Homotopy xed point spectra for closed subgroups of the Morava stabilizer groups

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1 Topology 43 (2004) Homotopy xed poit spectra for closed subgroups of the Morava stabilizer groups EthaS. Deviatz a; ;1, Michael J. Hopkis b;1 a Departmet of Mathematics, Uiversity of Washigto, Seattle, WA 98195, USA b Departmet of Mathematics, Massachusetts Istitute of Techology, Cambridge, MA 02139, USA Received 23 July 1999; received irevised form 12 October 2002; accepted 25 November 2002 Abstract Let G be a closed subgroup of the semi-direct product of the th Morava stabilizer group S with the Galois group of the eld extesio F p =F p. We costruct a homotopy xed poit spectrum E hg whose homotopy xed poit spectral sequece ivolves the cotiuous cohomology of G. These spectra have the expected fuctorial properties ad agree with the Hopkis-Miller xed poit spectra whe G is ite.? 2003 Elsevier Ltd. All rights reserved. MSC: 55P43; 55N22; 55T15 Keywords: Commutative S-algebra; Cotiuous homotopy xed poit spectra; Morava stabilizer group; Adams spectral sequece 1. Itroductio If a (discrete) group G acts oa spectrum Z, oe ca form the homotopy xed poit spectrum, oftedeoted Z hg. It is giveby the G xed poits of the fuctio spectrum F(EG; Z), where EG is a cotractible free G-space. There is the, for each spectrum X, a coditioally coverget spectral sequece H (G; Z X ) [X; Z hg ] ; obtaied from the usual ltratio of the bar costructio for EG. This spectral sequece is called the homotopy xed poit spectral sequece. Of course, the costructio of such a homotopy xed Correspodig author. Tel: ; fax: address: deviatz@math.washigto.edu (E.S. Deviatz). 1 Partially supported by the NSF /$ - see frot matter? 2003 Elsevier Ltd. All rights reserved. doi: /s (03)

2 2 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 poit spectrum requires that the group act i a appropriate poit-set category ad ot just up to homotopy. However, there are situatios i stable homotopy theory where group actios oly exist i the stable category; that is, up to homotopy. Ifact, the most importat group actioithe whole chromatic approach to stable homotopy theory the actioof the exteded Morava stabilizer group G othe p-local Ladweber exact spectrum E (see [22,5], ad Sectio 1 for a resume) arises ithis way. Yet the situatioithe case of this actiois ot hopeless. Ideed, H.R. Miller ad the secod author have proved that E is a A rig spectrum ad that the space of A rig maps from E to itself has weakly cotractible path compoets. Furthermore, the set of path compoets of this space is i bijective correspodece with the set of homotopy classes of rig spectrum maps from E to itself (see [27] for a accout of this theory.) Sice G acts o E by maps of rig spectra, it follows that the actiocabe taketo be oe of A maps, ad although this actio is still oly aactioup to homotopy, it is ahoest actioup to all higher A homotopies. Stadard techiques the allow oe to replace E by aequivalet spectrum owhich G acts othe ose. Hece, if G is a (ite) subgroup of G, there is a A homotopy xed poit spectrum E hg. These spectra have already had a umber of iterestig applicatios i stable homotopy theory (see e.g., [23]). Subsequetly, P.G. Goerss ad Hopkis [12 15] exteded the machiery of Hopkis-Miller to the E settig, ad thus E hg is a E rig spectrum. Ufortuately, this is still ot a etirely satisfactory state of aairs. G is a proite group, so oe might hope to dee, for G a closed subgroup of G, a cotiuous homotopy xed poit spectrum deoted abusively by E hg whose homotopy xed poit spectral sequece starts with the cotiuous cohomology of G. (This is why we restricted to ite subgroups i the previous paragraph.) Ideed, it is the cotiuous cohomology of G that is importat i stable homotopy theory by Morava s chage of rigs theorem, the K() -local E -Adams spectral sequece (see Appedix A) for L K() S 0 has the form H c (G ;E ) L K() S 0 : (1.1) Here K() deotes the th Morava K-theory, ad L K() deotes K() -localizatio. The spectral sequece (1.1) thus suggests that L K() S 0 should be the G homotopy xed poit spectrum of E i this cotiuous sese. The case = 1 provides further evidece for the existece of cotiuous homotopy xed poit spectra. Here we have that E 1 is the p-completio of the spectrum corepresetig complex K-theory ad that G 1 = Z p, the group of multiplicative uits i the p-adic itegers. The elemet a i Z p correspods to the Adams operatio a. Now the compoet 0 E 1 of the 0th space of E 1 cotaiig the base poit is just the p-completioof BU, ad, accordig to Quille [24], this space is equivalet to the p-completioof BGL( F l ) + for ay prime l ot equal to p. (As usual, BGL(R) + is the coected space represetig the algebraic K-theory of the rig R, ad F l is the algebraic closure of the eld with l elemets.) Uder this equivalece, the automorphism of BGL( F l ) + iduced by the frobeius automorphism of F l correspods to the Adams operatio l o BU p. More geerally, the proite group Ẑ = Gal( F l =F l ) acts o BGL( F l ) + ;ifg = Gal( F l =k) is ay closed subgroup of Ẑ, the i (BGL(k) + )=[ i (BGL( F l ) + )] G

3 for all i 1 [24; Corollary to Theorem 8]. Sice H s c (G; i BGL( F l ) + )=0 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) for all i ad all s 0 agai by the computatios of Quille this suggests that BGL(k) + should be regarded as the cotiuous homotopy G xed poit spectrum of BGL( F l ) +. Ithis paper, we costruct spectra E hg for G a closed subgroup of G, havig the desired properties of cotiuous homotopy xed poit spectra. Our costructio proceeds i two steps. First we costruct E hu for U a ope ad hece closed subgroup of G, thewe costruct E hg as a appropriate homotopy colimit of the E hu s, for G U. We eed to itroduce a little more otatio i order to state our mai results. Let R + G deote the category whose objects are cotiuous ite left G -sets together with the left G -set G. The morphisms are cotiuous G -equivariat maps, ad we deote by r g : G G the map giveby right multiplicatioby g G. Let E deote the category of commutative S 0 -algebras ithe category of S 0 -modules of May et al. [11]. (A comparisoof this category with the category of L-spectra for L the liear isometries operad ad hece of the category of E rig spectra is carried out i[11, II.4].) Fially, sice the atural umber will be xed throughout this paper, we write ˆL for the fuctor L K(). Here theis our rst mairesult. Theorem 1. There is a fuctor F :(R + G ) opp E with the followig properties. (i) F(S) is K() -local for each object S i R + G. (ii) F(G )=E ad F(r g ):E E is the actio of g G o E. (iii) There is a atural isomorphism ˆL(F(S) E ) Map(S; Map c (G ;E )) G of completed ˆL(E E )=Map c (G ;E )-comodules, where the actio of G o Map c (G ;E ), the set of cotiuous fuctios from G to E, is give by (gf)(g )=f(g 1 g ) for g; g G ad f Map c (G ;E ). I particular, F( ) ˆLS 0. (iv) Dee E hu = F(G =U), U a ope subgroup of G, ad let Z be ay CW -spectrum. There is a atural strogly coverget spectral sequece H c (U; E Z) (E hu ) Z which agrees with the spectral sequece obtaied by mappig Z ito a K() -local E -Adams resolutio of E hu. Remark 1.2. I what follows ad i the proof of Theorem 1 the ecessity of workig i a precise poit-set category of structured rig spectra will become apparet. However, may of our results, such as (iii) ad (iv) of Theorem 1, are statemets occurrig i the stable category. We shall therefore refer to objects as S 0 -module spectra or commutative S 0 -algebras whewe wish to emphasize that we eed to work at the poit-set level ad as CW-spectra or rig spectra whe our work takes place ithe stable category. Furthermore, [X; Y ] i will always deote the group of maps of degree i betwee X ad Y ithe stable category.

4 4 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 Despite our precautios, there are still some ambiguities. For example, suppose X ad Y are commutative S 0 -algebras, but we wish to uderstad the stable homotopy type of X Y. The X Y might deote the object which is the CW-approximatio to the actual poit-set level smash product of X ad Y. Or, X Y might deote the derived smash product; that is, the smash product of CW-approximatios to X ad Y. However, if X ad Y are cobrat objects i the closed model category structure o E called q-cobrat i [11] thethese two recipes give the same stable object (see [11, VII, 6]). Sice there is a fuctor E E sedig a object to a weakly equivalet q-cobrat oe i fact, a cell commutative S 0 -algebra ithe sese of [11] we may as well assume that F(S), for example, is always q-cobrat ad thus elimiate this ambiguity. We will ofte use this device of fuctorially approximatig by a cell object, eve if we do ot always metio it. Remark 1.3. Let I =(p; v 1 ;:::;v 1 ) E, ad, if Z is ay CW-spectrum, let {Z } be the directed system of ite CW-subspectra. The E Z = lim ;k E Z =I k E Z is a proite cotiuous G -module ad we dee H c (U; E Z) = lim H c (U; E Z =I k E Z ): ;k Igeeral, if G is a p-aalytic proite group ad M is a proite cotiuous Z p [[G]]-module, thelim Hc (G; M ) is idepedet of the presetatio M = lim M. Ifact, it is for example the cohomology of the usual cochaicomplex whose j-cochais are Map c (G j ;M) (cf. Proof of Lemma 4.21). We therefore use this as our deitio of Hc (G; M); see also [31] for a more geeral treatmet of this object. For the ext step, let G = U 0 ) U 1 ) U 2 ) ) U i ) (1.4) be a sequece of ormal ope subgroups of G with i U i = {e}. For example, usig the otatio at the begiig of Sectio 1 ad the descriptio of S i[7, 2.21], we may take U i = V i o Gal, where V i is the group of power series j 0 b jx pj with b j = 0 for 0 j i ad b 0 =1 if i 0. Deitio 1.5. Fix a sequece {U i } as above. For G a closed subgroup of G, dee E hg where holim E = ˆL holim E E h(uig) ; i deotes the homotopy colimit takeithe topological model category E. Remark 1.6. More precisely, oe should fuctorially replace holim E E h(uig) by a weakly equivalet i cell commutative S 0 -algebra before applyig ˆL. Thethe costructioe hg becomes fuctorial i E (see [11, VIII, 2]). We will prove the followig result i Sectio 6.

5 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) Theorem 2. The costructio E hg has the followig properties: (i) ˆL(E hg E ) is aturally isomorphic to Map c (G ;E ) G as completed Map c (G ;E )-comodules, where G acts o Map c (G ;E ) as i Theorem 1(iii). (ii) Let Z be ay CW-spectrum. The spectral sequece obtaied by mappig Z ito a K() -local E -Adams resolutio of E hg isomorphic to Hc (G; E Z). is strogly coverget to (E hg ) Z ad has E 2 -term aturally Remark 1.7. (i) Usig the rst part of this theorem, it s easy to see that E hg is caoically idepedet (up to weak equivalece i E) of the choice of sequece {U i }. (ii) Sice G is a closed subgroup of a p-aalytic proite group, it is itself p-aalytic (see [9, 10.7]), ad therefore Hc (G; E Z) is deed as i Remark 1.3. Now by Theorem 1, there are commutative diagrams ad for i 6 j ad g G, where the ulabeled arrows are iduced by the evidet projectios. There is thus a caoical map E hg E G, the G-xed poits of E. Let us deote by E h G the ordiary homotopy xed poit spectrum for the actio of G o E. Composig the above map with the caoical map E G E h G ([4, XI, 3.5]) yields a map E hg E h G. The ext result will be proved i Sectio 6. Theorem 3. Let G be a ite subgroup of G. The: (i) The map E hg E h G described above is a weak equivalece. (ii) Let Z be ay CW-spectrum. The homotopy xed poit spectral sequece H (G; E Z) (E h G ) Z =(E hg ) Z is aturally isomorphic to the spectral sequece of Theorem 2(ii). We also have a result o iterated homotopy xed poit spectra. Ideed, if K is a closed ad U is a ormal ope subgroup of G, thethe opposite of the group W (K)=N(K)=K acts o G =UK via xuk xhuk for x G, h N (K). This yields aactioof W (K) oe h(uk), ad hece, upo passig to the homotopy colimit, o E hk. Iparticular, if F is a ite subgroup of W (K), we may form (E hk ) hf ithe usual way. We caow state our ext result.

6 6 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 Theorem 4. Suppose G is a closed subgroup of G, K is a closed ormal subgroup of G, ad F G=K is ite. The E hg is aturally equivalet to (E hk ) hf. Aother sort of cosistecy result is give by examiig the case = 1. Sice the Galois actio o BGL( F l ) + correspods to the actio of G 1 = Z p o E 1, where oce agai l is a prime dieret from p, we would expect a relatioship betwee the cotiuous homotopy xed poits of BGL( F l ) + ad the cotiuous homotopy xed poit spectra of E 1. This is ideed the case. Let R be a commutative rig, ad let KR be the algebraic K-theory spectrum of R (so that KR Z BGL(R) + ). Quille s results deloop (see [20, VIII]); hece (K F l ) p is equivalet to the coective cover of E 1, ad the actio of t Ẑ o(k F l ) p correspods to the actio of l t Z p o E 1. Now choose s dividig p 1 such that l s 1 mod (p), ad dee a cotiuous group moomorphism Z p Z p by sedig a Z p to l sa. (The umber s is eeded to guaratee that l sa makes sese for all a Z p.) If G is a o-trivial closed subgroup of Z p, the G = p j Z p for some j 0. Let us also write G for the correspodig closed subgroup i Z p. We thehave the followig result. Theorem 5. With the otatio as above, E hg 1 L K(1) K(k), where k is the eld of ivariats of the actio of sp j Ẑ Ẑ o F l. Our al result, which will be proveas aapplicatioof the machiery developed here, was origially due to the secod author ad H.R. Miller, who suggested that this is the place where it should logically appear. Theorem 6 (Hopkis-Miller). Suppose c : G Z p is a cotiuous homomorphism, ad let c also deote the compositio G c Z p E. The c H 1 c (G ;E ) survives to ˆLS 0. Let us idicate our strategy for costructig E hu. We will costruct a cosimplicial E spectrum correspodig to the K() -local E -Adams resolutioof E hu ; thewe will dee E hu to be Tot of this cosimplicial spectrum. To do this, we eed oly determie the (expected) homotopy type of ˆL(E hu E ) together with the comodule structure map ˆL(E hu E )= ˆL(E hu Now we might expect a spectral sequece S 0 E ) ˆL(E hu E E ): H c (U; ˆL(E E )) ˆL(E hu E ); sice there is such a spectral sequece if U is replaced by a ite subgroup of G (see Theorem 5.3). But ˆL(E E ) = Map c (G ;E ), ad the actio of U is giveas itheorem 1(iii) (see Sectio 2). Sice Map c (G ;E )isu-acyclic (cf. proof of Lemma 4.20), we should have ˆL(E hu E ) = Map c (G ;E ) U : The comodule structure map is also determied. But, as E -modules, Map c (G ;E ) U = m i=1 E ;

7 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) where m is the cardiality of G =U. We are thus led to take X G=U = m i=1 E as a model for ˆL(E hu E ). We ca also dee the correspodig comodule structure map X G=U ˆL(X G=U E ). (Note that this map is ot except ia few very special cases the product of the maps E = ˆL(S 0 E ) ˆL(E E ).) With this costructio, we obtai a cosimplicial object, but oly i the stable category. However, the techology of Hopkis-Miller as expaded by Goerss-Hopkis is agaiavailable to allow us to coclude that the requisite diagrams ifact commute up to all higher homotopies i E. The origial cosimplicial object ca ow be replaced by a equivalet cosimplicial object i E, ad Tot ca be formed. The cotets of this paper are as follows. I Sectio 2, we recall the relevat parts of Morava s theory, ad i Sectio 3, we discuss the formatioof homotopy iverse limits for certaidiagrams commutative up to all higher homotopies. We costruct the cosimplicial Adams resolutioisectio 4 ad idetify the resultat spectral sequece as havig the form of a (cotiuous) homotopy xed poit spectral sequece. I Sectio 5 we compute ˆL(F(S) E ); this allows us to idetify the precedig spectral sequece as a K() -local E -Adams spectral sequece ad completes the proof of Theorem 1. We prove Theorems 2 ad 3 isectio6, Theorem 4 isectio7, ad, ally, Theorems 5 ad 6 isectio8. Aappedix summarizes the properties of K() -localizatios ad K() -local E -Adams spectral sequeces that we eed. I particular, we prove the strog covergece of these spectral sequeces. 2. Resume ofmorava s theory Let p be a xed prime, let 1, ad let E deote the spectrum with coeciet rig E = W F p [[u 1 ;:::;u 1 ]][u; u 1 ] obtaied via the Ladweber exact fuctor theorem for BP. W F p deotes the rig of Witt vectors with coeciets i the eld F p of p r elemets, ad the map BP E which also provides E with the structure of BP-algebra ithe stable category is giveby u i u 1 pi i ; r(v i )= u 1 p i = ; 0 i : Now let S deote the th Morava stabilizer group; i.e., the automorphism group of the height Hoda formal group law over F p. Let Gal Gal(F p =F p ), ad let G =S ogal. The Lubi-Tate theory of lifts of formal group laws provides aactioof S o E (see for example [7]), ad Gal acts o E via its actioow F p.ifh is a subgroup of Gal, let us write E H for the Ladweber exact spectrum with coeciet rig W (Fp H )[[u 1;:::;u 1 ]][u; u 1 ], where Fp H is the subeld of F p xed by the automorphism group H. We rst idetify the completed Hopf algebroid ˆL(E E ) with the split completed Hopf algebroid (E ; Map c (G ; Z p ) ˆ Zp E ); this piece of Morava s theory is crucial to all of our subsequet work. We start by observig that Map c (S ;WF p ) Gal is a completed Hopf algebra over Z p ; the diagoal map is give by Map c (S ;WF p ) Gal Map c (S S ;WF p ) Gal Mapc (S ;WF p ) Gal ˆ Map c (S ;WF p ) Gal ;

8 8 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 where the rst map is iduced by the multiplicatio o S ad the secod is a isomorphism by [5, AII.3]. There is also a map L : E Gal Map c (S ;WF p ) Gal ˆ Zp E Gal which is giveby L (x)(g)=g 1 x for x E Gal split completed Hopf algebroid (E Gal is the followig ideticatio. ; Map c (S ;WF p ) Gal ˆ Zp E Gal ) as com- Theorem 2.1. (E Gal; ˆL(EGal pleted Hopf algebroids. E Gal Mapc (S ;E ) Gal ad g S. With these structure maps, we obtaia ). A mairesult of Morava s theory ; Map c (S ;WF p ) Gal ˆ Zp E Gal )) is isomorphic to (E Gal Proof. We showed i[5, Sectio4] that (E Gal; Map c (S ;WF p ) Gal ˆ E Gal ) is isomorphic to a completed Hopf algebroid deoted (E Gal ;EGal ˆ U US ˆ U E Gal ). But we observed i[6, 3.4] that this Hopf algebroid is isomorphic to (E Gal ;EGal ˆ BP BP BP ˆ BP E Gal ), where the completed tesor product here deotes I -adic completio. Sice ˆL(E Gal E Gal )isthei -adic completioof E Gal EGal = E Gal BP BP BP BP E Gal (see [16]), the proof is complete. To derive the structure of the completed Hopf algebroid ˆL(E E ) from Theorem 2.1, we rst observe that ˆL(E E Gal )= ˆL(E Gal E Gal ) Zp W F p = Map c (S ;E ) Gal Zp W F p = Map c (S ;E ); where the last equality follows by [5, 5.4]. The ˆL(E E )= ˆL(E E Gal ) Zp W F p = Map c (S ;E ) Zp W F p Map c (G ;E ); where (f w)(s; )=w ( 1 (f(s))) for f Map c (S ;E ), w W F p, s S,ad Gal. Now cosider the split completed Hopf algebroid (E ; Map c (G ; Z p ) ˆ Zp E ); oce agai L : E Map c (G ;E ) is giveby L (x)(g)=g 1 x for x E ad g G. Observe that this is ot a Hopf algebroid over W F p, sice L ad R are dieret whe restricted to W F p. Upo chasig dow the ideticatios, the ext result follows from Theorem 2.1. Propositio 2.2. (E ; ˆL(E E )) is isomorphic to (E ; Map c (G ; Z p ) ˆ Zp E ) as completed Hopf algebroids. This isomorphism is used to dee the actio of G o E. Ideed, recall that if M is a completed left Map c (G ;E )-comodule, the M is a G -module with the actiogiveby g(m)= (m)(g 1 );

9 where E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) : M Map c (G ;E ) ˆ E M Map c (G ;M) is the comodule structure map. Iparticular, if X is a ite CW-spectrum, E X is aturally a G -module, ad the pairig E X E E Y E (X Y )isg -equivariat, where the left side is give the diagoal actio. Sice E is proite ieach degree, it follows that there exists, ithe stable category, a uique actio of G o E by rig spectrum maps iducig the above actio o E X. With this actio, it is immediate that the isomorphism of Propositio 2.2 is give by sedig x ˆL(E E )toh x Map c (G ;E ), where h x (g) is giveby the compositio S x ˆL(E E ) ˆL(g 1 E ) ˆL(E E ) E : (2.3) (Aideticatioof this sort rst appears ithe literature i[30].) Here suspesios have bee omitted from the otatio ad is the rig spectrum multiplicatio map. Moreover, it is easy to check usig (2.3) that the actioof G othe left factor of ˆL(E E ) correspods to the actio of G omap c (G ;E ) described itheorem 1. For later purposes, we will also eed to kow the formula for the actioof G othe right factor of ˆL(E E ). If we write g R x for this actioof g G o x ˆL(E E ), theit is agaieasy to see usig (2.3) that g R x correspods to the map sedig g G to gh x (g g) E. Remark 2.4. I[5], we used Theorem 2.1 to dee a atural W F p -liear actio of S o E X. There is also the evidet actio of Gal o E X = W F p Zp E Gal X, ad these actios piece together to give a atural actio of G o E X, whece aactioof G o E ithe stable category. This actiois the same as the actiodeed above. The ideticatios of Propositio 2.2 ad (2.3) cabe geeralized to iterated smash products of E ad beyod. Ideed, if X is a ite spectrum, where E ad thus ˆL(E (j+1) X )=E E ˆ E E E ˆ E ˆ E E E ˆ E E X; acts othe right of each factor E E ad o the left of the factor E X. But E E ˆ E ˆ E E E = (Map c (G ; Z p ) Zp E ) ˆ E ˆ E (Map c (G ; Z p ) Zp E ) = Map c (G ; Z p ) ˆ Zp ˆ Zp Map c (G ; Z p ) ˆ Zp E = Map c (G j ;E ); E E ˆ E ˆ E E E ˆ E E X = Map c (G j ;E ) ˆ E E X This isomorphism seds x ˆL(E (j+1) by the compositio S x ˆL(E (j+1) 1 ˆL(g X ) 1 g 1 j E X ) = Map c (G j ;E X ): X )toh x Map c (G j ;E X ), where h x (g 1 ;:::;g j ) is give ˆL(E (j+1) X ) ˆL( X ) E X: (2.5)

10 10 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 More geerally, if Z is ay spectrum, there is a atural trasformatio j :[Z; ˆL(E (j+1) )] Map c (G;E j Z) (2.6) such that j (x)(g 1 ;:::;g j ) is the composite Z x 1 ˆL(E (j+1) ˆL(g ) 1 g 1 j E ) ˆL(E (j+1) ) E : (2.7) (To show that j does ifact have its image imap c (G;E j Z), it suces, by the deitio of the topology o E Z, to show that this is the case whe Z is ite. But whe Z is ite, j is just the isomorphism described i(2.5) with X the Spaier-Whitehead dual of Z.) Sice Map c (G; j?) is exact othe category of proite groups (see [29, I, Theorem 3]), it follows that j is a atural trasformatio of cohomology theories satisfyig the product axiom. But j is aisomorphism with Z = S 0 ; therefore j is aisomorphism for ay CW-spectrum Z. 3. Homotopy iverse limits Let he deote the homotopy category of commutative S 0 -algebras. That is, two maps f;g: X Y i E are homotopic if f ad g lie i the same path compoet of E(X; Y ), the topological space of S 0 -algebra maps betwee X ad Y. Alteratively, E is a tesored category (over the category of ubased topological spaces), ad f ad g are homotopic if there exists a map h : X I Y restrictig to f ad g o the eds of the cylider (see [11, VII, 2]). Now let J be a small category ad suppose X : J he is a fuctor. I some cases, X ca be replaced by a homotopy equivalet strict diagram, ad the its homotopy iverse limit ca be formed. Deitio 3.1. Let X : J he be as above. X is said to be a h -diagram if for each : j 1 j 2 i J, E(Xj 1 ;Xj 2 ) X, the path compoet of E(Xj 1 ;Xj 2 ) cotaiig X, is weakly cotractible. The mairesult of this sectiois due to Dwyer, Ka, ad Smith [10]. We feel that the reader will appreciate aaccout of the proof. Theorem 3.2. Let X : J he be a h -diagram. The there exists a fuctor X : J E ad a atural trasformatio X X i he such that X (j) X (j) is a weak equivalece for each j Ob J. The proof makes use of a modicatioof the cosimplicial replacemet of a diagram (cf. [4, XI, 5]). We rst recall some otatio. If Z is a ubased topological space ad Y is a commutative S 0 -algebra, let F(Z; Y ) deote the cotesor product of Z ad Y i E; its uderlyig S 0 -module is just the fuctio S 0 -module of maps from Z + to Y ([11, VII, 2]). We shall also use the otatio F( ; ) to deote fuctio spectra i the stable category. With Z as above, let Z deote the geometric realizatio of the sigular simplicial set of Z. This is of course a fuctorial cobrat replacemet of Z; it also has the property that V W is aturally homeomorphic to (V W ) ad that this homeomorphism commutes with the projectios oto V ad W. Hece a pairig V W Z iduces a pairig V W Z.

11 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) Costructio 3.3. Let X : J he be a h -diagram. Dee a cosimplicial commutative S 0 -algebra h X by 0 h X = j J Xj; h X = J F( EX; Xj 0 ); where J is the set of diagrams i J,ad : j j1 j2 j 1 j EX = E(Xj 1 ;Xj 0 ) X1 E(Xj ;Xj 1 ) X : If 0 i + 1, the coface d i is deed via the commutative diagram (3.4) where deotes the projectio oto the factor idexed by : j j1 j2 j j+1 ; deotes the diagram ad j 1 i i+1 i j1 j i 1 ji+1 ji+2 j j+1 ; (d i g)(f 1 ;:::;f +1 )=g(f 1 ;:::;f i f i+1 ;:::;f +1 ) for (f 1 ;:::;f +1 ) EX. Here f i f i+1 deotes the image of (f i ;f i+1 ) uder the map E(Xj i ;Xj i 1 ) E(Xj i+1 ;Xj i ) E(Xj i+1 ;Xj i 1 ) iduced by the compositio pairig E(Xj i ;Xj i 1 ) E(Xj i+1 ;Xj i ) E(Xj i+1 ;Xj i 1 ): If i =0, d 0 is deed as i (3.4), although ow is the diagram j j2 j +1 ; ad d 0 : F( EX ;Xj 1 ) F( EX ;Xj 0 ) is deed by (d 0 g)(f 1 ;:::;f +1 )=f 1 (g(f 2 ;:::;f +1 )); where f 1 also deotes the image of f 1 E(Xj 1 ;Xj 0 )ie(xj 1 ;Xj 0 ).

12 12 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 ad Fially, if i = +1, is the diagram j 1 0 j1 j 1 j (d +1 g)(f 1 ;:::;f +1 )=g(f 1 ;:::;f ): As for the codegeeracies, s i is deed via the commutative diagram (3.5) where is the diagram ad j j1 j2 j i id i+1 i 1 ji j i +1 ji+1 j +1 (s i g)(f 1 ;:::;f +1 )=g(f 1 ;:::;f i ; id; f i+1 ;:::;f +1 ): Here id deotes the image of = ( ) i E(X ji ;X ji ) uder the evidet map. Recall that a cosimplicial S 0 -module Y is brat if the map s : Y +1 M Y, M Y {(y 0 ;:::;y ) Y Y : s i y j = s j 1 y i 0 6 i j6 }; giveby s(y)=(s 0 (y);:::;s (y)) is a q-bratio that is, a bratioithe Quilleclosed model sese for all 1. (Properly speakig, M Y should be deed as a equalizer; however we will cotiue to use this shorthad whe to do otherwise would result i more cofusio.) Lemma 3.6. Let X be a h -diagram. The h X is brat. Proof. Cosider rst the map s : 1 h X M 0 h X = 0 h X. The compositio 1 h X s 0 h X = j Xj Xj is giveby 1 h X = j 0 j 1 F( E(Xj 1 ;Xj 0 ) X ;Xj 0 ) F( E(Xj; Xj) id ;Xj) Xj; where the last map is evaluatioat id E(Xj; Xj). Sice id is a vertex of E(Xj; Xj), it follows that this last map is a q-bratio ad hece so is s. Now suppose 1. Let J;j k be the subset of J cosistig of those -tuples of maps with j 0 = j ad k = id, ad set D k ;jx = J k ; j EX:

13 Thelet D ;j X = 16k6 D k ;jx; E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) ad observe that the map s : h X M 1 h X restricts to aisomorphism j F(D ;jx; Xj) M 1 h X: But the iclusio D ;j X J ; j EX is the iclusio of a summad, where J ;j is the subset of J cosistig of those -tuples with j 0 = j. s is therefore a q-bratioithis case as well. Recall that, givea brat cosimplicial S 0 -module Y, Tot Y = F([ ];Y), the S 0 -module of (upoited) cosimplicial maps from [ ] to Y. Here [ ] is the cosimplicial space which i dimesio is the stadard -simplex [] with the usual coface ad codegeeracy maps. Let Sk s [ ] be the cosimplicial space which idimesio is the s-skeletoof [], ad dee Tot s Y = F(Sk s [ ];Y). The map Tot j+1 Y Tot j Y is a bratiowith ber F([j +1]= [j + 1], Y j+1 ker s 0 ker s j ). By mappig a CW-spectrum Z ito the tower {Tot j Y }, we obtaia spectral sequece E s;t 2 = s ([Z; Y ] t ) [Z; Tot Y ] t+s (cf : [4; X:6]): (3.7) This spectral sequece is strogly coverget i the sese of [4, IX.5.4] provided lim 1 Er s;t = 0 for j all s ad t. Iparticular, if X is a h -diagram, we obtaia spectral sequece E s;t 2 = lim J s ([Z; X ] t ) [ Z; Tot h X ] t+s : (3.8) This is proved as i[4, XI.7.1], usig the fact that EX is cotractible for all. With these costructios i had, we ca ow prove the mai result of this sectio. ProofofTheorem 3.2. If j is aobject of J, let J \ j be the category whose objects are morphisms j j i J ad whose morphisms are the evidet commutative diagrams. The evidet fuctor j : J \j J provides us with a h -diagram j X over J \j ad hece a fuctor X : J E deed by X (j) = Tot h j X. Now dee the map X (j) X (j) to be the compositio Tot ( h j X F [0]; ) 0 h j X = 0 h j X p Xj; (3.9)

14 14 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 where p is the projectio oto the factor idexed by the object j id j. To prove that commutes wheever f : j j, we must prove that (3.10) commutes, where the diagoal map is the projectio oto 0 h j X followed by the projectiooto the factor idexed by f : j j. First observe that the two compositios X (j) F ( [1]; 1 h j X ) F(d 0 ;id) F F(d 1 ;id) ( [0]; 1 h j X ) pf X (j ) are homotopic, where the last map is the projectio oto the factor idexed by f :(j id j) (j f j ). But these compositios are the same as ( X (j) F [0]; ) d 0 0 h j X 1 h j X pf X (j ): d 1 Now use the deitios of d 0 ad d 1 to check that these maps give the commutative diagram (3.10). By (3.8), there is a spectral sequece E s;t 2 = lim s t j X t s X (j): J\j But j j is aiitial object of J \ j; therefore { 0 s 0; lim s t j X = J\j t Xj s =0: Thus X (j)= X (j), ad a uravelig of the ideticatios shows that the map i (3.9) iduces the idetity o. 4. Costructio ofthe fuctor F We begi by statig the followig extesios of the Hopkis-Miller theory due to Goerss ad Hopkis. These are the results eeded to show that, for S Ob R + G, the cosimplicial object C S we will costruct i the stable category lifts to a h -diagram, ad hece, by Theorem 3.2, toa cosimplicial object i E. The F(S) is deed to be Tot C S. Theorem 4.1 (Goerss ad Hopkis [13]). Let E ad F be q-cobrat commutative S 0 -algebras with E Ladweber exact. Suppose that E is cocetrated i eve dimesios with a ivertible elemet

15 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) i degree 2. Suppose i additio that 0 E is m-adically complete for some ideal m, that E 0 =m is a algebra of characteristic p, ad that the relative frobeius for the homomorphism E 0 =m E 0 F=mE 0 F is a isomorphism. The each path compoet of E(F; E) is weakly cotractible, ad the Kroecker pairig 0 E(F; E) Hom E -alg(e F; E ) is a bijectio. Remark 4.2. Recall that if A is a commutative F p -algebra, the frobeius homomorphism A : A A is deed by A (x)=x p.iff : A B is a homomorphism of commutative F p -algebras, the relative frobeius is deed to be the map give by the dotted arrow i the followig diagram, where the square is a push-out ithe category of commutative F p -algebras: Remark 4.3. The coditio that E be Ladweber exact ca be weakeed. Ideed, E eed oly satisfy a coditio of the sort required by Adams [1, III, 13.3] i his costructio of uiversal coeciet spectral sequeces. For example, Property 1.1 of [6] suces. I particular, ay Ladweber exact spectrum satises this property [6, 1.3]. Theorem 4.4 (Goerss ad Hopkis [13]): Let H be a subgroup of Gal. The E H has a uique structure of commutative S 0 -algebra descedig to its ordiary rig spectrum structure. Remark 4.5. E H slightly abusive. is ifact the H homotopy xed poit spectrum of E, so our otatio is oly Iorder to costruct C S, we eed to apply Theorem 4.1 to spectra of the form ˆL(X (E ) (j) ), where X is a ite product of E s. The ext result eables us to do this. Propositio 4.6. Let E = ˆL(X E (j) ) ad let F = ˆL(Y E (k) ), where X ad Y are (o-empty) products of itely may copies of E ad j; k 0. The the pair F; E satises the coditios of Theorem 4.1; moreover, the fuctio 0 E(F; E) Hom alg (F ;E ) sedig a map F E to the iduced map o homotopy groups is oe-to-oe. Proof. First observe that E ad F are commutative S 0 -algebras the key poit here is that localizatiowith respect to a homology theory preserves such objects (see [11, VIII]). Aeasy reductio also shows that it suces to cosider the case where X = E ad Y = E. By Morava s theory, E = Map c (G;E j ) ad the actio of E othe right factor of E iduces right multiplicatio of

16 16 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 E omap c (G;E j ). It therefore follows that E (resp. F) is Ladweber exact for a appropriate map BP E (resp. BP F) ad hece, by the Ladweber exact fuctor theorem, E F is a at F -module. Iadditio, E 0 is complete with respect to the ideal m = Map c (G;I j ), where we also write I =(p; u 1 ;:::;u 1 ) (E ) 0. Now ˆL(F E) = Map c (G j+k+1 ;E ) ad, just as before, ˆL(F E) is Ladweber exact. Sice F E is at over E, it also follows that F E is Ladweber exact. Thus, if M(p i0 ;v i1 1 ;:::;vi 1 1 ) is a ite CW-spectrum whose Brow-Peterso homology is BP =(p i0 ;:::;v i 1 1 ), we have that F E=F E (p i0 ;:::;v i 1 1 )= (F E M(p i0 ;:::;v i 1 1 )) Therefore ( ˆL(F E) M(p i0 ;:::;v i 1 1 )) F E=F E I = ˆL(F E)= ˆL(F E) I = ˆL(F E)= ˆL(F E) (p i0 ;:::;v i 1 1 ): = Map c (G j+k+1 ; F p [u; u 1 ]): Clearly the frobeius is a isomorphism o E 0 =m, ad, from the above equality, it is also a isomorphism o E 0 F=mE 0 F. This implies that the relative frobeius for E 0 =m E 0 F=mE 0 F is a isomorphism. Fially, to prove that 0 E(F; E) Hom alg (F ;E ) is oe-to-oe, it suces to prove that the fuctio Hom E alg(e F; E ) Hom alg (F ;E ) giveby precompositiowith the map F = (S 0 F) (E F)=E F is oe-to-oe. But this follows easily from the commutative diagram ad the fact that the vertical maps are moomorphisms sice ay Ladweber exact spectrum is torsiofree. It will also be coveiet to kow that, for E ad F as above, the caoical map from 0 E(F; E) to the set of rig spectrum maps from F to E i the stable category is a bijectio. The ext result eables us to coclude this.

17 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) Lemma 4.7. Let E be as i Propositio 4.6, ad let F be a Ladweber exact commutative rig spectrum. The (i) E F (j) = E F E E F E E E F for ay j 1. }{{} j times (ii) The Kroecker pairig [F (j) ;E] Hom E (E F (j) ;E ) is a isomorphism for all j 1. Proof. Sice E ad F are Ladweber exact, E F is a at E -module. This immediately implies (i). As for (ii), begiby observig that F 1 F 2 is Ladweber exact if F 1 ad F 2 are. Hece F (j) satises the hypotheses of the lemma, so we may assume j = 1 without loss of geerality. There is a uiversal coeciet spectral sequece Ext E (E F; E ) E F (see Remark 4.3); it thus suces to show that Ext i E (E F; E ) = 0 for all i 0. We do this by a argumet similar to that i [27, 15.6]. Ideed, E F = E E E F ad E F is at over E ; hece Ext E (E F; E ) = Ext E (E F; E ): We thus eed oly show that Ext i (E ) 0 ((E ) 0 F; E 0 ) = 0 for all i 0, where ow E = ˆL(E (j+1) ) for some j 0. Let m be as ithe proof of Propositio3.6, ad write (E ) 0 F = M. Theagaiby at base chage Ext i (E ) 0 (M; E 0 =m) = Ext i F p (M=I M; E 0 =m): This last group is trivial for i 0 sice F p is a eld. Similarly, Ext i (E ) 0 (M; m t =m t+1 )=0 for i 0; ow use the fact that E 0 =lim t E 0 =m t to coclude that Ext i (E ) 0 (M; E 0 )=0 wheever i 0. Now let be the category whose objects are the oegative itegers a typical object will be deoted [] ad whose morphisms from [] to[m] are the mootoe odecreasig fuctios from {0; 1;:::;} to {0; 1;:::;m}. Recall the deitio of R + G from the Itroductio. We will costruct a h -diagram C :(R + G ) opp he such that, for each S Ob R + G, C S = C(S; ): he is a cosimplicial K() -local E -Adams resolutio for the ot yet costructed spectrum F(S). As described ithe Itroductio(for the case S =G =U), we seek a K() -local commutative S 0 -algebra

18 18 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 ad right E -module spectrum X S together with a E-map d : X S ˆL(X S E ) such that X S = Map G (S; Map c (G ;E )) (4.8) as right E -modules, such that ˆL(X S E ) X S ˆ E ˆL(E E ) = Map G (S; Map c (G ;E )) ˆ E Map c (G ;E ) = Map G (S; Map c (G G ;E )) (4.9) ad such that d correspods to the map iduced by the group multiplicatio G G G. Propositio There exists a h -diagram X :(R + G ) opp he ad a atural trasformatio d : X ˆL(X E ) satisfyig the above requiremets. Moreover, this costructio has the followig properties: (i) All maps are maps of right E -module spectra, where E acts o the right factor of ˆL(X E ). (ii) If S is ite, X X(S) is a product of a ite umber of copies of E. (iii) X G = ˆL(E E ) ad if r g : G G is right multiplicatio by g G, X(r g )= ˆL(g E ). (iv) d(g ) is give by the compositio ˆL(E E )= ˆL(E S 0 E ) ˆL(E E E ): Proof. If S is ite, the S is a ite disjoit uio of sets of the form G =U for various ope subgroups U of G. We thus eed oly dee X G=U ; X S cabe deed as aappropriate ite product of the X G=U s. Dee X G=U = G E =U ; i.e., X G=U is a ite product of copies of E,oe for each elemet of G =U. Eqs. (4.8) ad(4.9) are certaily satised. As for the map d(g =U), observe that stable E -module maps X G=U ˆL(X G=U E ) are i bijective correspodece with E -module maps X G=U ˆL(X G=U E ). Furthermore, by the results of Goerss-Hopkis together with Lemma 4.7, ay stable rig map X G=U ˆL(X G=U E ) lifts to a E map, uique up to E homotopy. Hece d(g =U) cabe choseto iduce the requisite map ohomotopy groups. If f : S 1 S 2 is a G -map with S 2 ite, the there exists a uique map Xf : X S2 X S1 of E -module spectra iducig the map f : Map G (S 2 ; Map c (G ;E )) Map G (S 1 ; Map c (G ;E )) ostable homotopy groups. As before, Xf has a represetative i E ad is uique up to E homotopy. Fially, the aturality of X ad d implies, by Propositio 4.6, that X is a fuctor ad d is a atural trasformatio. It also follows from 4.6 that X is a h -diagram. Costructio Let : S 0 E be the uit map, let : E E E be the multiplicatiomap, ad, for each S Ob R + G, let s : X S E X S be the module structure map. Dee a h -diagram by C :(R + G ) opp he C(S; [j]) C j S = ˆL(X S E (j) ):

19 The coface ad codegeeracy maps are give by d 0 (S)= ˆL(d(S) E (j) ):C j S Cj+1 S ; d i (S)= ˆL(X S E (i 1) s 0 (S)= ˆL(s E (j) ):C j+1 S C j S ; s i (S)= ˆL(X S E (i 1) E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) (E ) (j i+1) ):C j S Cj+1 S ; 1 6 i 6 j +1 E (j 1) ):C j+1 S C j S ; 1 6 i 6 j: That C is ifact a fuctor follows from Propositio4.6; this propositioalog with the Goerss- Hopkis results also implies that C is a h -diagram. Hece by Theorem 2.2, we may lift C to a diagram i E. From ow o, whe we refer to C, we actually meathis lift of C. Deitio The fuctor F :(R + G ) opp E of Theorem 1 is deed by F(S)=Tot( C S ), where, as before, C S is the cosimplicial replacemet of the (cosimplicial) diagram C S. Sice each C j S is K() -local, so is F(S). By (3.8), there is a spectral sequece lim s [Z; C S ] t [Z; F(S)] t+s for ay CW-spectrum Z. We ext claim that (4.13) lim s [Z; C S ] t = s [Z; C S ] t ; (4.14) where the last term deotes the sth cohomology group of the cosimplicial abeliagroup [Z; C S ] t. This result is of course well-kow; however we provide a quick proof for the coveiece of the reader. Propositio Let Ab be the category of cosimplicial abelia groups. The the -fuctors lim : Ab gr + Ab ad : Ab gr + Ab are aturally equivalet, where gr + Ab deotes the category of oegatively graded abelia groups. Proof. Sice lim 0 ad 0 are aturally equivalet, it suces to prove that s is eaceable. We begi this proof by makig a few recollectios. The forgetful fuctor U : Ab gr + Ab has a right adjoit V if D is aobject of gr + Ab, set (VD)[]= D m ; [] [m] where the product is takeover all morphisms [] [m] i. It thus follows that if D m is ijective for each m, the VD is ijective i Ab. Now let C be a cosimplicial abelia group, ad cosider the moomorphism C VUC. We will show that i (VUC) = 0 for all i 0. Ideed, rst observe that VD = V (D()), where { D m = ; D() m = 0 otherwise:

20 20 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 It thesuces to show that i (V (UC())) = 0 for all ad all i 0. But V (UC()) is just the simplicial cochaicomplex of the stadard -simplex with coeciets i C. Therefore its cohomology is trivial ipositive degrees. Igeeral, if J is a small category ad A is a J -diagram of abeliagroups, thelim J A may be computed as the cohomology of the cochaicomplex of the cosimplicial group A, the cosimplicial replacemet of A (see [4, XI, 6.2]). Thus the -fuctors C ad ( C) are aturally equivalet othe category of cosimplicial abeliagroups. The ext result is proved iappedix B; it will be eeded to prove Propositio A.5. Propositio Let C be a cosimplicial abelia group. There is a atural map T(C):C C of cochai complexes which iduces a isomorphism o. Moreover, T ca be chose so that the compositio C 0T(C) j Cj C j 0 seds x to d 0 d 0 x for ay x C 0, j 0 0. We cause spectral sequece (4.13) to verify Theorem 1(ii). This will follow easily from the ext result. Propositio Cosider the maps E ˆL(E E ) ad F(G ) ˆL(E E ) give by the maps ad E = E S 0E ˆL(E E ) F(G ) = Tot C G 0 C G C 0 G = ˆL(E E ) respectively. Let Z be ay CW-spectrum. The there is a uique bijectio E Z F(G ) Z such that the diagram commutes. Proof. The map E ˆL(E E ) is aaugmetatioof the cosimplicial object CG ithe stable category. Furthermore, CG is chaicotractible to E ithe stable category (cf. proof of Lemma 5.4); hece [Z; CG ] t = EZ t cocetrated i degree 0. The desired result ow follows from (4.13), (4.14), ad a uravelig of the ideticatios. ProofofTheorem 1(ii). The precedig propositio provides us with a caoical weak equivalece E F(G ). It also implies that the map is G -equivariat ad is a map of rig spectra (i the stable category). By Lemma 4.7 ad Theorem 4.1 this weak equivalece lifts to a E map.

21 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) Fially, we show that, for S = G =U, the spectral sequece (4.13) has the form of a homotopy xed poit spectral sequece. The proof requires some preparatio. Deitio Let M be aiverse limit of discrete G -modules, ad let H be a closed subgroup of G. Dee a cochai complex DH (M) by D j H (M) = (Map c (Gj+1 ;M)) H with dieretial : D j H (M) Dj+1 H (M) giveby j f(g 0 ;g 1 ;:::;g j+1 )= ( 1) i f(g 0 ;:::;ĝ i+1 ;:::;g j+1 ) i=0 +( 1) j+1 g 1 j+1 f(g 0g 1 j+1 ;:::;g jg 1 j+1 ): Warig The actioof H omap c (G j+1 ;M) is as elsewhere ithis paper; that is, if f Map c (G j+1 ;M)adh H, the(hf)(g 0 ;:::;g j+1 )=f(h 1 g 0 ;g 1 ;:::;g j+1 ). Lemma The -fuctor H (D H (?)) is equivalet to H c (H;?) o the abelia category of discrete G -modules. Proof. Sice H 0 (DH (M)) = M H, it suces to prove that H DH (?) is eaceable. To this ed, let N be a discrete G -module, ad cosider Map c (G ;N). Map c (G ;N) is a discrete G -module, ad there is a G -equivariat moomorphism N Map c (G ;N) deed by h, where h (g)=g 1. Now D j H (Map c (G ;N)) = (Map c (G j+1 G ;N)) H ; ad there is a cotractig homotopy q : DH +1 (Map c (G ;N)) D H (Map c (G ;N)) giveby (qf)(g 0 ;:::;g j ;t)=( 1) j+1 f(g 0 t;:::;g j t; t; 1); provig that H i (DH (Map c (G ;N))) = 0 for all i 0. We will idetify the cochai complex [Z; C G =U ]t cohomology of DH (?) for proite modules. with D U (Et Z). We must thus uderstad the Lemma Let H be a closed subgroup of G ad let M be a proite discrete G -module, say M = lim M, where rages over a directed set I. The (i) lim s D j H (M )= { D j H (M) s =0 0 s 0 (ii) H DH (M) = lim H DH (M ) = lim Hc (H; M ).

22 22 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 Proof. We have lim s D j H (M )= s D j H (M); where M is the I-diagram M ad D j H (M) is the cosimplicial replacemet of the I-diagram D j H (M). Now D j ( H (M)=Dj H M ) ; ad sice each M is ite, q M is proite. Moreover, D j H : proite G -modules Ab is exact; this agai follows from the existece of a cotiuous (set-theoretic) cross-sectio of a epimorphism of proite groups ([29, I, Theorem 3]). Therefore lim s D j H (M )=D j ( H s M ) { ( ) D j = D j H lim s H (M) s =0 M = 0 s 0 by the vaishig of lim s, s 0, for directed systems of proite groups. As for the secod part, cosider the double cochai complex DH (M). This yields two spectral sequeces lim s H t c(h; M ) H s+t ( D H (M) ) ; ) H (lim s t D H (M ) H s+t ( D H (M) ) : By (i), the secod spectral sequece collapses to give H ( D H (M) ) = H (D H (M)): Othe other had, H is a closed subgroup of a p-aalytic proite group ad is therefore itself a p-aalytic group (see [9, 10.7]). Hece H cotais a ope ormal subgroup U which is a Poicare pro-p-group ([19]; see also [28] for a summary). This implies that H t c(h; M ) is ite for each ad so lim s H t c(h; M ) = 0 for all s 0. The rst spectral sequece the collapses to give H ( D H (M) ) = lim H c (H; M ): This completes the proof.

23 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) Lemma There is a caoical isomorphism [Z; C G =U ]t D U Et Z of cochai complexes, where E Z is topologized as i Remark 1.3. Proof. The quotiet map G G =U iduces a map C G =U C G [Z; C G =U ] t [Z; C G ] t ad hece a map (4.23) of cochaicomplexes. Now X G = ˆL(E E ) (Propositio 4.10); therefore, the map j of (2.6) provides aisomorphism [Z; C j G ] t Map c (G j+1 ;EZ)=D t j {e} (Et Z): Moreover, this map is easily seeto be a cochaimap ad is G -equivariat, if the actio of g G o C j G is giveby ˆL(X(r g ) E E )= ˆL(g E E E ): Sice right multiplicatio by g U is trivial o G =U, the map i(4.23) is actually a map [Z; C G =U ] t ([Z; C G ] t ) U (D {e} (Et Z)) U = D U (E t Z) (4.24) of cochaicomplexes. Both [?;C j G =U ] ad D j U (E (?)) = Map c (G =U G j ;E (?)) are cohomology theories satisfyig the product axiom; it thus suces to prove that the map i (4.24) is aisomorphism whe Z = S 0. But, by (4.8) ad the fact that X G=U is a ite product of E s (Prop. 4.10), we have that C j G =U = X G=U E Map c (G j ;E ) We also have = Map c (G =U; E ) E Map c (G j ;E ): C j G = X G ˆ E Map c (G j ;E ) = Map c (G ;E ) ˆ E Map c (G j ;E ) ad the map X G=U X G correspods to the homomorphism Map c (G =U; E ) Map c (G ;E ) iduced by the quotiet map G G =U. From this it is clear that C j G =U ( C j G ) U ; completig the proof. Fially, we obtai our desired result. Propositio Let S = G =U. The spectral sequece (4.13) has E 2 -term caoically isomorphic to H s c (U; E t Z). This is the spectral sequece of Theorem 1(iv). It is strogly coverget because Hc s (U; EZ) t is a proite group for each s, t, ad therefore lim 1 =Er s;t =0. r

24 24 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) The Morava module of E hu Ithis sectiowe complete the proof of Theorem 1. The key step is the ideticatio of ˆL(F(S) E ) with X S this ot oly immediately implies Theorem 1(iii) but eables us to idetify the spectral sequece (4.13) with the K() -local E -Adams spectral sequece covergig to [Z; F(S)].IfS is ite, C S = i C G=U i for a ite umber of ope subgroups U i ; thus we may assume from the begiig that S = G =U. (If S = G, Theorem 1(iii) is a cosequece of 1(ii).) The techiques ivolved i our computatio of ˆL(F(S) E ) will be applied iother cotexts i Sectios 6 ad 7; we therefore proceed ia little more geerality. Notatio 5.1. Write C : J E for ay of the followig diagrams: (i) J = ad C = C G=U for U aopesubgroup of G. (ii) J = G for G a ite subgroup of G,adC: G E is the diagram giveby the actioof G o E. (iii) J = F, where F ad K are as itheorem 4 of the Itroductio, ad C : F E is the diagram giveby the actioof F o(the left factor of) ˆL(E hk E (j+1) ), j 0. Givea diagram C : J E, there is a diagram ˆL(C E ):J E deed by ˆL(C E )(j)= ˆL(C(j) E ); ˆL(C E )(f)= ˆL(C(f) E ) for j aobject ad f a morphism i J. There is also a caoical map ˆL [( Tot C ) E ] Tot ( ˆL(C E ) ) : (5.2) We will prove the followig result. Theorem 5.3. If C is oe of the diagrams i Notatio 5.1, the the map (5.2) is a weak equivalece. Before provig this theorem, we determie its cosequece for C = C G=U. The left side of (5.2) is ˆL(E hu E ). To idetify the right side, we examie the spectral sequece II Er ; (Z; C) obtaied by mappig a CW-spectrum Z ito the tower of bratios {Tot k ( ˆL(C E ))}. Lemma 5.4. Let C = C G=U. The II E s;t 2 (Z; C)=s [Z; ˆL(C E )] t = I particular, the map Tot ( ˆL(C E ) ) 0 ˆL(C E ) ˆL(C 0 E ) { 0 s 0; [Z; X G=U ] t s =0:

25 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) ˆL(X G=U E ) s X G=U is a weak equivalece. Proof. By Propositio 4.15, II E s;t 2 (Z; C)=s [Z; ˆL(C E )] t. Now let X G =U be the costat cosimplicial spectrum with X j G = X =U G =U. There are cosimplicial ithe stable category maps X G =U ˆL(C E )ad ˆL(C E ) X G giveo0-simplices by d : X =U G =U ˆL(X G=U E ) (see Propositio4.10) ad s : ˆL(X G=U E ) X G=U, respectively. Furthermore, these maps are chaihomotopy equivaleces; use the chai homotopy h : ˆL(C j E ) ˆL(C j 1 E ) deed by h =( 1) j X G=U (E ) (j 1). This theimplies that [Z; ˆL(C E )] t is as claimed. The weak homotopy equivalece Tot( ˆL(C E )) X G=U is obtaied by trackig dow the ideticatios. Corollary 5.5. There is a atural weak equivalece ˆL(E hu E ) X G=U of commutative S 0 -algebras ad right E -module spectra such that the diagram is homotopy commutative. I particular, Theorem 1(iii) holds. We ow tur to the proof of Theorem 5.3. Let I Er ; (Z; C) deote the spectral sequece obtaied by mappig a CW-spectrum Z ito the tower we have ˆL ( Tot k+1 ( C ) E ) ˆL ( Tot k ( C ) E ) ; I E s;t 1 (Z; C)=[Z; ˆL(F s (C) E )] t+s ; (5.6) where F s (C) is the ber of Tot s ( C) Tot s 1 ( C). There is a caoical stable map from the (uraveled exact couple of the) tower { ˆL(Tot k ( C) E )} to {Tot k ( ˆL(C E ))}; obers this map is the caoical map ˆL(F s (C) E ) F s ( ˆL(C E )): (5.7)

26 26 E.S. Deviatz, M.J. Hopkis / Topology 43 (2004) 1 47 Lemma 5.8. Let I E ; 2 (S 0 ;C) II E ; 2 (S 0 ;C) be the map of spectral sequeces described above. If C is oe of the diagrams i 5.1, the this map is a isomorphism. Proof. If C is a diagram of the form 5.1(ii) or 5.1(iii) thethe map (5.7) is a equivalece ad hece the desired result follows immediately. Now let C = C G=U ad examie I E ; 2 (S 0 ;C). By (4.14), H ([Z; F (C)] t+ )= [Z; C ] t for ay CW-spectrum Z, ad therefore Lemma 5.4 implies that H s t (F (C) E M(p i0 ;:::;v i 1 1 )) = s t (C E M(p i0 ;:::;v i 1 = 1 )) { t (X G=U M(p i0 ;:::;v i 1 1 )) s =0; 0 s 0: Iparticular, these cohomology groups are all ite. Here M(p i0 ;v i1 1 ;:::;vi 1 1 ) is as i the begiig of Sectio 4; the multi-idex I =(i 0 ;:::;i 1 ) varies over a coal sequece as i [6, Sectio4], so that ˆLY = holim (Y M(p i0 ;:::;v i 1 1 )) I for ay E() -local spectrum Y. We claim that that is holim I (F k (C) E M(p i0 ;:::;v i 1 1 )) = lim (F k (C) E M(p i0 ;:::;v i 1 I 1 )); lim 1 (F k (C) E M(p i0 ;:::;v i 1 I 1 ))=0: (5.9) Assumig this, it follows from the vaishig of lim 1 H t (F (C) E M(p i0 ;:::;v i 1 I 1 )) that H s t ˆL(F (C) E ) = lim H s t (F (C) E M(p i0 ;:::;v i 1 I 1 )) { t X G=U s =0 = 0 s 0: Furthermore, oe checks easily that the map I E s;t 2 (S0 ;C) II E s;t 2 (S0 ;C) is the idetity uder this ideticatio ad the ideticatio of Lemma 5.4. We ow verify the claim. F k (C) is equivalet to a product of C j G =U s ad is therefore Ladweber exact. Hece (F k (C) E )isaate -module, so (F k (C) E M(p i0 ;:::;v i 1 1 )) = (F k (C) E )=(p i0 ;:::;v i 1 1 ): Iparticular, the iverse system { (F k (C) E M(p i0 ;:::;v i 1 1 ))} is Mittag-Leer ad therefore 5.9 holds. This completes the proof.