mod* *ule and that we have (((E0)hU)k ^((E0)hH)kM)k ' *. We must show Mk is null. We use Lemma 7.1.2 to deduce * ' (((EhV )k ^(EhHV)k((E0)hH )k)k ^((E0)hH)kM)k ' ((EhV )k ^(EhHV)k^M)k. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 41 By Proposition 6.3.7, we deduce that (EhV )k is k-locally faithful over (EhHV )* *k, so we may conclude that Mk is null. Let L and L0be defined by the colimits L := colimV(EEhH\V )k, oG L0 := colimV(E(E0)hH\V )k. oG We have shown that the spectrum L0is a profaithful k-local H-Galois extension of ((E0)hH )k. Furthermore, the equivalence E ! E0 of discrete commutative G-A-algebras gives rise to Diagram (7.5). E is k-locally H-equivariantly equivalent to L. By Corollary 7.1.3, for each V * *we have ((EfG)H\V )k ' (EhH\V )k. Since EfG is a discrete H-spectrum, we have: (EfG)k = (colimV(EEfG)H\V )k oG ' (colimV(EEhH\V )k)k oG = Lk. The fibrant replacement map E ! EfG is an H-equivariant equivalence. L is consistent over (EhH )k. By Corollary 6.3.2, we just need to check that the map (7.6) (EhH )k ! (LhH )k is an equivalence. We have already seen that the map E ! L is a k-local equiva- lence. By Corollary 6.1.6, we see that the map of (7.6) is an equivalence. Proof of part (2).Let K be the colimit colimUEoG(EhHU )k. Since H is normal in G, the groups HU are open normal subgroups of G. By Proposition 6.3.7, the spectra (EhHU )k are k-local faithful G=HU-Galois extensions of A. Therefore, K is a k-local profaithful G=H-Galois extension of A. The spectrum K is k-locally equivalent to EhH by Theorem 7.1.1. Suppose that G=H is of finite vcd. We are left with showing that K is consist* *ent over A. By Corollary 6.3.2, it suffices to check that the map A ! (KhG=H )k 42 MARK BEHRENS AND DANIEL G. DAVIS is an equivalence. Using Corollary 6.1.3, Theorem 7.1.1, Corollary 7.1.3, and t* *he fact that E is consistent over A, we have: (KhG=H )k= ((colimUE(EhHU )k)hG=H )k oG ' ((colimUEEhHU )hG=H )k oG ' (((EfG)H )hG=H )k ' (EhG )k ' A. 7.3. Function spectra. In this section, we prove the following theorem. Theorem 7.3.1. Let H and K be closed subgroups of G. Then there is an equiv- alence FA ((EhH )k, (EhK )k) ' ((E[[G=H]])hK )k, where E[[G=H]] has the diagonal K-action. Corollary 7.3.2. If H and K are closed subgroups of G, and the left action of K on G=H is trivial, then there is an equivalence FA ((EhH )k, (EhK )k) ' (EhK [[G=H]])k. Proof.We have the following sequence of equivalences: FA ((EhH )k, (EhK )k)' ((E[[G=H]])hK )k ' (Hholim U(E[G=U])hK )k oG ' (Hholim UEhK [G=U])k oG ' (EhK [[G=H]])k. Remark 7.3.3. The conclusion of Corollary 7.3.2 is typically far from true for arbitrary H and K. For instance, let n be odd. It is shown in [40, Prop. 16] th* *at in the case of k = K(n), A = SK(n), E = Fn, G = Gn, H = {e}, and K = Gn, the K(n)-local Spanier-Whitehead dual of En is given by: 2 F (En, EhGnn) ' F (En, SK(n)) ' -n En 6' En. The remainder of this section will be spent proving Theorem 7.3.1. We first prove some technical lemmas. Recall that an A-module X is said to be k-locally F -small if the natural map colimiFA (X, Yi) ! FA (X, (colimiYi)k) is a k-local equivalence for every filtered diagram {Yi}i of k-local A-modules.* * Ob- serve that if X is a k-locally dualizable A-module, then it is k-locally F -sma* *ll, since THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 43 we have: (colimiFA (X, Yi))k' (colimi(DA (X) ^A Yi)k)k ' (colimi(DA (X) ^A Yi))k ' (DA (X) ^A colimiYi)k ' FA (X, (colimiYi)k). Lemma 7.3.4. Suppose that X is an A-module which is k-locally F -small, and that Y is a k-local A-module. Let T = limiTi be a profinite set. Then the natur* *al map Map c(T, FA (X, Y )) ! FA (X, Mapc(T, Y )k) is a k-local equivalence. Proof.We have: Map c(T, FA (X, Y ))k= (colimiMap(Ti, FA (X, Y )))k ~=(colimFA (X, Map(Ti, Y )))k i ' FA (X, (colimiMap(Ti, Y ))k) = FA (X, Mapc(T, Y )k). The following lemma is immediate from the definition of Map c. Lemma 7.3.5. Let {Yj}j be a filtered diagram of spectra and let T = limiTibe a profinite set. Then the natural map colimjMapc(T, Yj) ! Map c(T, colimjYj) is an isomorphism. Lemma 7.3.6. Let U be an open subgroup of G, and let V be an open normal subgroup of G, such that V U. Then there is a map of discrete G-A-modules , : A[G=U] ^A (EhU )k ! (EhV )k, where G acts on the source of , by acting only on A[G=U]. Proof.To produce the map ,, it suffices to construct the adjoint map of sets e,: G=U ! Mod A((EhU )k, (EhV )k). Observe that for g 2 G the G-action map g : E ! E descends to a map _ g: EhU ! EhV , which localizes to give e,(gU) = (_g)k : (EhU )k ! (EhV )k. It is easy to check that this map is independent of choice of coset representat* *ive. To show that , is G-equivariant we must show that e,is G-equivariant, where G acts on the morphism set Mod A((EhU )k, (EhV )k) by postcomposition. This is cl* *ear from the definition of e,. 44 MARK BEHRENS AND DANIEL G. DAVIS The map , gives rise to a map _V : (EhV )k[G=U] ! FA ((EhU )k, (EhV )k) as follows: the adjoint e_Vis given by the composite e_V: (EhV )k ^A (A[G=U] ^A (EhU )k)1^,--!(EhV )k ^A (EhV )k -~! (EhV ) k, where ~ : (EhV )k ^A (EhV )k ! (EhV )k is the multiplication map of the A-algeb* *ra (EhV )k. The map _V is easily checked to be G-equivariant, where G acts diagona* *lly on the source and acts on the target through its action on the term (EhV )k. Lemma 7.3.7. Let U be an open subgroup of G, and suppose that V is an open normal subgroup of G contained in U. Then the map _V : (EhV )k[G=U] '-!FA ((EhU )k, (EhV )k). is an equivalence. Proof.By Proposition 6.3.7, the spectrum (EhV )k is a k-local G=V -Galois exten* *sion of A, and by Proposition 3.3.1, there is an equivalence EhU ' (EhV )hU=V. By Proposition 6.3.5, the A-module (EhV )k is k-locally dualizable. Making use * *of Corollary 6.1.3 and Lemma 6.3.4, we have: ((EhV )k ^A (EhU )k)k' ((EhV )k ^A ((EhV )hU=V)k)k ' ((EhV )k ^A ((EhV )k)hU=V)k ' (((EhV )k ^A (EhV )k)hU=V)k ' Map (G=V, (EhV )k)hU=V ' Map (G=U, (EhV )k), where the last equivalence follows from the fact that the right U=V -action on * *G=V is free. Applying F(EhV)k(-, (EhV )k) to both sides, we have: FA ((EhU )k, (EhV )k)~=F(EhV)k((EhV )k ^A (EhU )k, (EhV )k) ' F(EhV)k(((EhV )k ^A (EhU )k)k, (EhV )k) ' F(EhV)k(Map (G=U, (EhV )k), (EhV )k) ' F(EhV)k((EhV )k, (EhV )k[G=U]) ' (EhV )k[G=U]. This sequence of equivalences may be checked to be compatible with the map _V . Lemma 7.3.8. Let U be an open subgroup of G. There is an equivalence of discrete G-spectra OE : E[G=U] '-!V colimE FA ((EhU )k, (EhV )k). oG,V U Here, G is acting diagonally on the left-hand side and acting only on each (EhV* * )k on the right-hand side. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 45 Remark 7.3.9. Let V be an open normal subgroup of G and let U be an open subgroup of G. Since EhV is a G=V -spectrum, the function spectrum FA ((EhU )k, (EhV )k) is a discrete G-spectrum, where G is acting only on the spectrum (EhV )k. The colimit V colimE FA ((EhU )k, (EhV )k) oG,V U is therefore a discrete G-spectrum. Proof of 7.3.8.The map OE is given as the composite E[G=U] -'! (colimV(EEhV )k)[G=U] oG -'! colim((EhV ) [G=U]) V EoG k -_! colim F ((EhU ) , (EhV ) ). V EoG,V UA k k The map _ is the colimit of the equivalences _V : (EhV )k[G=U] ! FA ((EhU )k, (EhV )k) of Lemma 7.3.7. Therefore, _ is an equivalence, so OE is an equivalence. Proof of Theorem 7.3.1.We have equivalences: ((E[[G=H]])hK )k' Hholim UholimMapc(Ko, E[G=U])k oG ' Hholim UholimMapc(Ko, colim FA ((EhU )k, (EhV )k))k oG V EoG,V U ' Hholim Uholim( colim Map c(Ko, FA ((EhU )k, (EhV )k)))k oG V EoG,V U ' Hholim Uholim( colim FA ((EhU )k, Mapc(Ko, (EhV )k)k))k oG V EoG,V U ' Hholim UholimFA ((EhU )k, ( colim Map c(Ko, (EhV )k))k) oG V EoG,V U ' Hholim UholimFA ((EhU )k, Mapc(Ko, colim (EhV )k)k) oG V EoG,V U ' Hholim UholimFA ((EhU )k, Mapc(Ko, E)k) oG ' Hholim UFA ((EhU )k, (holimMap c(Ko, E))k) oG ' Hholim UFA ((EhU )k, (EhK )k) oG ' FA (Hcolim U(EhU )k, (EhK )k) oG ' FA ((EhH )k, (EhK )k). 8. Applications to Morava E-theory 46 MARK BEHRENS AND DANIEL G. DAVIS 8.1. Morava E-theory as a profinite Galois extension. The general theory developed in this paper applies to Morava E-theory. In this setting, we have k = K(n), A = SK(n), G = Gn, where K(n) is the nth Morava K-theory spectrum, SK(n) is the K(n)-local sphere spectrum, and Gn is the nth extended Morava stabilizer group: Gn = Sn o Gal(Fpn=Fp). Let En be the nth Morava E-theory spectrum, where (En)* = W (Fpn)[[u1, ..., un-1]][u 1]. Here, the degree of u is -2 and the complete power series ring is in degree zer* *o. Goerss and Hopkins [14], building on work of Hopkins and Miller [35], showed th* *at Gn acts on En by maps of commutative S-algebras. Devinatz and Hopkins [8] constructed homotopy fixed point spectra EdhHn for closed subgroups H of Gn. (Here, we use the notation EdhHnto distinguish the Devinatz-Hopkins homotopy fixed point spectra from the homotopy fixed point spectra constructed in this paper.) Rognes [36, Thm. 5.4.4, Lem. 4.3.7] observed, for U an open normal subgroup of Gn, that the work of Devinatz and Hopkins [8] proves that EdhUnis a faithful K(n)-local Gn=U-Galois extension of SK(n). Therefore, the discrete commutative Gn-SK(n)-algebra Fn = colimUEEdhUn oGn is a profaithful K(n)-local profinite Gn-Galois extension of SK(n). The spectrum En is recovered by the equivalence (see [8, Def. 1.5, Thm. 3(i)]) En ' (Fn)K(n). With this in mind, we make the following definition. Definition 8.1.1. For H a closed subgroup of Gn, we define EhHn:= (FnhH)K(n), which is a commutative SK(n)-algebra, by Lemma 5.2.6. We note that the use of pro-spectra gives an alternative, but equivalent appr* *oach to defining EhHn. Let {MI}I be a cofinal collection of generalized Moore spectra [23, Prop. 7.10]. Then the pro-spectrum En = {Fn ^ MI}I is a continuous H-spectrum. Since Fn is E(n)-local and each MI is a finite spec- trum, the homotopy fixed points are identified by EhHn = holimI(Fn ^ MI)hH ' holimIFnhH^ MI ' (FnhH)K(n). THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 47 Thus, the homotopy fixed points of the continuous H-spectrum En coincide with the K(n)-localization of the homotopy fixed points of the discrete H-spectrum F* *n. In particular, Definition 8.1.1 is equivalent to the definition of EhHngiven in* * [5]. Proposition 8.1.2. The profaithful K(n)-local profinite Gn-Galois extension Fn of SK(n) is consistent and has finite vcd. Proof.There is a zig-zag ^SK(n)o+1 ' ^o+1 ' ^o+1 ^o+1 (Fn )K(n)- (Fn )K(n)-! ((Fn)K(n))K(n)' (En )K(n) of levelwise equivalence of cosimplicial objects. We therefore have equivalences ^SK(n)o+1 (SK(n))^K(n),Fn= holim(Fn )K(n) ' holim(E^o+1n)K(n) ' SK(n), where the last equivalence follows from the fact that the cosimplicial object (E^o+1n)K(n) is the K(n)-local En-Adams resolution for SK(n). Since Gn is a compact p-adic analytic group, it has finite virtual cohomologi* *cal dimension. Therefore, the extension Fn of SK(n) has finite vcd. Corollary 8.1.3. There is a weak equivalence EhGnn' SK(n). Proof.This follows immediately from Corollary 6.3.2. Remark 8.1.4. Because of the result of Devinatz and Hopkins that there is an equivalence EdhGnn' SK(n)[8, Theorem 1(iii)], it has been known for some time t* *hat the K(n)-local sphere behaves like a homotopy fixed point spectrum; Corollary 8* *.1.3 makes this idea precise. 8.2. Comparison with the Devinatz-Hopkins homotopy fixed points. Let H be a closed subgroup of Gn. The following theorem relates the homotopy fixed point construction EhHnof Definition 8.1.1 to the Devinatz-Hopkins homotopy fix* *ed point construction EdhHnof [8]. Theorem 8.2.1. If H is a closed subgroup of Gn, there is an equivalence EdhHn' EhHn. Proof.As explained in Subsection 8.1, Fn = colimUEoGnEdhUnis a consistent pro- faithful K(n)-local profinite Gn-Galois extension of SK(n) of finite vcd. Thus,* * by Lemma 6.3.6, for each U Eo Gn, there is a Gn=U-equivariant equivalence EdhUn' (FnhU)K(n). Therefore, given a generalized Moore spectrum MI, there is an equivalence (8.1) EdhUn^ MI ' FnhU^ MI. By Theorem 7.1.1, the natural map (H colimVFnhV)K(n)! (FnhH)K(n) oGn 48 MARK BEHRENS AND DANIEL G. DAVIS is an equivalence. Let V be any open subgroup of Gn. Then V contains a subgroup W , such that W is an open normal subgroup of Gn. We have the following chain of equivalences: EdhVn^ MI ' (EdhWn)hV=W ^ MI ' (EdhWn^ MI)hV=W ' (FnhW ^ MI)hV=W ' (FnhW)hV=W ^ MI ' FnhV^ MI, where the first equivalence is [8, Thm. 4], the second and fourth equivalences * *follow from the fact that MI is a finite spectrum, the third equivalence is because of* * (8.1), and the last equivalence is due to Proposition 3.3.1. Recall from Definition 8.1.1 that EhHn= (FnhH)K(n). Then the above observa- tions imply that EhHn ' (H colimVFnhV)K(n) oGn ' holimIcolimH V(FnhV^ MI) oGn ' holimIcolimH V(EdhVn^ MI) oGn ' (H colimVEdhVn)K(n) oGn ' EdhHn, where the last equivalence follows from [8, Def. 1.5]. Remark 8.2.2. As mentioned in the Introduction, Theorem 8.2.1 first appeared in the second author's thesis [4]. The arguments in [4] relied on a somewhat co* *m- plicated analysis of a K(n)-local En-Adams resolution of EdhHn, whereas our pro* *of makes use of Rognes's Galois theory, and, consequently, is more efficient. Corollary 8.2.3. If H is a closed subgroup of Gn and X is a finite spectrum, th* *en there is an equivalence EdhHn^ X ' (En ^ X)hH . Proof.By [5, Thm. 1.3, Rmk. 9.3], En ^ X is a continuous H-spectrum (in the sense of [5]). Then, by Theorem 8.2.1 and [5, Thm. 9.9], EdhHn^ X ' EhHn^ X ' (En ^ X)hH . Corollary 8.2.4. If X is a finite spectrum, there is an equivalence XK(n)' (En ^ X)hGn. Proof.By [5, Thm. 9.9] and Corollary 8.1.3, (En ^ X)hGn ' EhGnn^ X ' SK(n)^ X ' XK(n), where the last equivalence follows from the fact that X is finite. Let X be a finite spectrum. By [8, Thm. 2(ii)], there is a strongly convergent K(n)-local En-Adams spectral sequence that has the form (8.2) Hsc(H; sst(En ^ X)) ) sst-s(EdhHn^ X). THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 49 Also, by [5, Thm. 1.7], there is a descent spectral sequence (8.3) Hsc(H; sst(En ^ X)) ) sst-s((En ^ X)hH ). Theorem 8.2.5. If H is a closed subgroup of Gn and X is a finite spectrum, then spectral sequence (8.2) is isomorphic to spectral sequence (8.3), from the E2-t* *erms onward. Proof.By [18, proof of Prop. 7.4], spectral sequence (8.2) is the inverse limit* * over {I} of K(n)-local En-Adams spectral sequences that have the form (8.4) IEs,t2(I) = Hsc(H; sst(En ^ MI ^ X)) ) sst-s(EdhHn^ MI ^ X). Similarly, spectral sequence (8.3) is the inverse limit over {I} of conditional* *ly con- vergent descent spectral sequences that have the form (8.5) IIEs,t2(I) = Hsc(H; sst(En ^ MI ^ X)) ) sst-s(EhHn^ MI ^ X). Henceforth, we write the E2-terms IEs,t2(I) and IIEs,t2(I) as IEs,t2and IIEs,t2, respectively. Note that spectral sequence (8.4) is isomorphic to the strongly convergent K(* *n)- local En-Adams spectral sequence (8.6) IEs,t2~=Hsc(H; (En)-t(DX ^ DMI)) ) (EdhHn)-t+s(DX ^ DMI). Thus, to prove the theorem, it suffices to show that the spectral sequences in * *(8.5) and (8.6) are isomorphic to each other. Notice that IEs,t2~=IIEs,t2 ~= colimHs(H=(H \ N); sst(EdhN ^ MI ^ X)) NEoGn n ~= colimHs(NH=N; (EdhN )-t(DX ^ DMI)) NEoGn n = colimNEIIIEs,t2(N), oGn where IIIEs,t2(N) is the E2-term of the strongly convergent spectral sequence IIIE*,*r(N), which has the form (8.7) Hs(NH=N; (EdhNn)-t(DX ^ DMI)) ) (EdhNHn)-t+s(DX ^ DMI) and is the Adams spectral sequence constructed by Devinatz in [7, (0.1)]. By Lemma 8.2.7 below, there is a map from spectral sequence (8.7) to spectral sequence (8.6), such that the isomorphism IEs,t2~=colimIIIEs,t(N) NEoGn 2 implies that the spectral sequence of (8.6) is isomorphic to the spectral seque* *nce colimNEoGnIIIE*,*r(N). Thus, we only have to show that spectral sequences (8.5) and colimNEoGnIIIE*,*r(N) are isomorphic to each other. By [7, Thm. A.1], IIIE*,*r(N) is isomorphic to the usual descent spectral seq* *uence IVE*,*r(N) that has the form Hs(NH=N; (EdhNn)-t(DX ^ DMI)) ) ((EdhNn)hNH=N )-t+s(DX ^ DMI), 50 MARK BEHRENS AND DANIEL G. DAVIS since, in the notation of [7, App. A], the "homotopy fixed point spectral seque* *nce" with abutment [EdhNHn^ DX ^ DMI, (EdhNn)hNH=N ]*EdhNHn, which is isomorphic to [DX ^ DMI, (EdhNn)hNH=N ]*, is equivalent to IVE*,*r(N). Because of the isomorphism NcolimEIIIE*,*r(N) ~=colimIV E*,*r(N), oGn NEoGn our proof reduces to showing that (8.5) and colimNEoGnIVE*,*r(N) are isomorphic spectral sequences. The abutment of spectral sequence (8.5) is the homotopy of EhHn^ MI ^ X ' (Fn ^ MI ^ X)hH = holimMap c(Ho, colimNE(EdhNn^ MI ^ X)) oGn ~=holim colimMap c((NH=N)o, EdhN ^ MI ^ X), NEoGn n where the first equivalence is by [5, Cor. 9.8] and the second equivalence is * *an identification, given by Theorem 3.2.1. Since NH=N is a finite group, there is * *an identification (EdhNn^ MI ^ X)hNH=N = holimMap c((NH=N)o, EdhNn^ MI ^ X). Thus, for each U Eo Gn, the canonical map Mapc((UH=U)o, EdhUn^ MI ^ X) ! colimNEMapc((NH=N)o, EdhNn^ MI ^ X) oGn of cosimplicial spectra induces a map (EdhUn^ MI ^ X)hUH=U ! (Fn ^ MI ^ X)hH and a map _U :VE*,*r(U) ! IIE*,*r of conditionally convergent spectral sequences, where VE*,*r(U) is the descent * *spec- tral sequence that has the form Hs(UH=U; sst(EdhUn^ MI ^ X)) ) sst-s((EdhUn^ MI ^ X)hUH=U ). It will be helpful to note that the abutment of VE*,*r(U) can also be written as sst-s((EdhUn)hUH=U ^ MI ^ X). Since the map colimNEoGn_N induces an isomorphism colimNEVEs,t2(N) ~=IIEs,t2, oGn there is an isomorphism between spectral sequences (8.5) and colimNEoGnVE*,*r(N* *). Therefore, the proof is completed by showing that there is an isomorphism colimNEIVE*,*r(N) ~=colimV E*,*r(N) oGn NEoGn of spectral sequences; this follows from the fact that spectral sequences IVE*,* **r(N) and VE*,*r(N) are equivalent to each other. The following results are needed for the above proof. THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 51 Lemma 8.2.6. Suppose that A is a k-local commutative symmetric ring spectrum and that E is a k-local commutative A-algebra. Then the canonical k-local E- resolution of A in the category of A-modules * ! A ! E ! (E ^A E)k ! (E ^A E ^A E)k ! . . . is a k-local E-resolution of A in the category of S-modules. Proof.We use the terminology of [34], adapted to the category of k-local A-modu* *les as in [7, Sec. 2]. To prove the lemma, we will show that the associated k-local* * E- Adams resolution of A-modules (8.8) A ____A0 oo_______A1 oo______________A2 . . . 99 CC FFF x< 0, the spectra (E^As)k are k-local E-injective, and (2) that each of the ma* *ps ji in (8.8) is k-local E-monic. Claim (1) follows from the fact that the map (E^As)k = (S ^ E^As)k ! (E ^ E^As)k is split-monic. Claim (2) follows from the fact that every k-local E-monic map * *of A-modules is k-local E-monic as a map of S-modules. Indeed, if f : X ! Y is a k-local E-monic map of A-modules, then consider the following diagram. f X ____________//_Y u|| || fflffl| fflffl| (E ^A X)k_1^f_//(E ^A Y )k The map u is seen to be a k-local E-monic map of S-modules because the map (E ^ X)k 1^u--!(E ^ E ^A X)k is split-monic. Because f is a k-local E-monic map of A-modules, the map 1 ^ f * *is split-monic, and therefore 1 ^ f is a k-local E-monic map of S-modules. We dedu* *ce that f is a k-local E-monic map of S-modules. Lemma 8.2.7. Let H be a closed subgroup of Gn and let N be an open normal subgroup. Then, for any spectrum Z, there is a natural map from the Adams spectral sequence Hs(NH=N; (EdhNn)-t(Z)) ) (EdhNHn)-t+s(Z) of [7, (0.1)] to the Adams spectral sequence Hsc(H; (En)-t(Z)) ) (EdhHn)-t+s(Z) of [8, Thm. 2(ii)]. When Z = DMI ^ Z0, where Z0 is any finite spectrum, the induced map on E2-terms is the usual map in continuous group cohomology that is induced by the canonical maps H ! NH=N and EdhNn! colimUEoGnEdhUn. 52 MARK BEHRENS AND DANIEL G. DAVIS Proof.To ease our notation, we write EhKnin place of EdhKn, whenever K is closed in Gn. Also, we take implicit cofibrant replacements as needed. The first spect* *ral sequence is formed from the resolution * ! EhNHn ! EhNn! (EhNn^EhNHnEhNn)K(n)! . . . (see the discussions after [7, (0.1) and Prop. 3.6]). By Lemma 8.2.6, the canon* *ical K(n)-local En-resolution * ! EhHn! En ! (En ^EhHnEn)K(n)! (En ^EhHnEn ^EhHnEn)K(n)! . . . of EhHnin the category of EhHn-modules is also a K(n)-local En-resolution of Eh* *Hn in the category of S-modules. Thus, the second spectral sequence, which was originally constructed by using such a resolution in the category of S-modules * *(see [8, pg. 32, App. A]), can be regarded as a K(n)-local En-Adams spectral sequence in the category of EhHn-modules, so that we can also regard the second spectral sequence as being given by [7, (0.1)] through the resolution * ! EhHn! En ! (En ^EhHnEn)K(n)! (En ^EhHnEn ^EhHnEn)K(n)! . ... As in [7, (3.7)], there is a canonical map to the preceding resolution, from * *the resolution (UcolimEEhUHn)K(n)! ( colimEhUn)K(n)! ( colim(EhUn^EhUH EhUn))K(n)! . . . oGn UEoGn UEoGn n and this map is a levelwise weak equivalence (at the beginning of the last reso* *lution, the usual " * ! " was omitted for the sake of space). This last resolution rece* *ives the obvious map from the first resolution * ! EhNHn ! EhNn! 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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, U.S.A.