THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE
GROUPS
DANIEL G. DAVIS1
Abstract.Let G be a profinite group. We define an S[[G]]-module to be a
G-spectrum X that satisfies certain conditions, and, given an S[[G]]-mod*
*ule
X, we define the homotopy orbit spectrum XhG. When G is countably based
and X satisfies a certain finiteness condition, we construct a homotopy *
*orbit
spectral sequence whose E2-term is the continuous homology of G with coe*
*f-
ficients in the graded profinite bZ[[G]]-module ss*(X). Let Gn be the ex*
*tended
Morava stabilizer group and let En be the Lubin-Tate spectrum. As an app*
*li-
cation of our theory, we show that the function spectrum F(En, LK(n)(S0))
is an S[[Gn]]-module with an associated homotopy orbit spectral sequence.
1.Introduction
Let G be a finite group and let X be a (left) G-spectrum. Then the homotopy
orbit spectrum XhG is defined to be hocolimGX, the homotopy colimit of the G-
action on X (see, for example, [16, pg. 42]). Furthermore, there is a homotopy
orbit spectral sequence
Hp(G, ssq(X)) ) ssp+q(XhG),
where the E2-term is the group homology of G, with coefficients in the graded G-
module ss*(X) ([17, 5.1]). In this paper, under certain hypotheses, we extend t*
*hese
constructions to the case where G is a profinite group.
After making a few comments about notation, we summarize the contents of this
paper. We follow the convention that all of our spectra are in Spt, the categor*
*y of
Bousfield-Friedlander spectra of simplicial sets. We use (-)f to denote functor*
*ial
fibrant replacement in the category of spectra; for any spectrum Z, Z ! Zf is a
weak equivalence with Zf fibrant. Also, "holim" always denotes the version of t*
*he
homotopy limit of spectra that is constructed levelwise in the category of simp*
*licial
sets, as defined in [2] and [22, 5.6].
In Section 2, for a finite group G, we use the fact that the homotopy colimit*
* of
a diagram of pointed simplicial sets is the diagonal of the simplicial replacem*
*ent of
the diagram, to obtain an alternative model for homotopy orbits. Then, in Secti*
*on
3, we use this model to define the homotopy orbit spectrum for a profinite group
G and a certain type of G-spectrum X, which we now define. We point out that
the following definition was essentially first formulated by Mark Behrens.
Definition 1.1. Given a profinite group G, let {Ni} be a collection of open nor-
mal subgroups of G such that G ~=limiG=Ni. Then we let S[[G]] be the spec-
trum holimi(S[G=Ni])f. Also, let {Xi} be an inverse system of G-spectra and
____________
1The author was partially supported by an NSF grant.
1
2 DANIEL G. DAVIS
G-equivariant maps indexed over {i}, such that, for each i, the G-action on Xif*
*ac-
tors through G=Ni (so that Xi is a G=Ni-spectrum) and Xi is a fibrant spectrum.
Then we call the associated G-spectrum X = holimiXi an S[[G]]-module.
After defining the homotopy orbit spectrum XhG for an S[[G]]-module X, we
show that this construction agrees with the classical definition when G is fini*
*te.
Now let us recall a definition (see [4, Section 3] for more detail).
Definition 1.2. A discrete right G-spectrum X is a spectrum of pointed simplici*
*al
sets Xk, for k 0, such that each simplicial set Xk is a pointed simplicial di*
*screte
right G-set, and each bonding map S1^Xk ! Xk+1 is G-equivariant (S1 has trivial
G-action).
In Section 4, given a discrete right G-spectrum X and any spectrum Z, we show
that the function spectrum F (X, Z) is an S[[G]]-module, so that one can form
F (X, Z)hG. Interesting examples of F (X, Z) are the Brown-Comenetz dual cX of
X and the Spanier-Whitehead dual DX.
Now we summarize the main result of Section 5, in which we build a homotopy
orbit spectral sequence.
Recall that a topological space Y is countably based if there is a countable *
*family
B of open sets, such that each open set of Y is a union of members of B. In
particular, one can consider countably based profinite groups. Then, if G is a
countably based profinite group, by [24, Proposition 4.1.3], G has a chain
N0 N1 N2 . . .
of open normal subgroups, such that G ~=limi 0G=Ni. For example, a compact
p-adic analytic group is a countably based profinite group. (If a topological g*
*roup
G is compact p-adic analytic, then G is a profinite group with an open subgroup*
* H
of finite rank, by [7, Corollary 8.34]. Then, by [7, Exercise 1, pg. 58], G has*
* finite
rank, and thus, G is finitely generated ([7, pg. 51]), so that G is countably b*
*ased,
by [24, pg. 55].)
Let G be a countably based profinite group, with a chain {Ni}i 0of open normal
subgroups, such that G ~=limiG=Ni, and let {Xi} be a diagram of G-spectra, such
that holimiXi is an S[[G]]-module. Recall from [3, pg. 5] that a spectrum X is *
*an
f-spectrum, if, for each integer q, the abelian group ssq(X) is finite. Let bZd*
*enote
limn 1Z=nZ. Then, if each Xiis an f-spectrum, we show that there is a homotopy
orbit spectral sequence
Ep,q2~=Hcp(G, ssq(holimiXi)) ) ssp+q((holimiXi)hG),
where the E2-term is the continuous homology of G with coefficients in the grad*
*ed
profinite bZ[[G]]-module ss*(holimiXi).
We continue to assume that G is a countably based profinite group in Section
6, where we prove a result about the homotopy orbits of a tower of Eilenberg-Mac
Lane spectra that was suggested to the author by Mark Behrens. To be precise, l*
*et
A0 A1 A2 . .b.e a tower of G-modules and G-equivariant maps, such that,
for each i, the G-action on Aifactors through G=Ni, so that Aiis a G=Ni-module.*
* If
each abelian group Aiis finite, then we call {Ai} a nice tower of G-modules. Gi*
*ven
a nice tower {Ai} of G-modules, we show that holimiH(Ai) is an S[[G]]-module,
with
ss*((holimiH(Ai))hG) ~=Hc*(G, limiAi),
THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS 3
where H(Ai) is an Eilenberg-Mac Lane spectrum.
In Section 7, we consider an example that arises in chromatic homotopy theory.
Let ` be a prime and let n 1. Let En denote the Lubin-Tate spectrum, so that
ss*(En) = W (F`n)[[u1, ..., un-1]][u 1], where the degree of u is -2 and the po*
*wer
series ring over the Witt vectors has degree zero. Also, let Gn = Sn o Gal(F`n=*
*F`)
be the extended Morava stabilizer group, where Sn is the nth Morava stabilizer
group, and recall that the profinite group Gn is compact `-adic analytic. By wo*
*rk
of Paul Goerss, Mike Hopkins, and Haynes Miller (see [18], [10], and [6]), the *
*group
Gn acts on En.
Now let K(n) be the nth Morava K-theory, with K(n)* = F`[vn1], where the
degree of vn is 2(`n-1). Also, let LK(n)(-) denote the Bousfield localization f*
*unctor
with respect to K(n). Then we show that the K(n)-local Spanier-Whitehead dual
of En, the function spectrum F (En, LK(n)(S0)), is an S[[Gn]]-module and there *
*is
a homotopy orbit spectral sequence
Hcp(Gn, ssq(F (En, LK(n)(S0)))) ) ssp+q(F (En, LK(n)(S0))hGn).
When (`-1) - n, the use of the above spectral sequence to identify the spectr*
*um
F (En, LK(n)(S0))hGn more concretely is work in progress.
We want to point out that others have investigated homotopy orbits for profi-
nite groups. In [8], Halvard Fausk studies homotopy orbits in the setting of pr*
*o-
orthogonal G-spectra, where G is profinite. As pointed out in various places in*
* this
paper, Mark Behrens has worked on homotopy orbits for a profinite group. Also,
Dan Isaksen has thought about homotopy orbits for profinite groups in the conte*
*xt
of pro-spectra. Finally, [12], by Mike Hopkins and Hal Sadofsky, contains some
explorations of homotopy orbits involving En and Gn. It was the author's study *
*of
[12] that motivated him to begin work on the project represented by this paper.
Acknowledgements. I thank Hal Sadofsky for sparking my interest in homotopy
orbits for profinite groups. I thank Mark Behrens for helpful and encouraging
discussions about this work. His positive influence on this work is seen in the*
* many
places where his name is mentioned. Also, I thank Paul Goerss and Mark Hovey
for their encouragement.
2.Homotopy orbits when G is finite
In this section, let G be a finite group and let X be a (left) G-spectrum. We
use simplicial replacement to obtain an alternative model for the homotopy orbit
spectrum XhG, which will be useful for defining the homotopy orbit spectrum when
G is a profinite group. Given a spectrum Z, we let Zk denote the kth pointed
simplicial set of Z and we use Zk,lto signify the set of l-simplices of Zk.
By [2, XII, 5.2],
`
(XhG)k = hocolimGXk = diag( * (G ! Xk)),
where diag(-) is the functor that takes the diagonal of a bisimplicial set, and
G ! Xk is the diagram out of the groupoid G that is defined by the action of G
on Xk. The simplicial replacement is defined as follows:
` W
l(G ! Xk) = ul2IlXk,
where Ilis the indexing set that consists of all compositions of the form
ul= i0 ff1-. .f.fl-il,
4 DANIEL G. DAVIS
such that each ffj is a morphism in the groupoid G. `
The face and degeneracy maps of the pointed simplicial set l(G ! Xk) are
defined in the following way. Let 0 j < l and let dj(ul) be the map that, giv*
*en a
morphism ul2 Il, is defined by the composition
W
Xk -id!Xk -! Il-1Xk,
where the target of idis the copy of Xk that is indexed by the morphism
ul-1= i1 ff2-. .f.fl-il2 Il-1.
W W
Then the jth face map dj: IlXk ! Il-1Xk is the unique map that is induced
by all the dj(ul), where dj(ul) is the canonical map out of the copy of Xk that
is indexed by ul. Also, let dl(ul) be the map that, given a morphism ul 2 Il, *
*is
defined by the composition
W
Xk -ffl!Xk -! Il-1Xk,
where the target of fflis the copy of Xk that is indexed by the morphism
ul-1= i0 ff1-. .f.fl-1-il-12 Il-1.
W W
Then the lth face map dl: IlXk ! Il-1Xk is the unique map that is induced
by all the dl(ul), where dl(ul) is the canonical map out of the copy of Xk that*
* is
indexed by ul.
For 0 j l, let sj(ul) be the map that, given a morphism ul2 Il, is defined
by the composition
W
Xk -id!Xk -! Il+1Xk,
where the target of idis the copy of Xk that is indexed by the morphism
ul+1= i0 ff1-. .f.fl-il-id il2 Il+1.
W W
Then the jth degeneracy sj: IlXk ! Il+1Xk is the unique map that is induced
by all the sj(ul), where sj(ul) is the canonical map out of the copy of Xk that*
* is
indexed by ul.
Definition 2.1. Let K be a pointed simplicial set and let L be a set. Then
K[L] = K ^ L+ ,
where L+ is the constant simplicial set on L, together with a disjoint basepoin*
*t.
Similarly, if Z is a spectrum, then Z[L] is the spectrum with each (Z[L])k equa*
*l to
Zk ^ L+ .
Since each ul can be identified with an element of Gl, the l-fold product of *
*G,
Il= Gland
` W ~ l l
l(G ! Xk) = GlXk = Xk ^ (G )+ = Xk[G ].
Thus,
(XhG)k ~=diag(Xk[Go]),
where the l-simplices of the simplicial pointed simplicial set Xk[Go] are the p*
*ointed
simplicial set Xk[Gl].
We introduce some notation that organizes the above, allowing us to summarize
it in a theorem.
THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS 5
Definition 2.2. Let X be a G-spectrum. Then X[Go] is the simplicial spectrum
that is defined above, with (X[Go])k = Xk[Go] and l-simplices equal to the spec-
trum X[Gl]. Thus, X[Go] is the simplicial spectrum
X ~=X[*] oo__X[G]oo_ooX[G2]_oo_oo_...o.o_oo_oo_oo_
We let ssq(X[Go]) denote the simplicial abelian group associated to X[Go].
Definition 2.3. Let d(Zo) denote the spectrum that is the diagonal of the simpl*
*icial
spectrum Zo (see [14, pg. 100]). Thus, for all k 0,
(d(Zo))k = diag((Zo)k)
and, for all l 0,
(d(Zo))k,l= (Zl)k,l.
Theorem 2.4. If G is a finite group and X is a G-spectrum, then
XhG ~=d(X[Go]).
3. The homotopy orbit spectrum XhG
In this section, we use the model for the homotopy orbit spectrum that is giv*
*en
in Theorem 2.4 to help us define homotopy orbits for G a profinite group. We be*
*gin
with a comment about Definition 1.1.
Remark 3.1. Let X = holimiXi be an S[[G]]-module. The choice of the term
"S[[G]]-module" is motivated by the fact that, for each i, the action of G=Nion*
* Xi
yields a function G=Ni! Hom Spt(Xi, Xi) in
Q
Hom Sets(G=Ni, Hom Spt(Xi, Xi))~=G=NiHomSpt(Xi, Xi)
~=Hom Spt(Xi[G=Ni], Xi)
that corresponds to a map Xi^ (G=Ni)+ ! Xi. The analogy with actual modules
over the spectrum S[[G]] could be pursued further (as Mark Behrens has done);
however, for simplicity, we do not do this.
Using the author's preliminary work on homotopy orbits as a reference point, *
*the
following definition was basically given by Behrens, and then later independent*
*ly
formulated by the author.
Definition 3.2. Let X = holimiXi be an S[[G]]-module. For each l 0, the
diagrams {Xi} and {G=Ni} induce the diagram {Xi[(G=Ni)l]}, so that one can form
holimi(Xi[(G=Ni)l])f, which gives the simplicial spectrum holimi(Xi[(G=Ni)o])f.
Then we define XhG, the homotopy orbit spectrum of X with respect to the G-
action, to be the spectrum
XhG = d(holimi(Xi[(G=Ni)o])f).
Henceforth, we use the notation Xh0G to denote the classical homotopy orbit spe*
*c-
trum hocolimGX, when G is finite.
The definition of the homotopy orbit spectrum comes from imitating the model
in Theorem 2.4 and from the demands of the homotopy orbit spectral sequence
(see the proof of Theorem 5.3). For example, the reader might expect that the
definition involve a homotopy colimit of G-spectra, instead of the homotopy lim*
*it
of such; however, the homotopy limit is necessitated by the nature of continuous
group homology, which is the E2-term of the homotopy orbit spectral sequence.
6 DANIEL G. DAVIS
Remark 3.3. The functor (-)f appears in Definition 3.2 so that the homotopy
limit is well-behaved. Let us suppose that the functor (-)f does not appear in *
*the
definition of XhG, so that it reads instead as XhG = d(holimiXi[(G=Ni)o]). Reca*
*ll
that Xi[(G=Ni)o] is involved in the construction of (Xi)hG=Ni ~=d(Xi[(G=Ni)o]),
by Theorem 2.4. Thus, one might wonder if there is an isomorphism
d(holimiXi[(G=Ni)o]) ~=holimid(Xi[(G=Ni)o]),
so that, morally, XhG = holimi(Xi)hG=Ni. However, it is not hard to see that th*
*is
is not the case, since
(d(holimiXi[(G=Ni)o]))k,l= (diag(holimi(Xi)k[(G=Ni)o]))l
= (holimi(Xi)k[(G=Ni)l])l
= Hom S({i}op)( [l] x ({i}op)=-, {(Xi)k[(G=Ni)l]}i)
(see [2, Chapter XI, 2.2, 3.2]), where S is the category of simplicial sets, is*
* not
equivalent to
(holimid(Xi[(G=Ni)o]))k,l= (holimidiag((Xi)k[(G=Ni)o]))l
= Hom S({i}op)( [l] x ({i}op)=-, {diag((Xi)k[(G=Ni)o])}i).
We recall the following useful result.
Theorem 3.4 ([14, Corollary 4.22]). If Xo is a simplicial spectrum, then there *
*is
a spectral sequence
Ep,q2= Hp(ssq(X*)) ) ssp+q(d(Xo)),
where H*(ssq(X*)) is the homology of the Moore complex of the simplicial abelian
group ssq(Xo).
Lemma 3.5. Let Xo ! Yo be a map between simplicial spectra, such that, for each
n 0, the map Xn ! Yn is a weak equivalence between the n-simplices. Then the
induced map d(Xo) ! d(Yo) is a weak equivalence of spectra.
Proof.There is a spectral sequence
Hp(ssq(X*)) ) ssp+q(d(Xo)),
and a map to the spectral sequence
Hp(ssq(Y*)) ) ssp+q(d(Yo)).
Since ssq(Xn) ~=ssq(Yn), for each n 0, and ssq(X*) and ssq(Y*) are chain comp*
*lexes,
~=
there is an isomorphism Hp(ssq(X*)) ! Hp(ssq(Y*)) of E2-terms. Therefore, the
abutments of the above two spectral sequences are isomorphic, giving the conclu*
*sion
of the lemma.
Remark 3.6. Let G be finite and let X be any G-spectrum. Let {i} = {0},
N0 = {e}, and X0 = Xf. Then holim{0}X0 ~=Xf is an S[[G]]-module and
(holimX0)hG = d(holim(Xf[Go])f) ~=d((Xf[Go])f).
{0} {0}
Thus, there is a weak equivalence
Xh0G ~=d(X[Go]) -'!d((Xf[Go])f) ~=(holimX0)hG,
{0}
so that Definition 3.2 recovers the classical definition of homotopy orbits.
THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS 7
Now we show that (-)hG preserves weak equivalences of S[[G]]-modules, as de-
fined below.
Definition 3.7. Let {Xi} ! {Yi} be a natural transformation of diagrams of G-
spectra, such that Xi! Yi is a weak equivalence, for each i, and the induced map
X = holimiXi ! holimiYi = Y is a map between S[[G]]-modules. Then we say
that the weak equivalence X ! Y is a weak equivalence of S[[G]]-modules.
Theorem 3.8. If X ! Y , as in Definition 3.7, is a weak equivalence of S[[G]]-
modules, then XhG ! YhG is a weak equivalence.
Proof.Let i 0. Since Xi! Yi is a weak equivalence, the induced map
(Xi[(G=Ni)l])f ! (Yi[(G=Ni)l])f
is a weak equivalence between fibrant spectra, for each l 0. Thus,
holimi(Xi[(G=Ni)l])f ! holimi(Yi[(G=Ni)l])f
is a weak equivalence, so that, by Lemma 3.5, XhG ! YhG is a weak equivalence.
4.Examples of S[[G]]-modules
In this section, we consider a way that S[[G]]-modules arise naturally from d*
*is-
crete right G-spectra.
The following recollection will be helpful. There is a functor (-)c:Spt ! Spt,
such that, given Y in Spt, Yc is a cofibrant spectrum, and there is a natural t*
*rans-
formation (-)c ! idSpt, such that, for any Y , the map Yc ! Y is a trivial fibr*
*ation.
For example, if Y is a right K-spectrum for some group K, then Yc is also a rig*
*ht
K-spectrum, and the map Yc ! Y is K-equivariant.
Let Z be any spectrum and let X be a discrete right G-spectrum. Then
F (X, Z) ' F (colimiXNi, Zf) ' F (hocolimiXNi, Zf) ' holimiF ((XNi)c, Zf).
Thus, when X is a discrete right G-spectrum, we identify the left G-spectrum
F (X, Z) with the G-spectrum holimiF ((XNi)c, Zf). Under this identification, we
make the following observation.
Theorem 4.1. If X is a discrete right G-spectrum, then F (X, Z) is an S[[G]]-
module.
Proof.The spectrum (XNi)c is a right G=Ni-spectrum, because XNi is a right
G=Ni-spectrum. Thus, F ((XNi)c, Zf) is a G=Ni-spectrum. Also, since the source
and target of the function spectrum F ((XNi)c, Zf) are cofibrant and fibrant, r*
*espec-
tively, the function spectrum itself is fibrant. These facts imply that the spe*
*ctrum
holimiF ((XNi)c, Zf) is an S[[G]]-module.
We give two examples of F (X, Z) as an S[[G]]-module.
Example 4.2. Recall from [3] that, if cS0 is the Brown-Comenetz dual of S0, then
cX, the Brown-Comenetz dual of X, is F (X, cS0). Thus, if X is a discrete right
G-spectrum, then
cX = holimiF ((XNi)c, (cS0)f)
is an S[[G]]-module.
The following example is due to Mark Behrens.
8 DANIEL G. DAVIS
Example 4.3. If X is a discrete right G-spectrum, then the Spanier-Whitehead
dual of X,
DX = F (X, S0) = holimiF ((XNi)c, (S0)f),
is an S[[G]]-module.
5. The homotopy orbit spectral sequence for an S[[G]]-module
In this section, we construct a homotopy orbit spectral sequence for a certain
type of S[[G]]-module. We attempted to construct such a spectral sequence for a*
*ny
profinite group G. However, there were difficulties that we were able to get ar*
*ound
only by putting an additional hypothesis on the group G, that is, by assuming t*
*hat
G is countably based.
Now we recall the construction of the classical homotopy orbit spectral seque*
*nce
for Xh0G; the brief analysis below of its E2-term will be useful. Thus, for the
duration of this paragraph, let G be a finite group and let X be a G-spectrum. *
*By
Theorem 3.4, there is a spectral sequence
Ep,q2) ssp+q(d(X[Go])) = ssp+q(Xh0G),
where
(5.1) Ep,q2= Hp(ssq(X[G*])) ~=Hp(G, ssq(X)),
the pth group homology of G, with coefficients in ssq(X) (see, for example, [14,
(7.9)]). Note that, by (5.1),
(5.2) Hp(G, ssq(X)) ~=Hp(ssq(X)[G*]),
L
where, if A is an abelian group and K is a group, we use A[K] to denote K A.
Now we are ready to construct the homotopy orbit spectral sequence for a coun*
*t-
ably based profinite group.
Theorem 5.3. Let G be a countably based profinite group, and let
N0 N1 N2 . . .
be a chain of open normal subgroups of G, such that G ~=limiG=Ni. Also, let
X0 X1 X2 . . .
be a tower of G-spectra and G-equivariant maps such that X = holimiXi is an
S[[G]]-module. If, for all i, Xi is an f-spectrum, then there is a spectral seq*
*uence
of the form
Ep,q2~=Hcp(G, ssq(X)) ) ssp+q(XhG),
where the E2-term is the continuous homology of G with coefficients in the grad*
*ed
profinite bZ[[G]]-module ss*(X).
Proof.By Theorem 3.4, there is a spectral sequence
Ep,q2) ssp+q(d(holimi(Xi[(G=Ni)o])f)) = ssp+q(XhG),
where
Ep,q2= Hp(ssq(holimi(Xi[(G=Ni)*])f)).
We spend the rest of the proof identifying the E2-term as continuous group homo*
*l-
ogy.
THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS 9
Let l 0. Since {ssq+1(Xi)[(G=Ni)l]} is a tower of finite abelian groups,
lim1issq+1(Xi)[(G=Ni)l] = 0.
Thus,
ssq(holimi(Xi[(G=Ni)l])f) ~=limissq(Xi)[(G=Ni)l],
which implies that
Ep,q2~=Hp(limi(ssq(Xi)[(G=Ni)*])).
Since, for any l 0, the tower of abelian groups {ssq(Xi)[(G=Ni)l]}isatisfie*
*s the
Mittag-Leffler condition, by [23, Theorem 3.5.8], there is a short exact sequen*
*ce
0 ! lim1iHp+1(ssq(Xi)[(G=Ni)*]) ! Ep,q2! limiHp(ssq(Xi)[(G=Ni)*]) ! 0.
By (5.2), this short exact sequence can be written as
0 ! lim1iHp+1(G=Ni, ssq(Xi)) ! Ep,q2! limiHp(G=Ni, ssq(Xi)) ! 0.
Since G=Ni and ssq(Xi) are finite, Hp+1(G=Ni, ssq(Xi)) is a finite abelian gr*
*oup,
for each p 0 ([23, Corollary 6.5.10]). Thus, lim1iHp+1(G=Ni, ssq(Xi)) = 0. Th*
*ere-
fore, we obtain that
Ep,q2~=limiHp(G=Ni, ssq(Xi)) ~=Hcp(G, ssq(X)),
where the last isomorphism uses [19, Proposition 6.5.7].
6. Eilenberg-Mac Lane spectra and their homotopy orbits
Let the profinite group G be countably based, with a chain {Ni} of open normal
subgroups, such that G ~=limiG=Ni, and, as defined in the Introduction, let {Ai}
be a nice tower of G-modules. Let : Ch+ ! sAb be the functor in the Dold-Kan
correspondence from Ch +, the category of chain complexes C* with Cn = 0 for
n < 0, to sAb , the category of simplicial abelian groups (see, for example, [1*
*1,
Chapter III, Corollary 2.3]). Also, if A is an abelian group, let A[-n] be the *
*chain
complex that is A in degree n and zero elsewhere.
Given the tower {Ai}, we explain how to form {H(Ai)}, a tower of Eilenberg-
Mac Lane spectra (we follow the construction given in [15]), so that holimiH(Ai)
is an S[[G]]-module. By functoriality, for each k 0, { (Ai[-k])} is a tower *
*of
simplicial G-modules and G-equivariant maps, such that, for each i, (Ai[-k]) is
the Eilenberg-Mac Lane space K(Ai, k) and (Ai[-k]) is a simplicial G=Ni-module.
Furthermore, by taking 0 as the basepoint, each (Ai[-k]) is a pointed simplici*
*al
set.
For each i, we define the Eilenberg-Mac Lane spectrum H(Ai) by (H(Ai))k =
(Ai[-k]), so that ss0(H(Ai)) = Ai and ssn(H(Ai)) = 0, when n 6= 0. Then, by
functoriality, {H(Ai)} is a tower of G-spectra and G-equivariant maps, such that
each H(Ai) is a G=Ni-spectrum. Since each (H(Ai))k is a fibrant simplicial set
and each H(Ai) is an -spectrum (see, for example, [15, Example 21]), H(Ai) is a
fibrant spectrum. These facts imply the following result.
Lemma 6.1. The spectrum holimiH(Ai) is an S[[G]]-module.
Now we show that the homotopy orbit spectral sequence can be used to compute
ss*((holimiH(Ai))hG).
10 DANIEL G. DAVIS
Theorem 6.2. If G is a countably based profinite group with a chain {Ni} of
open normal subgroups, such that G ~=limiG=Ni, and, if {Ai} is a nice tower of
G-modules, then
ss*((holimiH(Ai))hG) ~=Hc*(G, limiAi).
Proof.By hypothesis, each H(Ai) is an f-spectrum. Then, by Theorem 5.3, there
is a homotopy orbit spectral sequence
Ep,q2~=Hcp(G, ssq(holimiH(Ai))) ) ssp+q((holimiH(Ai))hG).
Since lim1iAi= 0, we have:
( c
Ep,q2~=Hcp(G, limissq(H(Ai))) = Hp(G, 0) ifq 6= 0
Hcp(G, limiAi)ifq = 0
(
= 0 ifq 6= 0
Hcp(G, limiAi)ifq = 0.
Thus, the spectral sequence collapses, giving the conclusion of the theorem.
7.The Gn-homotopy orbits of F (En, LK(n)(S0))
Our main motivation for constructing homotopy orbits for a profinite group was
that we were interested in constructing the Gn-homotopy orbit spectrum of En, as
part of our effort to understand [12]. Though we have not been able to construct
such a spectrum, we are able to construct the Gn-homotopy orbits of the closely
related spectrum
2
F (En, LK(n)(S0)) ' -n En,
where the weak equivalence applies [20, Proposition 16] and the function spectr*
*um
F (En, LK(n)(S0)) is the K(n)-local Spanier-Whitehead dual of En. After explain-
ing why we have not been able to construct a homotopy orbit spectrum for En, we
consider F (En, LK(n)(S0))hGn.
Recall from the Introduction that, since Gn is a compact `-adic analytic grou*
*p,
it is a countably based profinite group. Then, following [6, (1.4)], weTfix a c*
*hain
Gn = N0 fl N1 fl N2 fl . .o.f open normal subgroups of Gn, such that iNi =
{e}, so that Gn ~= limiGn=Ni. Thus, an S[[Gn]]-module comes from a tower
X0 X1 X2 . .o.f Gn-spectra and Gn-equivariant maps.
To form the Gn-homotopy orbit spectrum of En, we have to show that En is
an S[[Gn]]-module. However, we do not know how to do this and we explain how
a natural way to try to do this fails. Let EdhNinbe the spectrum constructed by
Devinatz and Hopkins (see [6]) that behaves like the Ni-homotopy fixed points of
En with respect to a continuous Ni-action (see [4]). Let
MI0 MI1 MI2 . . .
be a tower of generalized Moore spectra such that
holimj(LK(n)(MIj))f ' LK(n)(S0)
([13, Proposition 7.10 (e)]). Also, recall from [4, Theorem 6.6] that
En ' holimj(colimi(EdhNin^ MIj)f).
THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS 11
Thus, one might attempt to present En as an S[[Gn]]-module by considering the
tower {colimi(EdhNin^MIj)f}j. However, by [4, Lemma 6.2], there is no open norm*
*al
subgroup N of Gn such that the Gn-action on colimi(EdhNin^MIj)f factors through
Gn=N. Thus, this attempt fails to present En as an S[[Gn]]-module.
We noted above that F (En, LK(n)(S0)) ' -n2En. Thus, one might like to assert
that, since we can construct the Gn-homotopy orbit spectrum of F (En, LK(n)(S0)*
*),
as done below, then we can also construct2the homotopy orbit spectrum of En, by
considering the homotopy orbits of n F (En, LK(n)(S0)). However, we have been
informed by Mark Behrens that, in general,2this can not be done, because the
equivalence F (En, LK(n)(S0)) ' -n En need not be Gn-equivariant. According
to Behrens, the S-n2 in -n2En, when properly interpreted, can have a non-trivi*
*al
Gn-action on it.
To form F (En, LK(n)(S0))hGn, we only need to show that F (En, LK(n)(S0)) is
an S[[Gn]]-module. As in [4], let Fn = colimiEdhNin. Then
F (En, LK(n)(S0))' F (LK(n)(Fn), holimj(LK(n)(MIj))f)
' F (Fn, holimj(LK(n)(MIj))f)
' F (hocolimi(EdhNin)c, holimj(LK(n)(MIj))f)
~=holimholimF ((EdhNi)c, (LK(n)(MI ))f)
i j n j
' holimiF ((EdhNin)c, (LK(n)(MIi))f),
where the first weak equivalence applies [6, Definition 1.5] and [4, Theorem 6.*
*3],
and the last weak equivalence uses the fact that {(i, i)}i is cofinal in {(i, j*
*)}i,j.
Thus, we make the identification
F (En, LK(n)(S0)) = holimiF ((EdhNin)c, (LK(n)(MIi))f).
Since EdhNinis a right Gn=Ni-spectrum, as in Section 4, the above identification
implies the following result.
Lemma 7.1. The spectrum F (En, LK(n)(S0)) is an S[[Gn]]-module.
The next result allows us to build the homotopy orbit spectral sequence for
F (En, LK(n)(S0))hGn.
Lemma 7.2. For each i 0, the spectrum F (EdhNin, LK(n)(MIi)) is an f-spectrum.
Proof.Our proof follows [5, proof of Lemma 3.5]. Since
LK(n)(MIi) ' LK(n)(S0) ^ MIi' EdhGnn^ MIi
(see [6, Theorem 1(iii)]), [6, Theorem 2(ii)] gives a strongly convergent K(n)-*
*local
En-Adams spectral sequence
Hsc(Gn, (En ^ MIi)-t(EdhNin)) ) sst-s(F (EdhNin, LK(n)(MIi))),
where the E2-term is continuous cohomology for profinite continuous Z`[[Gn]]-
modules (see [6, Remark 1.3]). By [9, Proposition 2.5],
L
(En ^ MIi)-t(EdhNin) ~= Gn=Nisst(En ^ MIi)
12 DANIEL G. DAVIS
is a finite abelian group. Thus, by [21, Proposition 4.2.2], the abelian group
Hsc(Gn, (En ^ MIi)-t(EdhNin)) is finite. The fact that E*,*1has a horizontal v*
*an-
ishing line (see [1, Theorem 6.10], [5, Proposition 2.3, proof of Lemma 3.5], a*
*nd
[6, Proposition A.3]) implies that ss*(F (EdhNin, LK(n)(MIi))) is finite in eac*
*h de-
gree.
Theorem 5.3 and Lemmas 7.1 and 7.2 give the following.
Theorem 7.3. There is a spectral sequence
Hcp(Gn, ssq(F (En, LK(n)(S0)))) ) ssp+q(F (En, LK(n)(S0))hGn).
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