Homotopy fixed point spectra for closed subgroups of
the Morava stabilizer groups
Ethan S. Devinatz and Michael J. Hopkinsy
Abstract
Let G be a closed subgroup of the semidirect product of the nth Morava*
* stabi
lizer group Sn with the Galois group of the field extension Fpn=Fp. We co*
*nstruct a
öh motopy fixed point spectrum" EkGnwhose homotopy fixed point spectral se*
*quence
involves the continuous cohomology of G. These spectra have the expected f*
*unctorial
properties and agree with the HopkinsMiller fixed point spectra when G is*
* finite.
0 Introduction
If a (discrete) group G acts on a spectrum Z, one can form the homotopy fixed p*
*oint
spectrum, often denoted ZhG. It is given by the G fixed points of the function*
* spectrum
F (EG, Z), where EG is a contractible free Gspace. There is then, for each spe*
*ctrum X, a
conditionally convergent spectral sequence
H*(G, Z*X) ) [X, ZhG]*,
obtained from the usual filtration of the bar construction for EG. This spectr*
*al sequence
is called the homotopy fixed point spectral sequence. Of course, the construct*
*ion of such
a homotopy fixed point spectrum requires that the group act in an appropriate p*
*ointset
category and not just up to homotopy.
However, there are situations in stable homotopy theory where group actions *
*only exist
in the stable category; that is, up to homotopy. In fact, the most important gr*
*oup action
in the whole chromatic approach to stable homotopy theory _ the action of the e*
*xtended
Morava stabilizer group Gn on the plocal Landweber exact spectrum En (see [22]*
*, [5], and x1
for a resum'e) _ arises in this way. Yet the situation in the case of this acti*
*on is not hopeless.
Indeed, H. R. Miller and the second author have proved that En is an A1 ring sp*
*ectrum and
that the space of A1 ring maps from En to itself has weakly contractible path c*
*omponents.
Furthermore, the set of path components of this space is in bijective correspon*
*dence with the
set of homotopy classes of ring spectrum maps from En to itself. (See [27] for *
*an account of
___________________________________
yBoth authors partially supported by the NSF.
1
this theory.) Since Gn acts on En by maps of ring spectra, it follows that the *
*action can be
taken to be one of A1 maps, and although this action is still only an action up*
* to homotopy,
it is an honest action up to ä ll higher A1 homotopies." Standard techniques *
*then allow
one to replace En by an equivalent spectrum on which Gn acts on the nose. Hence*
*, if G is a
(finite) subgroup of Gn, there is an A1 homotopy fixed point spectrum EhGn. The*
*se spectra
have already had a number of interesting applications in stable homotopy theory*
* (see e.g.,
[23]). Subsequently, P. G. Goerss and Hopkins [12], [13], [14], [15] extended t*
*he machinery
of HopkinsMiller to the E1 setting, and thus EhGnis an E1 ring spectrum.
Unfortunately, this is still not an entirely satisfactory state of affairs. *
*Gn is a profinite
group, so one might hope to define, for G a closed subgroup of Gn, a öc ntinuou*
*s homotopy
fixed point spectrum" denoted _ abusively _ by EhGnwhose öh motopy fixed point *
*spectral
sequence" starts with the continuous cohomology of G. (This is why we restricte*
*d to finite
subgroups in the previous paragraph.) Indeed, it is the continuous cohomology o*
*f Gn that
is important in stable homotopy theory _ by Morava's change of rings theorem, t*
*he K(n)*
local EnAdams spectral sequence (see Appendix I) for ß*LK(n)S0 has the form
(0.1) H*c(Gn, En*) ) ß*LK(n)S0.
Here K(n) denotes the nth Morava Ktheory, and LK(n) denotes K(n)*localization*
*. The
spectral sequence 0.1 thus suggests that LK(n)S0 should be the Gn homotopy fixe*
*d point
spectrum of En in this continuous sense.
The case n = 1 provides further evidence for the existence of continuous hom*
*otopy fixed
point spectra. Here we have that E1 is the pcompletion of the spectrum corepr*
*esenting
complex Ktheory and that G1 = Zxp, the group of multiplicative units in the p*
*adic integers.
The element a in Zxpcorresponds to the Adams operation _a. Now the component 1*
*0E1 of
the 0th space of E1 containing the base point is just the pcompletion_of BU, a*
*nd, according
to Quillen [24], this space is equivalent to the pcompletion of BGL(F l)+ for *
*any prime l
not equal to p. (As usual,_BGL(R)+_is the connected space representing the alge*
*braic K
theory of the ring R, and Fl is the algebraic_closure of the field with l eleme*
*nts.) Under
this_equivalence, the automorphism of BGL(F l)+ induced by the frobenius automo*
*rphism
of Fl corresponds_to the Adams_operation _l on_BUp._More generally, the profini*
*te group
bZ= Gal(F l=Fl) acts on BGL(F l)+; if G = Gal(F l=k) is any closed subgroup of *
*bZ, then
__ + G
ßi(BGL(k)+) = [ßi(BGL(F l) )]
for all i 1 [24; Corollary to Theorem 8]. Since
__ +
Hsc(G, ßiBGL(F l) ) = 0
for all i and all s > 0 _ again by the computations of Quillen _ this suggests_*
*that BGL(k)+
should be regarded as the continuous homotopy G fixed point spectrum of BGL(F l*
*)+.
In this paper, we construct spectra EhGnfor G a closed subgroup of Gn, havin*
*g the desired
properties of continuous homotopy fixed point spectra. Our construction procee*
*ds in two
2
hU for U an open _ and hence closed _ subgroup of G *
*, then
steps. First wehconstructGEn hU n
we construct En as an appropriate homotopy colimit of the En 's, for G U.
We need to introduce a little more notation in order to state our main resul*
*ts. Let
R+Gndenote the category whose objects are continuous finite left Gnsets togeth*
*er with the
left Gnset Gn. The morphisms are continuous Gnequivariant maps, and we denot*
*e by
rg : Gn ! Gn the map given by right multiplication by g 2 Gn. Let E denote the *
*category of
commutative S0algebras in the category of S0modules of May et al [11]. (A com*
*parison of
this category with the category of Lspectra for L the linear isometries operad*
* _ and hence
of the category of E1 ring spectra _ is carried out in [11; II.4].) Finally, s*
*ince the natural
number n will be fixed throughout this paper, we write bLfor the functor LK(n).*
* Here then
is our first main result.
Theorem 1 There is a functor F : (R+Gn)opp! E with the following properties.
i.F(S) is K(n)*local for each object S in R+Gn.
ii.F(Gn) = En and F(rg) : En ! En is the action of g 2 Gn on En.
iii.There is a natural isomorphism
ß*bL(F(S) ^ En) Map (S, Map c(Gn, En*))Gn
of completed ß*bL(En ^ En) = Map c(Gn, En*)comodules, where the action of*
* Gn on
Map c(Gn, En*), the set of continuous functions from Gn to En*, is given by
(gf)(g0) = f(g1g0)
for g, g02 Gn and f 2 Map c(Gn, En*). In particular, F(*) ' bLS0.
iv.Define EhUn= F(Gn=U), U an open subgroup of Gn, and let Z be any CW spect*
*rum.
There is a natural strongly convergent spectral sequence
H*c(U, E*nZ) ) (EhUn)*Z
which agrees with the spectral sequence obtained by mapping Z into a K(n)**
*local En
Adams resolution of EhUn.
Remark 0.2 In what follows_and in the proof of Theorem 1_the necessity of wor*
*king in a
precise pointset category of structured ring spectra will become apparent. How*
*ever, many of
our results, such as iii and iv of Theorem 1, are statements occurring in the s*
*table category.
We shall therefore refer to objects as "S0module spectraö r öc mmutative S0a*
*lgebras"
when we wish to emphasize that we need to work at the pointset level and as "C*
*Wspectra"
or "ring spectra" when our work takes place in the stable category. Furthermor*
*e, [X, Y ]i
will always denote the group of maps of degree i between X and Y in the stable*
* category.
Despite our precautions, there are still some ambiguities. For example, supp*
*ose X and
Y are commutative S0algebras, but we wish to understand the stable homotopy t*
*ype of
3
X ^ Y . Then X ^ Y might denote the object which is the CWapproximation to the*
* actual
pointset level smash product of X and Y . Or, X ^ Y might denote the derived*
* smash
product; that is, the smash product of CWapproximations to X and Y . However, *
*if X and
Y are cofibrant objects in the closed model category structure on E_called qc*
*ofibrant in
[11]_then these two recipes give the same stable object (see [11; VII, 6]). Si*
*nce there is
a functor E ! E sending an object to a weakly equivalent qcofibrant one_in fac*
*t, a cell
commutative S0algebra in the sense of [11]_we may as well assume that F(S), fo*
*r example,
is always qcofibrant and thus eliminate this ambiguity. We will often use thi*
*s device of
functorially approximating by a cell object, even if we don't always mention it.
Remark 0.3 Let In = (p, v1, . .,.vn1) En*, and, if Z is any CWspectrum, l*
*et {Zff} be
the directed system of finite CWsubspectra. Then E*nZ = lim E*nZff=IknE*nZffis*
* a profinite
ff,k
continuous Gnmodule and we define
H*c(U, E*nZ) = limH*c(U, E*nZff=IknE*nZff).
ak
In general, if G is a panalytic profinite group and M is a profinite continuou*
*s Zp[[G]]module,
then limH*c(G, Mff) is independent of the presentation M = limMff. In fact, it *
*is for example
ff ff
the cohomology of the usual cochain complex whose jcochains are Map c(Gj, M) (*
*cf. proof
of Lemma 3.21). We therefore use this as our definition of H*c(G, M); see also *
*[31] for a more
general treatment of this object.
For the next step, let
(0.4) Gn = U0 ) U1 ) U2 ) . .).Ui) . . .
T
be a sequence of normal open subgroups of Gn with iUi = {e}. For example, us*
*ing the
notation at the beginning of Section 1 and the description of Sn in [7; 2.21], *
*we may take
j
Ui= Vio Gal, where Vi is the group of power series jn 0bjxp with bj = 0 for 0 *
*< j < i and
b0 = 1 if i > 0.
Definition 0.5 Fix a sequence {Ui} as above. For G a closed subgroup of Gn, def*
*ine
EhGn= bLholimEEh(UiG)n,
!i
where holimE denotes the homotopy colimit taken in the topological model catego*
*ry E.
!
Remark 0.6 More precisely, one should functorially replace holim EEh(UiG)nby *
*a weakly
!i
equivalent cell commutative S0algebra before applying bL. Then the constructio*
*n EhGnbe
comes functorial in E (see [11; VIII, 2]).
We will prove the following result in x5.
4
hG has the following properties:
Theorem 2 The construction En
i.ß*bL(EhGn ^ En) is naturally isomorphic to Map c(Gn, En*)G as com*
*pleted
Map c(Gn, En*)comodules, where G acts on Map c(Gn, En*) as in Theorem 1ii*
*i.
ii.Let Z be any CWspectrum. The spectral sequence obtained by mapping Z int*
*o a
K(n)*local EnAdams resolution of EhGnis strongly convergent to (EhGn)*Z *
*and has
E2term naturally isomorphic to H*c(G, E*nZ).
Remark 0.7 i. Using the first part of this theorem, it's easy to see that EhG*
*nis canonically
independent (up to weak equivalence in E) of the choice of sequence {Ui}.
ii. Since G is a closed subgroup of a panalytic profinite group, it is itse*
*lf panalytic (see
[9; 10.7], and therefore H*c(G, E*nZ) is defined as in Remark 0.3.
Now by Theorem 1, there are commutative diagrams
Eh(UiG)n= F(Gn=UiG) _____//F(Gn)5=5Enk
kkk
  kkkk
  kkkk
fflffl fflfflkkk
Eh(UjG)n= F(Gn=UjG)
and
Eh(UiG)n= F(Gn=UiG)S_____//F(Gn) = En
SSSSS  g
SSSSS F(rg) 
SS))Sfflfflfflffl
F(Gn) = En
for i j and g 2 G, where the unlabeled arrows are induced by the evident proj*
*ections.
*
* 0G
There is thus a canonical map EhGn! EGn, the Gfixed points of En. Let us denot*
*e by Ehn
the ordinary homotopy fixed point spectrum for the action of G on En. Composin*
*g the
0G hG h*
*0G
above map with the canonical map EGn! Ehn ([4; XI, 3.5]) yields a map En ! En*
* . The
next result will be proved in x5.
Theorem 3 Let G be a finite subgroup of Gn. Then:
0G
i.The map EhGn! Ehn described above is a weak equivalence.
ii.Let Z be any CWspectrum. The homotopy fixed point spectral sequence
0G * hG *
H*(G, E*nZ) ) (Ehn ) Z = (En ) Z
is naturally isomorphic to the spectral sequence of Theorem 2ii.
5
We also have a result on iterated homotopy fixed point spectra. Indeed, if K*
* is a closed
and U is a normal open subgroup of Gn, then the opposite of the Weyl group W (K*
*) =
N(K)=K acts on Gn=UK via xUK 7! xhUK for x 2 Gn, h 2 N(K). This yields an
action of W (K) on Eh(UK)n, and hence, upon passing to the homotopy colimit, on*
* EhKn. In
particular, if F is a finite subgroup of W (K), we may form (EhKn)hF in the usu*
*al way. We
can now state our next result.
Theorem 4 Suppose G is a closed subgroup of Gn, K is a closed normal subgroup*
* of G,
and F G=K is finite. Then EhGnis naturally equivalent to (EhKn)hF.
Another sort_of consistency result is given by examining the case n = 1. Sin*
*ce the Galois
action on BGL(F l)+ corresponds to the action of G1 = Zxpon E1, where once agai*
*n l is a
prime different from_p,_we would expect a relationship between the continuous h*
*omotopy
fixed points of BGL(F l)+ and the continuous homotopy fixed point spectra of E1*
*. This is
indeed the case.
Let R be a commutative ring, and let KR be the algebraic Ktheory spectrum o*
*f_R (so
that 1 KR ' Z x BGL(R)+). Quillen's results deloop (see [20, VIII]); hence_(K*
*F l)p is
equivalent to the connective cover of E1, and the action of t 2 bZon (KF l)p co*
*rresponds to
the action of lt 2 Zxpon E1. Now choose s dividing p  1 such that ls 1 mod *
*(p), and
define a continuous group monomorphism Zp ! Zxpby sending a 2 Zp to lsa. (The n*
*umber
s is needed to guarantee that lsa makes sense for all a 2 Zp.) If G is a nont*
*rivial closed
subgroup of Zp, then G = pjZp for some j 0. Let us also write G for the corre*
*sponding
closed subgroup in Zxp. We then have the following result.
Theorem 5 With the notation as above, EhG1' LK(1)K(k), where k is the field o*
*f invariants
__
of the action of spjbZ bZon Fl.
Our final result, which will be proven as an application of the machinery de*
*veloped here,
was originally due to the second author and H. R. Miller, who suggested that th*
*is is the
place where it should logically appear.
Theorem 6 (HopkinsMiller) Suppose c : Gn ! Zp is a continuous homomorphism*
*, and
let c also denote the composition Gn c!Zp! En*. Then c 2 H1c(Gn, En*) survive*
*s to ß*bLS0.
Let us indicate our strategy for constructing EhUn. We will construct a cos*
*implicial E
spectrum corresponding to the K(n)*local EnAdams resolution of EhUn; then we *
*will define
EhUnto be Tot of this cosimplicial spectrum. To do this, we need only determi*
*ne the
(expected) homotopy type of bL(EhUn^ En) together with the comodule structure m*
*ap
bL(EhUn^ En) = bL(EhUn^ S0 ^ En) ! bL(EhUn^ En ^ En).
Now we might expect a spectral sequence
H*c(U, ß*bL(En ^ En)) ) ß*bL(EhUn^ En),
6
since there is such a spectral sequence if U is replaced by a finite subgroup o*
*f Gn (see Theorem
4.3). But ß*bL(En ^ En) = Map c(Gn, En*), and the action of U is given as in Th*
*eorem 1iii
(see x1). Since Map c(Gn, En*) is Uacyclic (cf. proof of Lemma 3.20) we should*
* have
ß*bL(EhUn^ En) = Map c(Gn, En*)U.
The comodule structure map is also determined. But, as En*modules,
Ym
Map c(Gn, En*)U = En*,
i=1
Q m
where m is the cardinality of Gn=U. We are thus led to take XGn=U = i=1En as*
* a
model for bL(EhUn^ En). We can also define the corresponding comodule structur*
*e map
XGn=U ! bL(XGn=U ^ En). (Note that this map is not _ except in a few very speci*
*al cases
_ the product of the maps En = bL(S0 ^ En) ! bL(En ^ En).) With this constructi*
*on, we
obtain a cosimplicial object, but only in the stable category. However, the te*
*chnology of
HopkinsMiller as expanded by GoerssHopkins is again available to allow us to *
*conclude
that the requisite diagrams in fact commute up to all higher homotopies in E. T*
*he original
cosimplicial object can now be replaced by an equivalent cosimplicial object in*
* E, and Tot
can be formed.
The contents of this paper are as follows. In x1, we recall the relevant par*
*ts of Morava's
theory, and in x2, we discuss the formation of homotopy inverse limits for cert*
*ain diagrams
commutative up to all higher homotopies. We construct the cosimplicial Adams re*
*solution in
x3 and identify the resultant spectral sequence as having the form of a (contin*
*uous) homotopy
fixed point spectral sequence. In x4 we compute ß*bL(F(S) ^ En); this allows us*
* to identify
the preceding spectral sequence as a K(n)*local EnAdams spectral sequence and*
* completes
the proof of Theorem 1. We prove Theorems 2 and 3 in x5, Theorem 4 in x6, and, *
*finally,
Theorems 5 and 6 in x7. An appendix summarizes the properties of K(n)*localiza*
*tions and
K(n)*local EnAdams spectral sequences that we need. In particular, we prove t*
*he strong
convergence of these spectral sequences.
Acknowledgment. The first author would like to thank P. G. Goerss for a hel*
*pful
discussion about Theorem 2.2.
1 Resum'e of Morava's theory
Let p be a fixed prime, let n 1, and let En denote the spectrum with coeffici*
*ent ring
En* = W Fpn[[u1, . .,.un1]][u, u1] obtained via the Landweber exact functor t*
*heorem for
BP . W Fpn denotes the ring of Witt vectors with coefficients in the field Fpn *
*of pn elements,
and the map BP*!r En* _ which also provides En with the structure of BP algebr*
*a in the
stable category _ is given by
8 i
< uiu1p i < n
r(vi) = u1pn i = n
:
0 i > n.
7
th Morava stabilizer group; i.e., the automorphism gr*
*oup of
Now let Sn denote the n
the height n Honda formal group law n over Fpn. Let Gal Gal(Fpn=Fp), and le*
*t Gn =
Sn o Gal. The LubinTate theory of lifts of formal group laws provides an acti*
*on of Sn
on En* (see for example [7]), and Gal acts on En* via its action on W Fpn. If*
* H is a
subgroup of Gal, let us write EHn for the Landweber exact spectrum with coeffic*
*ient ring
W (FHpn)[[u1, . .,.un1]][u, u1], where FHpnis the subfield of Fpn fixed by th*
*e automorphism
group H.
We first identify the completed Hopf algebroid ß*bL(En ^ En) with the split *
*completed
Hopf algebroid (En*, Map c(Gn, Zp)b ZpEn*); this piece of Morava's theory is cr*
*ucial to all of
our subsequent work. We start by observing that Map c(Sn, W Fpn)Galis a complet*
*ed Hopf
algebra over Zp; the diagonal map is given by
Map c(Sn, W Fpn)Gal! Map c(Sn x Sn, W Fpn)Gal Map c(Sn, W Fpn)GalbMap c(Sn, *
*W Fpn)Gal,
where the first map is induced by the multiplication on Sn and the second is an*
* isomorphism
by [5; AII.3]. There is also a map
jL : EGaln*! Map c(Sn, W Fpn)GalbZpEGaln*!Mapc(Sn, En*)Gal
which is given by jL(x)(g) = g1x for x 2 EGaln*and g 2 Sn. With these structur*
*e maps, we
obtain a split completed Hopf algebroid (EGaln*, Map c(Sn, W Fpn)GalbZpEGaln*).*
* A main result
of Morava's theory is the following identification.
Theorem 1.1 (EGaln*, ß*bL(EGaln^EGaln)) is isomorphic to (EGaln*, Map c(Sn, W*
* Fpn)GalbZpEGaln*)
as completed Hopf algebroids.
Proof. We showed in [5; x4] that (EGaln*, Map c(Sn, W Fpn)GalbEGaln*) is isomor*
*phic to a com
pleted Hopf algebroid denoted (EGaln*, EGaln*bUUS bUEGaln*). But we observed in*
* [6; 3.4] that
this Hopf algebroid is isomorphic to (EGaln*, EGaln*bBP*BP*BP bBP*EGaln*), wher*
*e the completed
tensor product here denotes Inadic completion. Since ß*bL(EGaln^ EGaln) is the*
* Inadic com
pletion of EGaln*EGaln= EGaln* BP* BP*BP BP* EGaln*(see [16]), the proof is co*
*mplete.
To derive the structure of the completed Hopf algebroid ß*bL(En^En) from The*
*orem 1.1,
we first observe that
ß*bL(En ^ EGaln)= ß*bL(EGaln^ EGaln) ZpW Fpn
= Map c(Sn, En*)Gal ZpW Fpn
= Map c(Sn, En*),
where the last equality follows by [5; 5.4]. Then
ß*bL(En ^ En) = ß*bL(En ^ EGaln) ZpW Fpn
= Map c(Sn, En*) ZpW Fpnfi!Mapc(Gn, En*),
8
where
ø(f w)(s, oe) = w . (oe1(f(s)))
for f 2 Map c(Sn, En*), w 2 W Fpn, s 2 Sn, and oe 2 Gal. Now consider the split*
* completed
Hopf algebroid (En*, Map c(Gn, Zp)b ZpEn*); once again jL : En* ! Map c(Gn, En**
*) is given
by jL(x)(g) = g1x for x 2 En* and g 2 Gn. Observe that this is not a Hopf alg*
*ebroid
over W Fpn, since jL and jR are different when restricted to W Fpn. Upon chasin*
*g down the
identifications, the next result follows from Theorem 1.1.
Proposition 1.2 (En*, ß*bL(En^ En)) is isomorphic to (En*, Map c(Gn, Zp)b ZpEn*
**) as com
pleted Hopf algebroids.
This isomorphism is used to define the action of Gn on En. Indeed, recall th*
*at if M is a
completed left Map c(Gn, En*)comodule, then M is a Gnmodule with the action g*
*iven by
g(m) = _(m)(g1),
where
_ : M ! Map c(Gn, En*)b En*M ! Map c(Gn, M)
is the comodule structure map. In particular, if X is a finite CWspectrum, En**
*X is naturally
a Gnmodule, and the pairing En*X En*En*Y ! En*(X ^ Y ) is Gnequivariant, whe*
*re the
left side is given the diagonal action. Since En* is profinite in each degree,*
* it follows that
there exists, in the stable category, a unique action of Gn on En by ring spect*
*rum maps
inducing the above action on En*X.
With this action, it is immediate that the isomorphism of Proposition 1.2 is*
* given by
sending x 2 ß*bL(En ^ En) to hx 2 Map c(Gn, En*), where hx(g) is given by the c*
*omposition
bL(g1^En) ~
(1.3) S x!bL(En ^ En) ! bL(En ^ En) !En.
(An identification of this sort first appears in the literature in [30].) Here *
*suspensions have
been omitted from the notation and ~ is the ring spectrum multiplication map. M*
*oreover, it
is easy to check using (1.3) that the action of Gn on the left factor of ß*bL(E*
*n^En) corresponds
to the action of Gn on Map c(Gn, En*) described in Theorem 1. For later purpose*
*s, we will
also need to know the formula for the action of Gn on the right factor of ß*bL(*
*En ^ En). If
we write gR x for this action of g 2 Gn on x 2 ß*bL(En ^ En), then it is again *
*easy to see
using (1.3) that gR x corresponds to the map sending g02 Gn to ghx(g0g) 2 En*.
Remark 1.4 In [5], we used Theorem 1.1 to define a natural W Fpnlinear actio*
*n of Sn on
En*X. There is also the evident action of Gal on En*X = W Fpn Zp EGaln*X, and*
* these
actions piece together to give a natural action of Gn on En*X, whence an action*
* of Gn on
En in the stable category. This action is the same as the action defined above.
9
The identifications of 1.2 and 1.3 can be generalized to iterated smash prod*
*ucts of En
and beyond. Indeed, if X is a finite spectrum,
ß*bL(E(j+1)n^ X) = En*Enb En*En*Enb En*. .b.En*En*Enb En*En*X,
where En* acts on the right of each factor En*En and on the left of the factor *
*En*X. But
En*Enb En*. .b.En*En*En = (Map c(Gn, Zp) ZpEn*)b En*. .b.En*(Map c(Gn, Zp) *
* ZpEn*)
= Map c(Gn, Zp)b Zp. .b.ZpMapc(Gn, Zp)b ZpEn*
= Map c(Gjn, En*),
and thus
En*Enb En*. .b.En*En*Enb En*En*X = Map c(Gjn, En*)b En*En*X
= Map c(Gjn, En*X).
This isomorphism sends x 2 ß*bL(E(j+1)n^ X) to hx 2 Map c(Gjn, En*X), where hx(*
*g1, . .,.gj)
is given by the composition
bL(g11^...^g1j^En^X) bL(~^X)
(1.5) S x!bL(E(j+1)n^ X) ! bL(E(j+1)n^ X) ! En ^ X.
More generally, if Z is any spectrum, there is a natural transformation
(1.6) øj : [Z, bL(E(j+1)n)]* ! Map c(Gjn, E*nZ)
such that øj(x)(g1, . .,.gj) is the composite
bL(g11^...^g1j^En) ~
(1.7) Z x!bL(E(j+1)n) ! bL(E(j+1)n) ! En.
(To show that øj does in fact have its image in Map c(Gjn, E*nZ), it suffices, *
*by the definition
of the topology on E*nZ, to show that this is the case when Z is finite. But wh*
*en Z is finite,
øj is just the isomorphism described in 1.5 with X the SpanierWhitehead dual o*
*f Z.) Since
Map c(Gjn, ?) is exact on the category of profinite groups (see [29; I, Theorem*
* 3]), it follows
that øj is a natural transformation of cohomology theories satisfying the produ*
*ct axiom. But
øj is an isomorphism with Z = S0; therefore øj is an isomorphism for any CWspe*
*ctrum Z.
2 Homotopy inverse limits
Let hE denote the homotopy category of commutative S0algebras. That is, two m*
*aps
f, g : X ! Y in E are homotopic if f and g lie in the same path component of E*
*(X, Y ),
the topological space of S0algebra maps between X and Y . Alternatively, E is *
*a tensored
category (over the category of unbased topological spaces), and f and g are hom*
*otopic if
10
there exists a map h : X I ! Y restricting to f and g on the ends of the cyl*
*inder (see
[11; VII, 2]).
Now let J be a small category and suppose X : J ! hE is a functor. In some *
*cases,
X can be replaced by a homotopy equivalent strict diagram, and then its homotop*
*y inverse
limit can be formed.
Definition 2.1 Let X : J ! hE be as above. X is said to be an h1 diagram if f*
*or each
ff : j1 ! j2 in J, E(Xj1, Xj2)Xff, the path component of E(Xj1, Xj2) containing*
* Xff, is
weakly contractible.
The main result of this section is due to W. G. Dwyer, D. M. Kan, and J. H. *
*Smith [10].
We feel that the reader will appreciate an account of the proof.
__
Theorem 2.2 Let X : J ! hE be_an h1 diagram. Then_there_exists a functor X *
*: J ! E
and a natural transformation X ! X in hE such that X (j) ! X(j) is a weak equi*
*valence
for each j 2 Ob J.
The proof makes use of a modification of the cosimplicial replacement of a d*
*iagram (cf.
[4; XI, 5]). We first recall some notation.
If Z is an unbased topological space and Y is a commutative S0algebra, let*
* F (Z, Y )
denote the cotensor product of Z and Y in E; its underlying S0module is just t*
*he function
S0module of maps from 1 Z+ to Y ([11; VII, 2]). We shall also use the notatio*
*n F ( , )
to denote function spectra in the stable category.
With Z as above, let Z denote the geometric realization of the singular sim*
*plicial set
of Z. This is of course a functorial cofibrant replacement of Z; it also has th*
*e property that
V x W is naturally homeomorphic to (V x W ) and that this homeomorphism comm*
*utes
with the projections onto V and W . Hence a pairing V x W ! Z induces a pai*
*ring
V x W ! Z.
ConstructionQ2.3 Let X : J ! hE be an h1 diagram. Define a cosimplicial commu*
*tative
S0algebra *hX by
Q 0 Q
hX = Xj
j2J
Q n Q
hX = F ( EXff, Xj0),
Jn
where Jn is the set of diagrams
ff : j0ff1j1ff2j2 . ..jn1ffnjn
in J, and
EXff = E(Xj1, Xj0)Xff1x . .x. E(Xjn, Xjn1)ffn.
11
i is defined via the commutative diagram
If 0 < i < n + 1, the coface d
Q di Q n+1
(2.4) h X ____________//h X
iff0 iff
fflffl di fflffl
F ( EXff0, Xj0)__ff//_F ( EXff, Xj0)
where ßffdenotes the projection onto the factor indexed by
ffn+1
ff : j0ff1j1ff2j2 . ..jn jn+1,
ff0denotes the diagram
ffiffi+1 ffi+2 ffn+1
j0ff1j1 . ..ji1 ji+1 ji+2 . ..jn jn+1,
and
(diffg)(f1, . .,.fn+1) = g(f1, . .,.fifi+1, . .,.fn+1)
for (f1, . .,.fn+1) 2 EXff. (Here fifi+1denotes the image of (fi, fi+1) under *
*the map
E(Xji, Xji1) x E(Xji+1, Xji) ! E(Xji+1, Xji1)
induced by the composition pairing
E(Xji, Xji1) x E(Xji+1, Xji) ! E(Xji+1, Xji1).)
If i = 0, d0 is defined as in 2.4, although now ff0is the diagram
ffn+1
j1ff2j2 . .. jn+1,
and d0ff: F ( EXff0, Xj1) ! F ( EXff, Xj0) is defined by
(d0ffg)(f1, . .,.fn+1) = f1(g(f2, . .,.fn+1)),
where f1 also denotes the image of f1 2 E(Xj1, Xj0) in E(Xj1, Xj0).
Finally, if i = n + 1, ff0is the diagram
j0ff1j1 . ..jn1ffnjn
and
(dn+1ffg)(f1, . .,.fn+1) = g(f1, . .,.fn).
As for the codegeneracies, si is defined via the commutative diagram
Q n+2 _____si____//Qn+1
(2.5) h X h X
iff0 iff
fflffl si fflffl
F ( EXff0, Xj0)__ff//_F ( EXff, Xj0)
12
0is the diagram
where ff
ffi+1 ffn+1
j0ff1j1ff2j2 . ..ji1ff1jiidji ji+1 . .. jn+1
and
(siffg)(f1, . .,.fn+1) = g(f1, . .,.fi, id, fi+1, . .,.fn+1).
Here id denotes the image of * = (*) in E(Xji, Xji) under the evident map.
Recall that a cosimplicial S0module Y is fibrant if the map
s : Y n+1! MnY {(y0, . .,.yn) 2 Y nx . .x.Y n: siyj = sj1yi 8 0 i < j *
* n}
given by s(y) = (s0(y), . .,.sn(y)) is a qfibration_that is, a fibration in th*
*e Quillen closed
model sense_for all n 1. (Properly speaking, MnY should be defined as an equ*
*alizer;
however we will continue to use this shorthand when to do otherwise would resul*
*t in more
confusion.)
Q *
Lemma 2.6 Let X be an h1 diagram. Then hX is fibrant.
Q 1 Q * Q 0
Proof. Consider first the map s : hX ! M0 hX = hX. The composition
Q 1 s Q 0 Q
hX ! hX = j Xj ! Xj
is given by
Q 1 Q
hX = F ( E(Xj1, Xj0)Xff, Xj0) ! F ( E(Xj, Xj)id, Xj) ! Xj,
j0ffj1
where the last map is evaluation at id 2 E(Xj, Xj). Since id is a vertex of E*
*(Xj, Xj), it
follows that this last map is a qfibration and hence so is s.
Now suppose n > 1. Let Jkn,jbe the subset of Jn consisting of those ntuple*
*s of maps
with j0 = j and ffk = id, and set
a
Dkn,jX = EXff.
ff2Jkn,j
Then let
[
Dn,jX = Dkn,jX,
1 k n
Q n Q *
and observe that the map s : hX ! Mn1 hX restricts to an isomorphism
Q n1Q *
F (Dn,jX, Xj) !M hX.
j
13
But the inclusion
a
Dn,jX ! EXff
Jn,j
is the inclusion of a summand, where Jn,jis the subset of Jn consisting of thos*
*e ntuples
with j0 = j. s is therefore a qfibration in this case as well.
Recall that, given a fibrant cosimplicial S0module Y , Tot Y = F ( [*], Y *
*), the S0
module of (unpointed) cosimplicial maps from [*] to Y . Here [*] is the cosim*
*plicial space
which in dimension n is the standard nsimplex [n] with the usual coface and c*
*odegeneracy
maps. Let Sks [*] be the cosimplicial space which in dimension n is the sskele*
*ton of [n],
and define Tot sY = F (Sks [*], Y ). The map Tot j+1Y ! TotjY is a fibration wi*
*th fiber
F ( [j + 1]= `[j + 1], Y j+1\ kers0 \ . .\.kersj). By mapping a CWspectrum Z i*
*nto the
tower {Tot jY }, we obtain a spectral sequence
(2.7) Es,t2= ßs([Z, Y ]t))=[Z, TotY ]t+s (cf. [4; X.6]).
This spectral sequence is strongly convergent in the sense of [4; IX.5.4] provi*
*ded lim1Es,tr= 0
*
*  J
for all s and t.
In particular, if X is an h1 diagram, we obtain a spectral sequence
Q * t+s
(2.8) Es,t2= lims([Z, X]t))=[Z, Tot hX] .
 J
This is proved as in [4; XI.7.1], using the fact that EXffis contractible for *
*all ff.
With these constructions in hand, we can now prove the main result of this s*
*ection.
Proof of Theorem 2.2. If j is an object of J, let J\j be the category whose obj*
*ects are
morphisms j ! j0 in J and whose morphisms are the evident commutative diagrams.*
* The
evident functor_~j : J\j ! J provides_us with an h1 diagram ~*jX over J\j and *
*hence a
Q *
functor X : J ! E defined_by X (j) = Tot h~*jX.
Now define the map X (j) ! X(j) to be the composition
Q * * Q 0 * Q 0 * p
(2.9) Tot h~jX ! F ( [0], h~jX) = h~jX ! Xj,
where p is the projection onto the factor indexed by the object j id!j. To pro*
*ve that
__ __Xf__
X (j)____//_X(j0)
 
 
fflfflXf fflffl
X(j) ____//_X(j0)
14
0, we must prove that
commutes whenever f : j ! j
__
(2.10) X (j)____//_X(j)
GGG 
GGG Xf
GG##fflffl
X(j0)
Q 0
commutes, where the diagonal map is the projection onto h~*jX followed by the*
* projection
onto the factor indexed by f : j ! j0. First observe that the two compositions
__ Q 1 * F(d0,id) Q 1 * pf 0
X (j) ! F ( [1], h~jX) !!F ( [0], h~jX) !X(j )
F(d1,id)
are homotopic, where the last map is the projection onto the factor indexed by *
*f : (j id!j) !
f 0
(j ! j ). But these compositions are the same as
__ Q 0 * d0Q 1 * pf 0
X(j) ! F ( [0], h~jX) !!h~jX ! X(j ).
d1
Now use the definitions of d0 and d1 to check that these maps give the commutat*
*ive diagram
2.10.
By 2.8, there is a spectral sequence
__
Es,t2= limsßt~*jX ) ßtsX (j).
J\j
But j ! j is an initial object of J\j; therefore
æ
0 s > 0
limsßt~*jX = .
J\j ßtXj s = 0
__
Thus ß*X (j) = ß*X(j), and an unraveling of the identifications shows that the *
*map in 2.9
induces the identity on ß*.
3 Construction of the functor F
We begin by stating the following extensions of the HopkinsMiller theory due t*
*o Goerss and
Hopkins. These are the results needed to show that, for S 2 Ob R+Gn, the cosimp*
*licial object
CS we will construct in the stable category lifts to an h1 diagram, and hence,*
* by Theorem
2.2, to a cosimplicial object in E. Then F(S) is defined to be TotCS.
Theorem 3.1 ([14]) Let E and F be qcofibrant commutative S0algebras with E *
*Landweber
exact. Suppose that ß*E is concentrated in even dimensions with an invertible *
*element in
15
degree 2. Suppose in addition that ß0E is madically complete for some ideal m,*
* that E0=m is
an algebra of characteristic p, and that the relative frobenius for the homomor*
*phism E0=m !
E0F=mE0F is an isomorphism. Then each path component of E(F, E) is weakly contr*
*actible,
and the Kronecker pairing
ß0E(F, E) ! Hom E*alg(E*F, E*)
is a bijection.
Remark 3.2 Recall that if A is a commutative Fpalgebra, the frobenius homomo*
*rphism
OEA : A ! A is defined by OEA(x) = xp. If f : A ! B is a homomorphism of commut*
*ative
Fpalgebras, the relative frobenius is defined to be the map given by the dotte*
*d arrow in
the following diagram, where the square is a pushout in the category of commut*
*ative Fp
algebras:
ffiB ___3B3_____________________________*
*__________________________??_KK
_____________________________________*
*__________________
_______________________________________*
*_________________________________
_________________________________________*
*_________________________
___________________________________________*
*______________________
BO____//_COOOf_______________________________*
*___
__________________
f  ____________________________________*
*_______
  ___________________
A _ffiA//_A
Remark 3.3 The condition that E be Landweber exact can be weakened. Indeed, E*
* need
only satisfy a condition of the sort required by Adams [1; III, 13.3] in his co*
*nstruction
of universal coefficient spectral sequences. For example, Property 1.1 of [6] *
*suffices. In
particular, any Landweber exact spectrum satisfies this property [6; 1.3].
Theorem 3.4 ([14]) Let H be a subgroup of Gal. Then EHn has a unique structu*
*re of
commutative S0algebra descending to its ordinary ring spectrum structure.
Remark 3.5 EHnis in fact the H homotopy fixed point spectrum of En, so our no*
*tation is
only slightly abusive.
In order to construct CS, we need to apply Theorem 3.1 to spectra of the for*
*m bL(X ^
(En)(j)), where X is a finite product of En's. The next result enables us to do*
* this.
Proposition 3.6 Let E = bL(X ^ E(j)n) and let F = bL(Y ^ E(k)n), where X and Y*
* are (non
empty) products of finitely many copies of En and j, k 0. Then the pair F, E *
*satisfies the
conditions of Theorem 3.1; moreover, the function
ß0E(F, E) ! Hom alg(F*, E*)
sending a map F ! E to the induced map on homotopy groups is onetoone.
16
0algebras _ the key point h*
*ere is that
Proof. First observe that E and F are commutative S
localization with respect to a homology theory preserves such objects (see [11;*
* VIII]). An
easy reduction also shows that it suffices to consider the case where X = En an*
*d Y = En.
By Morava's theory, E* = Map c(Gjn, En*) and the action of En on the right fact*
*or of E
induces "right multiplicationö f En* on Map c(Gjn, En*). It therefore follows *
*that E (resp.
F ) is Landweber exact for an appropriate map BP ! E (resp. BP ! F ) and hence,*
* by the
Landweber exact functor theorem. E*F is a flat F*module. In addition, E0 is co*
*mplete with
respect to the ideal m = Map c(Gjn, In), where we also write In = (p, u1, . .,.*
*un1) (En)0.
Now
ß*bL(F ^ E) = Map c(Gj+k+1n, En*)
and, just as before, bL(F ^ E) is Landweber exact. Since F*E is flat over E*, i*
*t also follows
that F ^E is Landweber exact. Thus, if M(pi0, vi11, . .,.vin1n1) is a finite *
*CWspectrum whose
BrownPeterson homology is BP*=(pi0, . .,.vin1n1), we have that
F*E=F*E . (pi0, . .,.vin1n1)=ß*(F ^ E ^ M(pi0, . .,.vin1n1))
! ß b i0 in1
*(L (F ^ E) ^ M(p , . .,.vn1))
= ß*bL(F ^ E)=ß*bL(F ^ E) . (pi0, . .,.vin1*
*n1).
Therefore
F*E=F*E . In = ß*bL(F ^ E)=ß*bL(F ^ E) . In
= Map c(Gj+k+1n, Fpn[u, u1]).
Clearly the frobenius is an isomorphism on E0=m, and, from the above equality, *
*it is also
an isomorphism on E0F=mE0F . This implies that the relative frobenius for E0=m*
* !
E0F=mE0F is an isomorphism.
Finally, to prove that ß0E(F, E) ! Hom alg(F*, E*) is onetoone, it suffice*
*s to prove that
the function
Hom E*alg(E*F, E*) ! Hom alg(F*, E*)
given by precomposition with the map
F* = ß*(S0 ^ F ) ! ß*(E ^ F ) = E*F
is onetoone. But this follows easily from the commutative diagram
Hom E*alg(E*F, E*)_______________//Homalg(F*, E*) ,
fflffl fflffl
 
fflffl 

Hom (E* Q)alg(E*F Q, E* Q) 

 
 fflffl
Hom (E* Q)alg(E* F* Q, E* Q)_____Homalg(F* Q, E* Q)
17
and the fact that the vertical maps are monomorphisms since any Landweber exact*
* spectrum
is torsion free.
It will also be convenient to know that, for E and F as above, the canonical*
* map from
ß0E(F, E) to the set of ring spectrum maps from F to E in the stable category i*
*s a bijection.
The next result enables us to conclude this.
Lemma 3.7 Let E be as in Proposition 3.6, and let F be a Landweber exact com*
*mutative
ring spectrum. Then
i.E*F (j)= E*F E* E*F E* . . .E*E*F for any j 1.
______________z_____________"
j times
ii.The Kronecker pairing
[F (j), E]*!Hom E*(E*F (j), E*)
is an isomorphism for all j 1.
Proof. Since E and F are Landweber exact, E*F is a flat E*module. This immedia*
*tely
implies i. As for ii, begin by observing that F1 ^ F2 is Landweber exact if F1 *
*and F2 are.
Hence F (j)satisfies the hypotheses of the lemma, so we may assume j = 1 withou*
*t loss of
generality.
There is a universal coefficient spectral sequence
Ext**E*(E*F, E*) ) E*F
(see Remark 3.3); it thus suffices to show that ExtiE*(E*F, E*) = 0 for all i >*
* 0. We do this
by an argument similar to that in [27; 15.6]. Indeed, E*F = E* En* En*F and En*
**F is flat
over En*; hence
Ext**E*(E*F, E*) = Ext**En*(En*F, E*).
We thus need only show that Exti(En)0((En)0F, E0) = 0 for all i > 0, where now *
*E = bL(E(j+1)n)
for some j 0. Let m be as in the proof of Proposition 3.6, and write (En)0F =*
* M. Then
again by flat base change
Exti(En)0(M, E0=m) = ExtiFpn(M=InM, E0=m).
This last group is trivial for i > 0 since Fpn is a field. Similarly, Exti(En)0*
*(M, mt=mt+1) = 0 for
i > 0; now use the fact that E0 = limE0=mt to conclude that Exti(E )(M, E0) = 0*
* whenever
 t n 0
i > 0.
18
Now let be the category whose objects are the nonnegative integers_a typi*
*cal object
will be denoted [n]_and whose morphisms from [n] to [m] are the monotone nondec*
*reas
ing functions from {0, 1, . .,.n} to {0, 1, . .,.m}. Recall the definition of *
*R+Gnfrom the
Introduction. We will construct an h1 diagram
C : (R+Gn)oppx ! hE
such that, for each S 2 Ob R+Gn,
CS = C(S, ) : ! hE
is a cosimplicial K(n)*local EnAdams resolution for the not yet constructed s*
*pectrum
F(S). As described in the Introduction (for the case S = Gn=U), we seek a K(n)*
**local
commutative S0algebra and right Enmodule spectrum XS together with an Emap d*
* :
XS ! bL(XS ^ En) such that
(3.8) ß*XS = Map Gn(S, Map c(Gn, En*))
as right En*modules, such that
(3.9) ß*bL(XS ^ En) ß*XS bEn*ß*bL(En ^ En)
= Map Gn(S, Map c(Gn, En*))b En*Map c(Gn, En*)
= Map Gn(S, Map c(Gn x Gn, En*))
and such that ß*d corresponds to the map induced by the group multiplication Gn*
*xGn ! Gn.
Proposition 3.10 There exists an h1 diagram X : (R+Gn)opp! hE and a natural t*
*ransfor
mation d : X ! bL(X ^ En) satisfying the above requirements. Moreover, this con*
*struction
has the following properties:
i.All maps are maps of right Enmodule spectra, where En acts on the right f*
*actor of
Lb(X ^ En).
ii.If S is finite, X X(S) is a product of a finite number of copies of En.
iii.XGn = bL(En ^ En) and if rg : Gn ! Gn is right multiplication by g 2 Gn, X*
*(rg) =
Lb(g ^ En).
iv.d(Gn) is given by the composition
bL(En ^ En) = bL(En ^ S0 ^ En) ! bL(En ^ En ^ En).
19
Proof. If S is finite, then S is a finite disjoint union of sets of the form Gn*
*=U for various open
subgroups U of Gn. We thus need only define XGn=U;QXS can be defined as an appr*
*opriate
finite product of the XGn=U's. Define XGn=U = Gn=UEn; i.e., XGn=U is a finite*
* product of
copies of En, one for each element of Gn=U. Equations 3.8 and 3.9 are certainly*
* satisfied. As
for the map d(Gn=U), observe that stable Enmodule maps XGn=U ! bL(XGn=U ^ En) *
*are in
bijective correspondence with En*module maps ß*XGn=U ! ß*bL(XGn=U^En). Further*
*more,
by the results of GoerssHopkins together with Lemma 3.7, any stable ring map X*
*Gn=U !
bL(XGn=U ^ En) lifts to an E map, unique up to E homotopy. Hence d(Gn=U) can be*
* chosen
to induce the requisite map on homotopy groups.
If f : S1 ! S2 is a Gnmap with S2 finite, then there exists a unique map Xf*
* : XS2 ! XS1
of Enmodule spectra inducing the map
f* : Map Gn(S2, Map c(Gn, En*)) ! Map Gn(S1, Map c(Gn, En*))
on stable homotopy groups. As before, Xf has a representative in E and is uniqu*
*e up to E
homotopy.
Finally, the naturality of ß*X and ß*d implies, by Lemma 3.6, that X is a fu*
*nctor and d
is a natural transformation. It also follows from 3.6 that X is an h1 diagram.
Construction 3.11 Let j : S0 ! En be the unit map, let ~ : En ^ En ! En be the
multiplication map, and, for each S 2 Ob R+Gn, let s : XS ^ En ! XS be the modu*
*le
structure map. Define an h1 diagram
C : (R+Gn)oppx ! hE
by
C(S, [j]) CjS= bL(XS ^ E(j)n).
The coface and codegeneracy maps are given by
d0(S) = bL(d(S) ^ E(j)n) : CjS! Cj+1S
di(S) = bL(XS ^ E(i1)n^ j ^ (En)(ji+1)) : CjS! Cj+1S, 1 i j*
* + 1
s0(S) = bL(s ^ E(j)n) : Cj+1S! CjS
si(S) = bL(XS ^ E(i1)n^ ~ ^ E(j1)n) : Cj+1S! CjS, 1 i j.
That C is in fact a functor follows from Proposition 3.6; this proposition alon*
*g with the
GoerssHopkins results also implies that C is an h1 diagram. Hence by Theorem*
* 2.2, we
may lift C to a diagram in E. From now on, when we refer to C, we actually mean*
* this lift
of C.
DefinitionQ3.12 The functor F :Q (R+Gn)opp ! E of Theorem 1 is defined by F(*
*S) =
Tot ( *CS), where, as before, *CS is the cosimplicial replacement of the (co*
*simplicial)
diagram CS. Since each CjSis K(n)*local, so is F(S).
20
By 2.8, there is a spectral sequence
(3.13) lims[Z, CS]t=) [Z, F(S)]t+s

for any CWspectrum Z. We next claim that
(3.14) lims[Z, CS]t= ßs[Z, C*S]t,

where the last term denotes the sth cohomology group of the cosimplicial abelia*
*n group
[Z, CS]t. This result is of course wellknown; however we provide a quick proo*
*f for the
convenience of the reader.
Proposition 3.15 Let Ab be the category of cosimplicial abelian groups. The*
*n the ffi
functors lim* : Ab ! gr+Ab and ß* : Ab ! gr+Ab are naturally equivalent, wher*
*e gr+Ab

denotes the category of nonnegatively graded abelian groups.
Proof. Since lim0 and ß0 are naturally equivalent, it suffices to prove that ßs*
* is effaceable.

We begin this proof by making a few recollections. The forgetful functor U : Ab*
* ! gr+Ab
has a right adjoint V _if D is an object of gr+Ab, set
Y
(V D)[n] = Dm ,
[n]![m]
where the product is taken over all morphisms [n] ! [m] in . It thus follows *
*that if Dm
is injective for each m, then V D is injective in Ab . Now let C be a cosimpl*
*icial abelian
group, and consider the monomorphism C !QV UC. We will show that ßi(V UC) = 0 f*
*or all
i > 0. Indeed, first observe that V D = V (D(n)), where
n
æ
Dn m = n
D(n)m = .
0 otherwise
It then suffices to show that ßi(V (UC(n))) = 0 for all n and all i > 0. But V *
*(UC(n)) is just
the simplicial cochain complex of the standard nsimplex with coefficients in C*
*n. Therefore
its cohomology is trivial in positive degrees.
In general, if J is a small category and A is a Jdiagram of abelian groups,*
* then lim*A
*
* Q J
may be computed as the cohomology of the cochain complex of the cosimplicial gr*
*oup *A,Q
the cosimplicial replacement of A (see [4; XI, 6.2]). Thus the ffifunctors ß*C*
* and ß*( *C)
are naturally equivalent on the category of cosimplicial abelian groups. The ne*
*xt result is
proved in Appendix II; it will be needed to prove Proposition AI.5.
21
PropositionQ3.16* Let C be a cosimplicial abelian group. There*is a natural ma*
*p T (C) :
C ! C of cochain complexes which induces an isomorphism on ß . Moreover, T c*
*an be
chosen so that the composition
T(C)Q j i j
C0 ! C ! C 0
j
sends x to d0. .d.0x for any x 2 C0, j0 0.
We can use spectral sequence 3.13 to verify Theorem 1ii. This will follow ea*
*sily from the
next result.
Proposition 3.17 Consider the maps En ! bL(En ^ En) and F(Gn) ! bL(En ^ En) gi*
*ven
by the maps
En^''
En = En ^ S0 !bL(En ^ En)
and
Q * Q 0 0
F(Gn) = Tot CGn ! CGn ! CGn = bL(En ^ En)
respectively. Let Z be any CWspectrum. Then there is a unique bijection E*nZ !*
* F(Gn)*Z
such that the diagram
E*nZ ________//_F(Gn)*Z
;; 
;; 
;; 
;ÆÆ ""
bL(En ^ En)*Z
commutes.
Proof. The map En ! bL(En ^ En) is an augmentation of the cosimplicial object *
*C*Gnin
the stable category. Furthermore, C*Gnis chain contractible to En in the stable*
* category (cf.
proof of Lemma 4.4); hence ß*[Z, C*Gn]t= EtnZ concentrated in degree 0. The des*
*ired result
now follows from 3.13, 3.14, and an unraveling of the identifications.
Proof of Theorem 1ii. The preceding proposition provides us with a canonical w*
*eak
equivalence En ! F(Gn). It also implies that the map is Gnequivariant and is *
*a map of
ring spectra (in the stable category). By Lemma 3.7 and Theorem 3.1 this weak e*
*quivalence
lifts to an E map.
Finally, we show that, for S = Gn=U, the spectral sequence 3.13 has the form*
* of a
homotopy fixed point spectral sequence. The proof requires some preparation.
22
Definition 3.18 Let M be an inverse limit of*discrete Gnmodules, and let H be *
*a closed
subgroup of Gn. Define a cochain complex DH (M) by
DjH(M) = (Map c(Gj+1n, M))H
with differential ffi : DjH(M) ! Dj+1H(M) given by
X j
ffif(g0, g1, . .,.gj+1)= (1)if(g0, . .,.bgi+1, . .,.gj+1)
i=0
+(1)j+1g1j+1f(g0g1j+1, . .,.gjg1j+1).
Warning 3.19 The action of H on Map c(Gj+1n, M) is as elsewhere in this paper*
*; that is, if
f 2 Map c(Gj+1n, M) and h 2 H, then (hf)(g0, . .,.gj+1) = f(h1g0, g1, . .,.gj+*
*1).
Lemma 3.20 The ffifunctor H*(D*H(?)) is equivalent to H*c(H, ?) on the abel*
*ian category
of discrete Gnmodules.
Proof. Since H0(D*H(M)) = MH , it suffices to prove that H*D*H(?) is effaceable*
*. To this
end, let N be a discrete Gnmodule, and consider Map c(Gn, N). Map c(Gn, N) is *
*a discrete
Gnmodule, and there is a Gnequivariant monomorphism N ! Map c(Gn, N) defined *
*by
n 7! hn, where hn(g) = g1n. Now
DjH(Map c(Gn, N)) = (Map c(Gj+1nx Gn, N))H ,
and there is a contracting homotopy
q : D*+1H(Map c(Gn, N)) ! D*H(Map c(Gn, N))
given by
(qf)(g0, . .,.gj, t) = (1)j+1f(g0t, . .,.gjt, t, 1),
proving that Hi(D*H(Map c(Gn, N))) = 0 for all i > 0.
We will identify the cochain complex [Z, C*Gn=U]t with D*U(EtnZ). We must th*
*us under
stand the cohomology of D*H(?) for profinite modules.
Lemma 3.21 Let H be a closed subgroup of Gn and let M be a profinite discret*
*e Gnmodule,
say M = limMff, where ff ranges over a directed set I. Then
ff
æ j
DH (M) s = 0
i.lim sDjH(Mff) =
ff 0 s > 0
ii.H*D*H(M) = limH*D*H(Mff) = limH*c(H, Mff).
ff ff
23
Proof. We have
Q * j
limsDjH(Mff) = ßs DH (M),
ff
Q * j
where M is the Idiagram ff 7! Mffand DH (M) is the cosimplicial replacement*
* of the
Idiagram DjH(M). Now
Q * j j Q *
DH (M) = DH ( M),
Q q
and since each Mffis finite, M is profinite. Moreover,
DjH: profiniteGnmodules ! Ab
is exact; this again follows from the existence of a continuous (settheoretic)*
* crosssection of
an epimorphism of profinite groups ([29; I, Theorem 3]). Therefore
Q *
limsDjH(Mff) = DjH(ßs M)
ff æ
DjH(M) s = 0
= DjH(limsMff) =
ff 0 s > 0
by the vanishing of lims, s > 0, for directed systems of profinite groups.
 Q
As for the second part, consider the double cochain complex *D*H(M). This *
*yields two
spectral sequences
Q * *
lim sHtc(H, Mff)=) Hs+t( DH (M))
ff Q
Hs(limtD*H(Mff) )= Hs+t( *D*H(M))
ff
By i, the second spectral sequence collapses to give
Q * * * *
H*( DH (M)) = H (DH (M)).
On the other hand, H is a closed subgroup of a panalytic profinite group and i*
*s therefore
itself a panalytic group (see [9; 10.7]). Hence H contains an open normal subg*
*roup U which
is a Poincar'e propgroup ([19]; see also [28] for a summary). This implies th*
*at Htc(H, Mff)
is finite for each ff and so limsHtc(H, Mff) = 0 for all s > 0. The first spect*
*ral sequence then
 ff
collapses to give
Q * * *
H*( DH (M)) = limHc(H, Mff).
ff
This completes the proof.
Lemma 3.22 There is a canonical isomorphism [Z, C*Gn=U]t D*UEtnZ of cochai*
*n com
plexes, where E*nZ is topologized as in Remark 0.3.
24
* ! C* and hence a map
Proof. The quotient map Gn ! Gn=U induces a map CGn=U Gn
(3.23) [Z, C*Gn=U]t! [Z, C*Gn]t
of cochain complexes. Now XGn = bL(En ^ En) (Proposition 3.10); therefore, the *
*map øj of
(1.6) provides an isomorphism
[Z, CjGn]t! Map c(Gj+1n, EtnZ) = Dj{e}(EtnZ).
Moreover, this map is easily seen to be a cochain map and is Gnequivariant, if*
* the action
of g 2 Gn on CjGnis given by
bL(X(rg) ^ En ^ . .^.En) = bL(g ^ En ^ En . .^.En).
Since right multiplication by g 2 U is trivial on Gn=U, the map in 3.23 is actu*
*ally a map
(3.24) [Z, C*Gn=U]t! ([Z, C*Gn]t)U (D*{e}(EtnZ))U = D*U(EtnZ)
of cochain complexes. Both [?, CjGn=U]* and DjU(E*n(?)) = Map c(Gn=U x Gjn, E**
*n(?)) are
cohomology theories satisfying the product axiom; it thus suffices to prove tha*
*t the map in
(3.24) is an isomorphism when Z = S0. But, by (3.8) and the fact that XGn=U is*
* a finite
product of En's (Prop. 3.10), we have that
ß*CjGn=U = ß*XGn=U En* Map c(Gjn, En*)
= Map c(Gn=U, En*) En* Map c(Gjn, En*).
We also have
ß*CjGn = ß*XGn bEn*Map c(Gjn, En*)
= Map c(Gn, En*)b En*Map c(Gjn, En*)
and the map ß*XGn=U ! ß*XGn corresponds to the homomorphism Map c(Gn=U, En*) !
Map c(Gn, En*) induced by the quotient map Gn ! Gn=U. From this it is clear that
ß*CjGn=U! (ß*CjGn)U,
completing the proof.
Finally, we obtain our desired result.
Proposition 3.25 Let S = Gn=U. The spectral sequence 3.13 has E2term canonic*
*ally
isomorphic to Hsc(U, EtnZ).
This is the spectral sequence of Theorem 1iv. It is strongly convergent beca*
*use Hsc(U, EtnZ)
is a profinite group for each s, t, and therefore lim1Es,tr= 0.
 r
25
hU
4 The Morava module of En
In this section we complete the proof of Theorem 1. The key step is the identi*
*fication
of bL(F(S) ^ En) with XS_this not only immediately implies Theorem 1iii but ena*
*bles
us to identify the spectral sequence 3.13 with theQK(n)*local EnAdams spectra*
*l sequence
converging to [Z, F(S)]*. If S is finite, CS = iCGn=Uifor a finite number of *
*open subgroups
Ui; thus we may assume from the beginning that S = Gn=U. (If S = Gn, Theorem 1i*
*ii is a
consequence of 1ii.)
The techniques involved in our computation of bL(F(S) ^ En) will be applied *
*in other
contexts in xx5, 6; we therefore proceed in a little more generality.
Notation 4.1 Write C : J ! E for any of the following diagrams:
i.J = and C = CGn=U for U an open subgroup of Gn.
ii.J = G for G a finite subgroup of Gn, and C : G ! E is the diagram given b*
*y the
action of G on En.
iii.J = F , where F and K are as in Theorem 4 of the Introduction, and C : F *
*! E is the
diagram given by the action of F on (the left factor of) bL(EhKn^ E(j+1)n)*
*, j 0.
Given a diagram C : J ! E, there is a diagram bL(C ^ En) : J ! E defined by
bL(C ^ En)(j)= bL(C(j) ^ En)
Lb(C ^ En)(f) = bL(C(f) ^ En)
for j an object and f a morphism in J. There is also a canonical map
(4.2) bL[(Tot Q *C) ^ En] ! Tot(Q *bL(C ^ En)).
We will prove the following result.
Theorem 4.3 If C is one of the diagrams in Notation 4.1, then the map 4.2 is *
*a weak
equivalence.
Before proving this theorem, we determine its consequence for C = CGn=U. The*
* left side of
4.2 is bL(EhUn^ En). To identify the right side, we examine the spectralQsequen*
*ce IIE*,*r(Z, C)
obtained by mapping a CWspectrum Z into the tower of fibrations {Tot k( *Lb(C*
* ^ En))}.
Lemma 4.4 Let C = CGn=U. Then
æ
IIEs,t s b * t 0 s > 0
2 (Z, C) = ß [Z, L(C ^ En)] = [Z, XG t .
n=U] s = 0
26
In particular, the map
Q * Q 0
Tot( bL(C ^ En)) ! bL(C ^ En)
! bL(C0 ^ En)
' bL(XGn=U ^ En)
s! X
Gn=U
is a weak equivalence.
Proof. By Proposition 3.15, IIEs,t2(Z, C) = ßs[Z, bL(C* ^ En)]t. Now let X*Gn*
*=Ube the
constant cosimplicial spectrum with XjGn=U= XGn=U. There are cosimplicial_in th*
*e stable
category_maps X*Gn=U! bL(C ^ En) and bL(C ^ En) ! X*Gn=Ugiven on 0simplices by*
* d :
XGn=U ! bL(XGn=U^En) (see Proposition 3.10) and s : bL(XGn=U^En) ! XGn=U respec*
*tively.
Furthermore, these maps are chain homotopy equivalences; use the chain homotopy
h : bL(Cj ^ En) ! bL(Cj1 ^ En)
defined by h = (1)jXGn=U ^ (En)(j1)^ ~. This then implies that ß*[Z, bL(C* ^*
* En)]t is
Q *
as claimed. The weak homotopy equivalence Tot ( Lb(C ^ En)) ! XGn=U is obtain*
*ed by
tracking down the identifications.
Corollary 4.5 There is a natural weak equivalence bL(EhUn^ En) !XGn=U of comm*
*utative
S0algebras and right Enmodule spectra such that the diagram
bL(EhUn^ En)___________________//XGn=U

 
 
 
bL(EhUn^ S0 ^ En) d

 
 
fflffl bL( ^E ) fflffl
Lb(EhUn^ En ^ En)_________n___//bL(XGn=U ^ En)
is homotopy commutative. In particular, Theorem 1iii holds.
We now turn to the proof of Theorem 4.3. Let IE*,*r(Z, C) denote the spectra*
*l sequence
obtained by mapping a CWspectrum Z into the tower
Q * Q *
. .!.bL(Tot k+1( C) ^ En) ! bL(Tot k( C) ^ En) ! . .;.
we have
IEs,t b t+s
(4.6) 1 (Z, C) = [Z, L(Fs(C) ^ En)] ,
27
Q * Q *
where Fs(C) is the fiber of Tots( C) ! Tots1( C). ThereQis*a canonical sta*
*bleQmap*
from the (unraveled exact couple of the) tower {bL(Tot k( C) ^ En)} to {Tot k*
*( Lb(C ^
En))}; on fibers this map is the canonical map
(4.7) bL(Fs(C) ^ En) ! Fs(bL(C ^ En)).
Lemma 4.8 Let IE*,*2(S0, C) ! IIE*,*2(S0, C) be the map of spectral sequence*
*s described
above. If C is one of the diagrams in 4.1, then this map is an isomorphism.
Proof. If C is a diagram of the form 4.1ii or 4.1iii, then the map 4.7 is an eq*
*uivalence and
hence the desired result follows immediately.
Now let C = CGn=U and examine IE*,*2(S0, C). By 3.14, H*([Z, F*(C)]t+*) = ß**
*[Z, C*]t
for any CWspectrum Z, and therefore Lemma 4.4 implies that
Hsßt*(F*(C) ^ En ^ M(pi0, . .,.vin1n1))=ßsßt(C* ^ En ^ M(pi0, . .,.vin1n*
*1))
æ i
0, . .,.vin1))s = 0
= ßt(XGn=U ^ M(p n1
0 s 6= *
*0.
In particular, these cohomology groups are all finite. Here M(pi0, vi11, . .,.v*
*in1n1) is as in the
beginning of x3; the multiindex I = (i0, . .,.in1) varies over a cofinal sequ*
*ence as in [6; x4],
so that
LbY = holim(Y ^ M(pi0, . .,.vin1))
I n1
for any E(n)*local spectrum Y .
We claim that
ß*holim(Fk(C) ^ En ^ M(pi0, . .,.vin1n1)) = limß*(Fk(C) ^ En ^ M(pi0, . .*
*,.vin1n1));
 I I
that is
(4.9) lim 1ß*(Fk(C) ^ En ^ M(pi0, . .,.vin1n1)) = 0.
I
Assuming this, it follows from the vanishing of lim1H*ßt*(F*(C) ^ En ^ M(pi0, *
*. .,.vin1n1))
 I
that
Hsßt*bL(F*(C) ^ En) = limHsßt*(F*(C) ^ En ^ M(pi0, . .,.vin1n1))
I
æ
ßtXGn=U s = 0
=
0 s 6= 0.
Furthermore, one checks easily that the map IEs,t2(S0, C) ! IIEs,t2(S0, C) is t*
*he identity
under this identification and the identification of Lemma 4.4.
28
j 's and is*
* therefore
We now verify the claim. Fk(C) is equivalent to a product of CGn=U
Landweber exact. Hence ß*(Fk(C) ^ En) is a flat En*module, so
ß*(Fk(C) ^ En ^ M(pi0, . .,.vin1n1)) = ß*(Fk(C) ^ En)=(pi0, . .,.vi*
*n1n1).
In particular, the inverse system {ß*(Fk(C) ^ En ^ M(pi0, . .,.vin1n1))} is M*
*ittagLeffler and
therefore 4.9 holds. This completes the proof.
Corollary 4.10 If C is one of the diagrams in Notation 4.1, then
Q * ' Q *
holimbL(Tot k( C) ^ En) !Tot ( bL(C ^ En)).
k
Proof. If C is a diagram of the form 4.1ii or 4.1iii, theQresult follows from t*
*heQfact that the
map 4.7 is an equivalence and hence the towers {bL(Tot k( *C) ^ En)} and {Tot *
*k( *Lb(C ^
En))} are equivalent.
If C = CGn=U, the result follows from Lemma 4.8 together with the fact that *
*both spectral
sequences are strongly convergent in the sense of [4; IX, 5.4].
The proof of Theorem 4.3 will now be completed by showing that
bL[(Tot Q *C) ^ En] '!holimbL(Tot k(Q *C) ^ En).
 k
We separate off the following key ingredient.
Let
. .!.Yk ! Yk1 ! . .!.Y0 ! *
be a tower of fibrations of S0modules, so that the canonical map limYk ! holim*
*Yk is a
 k  k
weak equivalence. Define Y kto be the fiber of limYi! Yk; there is then an inve*
*rse system
 i
of fibrations
Y k+1____//_limYi_//_Yk+1.
 i 
  
  
  
 
fflffl fflffl
Y k_____//_limYi__//_Yk
i
According to [4; XI, 5.5], the map
holim(limYi) ! holimYk
k i k
29
k. But the commutative diagram
is a fibration with fiber holimkY
holim(limYi)____//_holimYk
k i k
OO OO
' '
 
lim(limYi)__=___//_limYk
k i k
shows that this map is a weak equivalence, and thus holimY kis stably trivial. *
*In certain
 k
cases, we can say a good deal more.
Lemma 4.11 Let {Yk} be as above, and let E*,*r(Z) denote the spectral sequen*
*ce obtained
by mapping the CWspectrum Z into this tower. Suppose that there exist natural *
*numbers r0
and s0 such that Es,*r0(Z) = 0 for all spectra Z whenever s > s0. Then given k,*
* there exists
q such that the map Y k+q! Y kis stably trivial.
Proof. Let Fs be the fiber of Ys ! Ys1. There is a diagram
Y k?oo________Y=k+1Coo________Y=k+2oo_____.=.=.
??  CCCC 
???  CC 
?ØØ  C!! 
Fk+1 Fk+2
of exact triangles; let kE**r(Z) denote the spectral sequence obtained by mappi*
*ng Z into
this diagram. This spectral sequence is isomorphic to the spectral sequence ob*
*tained by
mapping Z into the tower {Yi}i k. Hence, kEs,*r0(Z) = 0 for s > max {r0  2, s*
*0  k  1}.
Since holimY i' *, kE**r(Z) is conditionally convergent (in the sense of [2]) t*
*o [Z, Y k]*, and
 i
thus the horizontal vanishing line implies that
_im([Z,_Y_k+s]*_!_[Z,_Y_k]*)_s,*
= kE1 (Z)
im([Z, Y k+s+1]* ! [Z, Y k]*)
and that {im([Z, Y k+s]* ! [Z, Y k]*)}s 0 is a complete Hausdorff filtration of*
* [Z, Y k]*. It
then follows that im([Z, Y k+s]* ! [Z, Y k]*) = 0 for s > max {r0  2, s0  k *
* 1}. But Z is
arbitrary; therefore Y k+s! Y kis trivial for these values of s, completing the*
* proof.
Lemma 4.12 Let {Yk} satisfy the hypotheses of Lemma 4.11. Then, if W and F a*
*re any
spectra, there is an equivalence
LF[(holim Yk) ^ W ] '!holimLF(Yk ^ W ).
k  k
30
Remark 4.13 The above map is of course chosen so that composition with the p*
*rojection
onto LF(Yk ^ W ) yields the canonical map
LF[(holim Yk) ^ W ] ! LF(Yk ^ W ).
k
We will show that lim1[Z, LF(Yk ^ W )]* = 0 for any spectrum Z, so the equival*
*ence of the
 k
lemma is uniquely determined.
Proof. We have diagrams (in the stable category)
(4.14) LF(Y k+1^ W )___//LF((holimYi) ^ W_)//_LF(Yk+1 ^ W )
 i 
 
  
  
  
fflffl fflffl
LF(Y k^ W ) ___//_LF((holimYi) ^ W_)//_LF(Yk ^ W )
i
of fiber sequences. By the previous lemma,
holimLF(Y k^ W ) ' *;
k
from this should follow the desired result. However, we prefer to avoid tryin*
*g to argue
that "the homotopy inverse limit of the fibers is the fiber of the homotopy li*
*mits", and
instead proceed less generally. Indeed, a diagram chase using 4.14 together wi*
*th the previous
lemma shows that the system {[Z, LF(Yk ^ W )]*} is MittagLeffler for any Z an*
*d therefore
lim1[Z, LF(Yk ^ W )]* = 0. A similar argument also shows that
 k
[Z, LF((holim Yi) ^ W )]*! lim[Z, LF(Yk ^ W )]*.
 i k
This completes the proof.
Proof of Theorem 4.3. Start with cases i and ii of Notation 4.1. By virtue of *
*the preceding
work, we need only show that
(4.15) bL[(Tot Q *C) ^ En] ^ X '!holimbL(Tot k(Q *C) ^ En) ^ X
 k
for some plocal finite spectrum X Bousfield equivalent to S0(p). Nilpotence *
*technology [8;
4.1] tells us that this is the same as requiring X to have torsion free Z(p)h*
*omology.
We will prove 4.15 by finding a torsion free X such that E*,*2(Z ^ DX, C) h*
*as a horizon
tal vanishing line independent of Z, whereQE*,*r(Z ^ DX, C) denotes the spectr*
*al sequence
obtained by mapping Z ^ DX into {Tot k *C}. But
Es,t2(Z ^ DX, C) = Hsc(K, Etn(Z ^ DX))
31
for K some closed subgroup of Gn. (K = U in case i, and K = G in case ii.) More*
*over,
Hsc(K, Etn(Z ^ DX)) = limHs,tc(K, En*X En* En*DZff),
ff
where Z = holimZffis a presentation of Z as a direct limit of finite CWspectra.
 ff
Now Hopkins and Ravenel have shown that there exists a finite spectrum X wit*
*h free Z(p)
homology_in fact, X can be taken to be S0 if p  1  n and a summand of an iter*
*ated smash
product of a finite complex projective space if p1  n_such that Hs,*c(K, En*X*
*=InEn*X) =
0 for s bigger than some s0. (This is proved for K = Gn in [26; 8.3.57]; from*
* this fol
lows the result for K closed in Gn. Or, one can observe that the proof for Gn *
*applies to
closed subgroups as well.) An easy induction then shows that Hs,*c(K, En*X) an*
*d hence
Hs,*c(K, En*X=Im En*X) vanish for s > s0 and m n, where Im = (p, v1, . .,.vm*
*1 ). Finally,
proceed by induction on a Landweber filtration of En*DZffto prove that Hs,*c(K,*
* En*X En*
En*DZff) = 0 for s > s0. (Note that the crosssection theorem [29; I, Theorem 3*
*] allows us
to conclude that H*c(K, ?) takes short exact sequences of profinite Kmodules t*
*o long exact
sequences.) Q
This vanishing line allows us to apply Lemma 4.12 to the tower {(Tot k *C) *
*^ X} to
complete the proof of 4.15.
As for case iii, we prove that E*,*2(Z, C) has a horizontal vanishing line i*
*ndependent of
Z. Indeed,
Es,t2(Z, C)= Hs(F, [Z, bL(EhKn^ E(j+1)n]t)
= Hs(F, Map c(Gj+1n, EtnZ)K )
by Proposition 5.3 and 5.5.
Now if M is any discrete Gnmodule, there is a spectral sequence
H*(F, H*c(K, Map c(Gj+1n, M))))=H*c(G, Map c(Gj+1n, M)).
But Map c(Gj+1n, M) = Map c(Gn, Map c(Gjn, M)) is both K and Gacyclic (see pro*
*of of Lemma
3.20); this implies that Map c(Gj+1n, M)K is F acyclic.
If M is profinite, say M = limMff, then there is a spectral sequence
ff
limiHsi(F, Map c(Gj+1n, Mff)K ))=Hs(F, Map c(Gj+1n, M)K ).
ff
But
H*(F, Map c(Gj+1n, Mff)K=) (Map c(Gj+1n, Mff)K )F
= Map c(Gj+1n, Mff)G
concentrated in degree 0, and by Lemma 3.21i, limiMap c(Gj+1n, Mff)G = 0 for i *
*> 0. Thus
ff
Es,t2(Z, C) = 0 for s > 0, and the proof concludes as before.
We conclude this section by proving Theorem 1iv.
32
Proposition 4.16 Let U be an open subgroup of Gn, and let S = Gn=U. The spect*
*ral
sequence 3.13 is naturally isomorphic to the K(n)*local EnAdams spectral sequ*
*ence con
verging to [Z, EhUn]*.
Proof. Consider the cosimplicial S0module CGn=U. By Corollary 4.5,
CjGn=U' bL(EhUn^ E(j+1)n)
and thus by Remark AI.9
* ! EhUn! C0Gn=Uffi! 1C1Gn=Uffi! 2C2Gn=U! . . .
P
is a K(n)*local Enresolution of EhUn, where ffi = (1)idi. The desired resu*
*lt now follows
from Proposition AI.5.
5 Homotopy fixed point spectra for closed subgroups
of Gn
We begin by recalling the construction of the homotopy direct limit in E for th*
*e case of a
direct sequence of commutative S0algebras.
Definition 5.1 Let
f0 f1 f2 fi1 fi
A0! A1! A2! . ..!Ai! . . .
__
be a direct sequence of commutative S0algebras. Then holimEAi= limEAj, where
!i !j
__ ` ` ` `
A j= A0 I f0A1 I f1. . .fj2Aj1 I fj1Aj
is a jfold mapping cylinder in E. That is, all limits (including tensor produc*
*ts) are to be
taken in E.
The next result is crucial in the homotopical analysis of EhGn.
f0 f1
Lemma 5.2 Let A0! A1! A2! . . .be a sequence of cellular algebra maps bet*
*ween cell
commutative S0algebras. Then there is a natural weak equivalence holimEAi' hol*
*imAi of
!i !i
spectra, where holim denotes the ordinary homotopy colimit of {Ai} regarded as *
*a sequence
!i
of spectra.
33
__ __
Proof. Let A jbe as in the previous definition. The evident map A_j!_Aj is a ho*
*motopy
equivalence in E and hence_in_the category of spectra. Moreover, A_jis a_relati*
*ve_cell com
mutative S0algebra under Aj1 for each j 1. This implies that Aj1 ! Aj is a*
* cofibration
of underlying spectra [11; VII, 4.14]. Thus, by [11; VII, 3.10], it follows that
__ __ __
holim EAi= limEA j= limA j' holimA j' holimAj.
!i !j !j !j !j
We can now identify the K(n)*local Enhomology of EhGnand thus prove Theore*
*m 2i.
Proposition 5.3 ß*bL(EhGn^ En) = Map c(Gn, En*)G as completed right ß*bL(En *
*^ En)
comodule algebras.
Proof. Just compute:
ß*(EhGn^ En ^ M(pi0, . .,.vin1n1))=limß*(Eh(UjG)n^ En ^ M(pi0, . .,.vin*
*1n1))
!j
= limMap c(Gn, En*=(pi0, . .,.vin1n1))*
*UjG
!j
= Map c(Gn, En*=(pi0, . .,.vin1n1))G,
since En*=(pi0, . .,.vin1n1) is discrete. The desired result follows easily.
It is now also a simple matter to identify EhGnwith the usual homotopy fixed*
* point
*
* 0G
spectrum when G is finite. As in the Introduction, we denote this spectrum by E*
*hn .
0G
Proposition 5.4 Let G be a finite subgroup of Gn. The map EhGn! Ehn described*
* in the
Introduction is a weak equivalence.
0G
Proof. Since EhGnand Ehn are both K(n)*local, it suffices to prove that the m*
*ap
0G
ß*bL(EhGn^ En) ! ß*bL(Ehn ^ En)
is an isomorphism. But we have a commutative diagram
0G
EhGn_______//_9Ehn,
99
999
9øø
En
and by the preceding proposition, ß*bL(EhGn^ En) injects into ß*bL(En ^ En) wit*
*h image
0G '
Map c(Gn, En*)G. On the other hand, Theorem 4.3 implies that bL(Ehn ^ En) ![*
*bL(En ^
0G
En)]h , where G acts on the left factor En. But ß*bL(En ^ En) = Map c(Gn, En**
*) is G
0G G
acyclic (see proof of Lemma 3.20); therefore ß*bL(Ehn ^En) = Map c(Gn, En*) as*
* well. This
completes the proof.
34
hGNow let G be a closed subgroup of Gn and form a K(n)*local EnAdams resolut*
*ion of
En as in Remark AI.9; that is, set
CjGn=G= bL(EhGn^ (En)(j+1)).
As in the proof of Lemma 3.22, there is a natural transformation
(5.5) [Z, C*Gn=G]t! D*GEtnZ
of cochain complexes; the proof of Proposition 5.3 generalizes to show that thi*
*s map is
an equivalence when Z = S0. But both [?, CjGn=G]* and DjGE*n(?) are cohomology *
*theories
satisfying the product axiom and hence the map in 5.5 is an isomorphism for all*
* Z. Theorem
2ii is thus a consequence of Lemmas 3.20 and 3.21.
Proposition 5.6 Let G be a closed subgroup of Gn. The K(n)*local EnAdams sp*
*ectral
sequence converging to [Z, EhGn]* is strongly convergent and has E2term natura*
*lly isomorphic
to H*c(G, E*nZ).
Proof. The only part which hasn't been proved above is the strong convergence. *
*But this
follows from Proposition AI.3.
We can also prove a öc variant" version of Proposition 5.6, although the nex*
*t result is
probably not the most general result which can be achieved in this direction. R*
*ecall that E(n)
denotes the Landweber exact spectrum with coefficient ring E(n)* = Z(p)[v1, . .*
*,.vn, v1n].
Proposition 5.7 Let X be a CWspectrum such that, for each E(n)module spectru*
*m M,
there exists a k with IknM*X = 0. Then the K(n)*local EnAdams spectral seque*
*nce con
verging to ß*bL(X ^ EhGn) has E2term naturally isomorphic to H*c(G, En*X).
Remark 5.8 The hypotheses imply that En*X is a discrete Gnmodule, so H*c(G, *
*En*X)
makes good sense.
The proof of this proposition requires a little preparation. Let Ln denote *
*the E(n)*
localization functor. There is a cofiber sequence
(5.9) nMnS0 ! LnS0 ! Ln1S0
(see [25; x5]) and
(5.10) MnS0 = Ln(holim nIM(pi0, vi11, . .,.vin1n1),
!I
n1X
nI = 2ir(p  1),
r=0
where I = (i0, i1, . .,.in1) ranges over a cofinal sequence of multiindices [*
*6; x4].
35
Lemma 5.11 Let X satisfy the hypotheses of Proposition 5.7, and let M be an *
*E(n)module
spectrum. Then
i.M ^ X is K(n)*local
ii.M ^ X ^ nMnS0 '!M ^ X ^ LnS0 ' M ^ X.
Proof. i. M is an E(n)module spectrum and is therefore E(n)*local. Since E*
*(n) is
smashing ([26; 7.5.6]), M ^ X is E(n)*local. Hence
bL(M ^ X) = F ( nMnS0, M ^ X),
and we must show that
M ^ X = F (LnS0, M ^ X) ! F ( nMnS0, M ^ X)
is an equivalence; i.e.
(M ^ X)*LnS0 '!(M ^ X)*( nMnS0).
But nMnS0 ! LnS0 is the composite of the maps Ln@i, 0 i n  1, in the cof*
*ibration
sequences
@i 0
NiS0 ! LiNiS0 MiS0 ! Ni+1S0 ! NiS ,
where N0S0 = S0; thus we need only show that (M ^ X)*MiS0 = 0 for 0 i n  1.
Consider the universal coefficient spectral sequence converging to (M ^ X)*MiS0*
* whose
E2term is
ExtE(n)*(v1iE(n)*=(p1 , . .,.v1i1), M*X) = ExtE(n)*(E(n)*MiS0, M*X).
Since there exists k with vkiM*X = 0, it follows that
ExtE(n)*(v1iE(n)*=(p1 , . .,.v1i1), M*X) = 0,
completing the proof.
ii. This follows from 5.9 and the fact that, since E(n  1) is smashing, we *
*have
(Ln1S0) ^ M ^ X ' Ln1(M ^ X) ' *.
Proof of Proposition 5.7. By Lemma 5.11, we have
bL(X ^ EhGn^ E(j+1)n)' X ^ EhGn^ E(j+1)n
' X ^ nMnS0 ^ EhGn^ E(j+1)n
0 i in1 hG (j+1)
' holim Xff^ nIM(p 0, . .,.vn1) ^ En ^ En
!
0 i in1 hG (j+1)
' holim bL(Xff^ nIM(p 0, . .,.vn1) ^ En ^ En ),
!
36
0 = n + n, and the homotopy colimit varies over the finite CWsubspectra*
* X of X
where nI I *
* ff
and the sequence of generalized V (n  1)'s of 5.10. But, by Proposition 5.6,
0 i in1 hG *
* (*+1)
H*ßtbL(X ^ EhGn^ (En)(*+1)) = limH*ßtbL(Xff^ nIM(p 0, . .,.vn1) ^ En ^ (En*
*) )
!
0 i in1
= limH*c(G, (En)t(Xff^ nIM(p 0, . .,.vn1)))
!
= H*c(G, (En)t(X ^ nMnS0))
= H*c(G, (En)tX).
This completes the proof.
Finally, if G is a finite subgroup of Gn, we identify the homotopy fixed poi*
*nt spectral
0G *
sequence converging to [Z, Ehn ] with the K(n)*local EnAdams spectral sequen*
*ce. Let us
first introduce some notation. Q
If G acts on (the commutative S0algebra) X, write *GX for the cosimplicia*
*l replace
ment of theQGdiagram definedQby the action of G on X. Also write Fs(G, X) for *
*the fiber
of Tots( *GX) ! Tots1( *GX). The next result proves Theorem 3ii.
Proposition 5.12 The sequence
0G
* ! Ehn ! F0(G, En) ! F1(G, En) ! . . .
0G
is a K(n)*local EnAdams resolution of Ehn .
Proof. First observe that Fs(G, En) is a product of En's and is therefore Enin*
*jective. We
thus need only show that
0G
(5.13) 0 ! [Z, bL(Ehn ^ En)] ! [Z, bL(F0(G, En) ^ En)] ! . . .
is exact for all CWspectra Z. Since G is finite, the cochain complex bL(F*(G, *
*En) ^ En) is
equivalent to F*(G, bL(En ^ En)). Hence
Hi[Z, bL(F*(G, En) ^ En)]= Hi(G, [Z, bL(En ^ En)])
= Hi(G, Map c(Gn, E*nZ))
æ
Map c(Gn, E*nZ)G i = 0
=
0 i > 0
by the proof of Lemma 3.20. But by Proposition 5.3,
0G * * G
[Z, bL(Ehn ^ En)] = Map c(Gn, EnZ) .
Tracking down the identifications completes the proof that sequence 5.13 is exa*
*ct.
37
6 Proof of Theorem 4
Let G be a closed subgroup of Gn, K a closed normal subgroup of G, and suppose *
*F = G=K
is finite. Then the canonical map EhGn! EhKnfactors through (EhKn)F, the F fix*
*ed points
of EhKn. The next result proves Theorem 4.
Proposition 6.1 The composition EhGn! (EhKn)F ! (EhKn)hF is a weak equivalence.
Proof. Let F also denote the category with one object * whose automorphism grou*
*p is F ,
and consider the functor Y : x F ! E with
Y([j], *) = bL(EhKn^ E(j+1)n).
F acts on EhKn, and Y maps morphisms in as in Remark AI.9. Write Y j Y([j]*
*, *).
Since
EhKn'!holimY j

by Corollary AI.8, we have that
(EhKn)hF ~! holim holimY j
F 
holim(Y j)hF.

Of course, there is a canonical augmentation (EhKn)hF ! (Y 0)hF and hence an au*
*gmentation
EhGn! (EhKn)hF ! (Y 0)hF. We claim that
* ! EhGn! (Y 0)hF ! 1(Y 1)hF ! . . .
is a K(n)*local EnAdams resolution of EhGn. Assuming this, it follows from Co*
*rollary AI.8
that
EhGn~!holim(Y j)hF

and hence that
EhGn~!(EhKn)hF.
To prove the claim, we must show that
0 ! [Z, bL(EhGn^ En)] ! [Z, bL((Y 0)hF ^ En)] ! [Z, bL((Y 1)hF ^ En)] !*
* . . .
is exact for any CWspectrum Z. (Since (Y i)hF is an Enmodule spectrum it is K*
*(n)*local
Eninjective.) Begin by recalling Theorem 4.3 which asserts that
Lb((Y i)hF ^ En) ~![bL(Y i^ En)]hF
38
for all i 0. But
[Z, bL(Y i^ En)]* = Map c(Gi+2n, E*nZ)K
by 5.5, and since Map c(Gi+2n, E*nZ)K is F acyclic (see proof of Theorem 4.3),*
* it follows that
[Z, bL((Y i)hF ^ En)]*= Map c(Gi+2n, E*nZ)G
= Map c(Gi+1n, Map c(Gn, E*nZ))G
= DiG(Map c(Gn, E*nZ)).
This is in fact an isomorphism of cochain complexes, so
æ
Map c(Gn, EtnZ)G i = 0
Hi([Z, bL((Y *)hF ^ En)]t) =
0 i > 0.
Since
[Z, bL(EhGn^ En)]t= Map c(Gn, EtnZ)G,
the claim is proved.
7 Two Applications
We begin with the proof of Theorem 6. Let c : Gn ! Zp be a surjective continuou*
*s homo
morphism, and consider the exact sequence of groups
0 ! K ! Gn c!Zp ! 0.
Then Zp acts_at least regarded as a discrete group_by S0algebra maps on EhK . *
*We have
the following result.
Proposition 7.1 Let t be the topological generator 1 of Zp. Then there is a fi*
*ber sequence
(in the stable category)
bLS0 bL''!EhKnidt!EhKn@! bLS0
where j : S0 ! EhKndenotes the unit map.
Proof. Let X be the fiber of id  t. Since t is an S0algebra map, the unit j f*
*actors to give
a commutative diagram
X ____________//EhKn``?.
? ==
? 
''0? ''
S0
39
0 is a K(n) equivalence, and thus bLS0 ~!X. To prove this, it *
*suffices to
We claim that j *
show that
ß*bL(j0^ En) : En* ! ß*bL(X ^ En)
is an isomorphism. There is a commutative diagram
(idt)*
ß*bL(EhKn^ En)______//ß*bL(EhKn^ En)
idfit K
Map c(Gn, En*)K____//_Mapc(Gn, En*) ,
where øt(f)(g) = f(t1g) for f 2 Map c(Gn, En*)K and g 2 Gn. (Here t also den*
*otes any
element of Gn whose image under c is t 2 Zp.) Since
Map c(Gn, En*)K = Map c(Gn=K, En*) = Map c(Zp, En*),
a standard argument shows that id  øt is surjective and hence
ß*bL(X ^ En) = ker(id  øt).
But ker(id  øt) just consists of the constant maps from Zp to En*; this implie*
*s that ß*bL(j0^
En) is an isomorphism, completing the proof.
The next result implies Theorem 6.
*
* P
Proposition 7.2 Let @ be as in Proposition 7.1. Then the composition @ O j : S*
*0 ! LbS0
is detected by c 2 H1c(Sn, En*)Galin the K(n)*local EnAdams spectral sequenc*
*e.
Remark 7.3 Let G be a profinite group and M a discrete Gmodule, and consider*
* the short
exact sequence
0 ! M !iMap c(G, M) ! Map c(G, M)=M ! 0,
where i(m)(g) = g1m. Then the coboundary map provides an epimorphism
(Map c(G, M)=M)G = H0c(G, Map c(G, M)=M) ! H1c(G, M).
If M is a trivial Gmodule, this map is an isomorphism; moreover, (Map c(G, M)=*
*M)G is just
the group of continuous group homomorphisms from G to M. This gives a canonical*
* identi
fication of H1c(G, M) with this group of homomorphisms. In particular, the homo*
*morphism
c : Gn ! Zp defines an element of limH1c(Gn, Z=(pj)) = H1c(Gn, Zp) and hence an*
* element
j
of H1c(Gn, En*).
40
bL(@ O j), and consider the cofiber sequence
Proof of Proposition 7.2. Write f =
f 0 i p 0
(7.4) . .!. 1bLS0 ! bLS ! C(f) ! bLS ! . ...
By Proposition AI.10, f is detected by ffi(1) 2 H1c(Gn, En*), where ffi is the *
*coboundary map
Hjc(Gn, En*) ! Hj+1c(Gn, En*) for the short exact sequence
p* * i* * 0
0 ! E*nS0 ! EnC(f) ! EnS ! 0
of Gnmodules, and 1 2 H0(Gn, En*) is just the unit in (En*)Gn = Zp. Now the s*
*equence
7.4 is selfdual; that is, applying the function spectrum functor F (?, bLS0) y*
*ields the same
sequence. Hence f is detected by ffi0(1) 2 H1c(Gn, En*), where ffi0 denotes th*
*e coboundary
map for the short exact sequence
p* ^ 0
0 ! E^n*S0 i*!E^n*C(f) ! En*S ! 0.
(By E^n*X, we here mean ß*bL(X ^ En).)
In addition, we have a diagram
p
. ._.___//bLS0i_//_C(f)___//_ØbLS0
 Ø 
 Ø bL''
 fflfflidtfflffl
. ._.___//bLS0___//EhKn___//_EhKn
of cofibration sequences; this yields the commutative diagram
(7.5) 0 ____//_E^n*S0_____//_E^n*C(f)__________//_E^n*S0_______//0
  
  ''*
 fflffl idfit fflffl
0 ____//_E^n*S0__//_Mapc(Gn, En*)K___//_Mapc(Gn, En*)K___//0
 
 
 
Map c(Zp, En*) Map c(Zp, En*)
This diagram is a diagram of Gnmodules; the action of Gn on Map c(Zp, En*) is *
*given by
(gh)(s) = g(h(s + c(g)))
for g 2 Gn, h 2 Map c(Zp, En*) and s 2 Zp. This follows by naturality from the *
*discussion
preceding Remark 1.4. There is also a commutative diagram
idfit
(7.6) 0 ____//_En*___//Mapc(Zp,OEn*)___//Mapc(Zp,OEn*)___//0OOOO
  
  
  idfit 
0 _____//Zp____//_Mapc(Zp, Zp)___//_Mapc(Zp, Zp)___//0
41
0(1) is the image of ffi00(1) 2 H1(G , Z ) in H1(G, E **
*), where ffi00is
of Gnmodules. Then ffi c n p c n
the coboundary map associated to the bottom exact sequence and 1 2 Map c(Zp, Zp*
*) is the
constant map with value 1. Finally, use the diagram
idfit
0_____//Zp___//_Mapc(Zp, Zp)____//_Mapc(Zp,ØZp)____//_0
  Ø
 Mapc(c,id) Ø
 fflffl fflffl
0_____//Zp___//_Mapc(Gn, Zp)__//_Mapc(Gn, Zp)=Zp___//_0
and Remark 7.3 to complete the proof.
We next turn to the proof of Theorem 5.
Lemma 7.7 Let G be as in the statement of Theorem 5; that is, G is the close*
*d subgroup
j hG spj
of Zxpgenerated by lsp . Then E1 is the fiber of id l : E1 ! E1.
Proof. Let F denote this fiber. Since the composition
spj
EhG1! E1idl!E1
is trivial, there is a commutative diagram
FO______//E1O==.
 
 
 
EhG1
To show that EhG1'!F , it suffices to show that
_____//_
(7.8) ß*bL(EhG1^ E1) ß*bL(F ^ E1) .
Map c(Zxp=G, E1*)
But there is a commutative diagram
(idlspj)*
ß*bL(E1 ^ E1)_____//ß*bL(E1 ^ E1)
Map c(Zxp, E1*)idfi//_Mapc(Zxp, E1*),
j x x
where here ø(f)(u) = f(lsp u) for f 2 Map c(Zp, E1*) and u 2 Zp. Clearly
ker(id ø) = Map c(Zxp=G, E1*);
42
the map in 7.8 is therefore an isomorphism providedxthat idø is surjective. Bu*
*t this follows
without difficulty from the facts that G Zp and Zp=G is finite.
The next result is well known.
Lemma 7.9 Let f : X ! Y be a map in the stable category such that ßif is an *
*isomorphism
for all i sufficiently large. Then LK(1)f : LK(1)X ! LK(1)Y is an equivalence.
Proof. Let M(pj) denote the mod (pj) Moore spectrum, with v1 selfmap . Then
f ^ 1M(pj) : X ^ 1M(pj) ! Y ^ 1M(pj)
is an equivalence, since ßi(f ^ M(pj)) is an isomorphism for all i sufficiently*
* large. But, if Z
is any spectrum,
L1Z ^ M(pj) = L1(Z ^ M(pj)) = Z ^ L1M(pj) = Z ^ 1M(pj)
by the telescope conjecture for n = 1 (see [26; 7.5.5]), and
LK(1)X = holimL1X ^ M(pj).
j
Hence LK(1)f is an equivalence as desired.
__
Proof of Theorem 5. Since K(k) is fixed by Gal(F l=k) = bZ=spjbZ, there is a co*
*mmutative
diagram
K(k) ____//_F__EhG1.
  fififi
  fifi
_fflffl_fflfflfifi
K(F l)____//E1
But Quillen showed [24; Theorem 7] that
BGL(k)+ = 10K(k) ! 10F
is an equivalence; hence
ßiK(k) !ßiEhG1
for all i 1. The result now follows from Lemma 7.9.
43
Appendix I. The K(n)*local En Adams spectral sequence
We begin with some generalities on the Adamstype spectral sequences that we wi*
*ll be
considering.
Let E be a commutative ring spectrum (in the stable category), and let F be *
*any spec
trum. Then one can construct the EAdams spectral sequence in the F*local cate*
*gory. In
more detail, we follow Miller [21] and define an injective class (see [18]) in *
*this category by
declaring an F*local spectrum X to be Einjective if it is a retract of LF(Y ^*
* E) for some
spectrum Y . A sequence X0 ! X ! X00is then Eexact if [X0, I] [X, I] [X00*
*, I] is
exact for every Einjective I. Given X, one may construct an Eexact sequence
(AI.1) * ! X ! I0 ! I1 ! . . .
such that Is is Einjective for all s. One may then construct a diagram
oio____ oo_i___ 2 oo_____
(AI.2) X = X0CC X18BB XB9B
CC 8888 999 . . .
jCC!!C k j øø8 k 999
0 1 øø
I I
of exact triangles; observe that the map
LF(j ^ E) : LF(Xi^ E) ! LF(Ii^ E)
is a split monomorphism. Conversely, a diagram of exact triangles as in AI.2 w*
*ith each
LF(j ^ E) split monic yields an Eexact sequence AI.1. Such a diagram (with ea*
*ch Ij
Einjective) is called an F*local EAdams resolution of X and is functorial up*
* to chain
homotopy.
By mapping a spectrum Z into an F*local EAdams resolution of X, we obtain *
*a spectral
sequence, called the F*local EAdams spectral sequence. The work of Bousfield *
*[3] comes
into play in dealing with the convergence question. We define the F*local En*
*ilpotent
spectra to be the smallest class C of (F*local) spectra such that
i.LFE 2 C
ii.LF(N ^ X) 2 C whenever N 2 C
iii.C is closed under retracts and cofibrations.
If X is F*local Enilpotent, then the proof of [3; Theorem 6.10] applies to sh*
*ow that the
F*local EAdams spectral sequence converges conditionally and strongly to [Z, *
*X]* for Z
any CWspectrum.
We now specialize to the case E = En and F = K(n). Here we have the followin*
*g result.
Proposition AI.3 If X is K(n)*local, then X is K(n)*local Ennilpotent.
44
0(n) with coefficient ring E0(n)*
* =
Proof. Consider the Landweberexact1spectrum0E *
Z(p)[u1, . .,.un1][u, u ]. Since E (n) is equivalent to a wedge of suspension*
*s of E(n) and
S0 is E(n)prenilpotent ([17; Theorem 5.3]), it follows that S0 is E0(n)prenil*
*potent. But
X is E0(n)*local; therefore X is E0(n)nilpotent. Since bLE0(n) is a retract o*
*f En, X is also
K(n)*local Ennilpotent.
We now examine some further properties of the K(n)*local EnAdams spectral *
*sequence
which have been used in the text.
Let X be K(n)*local, and let C be a cosimplicial S0module with an augmenta*
*tion
X ! C such that
* ! X ! C0 ffi! 1C1 ffi! 2C2 ! . . .
*
* P
is a K(n)*local Enresolution of X. (The suspensions appear so that each map f*
*fi = (1)idi
has degree 1.) Consider also the diagram
(AI.4) holimC = Tot 1oo_____ Tot0 oo_____Tot1 oo___. . .
 >>> AA;; AA
;;
>>> ;;;
OEOE> ÆÆ
F0 F1
Q * Q *
of exactQtriangles, whereQTot iis the fiber of Tot C ! Toti C and Fi is t*
*he fiber of
Tot i( *C) ! Toti1( *C) as in x4. Since Fi is the product of various Cj's, i*
*t is K(n)*
local Eninjective. Therefore, the canonical map h : X ! holimC extends to a di*
*agram

*_______/X/ _____/C0/__ffi/1C1/_ffi2C//__
   
h g0 h1 h2
fflffl fflffl fflffl fflffl
*____/holim/C____/F0/______F1//_____/F2/______././.

of augmented cochain complexes in the stable category, unique up to chain homot*
*opy. This
diagram is induced by a map of exact triangles and hence defines a map of spect*
*ral sequences.
Proposition AI.5 With the notation as above, {hi} induces an isomorphism ß*[Z,*
* C*] !
H*[Z, F*] for any spectrum Z. Hence the spectral sequence obtained by mapping *
*Z into
diagram AI.4 is isomorphic to a K(n)*local EnAdams spectral sequence.
Proof. The cochain complex
0 ! [Z, F0]t! [Z, F1]t+1! [Z, F2]t+2! . . .
Q *
is the normalized cochain complex of the cosimplicial abelian group [Z, C]t.*
* There is then
a natural cochain equivalence between these two complexes. Hence by Propositio*
*n 3.16,
45
*] to [Z, F ] inducing an isomo*
*rphism on
there is a cochain map, natural in Z, from [Z, C i * *
cohomology. This map is then induced by a cochain map {g } from C to F*. It no*
*w suffices
to show that
X ______C0//
 
h g0
(AI.6) fflffl fflffl
holimC ____F0//

commutes, for this implies that {gi} is chain homotopic to {hi} and thus induce*
*s the same
map on cohomology.
To prove the commutativity of AI.6, we need only show that
(AI.7) holimC ____//_C0
 

 0
 g
 
fflffl
holimC ____//_F0

commutes, where the top map is the canonical map
Q * Q * Q j 0
holimC = Tot( C) ! Tot0( C) = jC ! C .

Q
Now F0 = jCj, and by Proposition 3.16, the composition
g0Q j j0
C0 ! jC ! C
is given by (d0)j0. To prove the commutativity of AI.7, we must therefore prove*
* that
Q * Q 0 Q j j
Tot( C) ! C = jC ! C 0
is homotopic to
Q * Q 0 Q j 0 (d0)j0j
Tot( C) ! C = jC ! C !C 0
for each j0. But this follows by a standard argument (cf. proof of Theorem 2.2).
Corollary AI.8 The map h : X ! holimC is a weak equivalence.

46
i' *. Moreover, the s*
*pectral
Proof. By the discussion preceding Lemma 4.11, holimiTot
sequence E**r(S0, C) obtained by applying ß*(?) to AI.4 is isomorphic to the K(*
*n)*local En
Adams spectral sequence converging strongly to ß*X. It therefore follows that E*
***r(S0, C) is
strongly convergent and that ß*h is an isomorphism.
Remark AI.9 Given a K(n)*local S0module X there is a canonical choice of c*
*osimplicial
resolution C as above. Namely, define Cj = bL(X ^ E(j+1)n) with the coface and *
*codegeneracy
maps defined as in Construction 3.11, where XS is replaced by bL(X ^ En).
We conclude with a "geometric boundary theorem" which was used in x7.
Proposition AI.10 Let G be a closed subgroup of Gn, and let
f g
. .!. 1Z @!X ! Y ! Z ! . . .
be a cofibration sequence with E*n@ = 0. Suppose x 2 [X, EhGn] is detected by u*
* 2 H*c(G, EtnX)
in the K(n)*local EnAdams spectral sequence. Then xO@ is detected by ffi(u) 2*
* Hs+1c(G, EtnZ)
up to higher filtration, where ffi denotes the coboundary map in H*c(G, ?) asso*
*ciated to the
short exact sequence
0 ! E*nZ ! E*nY ! E*nX ! 0.
Remark AI.11 The functor D*G(?) of Definition 3.18 is exact on the category *
*of profinite
Gnmodules (see the proof of Lemma 3.21), so that the coboundary map ffi of the*
* proposition
can be defined.
Proof of AI.10. Let C be the cosimplicial resolution of EhGnof AI.9, and write
EhGn= W 0 oo_i____W 1oo___i_____W 2 oo___i_____
;; AA CCC ==CCCj
;; CC  CC . . .
j ;ÆÆ; k j C!!Ck CC!!
C0 1C1 2C2
for the associated diagram of exact triangles. Recall also from Lemma 3.22 that*
* [?, C*]t is
naturally isomorphic to D*G(Etn(?)).
Now x lifts to a map __x: X ! W ssuch that the composition
_x j
X ! W s! sCs
is a representative of u in DsG(E*nX). The composition j O __xO @ is trivial si*
*nce E*n@ = 0. We
may therefore construct a diagram
_@___//___f___// ____g____//
1ZØ X YØ ZØ
_zØ _x Ø Ø_
 Ø Øz
fflfflØi fflfflj fflfflk fflffl
W s+1 ____//_W_s__//_ sCs___//_ W s+1
47
__ s+1 * s+1*
* *
of cofibration sequences. It now follows_easily that j O z 2 [Z, C ] = D *
*(EnZ) is a
representative of ffi(u). But j O zrepresents x O @ up to higher filtration as *
*well, completing
the proof.
Appendix II. Proof of Proposition 3.16
The proof of this proposition requires some preparation.
Definition AII.1 If k 0, let D*kbe the cosimplicial abelian group with
M
Dnk= Z,
[k]![n]
the sum ranging over all morphisms [k] ! [n] in . Let 'k 2 Dkkdenote the elem*
*ent 1 in
the summand corresponding to the identity [k] ! [k].
D*khas a convenient universal property: If C is a cosimplicial abelian group*
* and x 2 Ck,
there exists a unique map øx : D*k! C of cosimplicial abelian groups with øx('k*
*) = x. We
can also put the D*k's together.
Definition AII.2 Let D**be the simplicial cosimplicial abelian group whose cos*
*implicial
group of ksimplices is D*k. If [m] ! [k] is a morphism in , the map D*k! D*m*
*is defined
by sending the summand indexed by [k] ! [n] to the summand indexed by [m] ! [k]*
* ! [n]
via the identity.
Q *
Lemma AII.3 There exists a sequence of maps Tk : D*k! D*kof cosimplicial *
*abelian
groups such that:
i.The composition
T0 Q j i j0
D00____//_jD0 ____//_D0
sends '0 to d0. .d.0'0 for all j0 0.
ii.The diagram
D*k+1___@___//_D*k
Tk+1 Tk
Q * fflffl Q fflffl
D*k+1_@__//_*D*k
Q *
commutes for all k 0, where D**and D**are here regarded as chain comp*
*lexes of
cosimplicial abelian groups and @ denotes their respective boundary maps.
48
Proof. We construct Tk by induction on k. There is a unique cosimplicial map T0*
* satisfying
i.QTo construct Tk+1, it suffices to prove that Tk(@'k+1) is a boundary in the *
*chain complex
k+1D*
*; we may thenQdefinekTk+1('k+1)+=1c, where @c = Tk('k+1).
We claim thatQHi( D**)Q= 0 for all i > 0. Indeed, for fixed i, Hi(D**)is*
* a cosimplicial
group and Hi( k+1D**) = k+1Hi(D**). But the chain complex Dj*is just the si*
*mplicial
chain complex of the standard jsimplex; therefore
æ
Z i = 0
Hi(Dj*) = .
0 otherwise
Since 0 = Tk1(@@'k+1) = @Tk(@'k+1) by the inductive hypothesis, it now foll*
*ows that
Tk(@'k+1) is a boundary if k > 0. If k = 0, use the fact that the maps
Tj0Q j i i
Dj0____//_D*0 ____//_D0
are augmentation preserving to conclude that T1(@'1) is a boundary as well. Thi*
*s completes
the induction and the proof.
Proof of Proposition 3.16. For x 2 Ck, define T (C)x to be the image of 'k und*
*er the
maps
Tk Q k Q*fixQk
Dkk____//_ D*k_____// C.
P k+1
Since @ : D*k+1! D*kmaps 'k+1 to i=0(1)idi'k, it follows that øffix= øx O @.*
* By AII.3ii,
we then have that T (C)(ffix) is given by the image of 'k+1 under the compositi*
*on
Tk Q k+1 Q *fixQk+1
Dk+1k+1@__//Dk+1k__//_ D*k____//_ C.
Q *
But ( øx) O Tk is a cosimplicial map; therefore
Q * Q * k+1X i i
(( øx) O Tk O @)('k+1)= (( øx) O Tk)( (1) d 'k)
i=0
k+1X Q
= di(( *øx) O Tk)('k)
i=0
= ffiT (C)(x) .
Thus T (C) is a cochain map.
Now it is also clear from the definition of T that T 0is as required and hen*
*ce that T is
an isomorphism on ß0. This implies that T is an isomorphism on ß*, completing t*
*he proof.
49
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52