THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS

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1 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS MARK BEHRENS 1 AND DANIEL G. DAVIS 2 Abstract. Let E be a k-local profte G-Galos exteso of a E -rg spectrum A ( the sese of Roges). We show that E may be regarded as producg a dscrete G-spectrum. Also, we prove that f E s a profathful k-local profte exteso whch satsfes certa extra codtos, the the forward drecto of Roges s Galos correspodece exteds to the profte settg. We show that the fucto spectrum F A ((E hh ) k, (E hk ) k ) s equvalet to the localzed homotopy fxed pot spectrum ((E[[G/H]]) hk ) k where H ad K are closed subgroups of G. Applcatos to Morava E-theory are gve, cludg showg that the homotopy fxed pots defed by Devatz ad Hopks for closed subgroups of the exteded Morava stablzer group agree wth those defed wth respect to a cotuous acto terms of the derved fuctor of fxed pots. Cotets 1. Itroducto 1 2. Dscrete symmetrc G-spectra 6 3. Homotopy fxed pots of dscrete G-spectra Cotuous G-spectra Modules ad commutatve algebras of dscrete G-spectra Profte Galos extesos Closed homotopy fxed pots of profte Galos extesos Applcatos to Morava E-theory 51 Refereces Itroducto I [34], Joh Roges develops a Galos theory of commutatve S-algebras whch mmcs Galos theory for commutatve rgs. Let k be a S-module, ad let ( ) k deote Bousfeld localzato wth respect to k. Gve a k-local cofbrat commutatve S-algebra A, ad a cofbrat commutatve A-algebra E that s k-local, Roges gves the followg defto of a fte k-local Galos exteso Mathematcs Subject Classfcato. Prmary 55P43; Secodary 55P91, 55Q51. 1 The frst author was supported by NSF grat DMS , the Sloa Foudato, ad DARPA. 2 Part of the secod author s work o ths paper was supported by a NSF VIGRE grat at Purdue Uversty, a vst to the Mttag-Leffler Isttute, ad a grat from the Lousaa Board of Regets Support Fud. 1

2 2 MARK BEHRENS AND DANIEL G. DAVIS Defto (Fte Galos exteso). The spectrum E s a k-local G-Galos exteso of A, for a fte dscrete group G, f t satsfes the followg codtos: (1) G acts o E through commutatve A-algebra maps. (2) The caocal map A E hg s a equvalece. (3) The caocal map (E A E) k Map(G, E) s a equvalece. E s sad to be k-locally fathful over A f (M A E) k mples that M k for every A-module M. I the cotext of k-local Galos extesos, we shall smply refer to such extesos as fathful. Remark Roges ([34, Prop ]; see also [1]) shows that a k-local G- Galos exteso s fathful f ad oly f the addtve form of Hlbert s Theorem 90 holds: (E tg ) k. We wll mostly cosder fathful Galos extesos, because these are the Galos extesos for whch the fudametal theorem of Galos theory holds. We refer the terested reader to [35] for a example, due to Be Welad, of a Galos exteso that s ot fathful. Let G be a profte group. Followg (ad slghtly modfyg) Roges s defto [34, Def ] of a k-local pro-g-galos exteso, we defe a (profathful) k-local profte G-Galos exteso E of A to be a colmt ( the category of commutatve A-algebras) of (fathful) k-local G/U α -Galos extesos E α of A, for a cofal system of ope ormal subgroups U α of G (see Defto 6.2.1). Sce a colmt of k-local spectra eed ot be k-local, the spectrum E s ot ecessarly k-local. I [6], the secod author developed a category of dscrete G-spectra ad defed ther homotopy fxed pots (see also [41], [24], [32], [14], [26]). I ths paper, we exame k-local profte G-Galos extesos E of A as objects the category of dscrete G-spectra, ad we study the spectra of A-module maps betwee the varous homotopy fxed pot spectra of E. Ufortuately, to say meagful thgs t seems that we must mpose more hypotheses o our profte Galos extesos. Assumpto I ths paper, we shall oly cocer ourselves wth localzatos ( ) k whch are gve as a composte of two localzato fuctors (( ) T ) M, where ( ) T s a smashg localzato ad ( ) M s a localzato wth respect to a fte spectrum M. The spectra S, HF p, E(), ad K() are all examples of such localzatos k (see [4], [19]). For a cofbrat commutatve S-algebra B ad a cofbrat commutatve B-algebra C, the k-local Amtsur derved completo Bk,C s the homotopy lmt of the cosmplcal spectrum C k (C B C) k (C B C B C) k (see, for example, [34, Def ]). Defto Let E be a k-local profte G-Galos exteso of A. (1) The exteso E s cosstet f the coaugmetato of the k-local Amtsur derved completo A A k,e s a equvalece.

3 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 3 (2) The exteso E s of fte vrtual cohomologcal dmeso (fte vcd) f the profte group G has fte vcd (.e., G has a ope subgroup U of fte cohomologcal dmeso: there exsts a d such that H s c (U; M) = 0 for each s > d ad each dscrete U-module M). Assumpto esures that f E has fte vcd, the the codto of E beg cosstet s equvalet to requrg that the map A (E hg ) k s a equvalece. Ths s prove as Corollary It the follows that the maps E α (E huα ) k are equvaleces (Lemma 6.3.6). The cosstecy hypothess may be uecessary, sce we do ot kow of ay profte Galos extesos whch are ot cosstet. The ma cocer of ths paper s the study of the termedate homotopy fxed pot spectra E hh wth respect to closed subgroups H of G. We prove the forward drecto of the Galos correspodece. Theorem (7.2.1). Suppose that E s a cosstet profathful k-local profte G- Galos exteso of A of fte vcd, ad that H s a closed subgroup of G. (1) The spectrum E s k-locally H-equvaratly equvalet to a cosstet profathful k-local H-Galos exteso of (E hh ) k of fte vcd. (2) If H s a ormal subgroup of G, the the spectrum E hh s k-locally equvalet to a profathful k-local G/H-Galos exteso of A. If the quotet G/H has fte vcd, the ths exteso s cosstet (ad of fte vcd) over A. Remark Note that the ope subgroups of a profte group G are precsely the closed subgroups of fte dex. Also, f G has fte vcd, the t easly follows from [38, I.3.3] that every closed subgroup also has fte vcd. We also detfy the fucto spectrum of A-module maps betwee ay two such homotopy fxed pot spectra. Theorem (7.3.1). Let E be a cosstet profathful k-local profte G-Galos exteso of fte vcd, ad let H ad K be closed subgroups of G. The there s a equvalece (1.1) F A ((E hh ) k, (E hk ) k ) ((E[[G/H]]) hk ) k. The spectrum E[[G/H]] that appears o the rght-had sde of (1.1) s the cotuous G-spectrum wth the dagoal acto. The case where K = H = {e} was hadled by Roges [34, (8.1.3)]. I the cotext of Morava E-theory, (1.1) was prove [15] uder the addtoal assumpto that K s fte, ad t was suggested by the authors of [15] that (1.1) should be true wth ths extra assumpto removed. Aother source of motvato for ths work arses from the fact that a specal case of (1.1) (Corollary 7.3.2) was eeded a essetal way by the frst author [2] (see [2, Thm , Cor ]). Oe mportat example of a profte Galos exteso s gve by Morava E- theory. Let k = K() be the th Morava K-theory spectrum ad let A = S K() be the K()-local sphere spectrum. Let G = G be the th exteded Morava stablzer group S Gal(F p /F p ). Let E be the th Morava E-theory spectrum, where

4 4 MARK BEHRENS AND DANIEL G. DAVIS (E ) = W (F p )[[u 1,..., u 1 ]][u ±1 ]. Goerss ad Hopks [16], buldg o work of Hopks ad Mller [33], have show that G acts o E by maps of commutatve S-algebras. Devatz ad Hopks [9] have gve costructos of homotopy fxed pot spectra E dhh for closed subgroups H of G. I partcular, they show that there s a equvalece E dhg S K(). Thus, the homotopy fxed pot spectra of E are tmately related to the th chromatc layer of the sphere spectrum. Roges [34, Thm , Prop ] proved for U a ope ormal subgroup of G, that the work of Devatz ad Hopks [8, 9] shows that E dhu s a fathful K()-local G /U-Galos exteso of S K(). Therefore, the dscrete G -spectrum F = colm E dhu U og s a profathful K()-local profte G -Galos exteso of S K(). Addtoally, the profte exteso F of S K() s cosstet ad has fte vcd (Proposto 8.1.2). The spectrum E s recovered by the equvalece [9] E (F ) K(). As metoed above, for ay closed subgroup H of G, Devatz ad Hopks [9] costructed the commutatve S-algebra E dhh. Further, they showed that E dhh behaves lke a homotopy fxed pot spectrum wth respect to a cotuous acto of H. I more detal, [9] showed that E dhh has the followg propertes: (a) there s a K()-local E -Adams spectral sequece H s c (H; π t (E )) π t s (E dhh ), where the E 2 -term s the cotuous cohomology of H, wth coeffcets the profte H-module π t (E ), ad ths spectral sequece has the form of a descet spectral sequece; (b) whe H s fte, there s a weak equvalece E dhh E hh, ad the descet spectral sequece for E hh s somorphc to the spectral sequece (a); ad (c) E dhh s a (N(H)/H)-spectrum, where N(H) s the ormalzer of H G. O the other had, whe H s ot fte, E dhh s ot kow to actually be the H-homotopy fxed pot spectrum of E, because (a) t s ot costructed wth respect to a cotuous H-acto, ad (b) t s ot obtaed by takg the total rght derved fuctor of fxed pots (ad homotopy fxed pots are, by defto, the total rght derved fuctor of fxed pots, some sese - see [6, Remark 8.4] for the precse defto the case of a cotuous H-spectrum that arses from a tower of dscrete H-spectra). To address ths stuato, [6], the secod author showed that H does act cotuously o E ad there s a actual H-homotopy fxed pot spectrum E hh, wth a descet spectral sequece H s c (H; π t (E )) π t s (E hh ). that they should be equvalet to each other, ad by a result the secod author s thess [5], they are. However, sce ths part of [5] was ever publshed, we use the machery of ths paper to prove the equvalece of these two spectra. I more detal, we gve proofs of the followg two results (whch orgally appeared [5]). From the above dscusso, we see from the propertes of E dhh ad E hh

5 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 5 Theorem (8.2.1). For every closed subgroup H of G, there s a equvalece E dhh E hh betwee the Devatz-Hopks costructo ad the homotopy fxed pots that are defed wth respect to the cotuous acto of H. The above theorem shows that E dhh ca be referred to as a homotopy fxed pot spectrum, whereas, prevously, E dhh was oly kow to behave lke a homotopy fxed pot spectrum. Theorem (8.2.3, 8.2.4, 8.2.5). Let H be a closed subgroup of G ad let X be a fte spectrum. The there s a equvalece E dhh X (E X) hh ad the K()-local E -Adams spectral sequece for π (E dhh X) s somorphc to the descet spectral sequece for π ((E X) hh ) from the E 2 -terms oward. I partcular, (X) K() (E X) hg. The paper s orgazed as follows. Our oto of homotopy fxed pot spectra uses the framework of equvarat spectra (wth respect to a profte group) as developed by the secod author [6]. The foudatos [6] use Bousfeld-Fredlader spectra. Sce we eed to work wth structured rg spectra to do Galos theory, t s essetal for ths paper that we reformulate portos of [6] the cotext of symmetrc spectra. A cocse summary of these foudatos appears Secto 2. I Secto 3, we descrbe propertes of the homotopy fxed pot fuctor. I Secto 4, we descrbe cotuous G-spectra, geeralzg somewhat the settg of [6]. I Secto 5, we expla how to exted our costructos to categores of modules ad commutatve algebras of spectra. I Secto 6, we expla how profte Galos extesos gve rse to dscrete G-spectra, ad we show that the homotopy fxed pots wth respect to ope subgroups of the Galos group gve rse to termedate fte Galos extesos. I Secto 7, we prove our results cocerg the homotopy fxed pot spectra wth respect to closed subgroups of the Galos group. I Secto 8, we show that the hypotheses o profte Galos extesos whch we requre are satsfed by Morava E-theory. We the apply our machery to show that the Devatz-Hopks homotopy fxed pots agree wth the secod author s homotopy fxed pots, ad deduce some corollares. Ackowledgmets. The frst author beefted from the put of Halvard Fausk, Paul Goerss, ad Dael Isakse. The secod author thaks Paul Goerss for helpful dscussos, whe he was a Ph.D. studet, regardg the results Sectos 8.1 ad 8.2. Also, the secod author s grateful to Paul for later helpful coversatos, to Mark Hovey for provdg some tuto related to Theorem 2.3.2, ad Joh Roges for a helpful dscusso regardg group actos. The secod author spet several weeks the summer of 2008 the Departmet of Math at Rce Uversty workg o ths paper, ad he thaks the departmet for ts hosptalty. The authors would also lke to express ther thaks to the referee, for suggestg umerous mprovemets to ths paper.

6 6 MARK BEHRENS AND DANIEL G. DAVIS 2. Dscrete symmetrc G-spectra Let G be a profte group. We beg ths secto by descrbg the basc categores of dscrete G-objects that wll be used ths paper. We the descrbe ad compare the model structures o the categores of dscrete G-objects Bousfeld- Fredlader ad symmetrc spectra. We ed ths secto wth descrptos of some basc costructos the category of dscrete G-spectra. More detaled accouts of some of these model categores ad costructos ca be foud [6] Smplcal dscrete G-sets. A G-set Z s sad to be dscrete f, for every elemet z Z, the stablzer Stab G (z) s ope G. We may express ths codto by sayg that Z s the colmt of ts fxed pots: Z = colm U og ZU, where the colmt s take over all ope subgroups. These codtos are equvalet to the codto that the acto map G Z Z s cotuous, whe Z s gve the dscrete topology. A smplcal dscrete G-set s a smplcal object the category of dscrete G-sets. Goerss showed that the category sset G of smplcal dscrete G-sets admts a model category structure [14]. Theorem (Goerss). The category sset G admts a model category structure where the cofbratos are the moomorphsms, the weak equvaleces are those morphsms whch are weak equvaleces o uderlyg smplcal sets, the fbratos are determed. Lemma The model structure o sset G s left proper ad cellular. Proof. The model structure s left proper because the cofbratos ad weak equvaleces are precsely the cofbratos ad weak equvaleces o the uderlyg smplcal sets, ad the model category structure o smplcal sets s left proper. The model structure [14] s cofbratly geerated, wth geeratg cofbratos I ad geeratg trval cofbratos J, where: I = {G/U G/U : U o G, 0}, { } J = A j j s a trval cofbrato, B :. #B α. Here, α s a fxed fte cardal greater tha the cardalty of G ad #B deotes the cardalty of the set of o-degeerate smplces of B (see the proof of Lemma 1.13 of [14]). The axoms of beg cellular are mmedately verfed from ths descrpto of the geeratg cofbratos. The category (sset G ) of poted smplcal dscrete G-sets, beg a udercategory, herts a model structure from sset G. The cofbratos, weak equvaleces, ad fbratos are detected o the level of uderlyg smplcal dscrete G-sets. If K ad L are poted smplcal dscrete G-sets, the ther smash product K L

7 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 7 s easly see to be a smplcal dscrete G-set. The smash product gves the category (sset G ) a symmetrc moodal structure. (It does ot exted to a closed symmetrc moodal structure.) Lemma The model category structure o (sset G ) s left proper ad cellular. Wth respect to the symmetrc moodal structure gve by the smash product, the model category (sset G ) s a symmetrc moodal model category. Proof. Left properess follows from the fact that sset G s left proper. The model structure o (sset G ) s cofbratly geerated wth geeratg cofbratos (respectvely geeratg trval cofbratos) I + (respectvely J + ). Here, I + ad J + are the sets of maps obtaed from I ad J by addg a dsjot basepot o whch G acts trvally. The axoms of beg a symmetrc moodal model category are easly verfed Dscrete G-spectra. Defe the category of dscrete G-spectra Sp G to be the category of Bousfeld-Fredlader spectra of smplcal dscrete G-sets. A object X Sp G cossts of a sequece {X } 0, where each X s a poted smplcal dscrete G-set, together wth G-equvarat maps σ : S 1 X X +1. Here, S 1 s gve the trval G-acto. A map f : X Y of dscrete G-spectra s a sequece of G-equvarat maps of poted smplcal sets f : X Y whch are compatble wth the spectrum structure maps. I [6], the secod author studed the followg model structure. Theorem The category Sp G admts a model structure where the cofbratos are the cofbratos of uderlyg Bousfeld-Fredlader spectra, the weak equvaleces are the stable weak equvaleces of the uderlyg Bousfeld-Fredlader spectra, the fbratos are determed. The method used [6] was to trasport a Jarde model structure o presheaves of spectra o a approprate ste. However, a alteratve approach s gve below usg the machery of M. Hovey [20]. Proof. Observe that (sset G ) satsfes the codtos of Defto 3.3 of [20]. Therefore, the category Sp G of spectra of smplcal dscrete G-sets admts a stable model category structure where the cofbratos are those morphsms A B where the duced maps A 0 B 0 A S 1 A 1 S 1 B 1 B 1 are cofbratos. the fbrat objects X are those spectra for whch (1) the spaces X are fbrat as smplcal dscrete G-sets, (2) the maps X ΩX +1 are weak equvaleces. the weak equvaleces f : X Y betwee fbrat objects are those f for whch the maps f : X Y are all weak equvaleces.

8 8 MARK BEHRENS AND DANIEL G. DAVIS Clearly the cofbratos of Sp G are the maps whch are cofbratos of uderlyg Bousfeld-Fredlader spectra. We are left wth verfyg that the weak equvaleces Sp G are precsely the stable equvaleces of uderlyg Bousfeld-Fredlader spectra. The forgetful fuctor U : sset G sset from smplcal dscrete G-sets to smplcal sets s a left Qulle fuctor (t preserves cofbratos ad trval cofbratos, ad s left adjot to the fuctor CoId G 1 of Secto 3.4). By Proposto 5.5 of [20], the duced forgetful fuctor U : Sp G Sp s a left Qulle fuctor. Let ( ) fg deote the fuctoral fbrat replacemet Sp G, so that there are atural trval cofbratos α G,X : X X fg. Suppose that φ : X Y s a morphsm Sp G, ad cosder the followg dagram: α G,X X φ Y X fg φfg Y fg We clam that φ s a stable equvalece Sp G f ad oly f the duced morphsm Uφ s a stable equvalece of uderlyg Bousfeld-Fredlader spectra. Because U s a left Qulle fuctor, ad the morphsms α G, are trval cofbratos, we may coclude that the morphsms Uα G, duce stable equvaleces of uderlyg Bousfeld-Fredlader spectra. Sce the uderlyg spectrum of a fbrat object Sp G s fbrat Sp, ad sce the stable equvaleces betwee fbrat objects Sp G ad Sp are precsely the levelwse equvaleces, we see that φ fg s a stable equvalece f ad oly f Uφ fg s a stable equvalece. Therefore, we may deduce that φ s a stable equvalece f ad oly f Uφ s a stable equvalece Dscrete symmetrc G-spectra. Let ΣSp deote the category of symmetrc spectra (see [21], [29] for accouts of symmetrc spectra). Defe the category of dscrete symmetrc G-spectra ΣSp G to be the category of symmetrc spectra of smplcal dscrete G-sets. Let Σ deote the th symmetrc group. A object X ΣSp G cossts of a sequece {X } 0, where each X s a poted smplcal dscrete G Σ -set, together wth sutably compatble G Σ Σ j -equvarat maps α G,Y σ,j : S X j X +j. Here, S = (S 1 ) s gve the trval G-acto, ad Σ permutes the factors of the smash product (S 1 ). Whe G s fte, a dscrete symmetrc G-spectrum s smply a aïve symmetrc G-spectrum, ad ot a geue equvarat symmetrc G-spectrum the sese of [28]. Maps f : X Y of dscrete symmetrc G-spectra are sequeces of G Σ - equvarat maps of poted smplcal sets f : X Y whch are compatble wth the spectrum structure maps.

9 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 9 For a cofbratly geerated model category C, let C Σj deote the dagram category of Σ j -equvarat objects C wth the projectve model structure ([17, Thm ]). Lemma I the projectve model category structure o (sset G ) Σj : the cofbratos are those maps that are projectve cofbratos the uderlyg category sset Σj, the weak equvaleces are those maps that are weak equvaleces the uderlyg category sset Σj, the fbratos are determed. Proof. The statemet cocerg weak equvaleces follows mmedately from the defto of the weak equvaleces the projectve model structure. The projectve cofbratos (sset G ) Σj are geerated by the set I Σj + = {(Σ j G/U ) + (Σ j G/U ) + : U o G, 0}. Usg the relatve skeletal fltrato, t s easy to see that the class of cofbratos geerated by the set I Σj + are the moomorphsms whch are relatve free Σ j - complexes. However, these are precsely the projectve cofbratos sset Σj. Theorem The category ΣSp G admts a left proper cellular model structure where the cofbratos are the cofbratos of uderlyg symmetrc spectra, the weak equvaleces are the stable weak equvaleces of uderlyg symmetrc spectra, the fbratos are determed. Proof. Observe that (sset G ) satsfes the codtos of Defto 8.7 of [20]. Therefore, the category ΣSp G of symmetrc spectra of smplcal dscrete G-sets admts a stable model category structure where the cofbratos are those morphsms A B where the duced maps A 0 B 0 A LA L B B 1 are projectve cofbratos (sset G ) Σj, where L s the latchg object of [20, Def. 8.4]. the fbrat objects X are those spectra for whch (1) the spaces X are fbrat as smplcal dscrete G-sets, (2) the maps X ΩX +1 are weak equvaleces. the weak equvaleces f : X Y betwee fbrat objects are those f for whch the maps f : X Y are all weak equvaleces of uderlyg smplcal sets. The cofbratos are mmedately see to be the cofbratos of uderlyg symmetrc spectra, usg Lemma The verfcato that the weak equvaleces are precsely the stable equvaleces of uderlyg symmetrc spectra s detcal to the argumet gve the proof of Theorem We have the followg proposto, whch helps to traslate results the category Sp G, to the category ΣSp G.

10 10 MARK BEHRENS AND DANIEL G. DAVIS Proposto There s a Qulle equvalece where U G s the forgetful fuctor. V G : Sp G ΣSp G : U G Proof. The fuctor V G s the left adjot of U G : t s explctly gve by (see [21, Sec. 4.3]) V G (X) = S T (G1S 1 ) GX where GX s the symmetrc sequece gve by (GX) = (Σ ) + X (where G acts through ts acto o X ), G 1 S 1 s the symmetrc sequece (, S 1,,, ) (wth trval G acto), T (G 1 S 1 ) s the free mood o G 1 S 1 wth respect to (whch gves the symmetrc moodal structure o symmetrc sequeces), ad S s the usual symmetrc sequece (S 0, S 1, S 2, ). We have the followg commutatve dagram of fuctors V G Sp G ΣSp G U Sp U G V U U Σ ΣSp where the bottom row s the Qulle equvalece of [21, Sec. 4]. The fuctor V G preserves cofbratos ad trval cofbratos because the fuctors U ad U Σ reflect ad detect cofbratos ad trval cofbratos, ad the fuctor V s a left Qulle fuctor. Therefore (V G, U G ) forms a Qulle par. To show that (V G, U G ) s a Qulle equvalece, we must show that for all cofbrat X Sp G ad all fbrat Y ΣSp G, a morphsm f : V G X Y s a weak equvalece f ad oly f ts adjot f : X U G Y s a weak equvalece. However, sce the fuctors U ad U Σ reflect ad detect weak equvaleces, t suffces to show that: { } { } UΣ f : VUX U Σ Y U f : UX UUΣ Y f ad oly f. s a weak equvalece s a weak equvalece Ths follows from the fact that U preserves cofbratos (Theorem 2.2.1), U Σ preserves fbrat objects (proof of Theorem 2.3.2), ad (V, U) form a Qulle equvalece [21, Thm ]. For the rest of ths paper, we shall be workg the world of symmetrc spectra, ad shall refer to a symmetrc spectrum as smply a spectrum. For a spectrum X, we shall always use π X to refer to ts true homotopy groups (the maps [S t, X] the stable homotopy category) ad ot the aïve homotopy groups (the colmt of the homotopy groups of X ) [36].

11 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS Mappg spectra. Let K ad L be dscrete G-sets. The the set of (oequvarat) fuctos Map(K, L) s a G-set wth G actg by cojugato. Ths acto s gve as follows: for g G ad f Map(K, L), g f s the map (g f)(z) = gf(g 1 z). Observe that Map(K, L) s ot geeral a dscrete G-set, but t s f K s fte. For a fte set K ad a spectrum X, we defe the mappg spectrum Map(K, X) to be the spectrum whose mth space s gve by Map(K, X) m = Map(K, X m ), where the -smplces of Map(K, X m ) s the set Map(K, (X m ) ). If X s a dscrete G-spectrum, ad K s a fte dscrete G-set, the the above deftos combe to gve that Map(K, X) s a dscrete G-spectrum. If K = lm α K α s a profte set ad X s a spectrum, the the spectrum of cotuous maps s the spectrum Map c (K, X) = colm Map(K α, X). α If K s a cotuous G-space, wth each K α a dscrete G-set, ad X s a dscrete G-spectrum, the Map c (K, X) s a dscrete G-spectrum. Lemma Let K = lm α K α be a profte set, where each of the K α s fte, ad each of the maps K α K β the pro-system s a surjecto. The fuctor preserves stable equvaleces. Map c (K, ) : ΣSp ΣSp Proof. I [37], t s show that stable equvaleces are preserved uder fte products. The argumet goes as follows: the caocal map from a fte wedge to a fte product s a π -somorphsm, hece a stable equvalece, ad stable equvaleces are preserved uder fte wedges (ths s easly checked from the defto of stable equvaleces, by mappg to jectve Ω-spectra). Therefore, for each α, the fuctor X Map(K α, X) = K α X preserves stable equvaleces. Sce we have assumed that the morphsms K α K β are surjectos, the duced morphsms Map(K β, X) Map(K α, X) are levelwse moomorphsms for every X. The category of symmetrc spectra possesses a jectve stable model structure, where the jectve cofbratos are the levelwse moomorphsms ad the weak equvaleces are the stable equvaleces (see [21, p. 199]). The drected system {Map(K α, X)} s a drected system of jectve cofbratos betwee jectvely cofbrat objects (every object s cofbrat the jectve model structure). The colmt Map c (K, X) = colm α Map(K α, X)

12 12 MARK BEHRENS AND DANIEL G. DAVIS may be computed o a cofal λ-sequetal subcategory of the dexg category of the system {K α }, for some ordal λ. We deduce, by [17, Prop ], that the fuctor preserves stable equvaleces. X Map c (K, X) = colm α Map(K α, X) If A s a abela group, edowed wth the dscrete topology, ad K = lm α K α s a profte set, the cotuous maps Map c (K, A) are gve by Map c (K, A) = colm Map(K α, A). α Lemma Let X be a object of ΣSp, ad let K be a profte set satsfyg the hypotheses of Lemma The there s a somorphsm π Map c (K, X) = Map c (K, π (X)) Proof. By Lemma t suffces to assume that X s fbrat. The result the follows from Corollary Permutato spectra. Let K be a dscrete G-set. The for X a dscrete G-spectrum, we may defe the permutato spectrum X[K] to be the spectrum whose th space s gve by X[K] = X K +. We let G act o the spectrum X[K] through the dagoal acto. Lemma The spectrum X[K] s a dscrete G-spectrum. Proof. Note that X K + = (colm N XN ) ( colm og N KN + ) og = colm N,N (XN K N + ) og s a smplcal dscrete G-set, wth G actg dagoally, sce the smplcal set X N K N + has a dagoal G/(N N )-acto ad the group G/(N N ) s fte. Thus, the spectrum X[K] s a dscrete G-spectrum Smash products. Gve dscrete G-spectra X ad Y, we defe ther smash product X Y to be the smash product of the uderlyg symmetrc spectra wth G actg dagoally. Sce the smash product commutes wth colmts, t follows, as the proof of Lemma 2.5.1, that X Y s a dscrete G-spectrum. Also, f K s a dscrete G-set, the there s a G-equvarat somorphsm X S[K] = X[K], where the sphere spectrum S has trval G-acto.

13 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS Homotopy fxed pots of dscrete G-spectra Much of the materal ths secto s assembled from [41], [25], [14], [26], [32], [27], ad [6]. Let G be a profte group. We beg ths secto wth a accout of the model category theoretc defto of G-homotopy fxed pots. We the descrbe the comparso wth hypercohomology spectra. Fte dex restrcto ad ducto fuctors, as well as terated homotopy fxed pots for fte dex subgroups are the dscussed. We expla how cotuous homomorphsms of groups duce varous chage of group fuctors, of whch ducto, coducto, fxed pots, ad restrcto fuctors are all specal cases. We the descrbe the varous techcal dffcultes related to the homotopy fxed pot costructo for closed subgroups of G. The techcal dffcultes are observed to vash f G has fte cohomologcal dmeso. As alluded to above, Sectos 3.3, 3.5, ad 3.6 dscuss the costructo of terated homotopy fxed pots. Much of ths materal overlaps wth portos of [7]: t was ecessary to repeat some of the materal from [7], so that certa ssues are clear ad to gve a cotext for the results of Secto 7.1. We ote that, as explaed Secto 2.3, spectrum meas symmetrc spectrum, so that, for example, a dscrete G-spectrum s a dscrete G-symmetrc spectrum The homotopy fxed pot spectrum. For a dscrete G-spectrum X, we defe the fxed pot spectrum by takg the fxed pots levelwse: (X G ) = (X ) G. The G-fxed pots fuctor s rght adjot to the fuctor trv, whch assocates to a spectrum X the dscrete G-spectrum X, where X ow has the trval G-acto: trv : ΣSp ΣSp G : ( ) G. Lemma The adjot fuctors (trv, ( ) G ) form a Qulle par. Proof. The fuctor trv preserves cofbratos ad weak equvaleces. Let α G,X : X X fg deote a fuctoral fbrat replacemet fuctor for the model category ΣSp G, where α G,X s a trval cofbrato of dscrete G-spectra. The homotopy fxed pot fuctor ( ) hg s the Qulle rght derved fuctor of ( ) G, ad s thus gve by X hg = (X fg ) G Hypercohomology spectra. The fuctor Γ G = Map c (G, ) s a coaugmeted comoad o the category of spectra, wth coproduct ψ : Γ G = Map c (G, ) Map c (G G, ) = Γ G Γ G duced from the product o G, cout Γ G = Map c (G, ) Map c (pt, ) = Id duced from the ut o G, ad coaugmetato Id Map c (G, ) gve by the cluso of the costat maps.

14 14 MARK BEHRENS AND DANIEL G. DAVIS Dscrete G-spectra are coalgebras over the comoad Γ G (ths follows from cosderg the map of spectra X Γ G (X), x (g g x), for ay dscrete G-spectrum X). Let C ad D be categores, ad suppose that Γ s a comoad C. Dualzg Defto 9.4 of [30], there s a oto of a Γ-fuctor F : C D. Let Y be a Γ-coalgebra. Dualzg Costructo 9.6 of [30], oe may assocate to (F, Γ, Y ) a cosmplcal object C (F, Γ, Y ) D (the comoadc cobar costructo), gve by C s (F, Γ, Y ) = F Γ s Y. If Γ s a coaugmeted comoad, the the detty fuctor Id C s a Γ-fuctor. We wll let Γ Y deote the cosmplcal object Γ Y = C (Id C, Γ, Y ) C. I [6], the homotopy fxed pot spectrum was show to have the followg alterate descrpto, provded G s suffcetly ce (see also [32], [14], [24]). Theorem Suppose that G has fte vcd, ad that X s a dscrete G- spectrum. The there s a equvalece X hg holm Γ GX = H c (G; X), where H c (G; X) s the hypercohomology spectrum. Proof. I [6, Thm. 7.4] t s prove that there s a equvalece X hg holm Γ GX fg. (The cosmplcal object defg the hypercohomology spectrum s dfferet, but somorphc to that appearg [6].) The result follows oce we establsh that the map duced from fbrat replacemet holm Γ GX holm Γ GX fg s a equvalece. Ths map s deduced to be a equvalece from the followg facts: (a) the fbrat replacemet map X X fg s a equvalece; (b) the fuctor Γ G preserves equvaleces, by Lemma 2.4.1; (c) the homotopy lmt costructo seds levelwse equvaleces to equvaleces, sce t s a Qulle derved fuctor Iterated homotopy fxed pots. Let U be a ope subgroup of G, so that G/U s fte. Proposto Let X be a dscrete G-spectrum. (1) If U s ormal G, the U-fxed pot spectrum (X fg ) U s fbrat as a dscrete G/U-spectrum. (2) The fbrat dscrete G-spectrum X fg s fbrat as a dscrete U-spectrum.

15 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 15 (3) If U s ormal G, the homotopy fxed pot spectrum X hu s a G/Uspectrum. (4) If U s ormal G, there s a equvalece X hg (X hu ) hg/u. Proof. To prove (1), observe that sce U s ormal, for ay dscrete G-spectrum Y, the U-fxed pot spectrum Y U s aturally a G/U-spectrum. There s a adjot par of fuctors (Res G G/U, ( ) U ) Res G G/U : ΣSp G/U ΣSp G : ( ) U, where Res G G/U s defed by restrcto alog the quotet homomorphsm G G/U. Sce Res G G/U preserves cofbratos ad weak equvaleces, the fuctor ( ) U preserves fbrat objects. We verfy (2) a smlar way (compare wth [26, Rmk. 6.26]). Defe the ducto fuctor o a dscrete U-spectrum Y to be Id G U Y = G + U Y. Here, G + U Y s formed by regardg G ad U as dscrete groups, but ths s easly see to produce a dscrete G-spectrum, sce U s a subgroup of fte dex. The ducto fuctor s the left adjot of a adjucto Id G U : ΣSp U ΣSp G : Res U G, where Res U G s restrcto alog the cluso U G. Sce o-equvaratly there s a somorphsm Id G U Y = G/U + Y, we see that Id G U preserves cofbratos ad weak equvaleces, from whch t follows that Res U G preserves fbrat objects. By (2), X fg s a fbrat dscrete U-spectrum. Also, X X fg s a trval cofbrato of spectra ad t s U-equvarat, so t s a trval cofbrato ΣSp U. Thus, X hu = (X fg ) U, whch s a G/U-spectrum. Ths proves (3). (4) s prove usg our fbracy results. There are equvaleces: X hg X G fg = (X U fg) G/U (X hu ) hg/u Homomorphsms of groups. If f : H G s a cotuous homomorphsm of profte groups, we may regard dscrete G-sets as dscrete H-sets. For a dscrete H-set Z, we defe the coduced dscrete G-set by f Z = CoId G H Z = Map c H(G, Z) = colm U og Map H(G/U, Z), where the G-acto s defed by the formula (g α)(g U) = α(g gu), for g G ad α Map H (G/U, Z). Ths costructo exteds to smplcal dscrete G-sets ad dscrete G-spectra the obvous maer to gve a fuctor f : ΣSp H ΣSp G.

16 16 MARK BEHRENS AND DANIEL G. DAVIS The fuctor f s the rght adjot of a adjot par (f, f ), where f = Res H G : ΣSp G ΣSp H s the restrcto fuctor alog the homomorphsm f. Sce f clearly preserves cofbratos ad weak equvaleces, we have the followg lemma. Lemma The adjot fuctors (f, f ) form a Qulle par. I partcular, f preserves fbratos ad weak equvaleces betwee fbrat objects. We make the followg observatos. (1) The Qulle par (f, f ) gves rse to a derved adjot par (Lf, Rf ). (2) Sce the fuctor f preserves all weak equvaleces, there are equvaleces Lf X f X for all dscrete G-spectra X. (3) If j : H G s the cluso of a closed subgroup, the for a dscrete H-spectrum X, we have a o-equvarat somorphsm j X = Map c H(G, X) = Map c (G/H, X). By Lemma 2.4.1, we see that j preserves weak equvaleces, ad therefore there s a equvalece j X Rj X. (4) The adjot par (trv, ( ) G ) of Secto 3.1 agrees wth the adjot par (r, r ) whe r : G {e} s the homomorphsm to the trval group. Therefore, the homotopy fxed pot fuctor s gve by ( ) hg = Rr. (5) Gve cotuous homomorphsms H f G g K, there are atural somorphsms (g f) = g f ad (g f) = f g. We get smlar formulas o the level of derved fuctors. (6) If : U G s the cluso of a ope subgroup, the the ducto fuctor! = Id G U (Proposto 3.3.1) s the left adjot of the Qulle par (!, ). We use these derved fuctors to prove a verso of Shapro s Lemma. Lemma Let X be a dscrete G-spectrum, ad suppose that H s a closed subgroup of G. The there s a equvalece Map c (G/H, X) hg X hh. Proof. Cosder the followg dagram of groups. H j G s r {e} If Z s a dscrete G-set, there s a G-equvarat bjecto δ : j j Z = Map c H(G, Z) = Map c (G/H, Z). The map δ seds a map α Map c H(G, Z) to the map δ(α) : gh gα(g 1 ). The verse δ 1 seds a map β Map c (G/H, Z) to the map δ 1 (β) : g gβ(g 1 H).

17 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 17 The somorphsm δ duces for a dscrete G-spectrum Y a somorphsm δ : j j Y = Map c (G/H, Y ), ΣSp G. By Lemma 3.4.1, the fuctor j seds H-fbrat objects to G-fbrat objects. Therefore we have equvaleces: Map c (G/H, X) hg = Rr j j X Rr Rj j X Rs j X = X hh Iterated fxed pots for closed subgroups. We wsh to exted the results of Secto 3.3 to closed subgroups. The followg proposto may be compared to [26, Lem. 6.35]. Proposto Let N be a closed ormal subgroup of G, ad let X be a dscrete G-spectrum. The there s a equvalece Proof. Cosder the followg dagram. ((X fg ) N ) hg/n X hg. G q G/N r s {e} There s a equvalece ((X fg ) N ) hg/n = Rs Rq X Rr X = X hg. Let H be a closed subgroup of G. The reader mght woder f X fg s fbrat as a dscrete H-spectrum, but ths does ot appear to hold for otrval closed subgroups H that are ot ope. We dscuss these dffcultes Secto 3.6. Though (X fg ) H s ot kow to always equal X hh, the followg result detfes (X fg ) H wth a caocal colmt that always maps to X hh. Corollary Let H be a closed subgroup of G. There s a equvalece (X fg ) H colm H U og XhU. Proof. Sce H acts dscretely o X fg, we have a caocal somorphsm (X fg ) H = colm H U og (X fg) U. By Proposto 3.3.1, the spectrum X fg s fbrat as a dscrete U-spectrum, so there are equvaleces (X fg ) U X hu.

18 18 MARK BEHRENS AND DANIEL G. DAVIS By Corollary 3.5.2, gve a dscrete G-spectrum X ad a closed subgroup H, there s a atural map (X fg ) H colm H U og XhU X hh. If we restrct ourselves to the case where G has fte cohomologcal dmeso, the, as show below, the terated homotopy fxed pot spectrum behaves a more satsfactory way. Proposto Suppose that G has fte cohomologcal dmeso, ad suppose that X s a dscrete G-spectrum. Let H be a closed subgroup of G. The the atural map colm H U og XhU X hh s a equvalece. Proof. For K a profte group of fte vcd, ad Y a dscrete K-spectrum, let E r (K; Y ) deote the codtoally coverget descet spectral sequece There s a map of spectral sequeces E 2 (K; Y ) = H c (K; π (Y )) π (Y hk ). E r(h; X) := colm E r(u; X) E r (H; X), H U og whch s a somorphsm o the level of E 2 -terms by [42, Thm ]. The proposto ow follows from [32, Prop. 3.3]. I Secto 7.1, we wll see that we may exted Proposto to groups of fte vrtual cohomologcal dmeso provded that we are takg homotopy fxed pots of a cosstet profathful k-local profte Galos exteso. Corollary Let G be of fte cohomologcal dmeso, ad let X be a dscrete G-spectrum. Suppose that H s a closed subgroup of G. The there s a equvalece (X fg ) H X hh. Theorem Let G be of fte cohomologcal dmeso, ad let X be a fbrat dscrete G-spectrum. Suppose that H s a closed subgroup of G. The X s fbrat as a dscrete H-spectrum. Proof. By the proof of Theorem 2.3.2, the fbrat objects Y of ΣSp H are precsely the objects for whch Y are fbrat as smplcal dscrete H-sets, ad the maps Y ΩY +1 are weak equvaleces. Sce X s fbrat as a dscrete G-spectrum, each X s fbrat as a smplcal dscrete G-set, ad the maps X ΩX +1 are weak equvaleces. The oly thg remag to check s that each space X s fbrat whe regarded as a smplcal dscrete H-set. A crtero for fbracy s establshed [3, ]. We remark that [3] deals wth the more geeral class of locally compact totally dscoected groups G, actg o smplcal smooth G-sets. However, the case where G s a profte group, the category of smooth G-sets s the category of dscrete G-sets. We eed to check (1) the V -fxed pots X V s a Ka complex for every ope subgroup V H,

19 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 19 (2) for every ope subgroup V H ad every hypercover {H/V α, } α I of H/V, the duced map X V holm s a weak equvalece. Let V be a ope subgroup of H. We have X Vα, α I X V = colm V U og XU. Now X U s a Ka complex, sce X s fbrat as a smplcal dscrete G-set. Sce fltered colmts of Ka complexes are Ka complexes, we deduce that X V s a Ka complex. Ths verfes codto (1). We furthermore pot out that for each ope subgroup U G cotag V, Proposto mples that the fxed pot spectrum X U s fbrat. Therefore, the maps X U ΩX U +1 are weak equvaleces. Sce the fuctor Ω commutes wth fltered colmts, ad because fltered colmts of weak equvaleces are weak equvaleces, the maps X V ΩX V +1 are weak equvaleces, ad we deduce that the spectrum X V s fbrat (as a symmetrc spectrum). By Proposto 3.3.1, we have (X fh ) V X hv. Corollary therefore mples that the map X V (X fh ) V s a weak equvalece. Sce weak equvaleces betwee fbrat symmetrc spectra are levelwse equvaleces, we deduce that each map X V (X fh ) V s a weak equvalece. We ow verfy codto (2). Suppose that {H/V α, } α I s a hypercover of H/V. Cosder the followg dagram. X V holm α I X Vα, (X fh ) V holm α I (X fh ) Vα, The bottom map s a weak equvalece sce X fh s fbrat as a dscrete H- spectrum. We have show that the vertcal maps are weak equvaleces. We deduce that the top map s a weak equvalece. Corollary Let X be a dscrete G-spectrum. Suppose that G has fte cohomologcal dmeso, ad suppose that N s a closed ormal subgroup of G. The the homotopy fxed pot spectrum X hn s a dscrete G/N-spectrum, ad there s a equvalece (X hn ) hg/n X hg.

20 20 MARK BEHRENS AND DANIEL G. DAVIS Proof. Sce X fg s fbrat as a dscrete N-spectrum, we have X hn = (X fg ) N ad (X fg ) N s a dscrete G/N-spectrum. Furthermore, by Proposto 3.5.1, we have X hg ((X fg ) N ) hg/n (X hn ) hg/n The dffcultes cocerg arbtrary closed fxed pots. Let H be a closed subgroup of a arbtrary profte group G. We would be able to remove the fte cohomologcal dmeso hypothess Secto 3.5, f we kew that the restrcto fuctor Res H G : ΣSp G ΣSp H seds G-fbrat objects to H-fbrat objects. Whle we kow of o couterexamples to ths asserto, we also doubt that t ths s true geeral. We saw Proposto that for U a ope subgroup of G, the presece of a ducto fuctor Id G U whch was a left Qulle adjot to Res U G allowed us to prove that Res U G preserves fbrat objects. However, as poted out to the secod author by Jeff Smth, Res H G caot possess a left adjot, geeral, sce t does ot preserve lmts. Ths ca be see as follows. For a profte group K ad a dagram {X α } the category ΣSp K, let lm α K X α deote the lmt computed the category ΣSp K. Ths lmt s gve by the followg formula: lm K X α = colm (lm Sp X α ) U. α U ok α Here, the lmt lm Sp s the lmt computed the uderlyg category of symmetrc spectra. Thus, gve a dagram {X α } ΣSp G, the restrcto of the lmt s gve by Res H G lm G X α = colm (lm Sp X α ) U. α U og α However, the lmt of the restrcto s computed to be lm H Res H G X α = colm (lm Sp X α ) V. α V oh α Whe H s ot ope G, the lack of cofalty mples that these subspectra of lm Sp α X α geeral do ot agree. Oe mght suspect that oe could stll prove that the map colm H U XhU X hh og s a equvalece f G has fte vrtual cohomologcal dmeso, by a comparso of descet spectral sequeces. Ths approach, however, also presets dffcultes. As the proof of Proposto 3.5.3, there s a map of spectral sequeces E r(h; X) = colm E r(u; X) E r (H; X) H U og

21 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 21 whch s a somorphsm o the level of E 2 -terms (see [42, Thm ]). The problem s that the colmt of the spectral sequeces does ot coverge to the colmt of the abutmets geeral. 4. Cotuous G-spectra I ths paper, a cotuous G-spectrum s a pro-object the category of dscrete G-spectra. I ths secto, we exted some of our costructos for ΣSp G to the category of cotuous G-spectra. For cotuous G-spectra that are dexed over {0 1 2 }, part of ths materal appears more detal [6] Pro-objects dscrete G-spectra. Followg stadard usage, a pro-object a category C s a cofltered dagram C. We defe the category of cotuous G-spectra ΣSp c G to be the category of pro-objects ΣSp G. Thus, a cotuous G-spectrum s a cofltered dagram X = {X } I of dscrete G-spectra. Maps the category of cotuous G-spectra are gve by ΣSp c G(X, Y) = lm j colm ΣSp G (X, Y j ). Ay pro-spectrum X = {X } gves rse to a spectrum X va the homotopy lmt fuctor: X = holm X. We shall always deote our pro-spectra by boldface type ad ther homotopy lmts wth o-boldface type. Remark A more geeral theory of pro-spectra, cludg a model category structure, has bee developed by Isakse (see [23] ad [13, Secto 1.1]). Fausk has developed a category of cotuous geue G-spectra where G s a compact Hausdorff topologcal group [12]. The oto of cotuous G-spectrum ths paper (that s, a pro-dscrete G-spectrum) correspods roughly, [12], to a pro- G-spectrum that s the full subcategory of cofbrat objects the Postkov Le(G)-model structure o pro-m S. For more detal, we refer the reader to [12, Secto 11.3], especally the dscusso cetered aroud [op. ct., Eq. (11-15)] Cotuous mappg spectra. Let K = lm K be a profte G-set. Gve a cotuous G-spectrum X, the cotuous mappg spectrum Map c (K, X) s defed to be the cotuous G-spectrum {Map c (K, X j )} j. We deote the homotopy lmt of Map c (K, X) by Map c (K, X). If K satsfes the hypotheses of Lemma 2.4.1, ad the derved fuctors lm s j Map c (K, π t (X j )) = 0, for all s > 0 ad all t Z, the the Bousfeld-Ka spectral sequece collapses, ad thus, lm s j Map c (K, π t (X j )) π (Map c (K, X)) π (Map c (K, X)) = Map c (K, lm j π (X j )).

22 22 MARK BEHRENS AND DANIEL G. DAVIS 4.3. Cotuous permutato spectra. Let K = lm K be a profte G-set, ad let each fte set K j, for each j the dexg set for K, be a dscrete G-set. Also, let X = {X } be a cotuous G-spectrum. Defe the permutato spectrum X[[K]] to be the cotuous G-spectrum gve by {X [K j ]},j. We deote the homotopy lmt of X[[K]] by X[[K]] = holm X [K j ].,j Note that f lm s,j π t (X )[K j ] = 0, for all s > 0 ad all t Z (where π t (X )[K j ] s a abela group), the π (X[[K]]) = lm,j π (X )[K j ]. If E s a dscrete G-spectrum, we use E[[K]] to deote the cotuous G-spectrum {E[K j ]} j Cotuous homotopy fxed pots. For a cotuous G-spectrum X, we defe the homotopy fxed pot pro-spectrum X hg to be {X hg }. We deote the homotopy lmt of X hg by X hg, ad we refer to X hg as the homotopy fxed pot spectrum Cotuous hypercohomology spectra. We defe Γ G : (pro ΣSp) (pro ΣSp) to be the coaugmeted comoad gve by Γ G (X) = Map c (G, X). Let Γ G (X) be the homotopy lmt of Γ G (X). If X s a cotuous G-spectrum, the t s a coalgebra over Γ G. Let H c (G; X) deote the pro-spectrum obtaed by takg hypercohomology levelwse: H c (G; X) = {H c (G; X )}. Let H c (G; X) deote the homotopy lmt of the pro-spectrum H c (G; X). The followg result follows mmedately from Theorem Theorem Suppose that X = {X } s a cotuous G-spectrum. If G has fte vcd, the there s a equvalece X hg H c (G; X) Homotopy fxed pot spectral sequece. Let G have fte vcd. The Theorem mples that X hg holm holm Γ GX, ad, hece, the assocated Bousfeld-Ka spectral sequece has the form E s,t 2 (G; X) = πs π t (holm Γ GX ) π t s (X hg ), gvg the codtoally coverget homotopy fxed pot spectral sequece for X hg.

23 THE HOMOTOPY FIXED POINT SPECTRA OF PROFINITE GALOIS EXTENSIONS 23 Observe that there s a atural somorphsm Γ k G( ) = Map c (G k, ). If lm s Map c (G k, π t (X )) = 0 for all s > 0, all k 0, ad all t Z, the for each k 0, the Bousfeld-Ka spectral sequece collapses, ad thus, lm s Map c (G k, π t (X )) π (holm Map c (G k, X )) E s,t 2 (G; X) = π s (lm Map c (G, π t (X ))) = H s (Map c (G, lm π t (X ))) = H s (Map c (G, π t (X))) = H s c (G; π t (X)). Here, H s c (G; π t (X)) deotes the cotuous cohomology of cotuous cochas, wth coeffcets the topologcal G-module π t (X) = lm π t (X ) Completed smash product. If X ad Y are cotuous G-spectra, we defe the completed smash product X c Y to be the cotuous G-spectrum {X Y j },j. The completed smash product gves ΣSp c G a symmetrc moodal product, where the ut s {S 0 } (the sphere spectrum regarded as a dagram dexed by a sgle elemet). 5. Modules ad commutatve algebras of dscrete G-spectra Let A be a commutatve symmetrc rg spectrum ad let G be a profte group. I ths secto, we descrbe the model categores of dscrete G-A-modules ad dscrete commutatve G-A-algebras. We show that the homotopy fxed pots of a dscrete G-A-module s a A-module ad the homotopy fxed pots of a dscrete commutatve G-A-algebra s a commutatve A-algebra. These structured homotopy fxed pot costructos are show to agree, the stable homotopy category, wth the usual homotopy fxed pots of the uderlyg dscrete G-spectrum. We the make comparsos betwee fltered homotopy colmts ad fltered colmts of modules ad commutatve algebras, ad coclude that, whe properly terpreted, they all cocde the stable homotopy category. We coclude ths secto by descrbg how to make the hypercohomology spectra of dscrete commutatve G- A-algebras take values the category of commutatve A-algebras Modules of dscrete G-spectra. Let A be a commutatve symmetrc rg spectrum. By a dscrete G-A-module, we shall mea a dscrete G-spectrum X that also possesses the structure of a A-module. We requre these structures to be compatble the followg sese: for every elemet g G, the followg dagram must commute. A X ξ X A g A X ξ X g

24 24 MARK BEHRENS AND DANIEL G. DAVIS Here, ξ s the A-module structure map. Let Mod G,A deote the category of dscrete G-A-modules, wth morphsms beg the G-equvarat maps that are also maps of A-modules. Note that, gve dscrete G-A-modules X ad Y, ther smash product X A Y s easly see to be a dscrete G-A-module wth the dagoal acto. The followg smplfed varat of D.M. Ka s lftg theorem wll be used repeatedly to provde the desred model structures o structured categores lke Mod G,A. Lemma Suppose that M s a cofbratly geerated model category wth geeratg cofbratos I ad geeratg trval cofbratos J. Furthermore, assume that the domas of I ad J are α-small for some cardal α. Suppose that we are gve a complete ad cocomplete category N ad a adjot par (F, G) F : M N : G, where: (1) G commutes wth fltered colmts; ad (2) G takes relatve F J-cell complexes to weak equvaleces. The N admts a duced model category structure where the fbratos ad weak equvaleces are those morphsms whch get set to fbratos ad weak equvaleces by G, ad the cofbratos are determed. Ths model category structure s cofbratly geerated wth geeratg cofbratos F I ad geeratg trval cofbratos F J. The domas of F I ad F J are α-small N. Proof. Ths lemma s a specal case of Theorem of [17]. To see that the hypotheses of ths theorem are met our stuato, we must verfy that the domas of F I ad F J are α-small wth respect to relatve F I- ad F J-cell complexes, respectvely. However, our hypotheses mply that F preserves all α-small objects: gve a α-small object X M, ad a λ-sequece (λ α) N, we have: Y 1 Y 2 Y 3 Y β (β < λ) colm Hom N (F X, Y ) = colm Hom M (X, GY ) = Hom M (X, colm GY ) = Hom M (X, G colm Y ) = Hom N (F X, colm Y ). I [29], a model category structure s defed o the category of A-modules. The fbratos ad weak equvaleces of ths model structure are the fbratos ad weak equvaleces of uderlyg symmetrc spectra, ad the cofbratos are determed. Proposto The category Mod G,A s a model category, where the fbratos ad weak equvaleces are the fbratos ad weak equvaleces of the uderlyg dscrete G-spectra, ad the cofbratos are the cofbratos of the uderlyg A- modules. Proof. We apply Lemma to the adjot par A : ΣSp G Mod G,A : U,