A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE FOR CERTAIN HOMOTOPY FIXED POINT SPECTRA ETHAN S. DEVINATZ Abstract.Let H and K be closed subgroups of the extended Morava stabi- lizer group Gn and suppose that H is normal in K. We construct a strongly convergent spectral sequence H*c(K=H, (EhHn)*X) ) (EhKn)*X, where EhHnand EhKnare the continuous homotopy fixed point spectra of Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in the category of K(n)*-local EhKn-modules. ____________ 2000 Mathematics Subject Classification. Primary 55N20; Secondary 55P43, 55T* *15. Key words and phrases. Adams spectral sequence, continuous homotopy fixed po* *int spectra, Morava stabilizer group. The author was partially supported by a grant from the National Science Foun* *dation. 1 2 ETHAN S. DEVINATZ Introduction Suppose that Z is a spectrum and that G is a group acting on Z in some good point-set category of spectra. If G is, say, profinite, one might hope to const* *ruct, for each closed subgroup H of G, a öc ntinuous homotopy fixed point spectrum," denoted (abusively) by ZhH . There should be a öc ntinuous homotopy fixed point spectral sequence" H*c(H, Z*X) ) [X, ZhH ]* for all CW-spectra X, where H*c(H, ?) denotes the continuous cohomology of the profinite group H. This of course requires that Z*X has an appropriate topology* * on which G acts continuously (see [19] or [6, Remark 0.7]. In addition, ZhH should* * have the expected functorial properties and should agree with the ordinary homotopy fixed point spectrum when H is discrete. We are interested in one very important example where such a construction is possible_namely, the action of the Morava stabilizer group Gn on the Landweber exact spectrum En. First recall the definitions. For a fixed prime number p, the coefficient ring En* is W Fpn[[u1, . .,.un-1]][u, u-1], where |ui| = 0, |u| = -* *2, and as usual W Fpn denotes the ring of Witt vectors with coefficients in the field * *Fpn of pn elements. Gn = Sn o Gal, where Sn denotes the automorphism group of the height n Honda formal group law over Fpn, and Gal Gal(Fpn=Fp) denotes the Galois group of the field extension Fpn=Fp. Morava's theory implies that Gn acts on En by ring spectrum maps in the stable category, and technology developed by Goerss, Hopkins, and Miller (see [9], [10], [11], [12], [16]) then implies that* * En has a model on which Gn acts before passage to the stable category, so that homotopy fixed point spectra may be formed. Moreover, Gn is a profinite group_even a p-adic analytic group_and we constructed in [6] good continuous homotopy fixed point spectra for the action of this group on En. These continuous homotopy fix* *ed point spectra are the homotopically significant spectra in this situation, sinc* *e, for example, the continuous homotopy fixed point spectral sequences are generalized Adams spectral sequences and EhGnn= LK(n)S0, the K(n)*-localization of S0. Returning to the general situation, naturality of the continuous homotopy fix* *ed point spectrum ZhH implies that it is acted upon by the group N(H)=H. If F = K=H is a finite subgroup of N(H)=H, we can form the ordinary homotopy F fixed point spectrum (ZhH )hF, and we would expect this spectrum to be equivalent to the continuous homotopy fixed point spectrum ZhK . This was proved for the action of Gn on En in [6]. If F is closed but not finite, the situation becomes* * more problematic. Here the relevant spectrum is the continuous homotopy F fixed poin* *ts of EhHn. However, we don't have an intrinsic construction of such a spectrum_our construction of continuous homotopy fixed point spectra was specific to the act* *ion of Gn on En. Yet it is clear that (EhHn)hF ought to be just EhKn. We can then a* *sk whether there is a convergent spectral sequence (0.1) H*c(K=H, (EhHn)*X) ) [X, EhKn]*; such a spectral sequence provides a way of using information about EhHnto gain information about EhKn. In this paper, we construct the spectral sequence 0.1 and prove that it is al* *ways strongly convergent. Our construction makes use of the highly structured results of [6]: if G is a closed subgroup of Gn, EhGnis a commutative S0-algebra in the sense of [8] and the maps between these continuous homotopy fixed point spectra A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 3 arising from functoriality are maps of commutative S0-algebras. In particular, * *the inclusion H ! K makes EhHna commutative EhKn-algebra. We can then form a K(n)*-local EhHn-Adams resolution of EhKnin the stable category of EhKn-modules, again in the sense of [8]. By neglect of structure, this Adams resolution gives* * us a diagram of cofibrations in the stable category. Map the CW-spectrum X into this diagram to obtain the desired spectral sequence. This paper is organized as follows. In x1, we recall the definition and prope* *rties of the stable category of R-modules, for R a commutative S0-algebra, and deal with some subtleties which will arise later. In x2, we give a general discussio* *n of Adams spectral sequences in the stable category of R-modules. This material is a straightforward adaptation of work of Miller [15] and Bousfield [3]; these Adams spectral sequences have also been considered by Baker and Lazerev [1]. In x3, we identify the E2-term of the aforementioned Adams spectral sequence with the con- tinuous cohomology of K=H. Actually, we prove a more general result valid for EhKn-modules X; this is stated as Theorem 3.1. Along the way, we identify the a* *l- gebra ß*LK(n)(EhHn^EhKnEhHn) of homology cooperations. The strong convergence of this spectral sequence is established in x4, and finally, in an Appendix, we* * prove that, if K=H is finite, the spectral sequence we have constructed agrees with t* *he ordinary homotopy fixed point spectral sequence. 4 ETHAN S. DEVINATZ 1. Categories of module spectra We use the framework of [8] in this paper. Our basic category of spectra is thus the category of S0-modules MS0; this category becomes the usual stable category, denoted DS0, upon taking its homotopy category and inverting the weak equivalences. The advantage of the category of S0-modules is that it is symmetr* *ic monoidal; that is, there is an associative, commutative, and unital smash produ* *ct, denoted ^, and so one has the notion of S0-algebras or commutative S0-algebras. (These correspond to the earlier notions of A1 and E1 spectra respectively.) If* * R is an S0-algebra, there is then the evident notion of a (left) R-module, and on* *e can form the category MR of such R-modules and R-module maps between them. If R is commutative, a left R-module is the same as a right R-module, and one can define the smash product M ^R N of two R-modules to be the R-module given by the coequalizer diagram ~M_^N__// M ^ R ^ N M^~____//M ^ N_______//M ^R N, N where ~M and ~N are the module structure maps for M and N. This smash product makes MR a symmetric monoidal category with R as the unit. The stable category DR of R-modules, called the derived category in [8], is formed from the homotopy category of MR by inverting the weak equivalences; i.e., those maps of R-modules, which, regarded as maps of spectra, induce isomorphisms of homotopy groups. Equivalently, DR is the homotopy category of cell R-modules, again as defined in [8]. The categories MR and DR have the same formal properties as the categories MS0 and DS0, and we shall use these for the most part without comment. However, the (derived) smash product in DR will be the source of some complications later, and thus we give a more detailed discussion now. If M and N are in MR , the derived smash product M^_RN in DR is defined to be M ^R N, where M and N are cell R-modules weakly equivalent to M and N in MR . Of course, the canonical map M^_RN ! M ^R N need not be an equivalence in DR . If f : R ! R0is a map of commutative S0-algebras, then ther* *e is an evident functor f* : MR0 ! MR . Since f* preserves weak equivalences, we also have f* : DR0 ! DR and a natural transformation f*M^_Rf*N ! f*(M^_R0N) in DR for M and N in MR0. There are, however, certain situations where the map M^_RN ! M ^R N is an equivalence. These will prove quite convenient in what follows. Proposition 1.1 ([8, III, Theorem 3.8]).If M is a cell R-module and OE : N ! N0 is a weak equivalence in MR , then M ^R OE : M ^R N ! M ^R N0 is a weak equivalence. In particular, M ^R N ' M^_RN whenever M is a cell R-module. To state the next condition, we need a little more preparation. Recall that t* *he category of commutative R-algebras has the structure of a model category (see [* *8, VII]); we define a q-cofibrant commutative R-algebra to be a cofibrant object in this category. There is also the notion of a cell commutative R-algebra ([8, V* *II, Definition 4.11]), and any such object is q-cofibrant. These cell objects, howe* *ver, do not_in_general have the homotopy type of cell R-modules. __ Let ER denote the class of R-modules defined in [8, VII, 6]. Included in ER a* *re all R-modules having the homotopy type of cell R-modules. The main reasons for interest in this class are the following results. A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 5 Theorem 1.2 ([8, VII, Theorem_6.7]).Suppose_that R is a q-cofibrant commuta- tive S0-algebra and that Mi2 ER for i = 1, . .,.k. Then M1^_R. .^._RMk ! M1 ^R . .^.RMk is an equivalence in DR . Theorem 1.3 ([8, VII, Theorem_6.5]).Suppose_that A is a q-cofibrant commuta- tive R-algebra. Then A is in ER. __ Since ER is closed under homotopy equivalences, finite ^R -products, pushouts along cofibrations, and colimits of (countable) sequences of cofibrations, the * *next result is an easy consequence of 1.3. Proposition 1.4.Let R0be a q-cofibrant commutative R-algebra, and suppose the R-module M_has the homotopy type of a cell R0-module (regarded as R-modules). Then M 2 ER. These results will all be relevant to our work, since, in [6], we constructed* * the continuous homotopy fixed point spectra EhGnto be cell commutative S0-algebras. We will also need some results from [8] on the Bousfield localization of R-mo* *dules. Suppose then that F is a cell R-module. A map f : X ! Y of R-modules is an F*R-equivalence if F ^f : F ^R X ! F ^R Y is a weak equivalence, and an R-module W is F*R-acyclic if F ^R W is weakly contractible. (We assume F is a cell R-mod* *ule so that we can work with the ordinary, as opposed to the derived, smash product* *.) An R-module Y is F*R-local if [W, Y ]R*is trivial whenever W is F*R-acyclic. He* *re [W, Y ]Ridenotes the group of maps iW ! Y in DR ; we will also denote this group by [W, Y ]-iR. Finally, f : X ! Y is called the F*R-localization of X if * *f is an F*R-equivalence and Y is F*R-local. If such a localization exists, it is uni* *que up to canonical isomorphism in DR . It is proved in [8, VIII, 1] that X always has* * an F*R-localization, which we denote ~ : X ! LRFX. Moreover, if R is a q-cofibrant S0-algebra_e.g., R = S0_and X is a cell commutative R-algebra, then ~ can be constructed to be a map of cell commutative R-algebras and even to be natural in this category; i.e., the full subcategory of the category of commutative R-alge* *bras and R-algebra maps whose objects are the cell commutative R-algebras. If K is a cell S0-module, and F = R ^ K as R-modules, then a map f : X ! Y of R-modules is an F*R-equivalence if and only if it is a K*-equivalence when regarded as a map of S0-modules. Hence there is a canonical map LRFX ! LK X in DS0, where we write LK X for LS0KX. Furthermore, an S0-module W is K*- acyclic if and only if R ^ W is F*R-acyclic; this implies that LRFX is K*-local* * and therefore that LRFX ! LK X is a weak equivalence. In particular, K*-localization defines a functor and natural transformation on DR , and the K*-localization of a commutative R-algebra can be taken to be a commutative R-algebra (for R a q-cofibrant commutative S0-algebra). The case with K = K(n) and R = EhGn, for G a closed subgroup of Gn, will be the case of interest in this paper. 6 ETHAN S. DEVINATZ 2.Adams spectral sequences in categories of module spectra In this section we set up and discuss the convergence of Adams spectral seque* *nces in localized categories of R-modules. Since we will be exclusively working in (* *full subcategories of) DR , we will, in this section only, write ^R _instead of usin* *g the notation ^_Rintroduced in x1_for the derived smash product. Say that an R-module E is a commutative homotopy R-algebra if it is provided with maps j : R ! E and ~ : E ^R E ! E (in DR ) such that the expected diagrams commute. Let F be an R-module. Following Miller [15], we construct the F*R-local E-Adams resolution of an (F*R-local) R-module Y as an injective resolution of Y for an appropriate injective class. Definition 2.1.With notation as above, an R-module is F*R-local E-injective if * *it is a retract of LRF(Z ^R E) for some R-module Z. Such an R-module will also be called E-injective in LRFDR , where LRFDR is the category of RF*-local R-module* *s. A sequence of F*R-local R-modules X0! X ! X00is E-exact in LRFDR if [X0, I]R*oo_[X, I]R*oo__[X00, I]R* is exact for every F*R-local E-injective R-module I. One can easily check that this defines an injective class in the sense of [14* *] in the category LRFDR . Thus each object Y in LRFDR has a resolution by F*R-local E-injective R-modules, unique up to chain homotopy. If (2.1) *_____//Y____//_I0___//I1____//I2___//_. . . is such a resolution, then we may construct a diagram (2.2) Y ____Y 0Booi___Y=1Boo_i___Y=2oo____=.=. . BB ___ BBB ___ j !!B_ k j !! _k I0 I1 of exact triangles; in this diagram the map LRF(j ^ E) : LRF(Y i^R E) ! LRF(Ii^R E) is a split monomorphism. Conversely, a diagram of exact triangles as in 2.2 with each Ij F*R-local E-injective and LRF(j ^R E) split monic yields an injective r* *eso- lution 2.1. Such a diagram is called an F*R-local E-Adams resolution of Y or an E-Adams resolution of Y in LRFDR and is functorial up to chain homotopy. By mapping an R-module X into an F*R-local E-Adams resolution of Y , we obtain a spectral sequence, called the F*R-local E-Adams spectral sequence, with Es,t1= [X, Is]Rt-s. (This indexing follows the convention that i and j are maps of degree 0, and k * *is a map of degree -1.) When R and F are understood, we will write this spectral sequence as Es,tr(X, Y ; E). Naturally, we hope that this spectral sequence con* *verges strongly to Y ; the rest of this section is devoted to establishing a criterion* * which guarantees this. Following Bousfield [3, Definition 3.7], we define the class of F*R-local E-n* *ilpotent R-modules to be the smallest class C of objects in LRFDR such that i.LRFE 2 C A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 7 ii.LRF(N ^R X) 2 C whenever N 2 C and X 2 DR iii.If X ! Y ! Z ! X is a cofiber sequence in LRFDR and two of X, Y, Z are in C, then so is the third iv.C is closed under retracts in LRFDR . Next, say that Z 2 LRFDR is LRF-ER*-local if [W, Z]R*is trivial whenever LRF(* *E ^R W ) ' *. Observe that this is the same as saying that Z is (F ^R E)R*-local. The following result is proved just as in [3, Lemma 3.8]. Proposition 2.2.If Z is F*R-local E-nilpotent, then Z is LRF-ER*-local. The notion of an F*R-local E-nilpotent resolution of an object Y 2 LRFDR may also be defined along the lines of [3], and its uniqueness in an appropriate "p* *ro- category" may also be established. These resolutions are related to Adams spectral sequences as follows. Consider the Adams resolution 2.2, and let Ys be the cofiber of i(s+1): Y s+1! Y . We can then construct commutative diagrams Y s+1_____//Y_____//Ys_____// Y s+1___//_. . . |i| |||| || |i| fflffl| || fflffl| fflffl| Y s______//Y____//Ys-1_____// Y s____//_. . . of cofibration sequences, as well as a diagram (2.3) *____Y1oo________Y0oo________Y1oo____. . . DD zz<< DDD zz<< D""D zz D"" zz I0 I1 of exact triangles in DR . The spectral sequence obtained by mapping an R-module X into this diagram yields the F*R-local E-Adams spectral sequence. Moreover, t* *he tower {Ys}s 0 under Y is an F*R-local E-nilpotent resolution of Y (see [3, Lemma 5.7]). Thus, if Y is itself F*R-local E-nilpotent, then the constant tower {Y }* * is pro- isomorphic to {Ys}. This implies the following convergence result (see [3, Theo* *rem 6.10]). Proposition 2.3.If Y is F*R-local E-nilpotent, then the F*R-local E-Adams spec- tral sequence converges strongly and conditionally to [X, Y ]R*, for any R-modu* *le X. In addition, there exists s0 such that Es,*1(X, Y ; E) = 0 for all s > s0 and a* *ny R-module X. Remark 2.4.We are using the notions of strong and conditional convergence as given in Boardman [2] applied to the unrolled exact couple [X, Y ]R*__[X, Y 0]oo_________[X, Y 1]R*oo________ [X, Y 2]R*oo_ . . . MMM ppp88p NNNNN ppp88p MM&& pp N&& pp [X, I0]R* [X, I1]R* Note that in the presence of conditional (resp. strong) convergence, strong (re* *sp. conditional) convergence is equivalent to the condition that lim1rEs,tr(X, Y ; * *E) = 0 for all s, t ([2, Theorem 7.3]). The next result is a partial converse to Proposition 2.3 and will be useful t* *o us later. 8 ETHAN S. DEVINATZ Proposition 2.5.Suppose that Y is LRF-ER*-local and that there exist s0 and r0 such that for all R-modules X, the F*R-local E-Adams spectral sequence satisfies Es,*r0(X, Y ; E) = 0 whenever s > s0. Then Y is F*R-local E-nilpotent. Proof.Since lim1Ers,tr(X, Y ; E) = 0 for all s, t, it follows from [2, Theorem * *7.4], applied to the unrolled exact couple 0 ____[X, Y-1]R*oo________[X, Y0]R*oo_________ [X, Y1]R*oo__. .,. OOO ppp88 NNNN ppp88 OO''O pp N&& pp [X, I0]R* [X, I1]R* that i.lim1[sX, Ys]R*= 0 for all R-modules X ii.Fs,t(X, Y ; E)=Fs+1,t+1(X, Y ; E) -! Es,t1(X, Y ; E), where Fs,t(X, Y ; E)= ker(lim[iX, Yi]Rt-s-! [X, Ys-1]Rt-s) = ker([X, holimYii]Rt-s-! [X, Ys-1]Rt-s). The horizontal vanishing line now implies that Fs0+1,*(X, Y ; E) = 0. Since X is arbitrary, this in turn implies that the map p : holimYii! Ys0is the inclusion * *of a summand in DR ; in particular, holimYiiis F*R-local E-nilpotent. Now consider the map ' : Y ! holimYsswhich lifts the canonical maps Y ! Ys. (This map is unique since lim1[Y, Ys]R*= 0.) ' is a map between LRF-ER* local objects; it is therefore an equivalence if and only if LRF(E ^R ') : LRF(E ^R Y ) ! LRF(E ^R holimYss) is an equivalence. We will prove that LRF(E ^R ') is an equivalence, and hence * *Y is F*R-local E-nilpotent. Begin by observing that LRF(E ^R Y ) is F*-local E-injective, so that æ R R Es,t1(X, LRF(E ^R Y ); E) = Es,t2= [X, LF (E0^R Y )]tss=60=.0 Since LRF(E^R ?) applied to an F*R-local E-Adams resolution of Y is an F*R-loca* *l E- Adams resolution of LRF(E^R Y ), it follows as above that lim1[sX, LRF(E^R Ys)]* *R*= 0 for all X. Hence there is a canonical map LRF(E ^R Y ) ! holimLsRF(E ^R Ys), and this map is an equivalence. Now there is also a canonical map j : LRF(E ^R holimYss) ! holimLsRF(E ^R Ys), whence the composition R(E^') j LF (E ^R Y ) LF-----!LRF(E ^R holimYss) -! holimLsRF(E ^R Ys) is an equivalence. Thus ß*j is an epimorphism. A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 9 On the other hand, choose r0 : Ys0! holimYssso that r0 O p = id, and let r be the composition R(E^r0) holimLsRF(E ^R Ys) ! LRF(E ^R Ys0) LF-----!LRF(E ^R holimYss). Then r O j is the identity, so ß*j is a monomorphism and therefore an isomorphi* *sm._ This implies that LRF(E ^R ') is an equivalence, completing the proof. * *|__| 10 ETHAN S. DEVINATZ 3. Identification of the E2-term For the rest of this paper, the integer n 1 will be fixed, so we will delet* *e the subscript n from En or any of its continuous homotopy fixed point spectra. We w* *ill also write LK(n) as bL, but will retain the notation K(n), reserving the notati* *on K for closed subgroups of Gn. Let H and K be closed subgroups of Gn with H normal in K. The main result of [6] constructs K(n)*-local cell commutative S0-algebras EhH and EhK which are to be interpreted as continuous homotopy fixed point spectra. The inclusion of H in K induces a map EhK ! EhH of commutative S0-algebras, so that EhH is a commutative EhK -algebra. Moreover, there is an action of K=H on EhH by EhK -algebra maps, whose construction we now recall. If (3.1) Gn = U0 ) U1 ) U2 ) . .).Ui) . . . T is a sequence of open normal subgroups of Gn with Ui= {e}, then i EhG = bL(holim!iEh(UiG)) for any closed subgroup G of Gn. Each left Gn-map æ : Gn=HUi ! Gn=HUi making the diagram Gn=HUi ______æ______//_Gn=HUi OOO ooo OO''O wwooo Gn=KUi commute induces an Eh(KUi)-algebra map F(æ) : Eh(UiH)! Eh(UiH)[6, Theorem 1]. If gH is an element of K=H, define such an automorphism of Gn=HUi by sending xHUi to xgHUi. Then pass to homotopy colimits to obtain an EhK - algebra automorphism of EhH . Under the isomorphism ß*bL(EhH ^ E) -! Map c(H\Gn, E*) of ß*bL(E ^ E) Map c(Gn, E*)-comodules ([6, Theorem 2])_here H\Gn denotes the space of right cosets Hx in Gn_this action of gH _ginduces the map _g *: Map c(H\Gn, E*) ! Map c(H\Gn, E*) given by _g -1 (3.2) *(f)(Hx) = f(Hg x). We can now state the main results of this paper. To make the notation less cumbersome, we will write E(G) for EhG when this object appears as a sub or superscript. Theorem 3.1. Let X be an EhK -module. The K(n)*-local EhH -Adams spec- tral sequence Es,tr(X, EhK ; EhH ) in DE(K) has E2-term naturally isomorphic to Hsc(K=H, [X, EhH ]E(K)t). Remark 3.2.By K(n)*-local, we really mean (K(n) ^ EhK )E(K)*-local, but, by the discussion in x1, the functors LK(n)and LE(K)K(n)^E(K)are naturally equivalent * *when restricted to EhK -modules. A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 11 By Proposition 2.3, the next result implies that this spectral sequence conve* *rges strongly to [X, EhHn]E(K)*. Theorem 3.3. Every EhK -module is K(n)*-local EhH -nilpotent in DE(K). Corollary 3.4.There is a strongly convergent spectral sequence Hsc(K=H, (EhH )*Z) ) (EhK )t+s(Z), valid for any object Z in the stable category. Proof.If Z is a cell S0-module, apply Theorem 3.1 to X = EhK ^ Z and use the fact that [X, M]E(K) = [Z, M] whenever M is an EhK -module. |___| To prove Theorem 3.1, we must first make sense of Hsc(K=H, [X, EhH ]E(K)t). Since K=H is a p-analytic profinite group [7, Theorem 9.6], Hsc(K=H, M) makes good sense whenever M is a profinite continuous Zp[[K=H]]-module: it can be defined as limHffsc(K=H, Mff), where M = limMffffwith each Mffa finite discrete K=H-module (see [6, Remark 0.3]). The action of K=H on EhH induces an action on [X, EhH ]E(K)t; we thus need to provide [X, EhH ]E(K)twith a natural topology so that it becomes a profinite continuous Zp[[K=H]]-module. If I is the multi-index (i0, . .,.in-1), let M(I) denote a finite spectrum wi* *th BP*M(I) = BP*=(pi0, vi11, . .,.vin-1n-1), provided such a spectrum exists. Lemma 3.5. Let X be a finite cell EhK -module. Then [X, EhH ^ M(I)]E(K)*is a finite discrete K=H-module in each degree. Proof.It suffices to show that [X, EhH ^M(I)]E(K)*is a finite discrete K=H-modu* *le in each degree when X is a "sphere EhK -module" ([8, III,2]). Hence we need only show that ß*(EhH ^ M(I)) is a finite discrete K=H-module in each degree. Consider the continuous homotopy fixed point spectral sequence E*,*2= H*c(H, E*=(pi0, . .,.vin-1n-1)) ) ß*(EhH ^ M(I)) from [6, Theorem 2]. Since E*=(pi0, . .,.vin-1n-1) is finite in each degree, it* * follows that H*c(H, E*=(pi0, . .,.vin-1n-1)) is finite in each bidegree (see e.g. [6, * *proof of Lemma 3.21]. But E*,*1has a horizontal vanishing line since EhH is K(n)*-local E nilpotent (in the ordinary stable category) [6, Proposition AI.3], and this spe* *ctral sequence is the K(n)*-local E-Adams spectral sequence. This allows us to conclu* *de that ß*(EhH ^ M(I)) is finite in each degree. Furthermore, ß*(EhH ^ M(I)) = lim!iß*(EhHUi ^ M(I)), where the Ui's are as in 3.1. But ß*(EhHUi ^M(I)) is a discrete K=H module, sin* *ce K \ Uiacts trivially on EhHUi. Hence ß*(EhH ^ M(I)) is a discrete K=H-module,_ and the proof is complete. |__| 12 ETHAN S. DEVINATZ Proposition 3.6.If X is any cell EhK -module, then [X, EhH ]E(K)*= lim ff,I[Xff, EhH ^ M(I)]E(K)*, where the Xffrange over the finite EhK -cell subcomplexes of X, and the M(I) ra* *nge over a sequence of multi-indices such that bLS0 = holimMI(I). With the topology* * of this inverse limit of discrete spaces, [X, EhH ]E(K)tis therefore a profinite c* *ontinuous K=H-module. Proof.Since EhH is K(n)*-local, EhH -! LbEhH = holimEIhH^ M(I). It therefore follows from the previous lemma that [Xff, EhH ]E(K)*= lim[IXff, EhH ^ M(I)]E(K)* and in particular that [Xff, EhH ]E(K)*is a profinite group in each degree. Hen* *ce [X, EhH ]E(K)*= lim[ffXff, EhH ]E(K)* = limff,I[Xff, EhH ^ M(I)]E(K)*, completing the proof. |___| Now consider the EhH -injective resolution * ! EhK !jI0 d!I1 d!. .i.n bLDE(K) given by (3.3) Ij = bL(EhH ^_E(K). .^._E(K)EhH) __________-z_________" j+1 copies P j+1 with j : EhK ! EhH the inclusion and d : Ij ! Ij+1 given by d = i=0(-1)idi, where di= bL((EhH )(i)^_E(K)j^_E(K)(EhH )(j+1-i)). (Recall from x1 that ^_E(K)denotes the smash product in DE(K).) Since iterated smash products will be ubiquitous, let us write j(A; R) A^_RA^_R._.^._RA_-z______" j+1 copies whenever A is a commutative R-algebra (and R is a commutative S0-algebra). Thus bL j(EhH ; EhK ) = Ij. Theorem 3.1 will follow easily from the next result. Theorem 3.7. Let X be a cell EhK -module, and let Tj : [X, Ij]E(K)*! Map c(K=H_x_._.x.K=H_-z_______", [X, EhH ]E(K)*) j copies be defined by taking Tj(f)(_g1, . .,._gj) to be the composition _g-1^...^_g-1^EhH ~ X f!Ij -1--------j---!Ij ! EhH , where we write _g: EhH ! EhH for the action of _g2 K=H on EhH . Then Tj is an isomorphism. A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 13 Remark 3.8.It is not obvious that Tj(f) is even a continuous map from (K=H)j to [X, EhH ]E(K)*. This will be proven along with the theorem. Assuming Theorem 3.7, we can now prove half of our main result. Proof of Theorem 3.1. With Tj as in 3.7, we have (3.4) P j _ _ _ Tj+1(df)(_g1, . .,._gj+1)=i=0(-1)iTj(f)(g1, . .,.bgi+1, . .,.gj+1) +(-1)j+1_g-1j+1[Tj(f)(_g1, _g-1j+1, . .,._gj_g-1* *j+1)] for all f 2 Ij. Define, for each discrete K=H-module M, a cochain complex C*(K=H, M) by Cj(K=H, M) = Map c((K=H)j, M) with differential d : Cj(K=H, M) ! Cj+1(K=H, M) given by P j _ _ _ (df)(_g1, . .,._gj+1)= i=0(-1)if(g1, . .,.bgi+1, . .,.gj+1) +(-1)j+1_g-1j+1[f(_g1_g-1j+1, . .,._gj_g-1j+1)]. Then H*(C*(K=H, ?)) defines an effaceable ffi-functor, and, since H0(C*(K=H, M)) = MK=H , it follows that H*(C*(K=H, M)) = H*c(K=H, M). Of course, C*(K=H, M) can also be defined if M = limMffffis a profinite conti* *n- uous Zp[[K=H]]-module, and the same proof as in [6, Lemma 3.21] applies to show that H*(C*(K=H, M)) = limHff*(C*(K=H, Mff)) = limHff*c(K=H, Mff) = H*c(K=H, M). Since 3.4 identifies the cochain complex [X, I*]E(K)*with C*(K=H, [X, EhH ]E(* *K)*), it therefore follows that Es,t2(X, EhK ; EhH ) = Hsc(K=H, [X, EhH ]E(K)*), as desired. |___| Another consequence of Theorem 3.7 is the identification of the ring of homol* *ogy cooperations (in the K(n)*-local category) of EhH regarded as an EhK -algebra. Corollary 3.9.Let K and H be closed subgroups with Gn with H normal in K. Then ß*bL(EhH ^_E(K)EhH ) = Map c(K=H, EhH*). The action _gL*of _g2 K=H on the left factor of ß*bL(EhH ^_E(K)EhH ) is given by _gL __ _-1__ *(f)(x) = f(g x), and the action _gR*of _gon the right factor of ß*bL(EhH ^_E(K)EhH ) is given by _gR __ _ ___ *(f)(x) = g*f(x g), 14 ETHAN S. DEVINATZ for any f 2 Map c(K=H, EhH*). The rest of this section is devoted to proving Theorem 3.7. We begin by com- puting ß*bL(Ij^_E). (Note here that, in taking the derived smash product with E, Ij is regarded as an object in DS0.) This will require a number of lemmas; we f* *irst identify bL(Ij^_E) with bL[ j(bL(EhH ^ E); bL(EhK ^ E))]. Remark 3.10.The proof of 3.7 will be seen to reduce to the case where H and K are open in Gn, and thus we only need to understand bL(Ij^_E) in this case. However, we will carry out our analysis of bL(Ij^_E) in the general case, since* * some of our work along the way will be used in the next section. Lemma 3.11. There is a canonical equivalence Ij^_E ' j(EhH ^ E; EhK ^ E) in DS0. Proof.This result would be essentially immediate if derived smash products were not involved; their presence, however, forces us to do a little more work._ Replace EhH by a weakly equivalent q-fibrant commutative EhK -algebra EhH . Then, since EhK is a q-cofibrant commutative S0-algebra, it follows by Theo- rems_1.2 and 1.3 that_Ij_is represented in_DEhK by (the ordinary smash product) EhH ^E(K). .^.E(K)EhH. E hH^E(K). .^.E(K)EhH is the coproduct of j +1 copies __hH of E in the category of commutative EhK -algebras and is therefore q-cofibra* *nt as well. Again using_the fact that EhK_ is a q-cofibrant commutative S0-algebra, we conclude that EhH ^E(K) . .^.E(K)EhH is also a q-cofibrant commutative S0- algebra._Since E is a q-cofibrant_commutative S0-algebra, Ij^_E is thus represe* *nted by (E hH ^E(K) . .^.E(K)EhH) ^ E in DS0. But __hH __hH __hH __hH (E ^E(K) . .^.E(K)E ) ^ E=(E ^E(K) . .^.E(K)E ) ^ (E ^E . .^.EE) __hH __hH = (E ^ E) ^E(K)^E . .^.E(K)^E(E ^ E), __hH and E ^E is a q-cofibrant commutative_algebra over_the q-cofibrant commutative S0-algebra E(K)^E. Therefore, (E hH^E(K). .^.E(K)EhH)^E represents j(EhH ^ E; EhK ^ E) in DS0, completing the proof. |___| Lemma 3.12. The canonical map j(EhH ^E; EhK ^E) ! j(bL(EhH ^E); bL(EhK ^ E)) is a K(n)*-equivalence in DS0. Therefore, Lb(Ij^_E) ' bL( j(bL(EhH ^ E); bL(EhK ^ E))]. Proof.First recall from x1 that EhH ^ E and EhK ^ E are cell commutative S0- algebras, so that bL(EhH ^ E) and bL(EhK ^ E) are cell commutative S0-algebras, and bL(EhH ^ E) is a commutative bL(EhK ^ E)-algebra. In addition, bL(EhH ^ E) is a commutative (EhH ^ E)-algebra, so we have the factorization j(EhH ^ E; EhK ^ E) v! j(bL(EhH ^ E); EhK ^ E) w! j(bL(EhH ^ E); bL(EhK ^ E). K(n) is Bousfield equivalent to K(n)(j+1); we will therefore first prove that* * v is a K(n)*-equivalence by proving that it is a K(n)(j+1)*-equivalence. Now j(EhH ^ E; EhK ^ E) ^ K(n)(j+1)= j(EhH ^ E ^ K(n); EhK ^ E) A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 15 with a similar statement for j(bL(EhH ^E); EhK ^E). (We are of course assuming that K(n) is a cell S0-module.) Since EhH ^ E ^ K(n) '!bL(EhH ^ E) ^ K(n), it is now immediate that v is a K(n + 1)(j+1)*-equivalence. That w is a K(n)*-equivalence follows by induction from the next result,_com- pleting the proof. |__| Lemma 3.13. Let F be a cell S0-module, let R be a cell commutative S0-algebra, and let LR be the F*-localization of R (constructed to be an R-algebra and a ce* *ll commutative S0-algebra.) If M and N are any LR-modules, then the canonical map M^_RN ! M^_LRN is an F*-equivalence. Proof.We will prove that M^_R(N ^ F ) ! M^_LR(N ^ F ) is an equivalence; this is equivalent to the conclusion of the lemma. Let M ! M (resp. N ! N) be a weak equivalence of LR-modules, where M (resp. N) is a cell LR-module, and let 0M ! M (resp. 0N ! N) be a weak equivalence of R-modules, where 0M (resp. 0N) is a cell R-module. Then, according to [8, IV, 7.5], we have a commutative diagram B( 0M, R, 0N ^ F )____//_ 0M ^R ( 0N ^ F') M^_R(N ^ F ) | | | | | | | | | fflffl| |fflffl fflffl| B( M, LR, N ^ F )_____//_ M ^LR ( N ^ F )' M^_LR(N ^ F ), and the horizontal arrows are weak homotopy equivalences. Here B(X, Y, Z) de- notes the geometric realization of the simplicial S0-module B*(X, Y, Z), the si* *m- plicial bar construction for a right Y -module X and a left Y -module Z. Thus we need only show that the left vertical arrow is also a weak equivalence. Since R and LR are q-cofibrant S0-algebras, B*( 0M, R, 0N ^ F ) and B*( M, LR, N ^ F ) are proper simplicial S0-modules [8, IV, 7.6], and it suffi* *ces to show that (3.5) Bq( 0M, R, 0N ^ F )_____//_Bq( M, LR, N ^ F ) || || || || || || 0M ^ R(q)^ 0N ^ F_____// M ^ (LR)(q)^ N ^ F is a weak equivalence. The map 0M^_R^_. .^._R^_( 0N ^ F ) ! M^_LR^_. .^._LR^_( N ^ F ) is certainly a weak equivalence; we thus only need to show that these derived s* *mash products are represented by the point-set level smash products of 3.5. By Theor* *em 1.2, we_are_reduced to verifying that_ 0M, M, R, LR, 0N ^ F , and N ^ F are all in ES0. But R and LR are in ES0_because they are q-cofibrant commutative S0-algebras, and the others are in ES0 because they are either cell R-modules o* *r_ cell LR-modules (see Proposition 1.4). This completes the proof. |_* *_| We will use a Künneth type spectral sequence to compute ß*bL[ j(EhH ^E); bL(E* *hK ^ E))]. This requires understanding the structure of ß*bL(EhH ^E) = Map c(H\Gn, E* **) 16 ETHAN S. DEVINATZ as a module over ß*bL(EhK ^ E) = Map c(K\Gn, E*). It will be useful to proceed a little more generally. Say that a commutative ring C is a complete ring if it is provided with a com* *plete decreasing filtration {F sC}s 0 by ideals such that F sC . F tC F s+tC for all s, t. C=F sC is given the discrete topology, and C = limCs=F sC is then given t* *he topology of the inverse limit. Lemma 3.14. Let S and T be profinite sets, and let C be a complete commutative ring. Then there is a canonical isomorphism Map c(S, C)b CMap c(T, C) ! Map c(S x T, C). Proof.By definition, Map c(S, C)b CMap c(T, C) = limMjapc(S, C) C Map c(T, C)=F j, where F j= + Mapc(S, F sC) C Map s(T, F tC). s+t=j Moreover, if X is any profinite set, Map c(X, C) = limMsapc(X, C=F sC) = limMsapc(X, C)=Map c(X, F sC). It therefore suffices to show that the canonical map Mapc(S, C=F sC) C Map c(T, C=F sC) ! Map c(S x T, C=F sC) is an isomorphism. But this follows by reduction to the case where S and_T are finite. |__| Lemma 3.15. Let G be a profinite group, and let H and K be closed subgroups of G with H normal in K. If C is any complete commutative ring, then Map c(H\G, C) Map c(H\K, C)b cMapc(K\G, C) as Map c(K\G, C)-algebras. Proof.Since the above lemma implies that Map c(H\K, C)b cMapc(K\G, C) = Map c(H\K x K\G, C), it suffices to show that there exists a homeomorphism H\K x K\G ! H\G such that the diagram H\K x K\G ________//H\G JJ " JJJ """ JJ%% """" K\G commutes. By [17, I x1, Proposition 1], there exists a continuous section_s : K* *\G ! H\G. Then define H\K x K\G ! H\G by sending (Hx, Kg) to Hx . s(Kg). |__| We can now make the desired computation of ß*bL[ j(bL(EhH ^E); bL(EhK ^E))]. A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 17 Proposition 3.16.Let M(I) be a generalized Toda V (n - 1) corresponding to the multi-index (i0, . .,.in-1) as in the statement of Lemma 3.5, and let H and K be closed subgroups of Gn with H normal in K. Write A = ß*bL(EhK ^ E), B = ß*bL(EhH ^ E), and J = (pi0, vi11, . .,.vin-1n-1) E*. Then B A . . .AB=J(B A . . .AB) ! ß*[ j(bL(EhH ^ E); bL(EhK ^ E)) ^ M(I)]. Proof.We will prove this by induction on j, the result being trivial for j = 0.* * Begin by assuming that B_bA._.b.AB_-z_____"!ß*bL[ j-1(bL(EhH ^ E); bL(EhK ^ E))], j copies where the completed tensor product is with respect to the J-adic filtration on * *B. Now B bA. .b.AB = Map c(H\Gn xK\Gn . .x.K\GnH\Gn , E*) _____________-z____________" j copies (cf. Lemma 3.14); in particular, B bA. .b.AB is flat over E*. Write N = B bA. .b.AB. By [8, IV, Theorem 4.1], there is a spectral sequence TorAs,t(N, ß*[bL(EhH ^ E) ^ M(I)]) ) ßs+t[ j(bL(EhH ^ E); bL(EhK ^ E)) ^ M(I)], so we need only show that TorAs,*(N, Mapc(H\Gn, E*=J)) = 0 for all s > 0. N and A are both flat over E*; therefore TorAs,*(N, Mapc(H\Gn, E*=J)) = TorA=JAs,*(N=JN, Mapc(H\Gn, E*=J)). Moreover, by the preceding lemma, Map c(H\Gn, E*=J) Map c(K=H, E*=J) E*=JA=JA. Since Map c(K=H, E*=J) is flat over E*=J, we then have TorA=JAs,*(N=JN, Mapc(H\Gn, E*=J))= TorE*=Js,*(N=JN, Mapc(K=H, E*=J)) = 0 whenever s > 0. This completes the induction and the proof. |__* *_| With the computation of ß*bL(Ij^_E) in hand, we can now begin the proof of Theorem 3.7. We start by considering a special case; namely we assume that K=H is finite. Define an EhK -module map ` (3.6) øj : Ij ! EhH (K=H)j by requiring that the projection onto the summand indexed by (_g1, . .,._gj) be* * given by the composition bL(_g-11^...^_g-1j^EhH)~ Ij -------------! Ij -! EhH . The next result implies immediately that Tj is an isomorphism in this case. Proposition 3.17.If K and H are closed subgroups of Gn with H normal in K and K=H finite, then øj is a weak equivalence. 18 ETHAN S. DEVINATZ Proof.First assume that K and H are open in Gn. We will prove that W hH bL(øj^ E)* : ß*bL(Ij^_E)__//_ß* bL(E ^ E) (K=H)j L Map (H\Gn, E*) (K=H)j is an isomorphism. By Lemma 3.12 and Proposition 3.16, ß*bL(Ij^_E) = Map (H\Gn, E*) Map(K\Gn,E*). . .Map(K\Gn,E*)Map(H\Gn, E*). Tracking down the identifications, it follows from 3.2 that the projection p(_g* *1,...,_gj) of bL(øj^ E)* onto the summand indexed by (_g1, . .,._gj) 2 (K=H)j is given by p(_g1,...,_gj)(f1, . .,.fj+1)(Hg) = f1(_g1g)f2(_g2g) . .f.j(_gjg)fj+1(Hg* *). This formula also defines a map M zj : Map (H\Gn, Zp) Map(K\Gn,Zp). . .Map(K\Gn,Zp)Map(H\Gn, Zp) ! Map (H\G* *n, Zp) (K=H)j and zj E* = bL(øj ^ E)*. zj is easily seen to be an epimorphism; we claim that this implies that zj is in fact an isomorphism. Indeed, by Lemma 3.15, Map (H\Gn, Zp) = Map (K=H, Zp) ZpMap (K\Gn, Zp), so zj becomes a map M Map (K=H, Zp) Zp. . .ZpMap(K=H, Zp) ZpMap (K\Gn, Zp) ! Map (H\Gn, Zp). __________________-z_________________" j j+1 copies (K=H) Both sides are free Zp-modules of the same finite rank and therefore zj must be* * a monomorphism as well. For the general case, recall the sequence {Ui} of open normal subgroups of 3.* *1, and let Hi= HUi, Ki= KUi. Then (3.7) bL(holim j(EhHi; EhKi)) '!bL( j(EhH ; EhK )). !i There are several ways to see this. The most painless is to use Proposition 3.1* *6 to verify that ß*bL(holim!i j(EhHi; EhKi)^_E) ! ß*bL( j(EhH ; EhK )^_E). Alternatively, one can use techniques such as Proposition 1.4 to replace derived smash products by ordinary smash products and then use their good properties with respect to colimits. Lemma 3.13 will also be involved at the end. We have already proved that ` øj : bL j(EhHi; EhKi) ! EhHi (Ki=Hi)j is an equivalence; the result in general now follows from 3.7, the naturality o* *f øj,_ and the fact that K=H ! Ki=Hi is a bijection for i sufficiently large. * * |__| A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 19 Proof of Theorem 3.7. First observe that it suffices to show that (3.8) Tj : [Xff, Ij ^ M(I)]E(K)*! Map c((K=H)j, [Xff, EhH ^ M(I)]E(K)*) is an isomorphism for any finite cell EhK -module Xffand multi-index I. For the* *n, since [Xff, EhH ^M(I)]E(K)*is finite in each degree (Lemma 3.5) and Mapc((K=H)j* *, ?) is exact on the category of profinite abelian groups (see [17, Ix1, Proposition* * 1]), all the relevant limiterms vanish, and we have [X, Ij]E(K)*= lim[Xff, Ij ^ M(I)]E(K)* Map c((K=H)j, [X, EhH ]E(K)*)= limMapc((K=H)j, [Xff, EhH ^ M(I)]E(K)*). Let {Ui} be as in the previous proof, and let Hi = K \ HUi. Then K=Hi is finite, so from above, Tj : [Xff, j(EhHi; EhK ) ^ M(I)]E(K)*! Map ((K=Hi)j, [Xff, EhHi ^ M(I)]E(K)*). But, as in the proof above, holim!i j(EhHi; EhK ) ^ M(I) ! j(EhH ; EhK ) ^ M(I) = Ij ^ M(I) is an equivalence in DE(K); thus [Xff, Ij ^ M(I)]E(K)*= lim!i[Xff, j(EhHi; EhK ) ^ M(I)]E(K)*. Moreover, Map c((K=H)j, [Xff, EhH ^ M(I)]E(K)*) = lim!iMap((K=Hi)j, [Xff, EhHi ^ M(I)]E(K* *)*), and again it follows from the naturality of Tj that the map in 3.8 is_an_isomor* *phism. |__| 20 ETHAN S. DEVINATZ 4.Convergence Our proof of Theorem 3.3 will involve two main steps. We will first prove that EhK is bL- EE(K)*local. We will then establish a vanishing line in the E-Adams spectral sequence in bLDE(K); by virtue of Proposition 2.5, this implies that E* *hK is E-nilpotent in bLDE(K). But E is an EhH -module in DE(K); therefore E is EhH - nilpotent in bLDE(K), and hence so is EhK . This immediately implies the conclu* *sion of Theorem 3.3. The proof of the first step requires an analysis of bL(E ^ EhK ). Recall that* * E and EhK (and hence EhK ^ E and E ^ E) are cell commutative S0-algebras; thus the maps EhK = EhK ^ S0 ! EhK ^ E ! bL(EhK ^ E) EhK = EhK ^ S0 ! E ^ E ! bL(E ^ E) are algebra maps. In particular, these maps give bL(EhK ^ E) and bL(E ^ E) the structure of EhK -modules. Lemma 4.1. bL(EhK ^ E) is a retract of bL(E ^ E) in DE(K). Proof.We will prove that bL(EhK ^ E) is a retract of bL(E ^ E) in DbL(E(K)^E); * *that is, there exists a map r : bL(E ^ E) ! bL(EhK ^ E) in DbL(E(K)^E)such that the composition Lb(EhK ^ E) ! bL(E ^ E) r!bL(EhK ^ E) is a weak equivalence. The desired result then follows upon applying the functor DbL(E(K)^E)! DE(K). First observe that there is a retraction h : ß*bL(E ^ E) ! ß*bL(EhK ^ E) of ß*bL(EhK ^ E) modules. Indeed, let h be the map ß*bL(E ^ E) = Map c(Gn, E*) Mapc(s,E*)------!Mapc(K\Gn, E*) = ß*bL(EhK ^ E), where s : K\Gn ! Gn is a continuous cross-section of the projection Gn ! K\Gn ([17, Ix1, Proposition 1]). It is clear that h is a retraction; we thus only ne* *ed to show that it is a map of ß*bL(EhK ^ E)-modules. This amounts to showing that the diagram Mapc(K\Gn,E*) h Map c(K\Gn, E*) Mapc(Gn, E*)_______________//_Mapc(K\Gn, E*) Mapc(K\Gn, E*) | | | | | | | | fflffl| Mapc(K\Gnxs,E*) fflffl| Map c(K\Gn x Gn, E*)________________________//_Mapc(K\Gn x K\Gn, E*) | | | | | | | | fflffl| Mapc(s,E*) fflffl| Map c(Gn, E*)_______________________________//Mapc(K\Gn, E*) commutes, where the lower vertical maps are induced by diagonal maps. But this is easy to check. A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 21 Now write A = ß*bL(EhK ^E) and B = ß*bL(E ^E). There is a spectral sequence bL(E(K)^E) Ext**A(B, A=JA) ) [bL(E ^ E), bL(EhK ^ E ^ M(I)]* [8, IV, Theorem 4.1]. As usual, I is the multi-index (i0, . .,.in-1), and J is* * the ideal (pi0, vi11, . .,.vin-1n-1) in E*. We will show that Exts,*A(B, A=JA) = 0* * for all s > 0; this implies that there is a retraction r in DbL(E(K)^E)with ß*r = h. Since B and A are both flat over E*, Ext**A(B, A=JA) = ExtA=JA(B=JB, A=JA). By Lemma 3.15 with H = {e}, B=JB Map c(K, E*=J) E*=JA=JA as A=JA-algebras. Since Map c(K, E*=J) is a flat E*=J-module, it then follows t* *hat Ext **A=JA(B=JB, A=JA) = Ext**E*=J(Map c(K, E*=J), A=JA) and that Ext**E*=J(Map c(K, E*=J), N) = Ext**Fpn[u,u-1](Map c(K, E*=In), N) whenever N is an E*-module annihilated by the ideal In = (p, v1, . .,.vn-1). He* *nce Exts,*E*=J(Map c(K, E*=J), N) = 0 for all s > 0. Now consider the short exact sequences k + J)A A A 0 ! _(In_______k+1! ___________k+1! _________k! 0, (In + J)A (In + J)A (In + J)A and take N = (Ikn+ J)A=(Ik+1n+ J)A. Using induction on k, we obtain that Exts,*E*=J(Map c(K, E*=J), A=(Ikn+ J)A) = 0 for all s > 0 and k 0. But Ikn J for k sufficiently large; therefore Exts,*A(B, A=JA) = Exts,*E*=J(Map c(K, E*=J), A=JA) = 0 whenever s > 0. |___| Lemma 4.2. EhK is bL- EE(K)*local. Proof.Let X be a cell EhK -module. We must show that if X ^E(K)(E ^K(n)) ' * in DE(K), then [X, EhK ]E(K)*= 0. As usual, we take K(n) to be a cell S0-module. Let E be a cell EhK -module and E ! E a weak equivalence of EhK -modules. Then X ^E(K)(E ^ K(n)) is trivial in DE(K) if and only if (X ^E(K) E) ^ K(n) is contractible in ME(K). This implies that [X ^E(K)( E ^E)]^K(n) is contractible, where EhK acts on the left factor of E ^ E. But E ^ E ! E ^ E is a weak equivalence since E is a q-cofibrant commutative S0-algebra and E is a cell mo* *dule over a q-cofibrant commutative S0-algebra (see 1.2-1.4). Hence [X ^E(K)(E ^E)]^ K(n), and therefore [X ^E(K) bL(E ^ E)] ^ K(n), is weakly contractible. (Here we have applied Proposition 1.1 a number of times.) By the preceding lemma, it fol* *lows that (X ^E(K) bL(EhK ^ E)) ^ K(n) is also weakly contractible. Thus [X ^E(K) (EhK ^ E)] ^ K(n) = X ^ E ^ K(n) is weakly contractible. Now apply 1.1-1.4 again to obtain that X^_E^_K(n) is weakly equivalent to X ^E ^K(n); since E^_K(n) is a wedge of K(n)'s, this impli* *es 22 ETHAN S. DEVINATZ that X is K(n)*-acyclic. But EhK is K(n)*-local, and the K(n)*-localization of an EhK -module is the same as the (EhK ^ K(n))E(K)*-localization; therefore [X, EhK ]E(K)*= 0. This completes the proof. |___| The next result provides us with the requisite vanishing line. Lemma 4.3. Let K be a closed subgroup of Gn, let P be a p-Sylow subgroup of K, and let X be the p-localization of a finite spectrum with free Z(p)-homology* * such that Hs,*c(P, E*X=InE*X) = 0 for all s bigger than some s0. Then, if Z is any EhK -module, Hs,*c(K, [Z, E ^ X]*E(K)) = 0 for all s > s0. Remark 4.4.The topology on [Z, E ^ X]*E(K)= [Z ^ DX, E]*E(K)is just that of Proposition 3.6 with H = {e}. Once this lemma is proved, we easily obtain the following result, which, by t* *he remarks at the beginning of this section, implies Theorem 3.3. Lemma 4.5. EhK is E-nilpotent in bLDE(K). Proof.By the discussion in [6, proof of Theorem 4.3], there exists an X satisfy* *ing the hypotheses of the previous lemma. It then follows from Proposition 2.5 that EhK ^ X is E-nilpotent in bLDE(K). Now consider the class N consisting of all finite p-local spectra Y such that EhK ^ Y is E-nilpotent in bLDE(K). Nilpotence technology [13, Theorem 7] tells us that N is the collection of K(m - 1)*-acyclic spectra for some m 0. Since X 2 N , we must have that N consists of all finite p-local spectra. In particul* *ar,_ S0(p)2 N and thus EhK is E-nilpotent in bLDE(K). |__| We now turn to the proof of 4.3. Say that an Fpn-vector space M is a twisted K-module over Fpn if it is a Z[K]-module such that g(cm) = (ß(g)c) . gm for all g 2 K, c 2 Fpn, and m 2 M, where ß : K ! Galis the usual projection. We then have the following preliminary result. Our notation is as in 4.3. Lemma 4.6. Suppose that N is a finite discrete twisted K-module over Fpn with the property that Hsc(P, N) = 0 for all s bigger than some s0. Then Hsc(K, M F* *pn N) = 0 for all s > s0 whenever M is a finite discrete twisted K-module over Fpn and K acts diagonally on M FpnN. Proof. H*c(K, M FpnN) ! H*c(P, M FpnN) is a monomorphism, so we need only show that Hsc(P, M FpnN) = 0 whenever s > s0. Now let L be the image of the projection ß : P ! Gal, so that L = Gal(Fpn=Fpm) for some m 1, and let P0 be the kernel of ß. Since P0 is p-analytic, Hjc(P0, * *M Fpn N) is a finite dimensional Fpn-vector space Vj. Vj is acted upon by L, and oe(c* *v) = A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 23 oe(c)oe(v) for all oe 2 L, c 2 Fpn, and v 2 Vj. It then follows (cf. [5, Lemma * *5.4]) that Vj = VjL Fpm Fpn as Fpn-vector spaces and as L-modules, so that æ j L Hi(L, Hjc(P0, M FpnN)) = Hc(P0, M0 FpnN) ii=60=.0 Therefore, by the Lyndon-Hochschild-Serre spectral sequence for profinite groups (see [18, II, x4]), Hsc(P, M FpnN) = 0 , Hsc(P0, M FpnN) = 0. The point here is that M and N are both P0-modules over Fpn; there is no Galois action on Fpn to worry about. Since P0 is a profinite p-group, the only finite simple discrete P0-module ov* *er Fpn is Fpn with the trivial action (cf. [18, Propositon 17]). By considering a comp* *osition series for M, it now follows easily that Hsc(P0, M FpnN) = 0 whenever s >_s0. This completes the proof. |__| Proof of Lemma 4.3. By Proposition 3.6 and the definition of cohomology, it suf* *fices to prove that Hs,*c(K, [Z ^ M(I), E ^ X]*E(K)) = 0 for all s > s0 and multi-indices I, whenever Z is a finite cell EhK -module. Consider now the map E*E(K)(Z ^ M(I)) E* E*X ! [Z ^ M(I), E ^ X]*E(K) which sends f c to the composition Z ^ M(I) = Z ^ M(I) ^ S0 f^c-!E ^ E ^ X ~^X-!E ^ X. Using induction on the EhK -cells of Z ^ M(I), together with the fact that E*X * *is a free E*-module, it's easy to check that this map is an isomorphism. Moreover, E*E(K)(Z ^ M(I)) is annihilated by a finite power of In; it therefore suffices * *to show that Hs,*c(K, IknE*E(K)(Z ^ M(I))=Ik+1nE*E(K)(Z ^ M(I)) E*=InE*X=InE*X) = 0 for all s > s0 and k 0. But this follows from Lemma 4.6. |_* *__| 24 ETHAN S. DEVINATZ Appendix A. A consistency result Let R be a commutative S0-algebra, and let X be an R-module on which a finite group G acts by R-module maps. This action defines an evident functor from the category with one object and morphism group G to the category of R-modules; we write *X for the cosimplicial replacement of this diagram. (Here, and in what follows, we use the terminology and notation of [4, Chapters X and XI].) Tot( *X) = XhG , and the homotopy fixed point spectral sequence converging to [Z, XhG ]*Ris obtained by mapping Z into the tower of fibrations {Totk *X}. If K and H are closed subgroups of Gn with H normal in K and K=H finite, then we may specialize to the case where R = EhK , X = EhH , and G = K=H. Theorem A.1. With the notation as above, the homotopy fixed point spectral se- quence agrees with the Adams spectral sequence E*,*r(Z, EhK ; EhH ) in bLDE(K). Proof.Let Fk denote the fiber of Totk( *EhH ) ! Totk-1( *EhH ). It suffices to prove that the sequence * ! EhK ! F0 ! F1 ! . . . is an EhH -Adams resolution of EhK in bLDE(K). Each Fk is a product of suspensi* *ons of copies of EhH and so is EhH -injective in bLDE(K). We are therefore reduced * *to showing that (A.1) 0 ! [Z, bL(EhK ^_E(K)EhH ]E(K)*! [Z, bL(F0^_E(K)EhH ]E(K)*! . . . is exact for all cell EhK -modules Z. But bL(Fk^_E(K)EhH ) is equivalent to the* * fiber of (A.2) Totk( *bL(EhH ^_E(K)EhH )) ! Totk-1( *bL(EhH ^_E(K)EhH )), where K=H acts on the left factor of bL(EhH ^_E(K)EhH ). Hence H*[Z, bL(F*^_E(K)EhH )]*E(K)= H*(K=H, [Z, bL(EhH ^_E(K)EhH )]*E(K)). By Corollary 3.9, [Z, bL(EhH ^_E(K)EhH )]*E(K)= Map c(K=H, [Z, EhH ]*E(K)) as K=H-modules, and therefore H*(K=H, [Z, bL(EhH ^_E(K)EhH )]iE(K)) = [Z, EhH ]iE(K) concentrated in homological degree 0. This proves that the desired sequence_is exact. |__| Remark A.2. The alert reader may have noticed that bL(EhH ^_E(K)EhH ) in A.2 is only a homotopy object and therefore the action of K=H is only an action up to homotopy. However, an honest action may be constructed by observing that EhH ^_E(K)EhH is equivalent to EhH ^E(K) EhH , where EhH is a cell EhK - module weakly equivalent to EhH , and that bLcan be constructed to be natural on ME(K) (see [8, Chapter VIII]). A LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE 25 References [1]A. Baker and A. Lazerev, On the Adams spectral sequence for R-modules, Algeb* *r. Geom. 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DEVINATZ Department of Mathematics, University of Washington, Seattle, Washington 98195 Current address: Department of Mathematics, University of Washington, Seattle* *, Washington 98195 E-mail address: devinatz@math.washington.edu