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
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26 ETHAN S. 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