HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) USING THE
CONTINUOUS ACTION1
DANIEL G. DAVIS2
Abstract.Let K(n) be the nth Morava K-theory spectrum. Let En be the
Lubin-Tate spectrum, which plays a central role in understanding LK(n)(S*
*0),
the K(n)-local sphere. For any spectrum X, define E_(X) to be the spectr*
*um
LK(n)(En ^ X). Let G be a closed subgroup of the profinite group Gn, the
group of ring spectrum automorphisms of En in the stable homotopy catego*
*ry.
We show that E_(X) is a continuous G-spectrum, with homotopy fixed point
spectrum (E_(X))hG. Also, we construct a descent spectral sequence with
abutment ss*((E_(X))hG).
1.Introduction
Let p be a fixed prime. For each n 0, let K(n) be the nth Morava K-theory
spectrum, where K(0) is the Eilenberg-Mac Lane spectrum HQ, and, for n 1,
K(n)* = Fp[vn, v-1n], where the degree of vn is 2(pn-1). Let X be a finite spec*
*trum.
There are maps LnX ! Ln-1X, where Ln denotes Bousfield localization with
respect to K(0) _ K(1) _ . ._.K(n). Then the chromatic convergence theorem [33,
Theorem 7.5.7] says that X(p)' holimn 0LnX, where X(p)is the p-localization of
X. Thus, to understand X(p), it is important that one understands each localiza*
*tion
LnX.
Henceforth, let n 1, and let ^Ldenote Bousfield localization with respect to
K(n). Then there is a homotopy pullback square [15]
LnX _________//_^L(X)
| |
| |
fflffl| fflffl|
Ln-1(X(p))_____//Ln-1L^(X),
which shows that, to understand the localizations LnX, it is very helpful to un*
*der-
stand each ^L(X).
In attempting to understand ^L(X), one of the main tools is a certain spectral
sequence, which we now recall. Let En be the Lubin-Tate spectrum with En* =
____________
Date: July 26, 2005.
1This version of the paper is identical to the version accepted for publicat*
*ion in the Journal of
Pure and Applied Algebra (except for minor changes required by a change in docu*
*ment format).
The main differences between this version and the version dated 1/12/05 are: th*
*e present version
is a little shorter; pg. 5 begins with two new paragraphs; Cor. 3.7 was added; *
*Hscont(G; -) is
used in places where Hscts(G; -) was used earlier (this is only a change in pre*
*sentation and not a
correction - see the last sentence of Def. 2.15); and there are some changes in*
* notation.
2The author was supported by an NSF grant. Some revision of this paper was d*
*one while the
author was visiting the Notre Dame math department.
1
2 DANIEL G. DAVIS
W [[u1, ..., un-1]][u 1], where the degree of u is -2, and the complete power s*
*eries
ring over the Witt vectors W = W (Fpn) is in degree zero. Let Gn be the profini*
*te
group of ring spectrum automorphisms of En in the stable homotopy category [17,
Thm. 1.4]. (There is an isomorphism Gn ~=Sn o Gal, where Sn is the nth Morava
stabilizer group - the automorphism group of the Honda formal group law n of
height n over Fpn, and Gal is the Galois group Gal(Fpn=Fp) (see [38, Prop. 4]).)
By [32] and [14, Prop. 7.4], Morava's change of rings theorem yields a spectral
sequence
(1.1) H*c(Gn; ss*(En ^ X)) ) ss*^L(X),
where the E2-term is the continuous cohomology of Gn, with coefficients in the
continuous Gn-module ss*(En ^ X) (see Definition 2.17). Thus, we see that it is
critical to study the relationship between En and Gn.
The above action of Gn on ss*(En^X) is induced by a point-set level action of*
* Gn
on En (work of Goerss and Hopkins ([12], [9]), and Hopkins and Miller [34]). Let
G be a closed subgroup of Gn. Using the Gn-action on En, Devinatz and Hopkins
[5] construct spectra EdhGnwith spectral sequences
(1.2) Hsc(G; sst(En ^ X)) ) sst-s(EdhGn^ X).
Also, they show that EdhGnn' ^L(S0), so that EdhGnn^ X ' ^L(X), and (1.1) is a
special case of (1.2).
We compare the spectrum EdhGnand spectral sequence (1.2) with constructions
for homotopy fixed point spectra. When K is a discrete group and Y is a K-
spectrum of topological spaces, there is a homotopy fixed point spectrum Y hK =
Map K(EK+ , Y ), where EK is a free contractible K-space. Also, there is a desc*
*ent
spectral sequence
Es,t2= Hs(K; sst(Y )) ) sst-s(Y hK),
where the E2-term is group cohomology [29, x1.1].
Now let K be a profinite group. If S is a K-set, then S is a discrete K-set if
the action map K x S ! S is continuous, where S is given the discrete topology.
Then, a discrete K-spectrum Y is a K-spectrum of simplicial sets, such that each
simplicial set Yk is a simplicial discrete K-set (that is, for each l 0, Yk,l*
*is a
discrete K-set, and all the face and degeneracy maps are K-equivariant). Then,
due to work of Jardine (e.g. [21], [22], [23], [24]) and Thomason [41], as expl*
*ained
in Sections 5 and 7, there is a homotopy fixed point spectrum Y hK defined with
respect to the continuous action of K, and, in nice situations, a descent spect*
*ral
sequence
Hsc(K; sst(Y )) ) sst-s(Y hK),
where the E2-term is the continuous cohomology of K with coefficients in the
discrete K-module sst(Y ).
Notice that we use the notation EdhGnfor the construction of Devinatz and
Hopkins (which they denote as EhGnin [5]), and (-)hK for homotopy fixed points
with respect to a continuous action, although henceforth,0when K is finite and Y
is a K-spectrum of topological spaces, we write Y h Kfor holimK Y , which is an
equivalent definition of the homotopy fixed point spectrum Map K(EK+ , Y ).
After comparing the spectral sequence for EdhGn^ X with the descent spectral
sequence for Y hK, En ^ X appears to be a continuous Gn-spectrum with "descent"
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 3
spectral sequences for "homotopy fixed point spectra" EdhGn^ X. Indeed, we apply
[5] to show that En ^ X is a continuous Gn-spectrum; that is, En ^ X is the
homotopy limit of a tower of fibrant discrete Gn-spectra. Using this continuous
action, we define the homotopy fixed point spectrum (En ^ X)hG and construct its
descent spectral sequence.
In more detail, Gn acts on the K(n)-local spectrum En through maps of com-
mutative S-algebras. The spectrum EdhGn, a K(n)-local commutative S-algebra,
is referred to as a "homotopy fixed point spectrum" because it has the following
desired properties: (a) spectral sequence (1.2), which has the form of a descen*
*t spec-0
tral sequence, exists; (b) when G is finite,0there is a weak equivalence EdhGn!*
* EhnG,
and the descent spectral sequence for EhnG is isomorphic to spectral sequence (*
*1.2)
(when X = S0) [5, Thm. 3]; and (c) EdhGnis an N(G)=G-spectrum, where N(G)
is the normalizer of G in Gn [5, pg. 5]. These properties suggest that Gn acts *
*on
En in a continuous sense.
However, in [5], the Gn-action on En is not proven to be continuous, and EdhGn
is not defined with respect to a continuous G-action. Also, when G is profinite*
*, ho-
motopy fixed points should always be the total right derived functor of fixed p*
*oints,
in some sense, and [5] does not show that the "homotopy fixed point spectrum"
EdhGncan be obtained through such a total right derived functor.
After introducing some notation, we state the main results of this paper. Let
BP be the Brown-Peterson spectrum with BP* = Z(p)[v1, v2, ...], where the degree
of vi is 2(pi - 1). The ideal (pi0, vi11, ..., vin-1n-1) BP* is denoted by I;*
* MI is
the corresponding generalized Moore spectrum M(pi0, vi11, ..., vin-1n-1), a spe*
*ctrum
with trivial Gn-action. Given I, MI need not exist; however, enough exist for o*
*ur
constructions. Each MI is a finite type n spectrum with BP*(MI) ~=BP*=I. The
set {i0, ..., in-1} of superscripts varies so that there is a family of ideals *
*{I}. ([3, x4],
[19, x4], and [27, Prop. 3.7] provide details for our statements about the spec*
*tra
MI.) The map r :BP* ! En* - defined by r(vi) = uiu1-pi, where un = 1 and
ui = 0, when i > n - makes En* a BP*-module. By the Landweber exact functor
theorem, ss*(En ^ MI) ~=En*=I.
The collection {I} contains a descending chain of ideals {I0 I1 I2 . .}*
*.,
such that there exists a corresponding tower of generalized Moore spectra
{MI0 MI1 MI2 . .}..
In this paper, the functors limIand holimIare always taken over the tower of id*
*eals
{Ii}, so that limIand holimIare really limIiand holimIi, respectively. Also, in*
* this
paper, if {Xff}ffis a diagram of spectra (even if each Xffhas additional struct*
*ure),
then holimffXffalways denotes the version of the homotopy limit of spectra that
is constructed levelwise in S, the category of simplicial sets, as defined in [*
*2] and
[41, 5.6].
As in [5, (1.4)], let Gn =TU0 fl U1 fl . .f.l Uifl . .b.e a descending chain *
*of open
normal subgroups, such that iUi= {e} and the canonical map Gn ! limiGn=Ui
is a homeomorphism. We define
Fn = colimiEdhUin.
Then the key to getting our work started is knowing that
En ^ MI ' Fn ^ MI,
4 DANIEL G. DAVIS
and thus, En ^ MI has the homotopy type of the discrete Gn-spectrum Fn ^ MI.
This result (Corollary 6.5) is not difficult, thanks to [5].
Given a tower {ZI} of discrete Gn-spectra, there is a tower {(ZI)f}, with Gn-
equivariant maps ZI ! (ZI)f that are weak equivalences, and (ZI)f is a fibrant
discrete Gn-spectrum (see Definition 4.1). For the remainder of this section, X*
* is
any spectrum with trivial Gn-action, and, throughout this paper,
E_(X) = ^L(En ^ X).
We use ~=to signify an isomorphism in the stable homotopy category.
Theorem 1.3. As the homotopy limit of a tower of fibrant discrete Gn-spectra,
En ~=holimI(Fn ^ MI)f is a continuous Gn-spectrum. Also, for any spectrum X,
E_(X) ~=holimI(Fn ^ MI ^ X)f is a continuous Gn-spectrum.
We define homotopy fixed points for towers of discrete G-spectra; we show that
these homotopy fixed points are the total right derived functor of fixed points*
* in the
appropriate sense; and we construct the associated descent spectral sequence. T*
*his
enables us to define the homotopy fixed point spectrum (E_(X))hG, using the con-
tinuous Gn-action, and construct its descent spectral sequence. More specifical*
*ly,
we have the following results.
Definition 1.4. Given a profinite group G, let OG be the orbit category of G. T*
*he
objects of OG are the continuous left G-spaces G=K, for all K closed in G, and *
*the
morphisms are the continuous G-equivariant maps.
Let Sptbe the model category (spectra)stableof Bousfield-Friedlander spectra.
Theorem 1.5. There is a functor P :(OGn)op ! Spt, defined by P (Gn=G) = EhGn,
where G is any closed subgroup of Gn.
We also show that the G-homotopy fixed points of E_(X) can be obtained by
taking the K(n)-localization of the G-homotopy fixed points of the discrete G-
spectrum (Fn ^ X). This result shows that the spectrum Fn is an interesting
spectrum that is worth further study.
Theorem 1.6. For any closed subgroup G and any spectrum X, there is an iso-
morphism (E_(X))hG ~=^L((Fn ^ X)hG). In particular, EhGn~=^L(FnhG).
Theorem 1.7. Let G be a closed subgroup of Gn and let X be any spectrum. Then
there is a conditionally convergent descent spectral sequence
(1.8) Es,t2) sst-s((E_(X))hG).
If the tower of abelian groups {sst(En ^ MI ^ X)}I satisfies the Mittag-Leffler*
* con-
dition, for each t 2 Z, then Es,t2~=Hscont(G; {sst(En ^ MI ^ X)}) (see Definiti*
*on
2.15). If X is a finite spectrum, then (1.8) has the form
(1.9) Hsc(G; sst(En ^ X)) ) sst-s((En ^ X)hG),
where the E2-term is the continuous cohomology of (1.2).
Also, Theorem 9.9 shows that, when X is finite, (En ^ X)hG ~=EhGn^ X, so that
descent spectral sequence (1.9) has the same form as spectral sequence (1.2). *
*It
is natural to wonder if these two spectral sequences are isomorphic to each oth*
*er.
Also, the spectra EdhGnand EhGnshould be the same. We plan to say more about
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 5
the relationship between EdhGnand EhGnand their associated spectral sequences in
future work.
While reading this Introduction (and taking a quick look at x9), the reader m*
*ight
notice that, due to the definition of Fn, we use the "homotopy fixed point spec*
*tra"
EdhUinto construct the homotopy fixed point spectra EhGn. We discuss the degree
to which this method is circular. To obtain the results of this paper, we requi*
*re a
tower {En=I}I of discrete Gn-spectra such that holimI(En=I)f is the Gn-spectrum
En, and, for each I, En=I and En ^ MI have the same stable homotopy type. Any
tower with the stated properties will work (and, given such a tower, one defines
Fn = colimiholimI(En=I)hUi). We obtained such a tower by using the spectra
EdhUinto form the tower {Fn ^ MI}I.
We believe that one could probably use obstruction theory to construct the
tower {En=I}I. This would yield the above results independently of [5], so that,
presumably, [5] is not required to build (E_(X))hG. However, to date, no one has
obtained the requisite tower using obstruction theory, and we suspect that such
work would be quite difficult.
We outline the contents of this paper. In x2, we establish some notation and
terminology, and we provide some background material. In x3, we study the model
category of discrete G-spectra. In x4, we study towers of discrete G-spectra a*
*nd
give a definition of continuous G-spectrum. Homotopy fixed points for discrete
G-spectra are defined in x5, and x6 shows that En is a continuous Gn-spectrum,
proving the first half of Theorem 1.3. In x7, two useful models of the G-homoto*
*py
fixed point spectrum are constructed, when G has finite virtual cohomological d*
*i-
mension. In x8, we define homotopy fixed points for towers of discrete G-spectr*
*a,
build a descent spectral sequence in this setting, and show that these homotopy
fixed points are a total right derived functor, in the appropriate sense. In x9*
*, we
complete the proof of Theorem 1.3, study (E_(X))hG, and prove Theorems 1.5 and
1.6. In x10, we consider the descent spectral sequence for (E_(X))hG and prove
Theorem 1.7.
Acknowledgements. This paper is a development of part of my thesis. I am
very grateful to my thesis advisor, Paul Goerss, for many helpful conversations*
* and
useful suggestions regarding this paper. Also, I thank Ethan Devinatz for very
helpful answers to my questions about his work [5] with Mike Hopkins. I am grat*
*e-
ful to Halvard Fausk, Christian Haesemeyer, Rick Jardine, and Charles Rezk for
useful conversations. I extend my appreciation to Charles Weibel and the refere*
*es
for many helpful comments.
2. Notation, Terminology, and Preliminaries
We begin by establishing some notation and terminology that will be used
throughout the paper. Ab is the category of abelian groups. Outside of Ab ,
all groups are assumed to be profinite, unless stated otherwise. For a group G,
we write G ~=limN G=N, the inverse limit over the open normal subgroups. The
notation H m + 1. We
express this conclusion by saying that G has finite virtual cohomological dimen*
*sion
and we write vcd(G) m. Also, if K is a closed subgroup of G, H \ K is an open
pro-p subgroup of K with cdp(H \ K) m, so that vcd(K) m, and thus, m is a
uniform bound independent of K.
Now we state various results related to towers of abelian groups and continuo*
*us
cohomology. The lemma below follows from the fact that an exact additive functor
preserves images.
Lemma 2.10. Let F :Ab ! Ab be an exact additive functor. If {Ai}i 0 is a
tower of abelian groups that satisfies the Mittag-Leffler condition, then so do*
*es the
tower {F (Ai)}.
Remark 2.11. Let G be a profinite group. The functor Map c(G, -): Ab ! Ab ,
which sends A to Map c(G, A), is definedQby giving A the discrete topology. The
isomorphism Map c(G, A) ~=colimN G=N A shows that Map c(G, -) is an exact
additive functor. Later, we will use Lemma 2.10 with this functor.
The next lemma is a consequence of the fact that limits in Ab and in topologi*
*cal
spaces are created in Sets.
Lemma 2.12. Let M = limffMffbe an inverse limit of discrete abelian groups,
so that M is an abelian topological group. Let H be any profinite group. Then
Map c(H, M) ! limffMapc(H, Mff) is an isomorphism of abelian groups.
The lemma below follows from the fact that sst(En ^ MI) ~=sst(En)=I is finite.
Lemma 2.13. If X is a finite spectrum and t is any integer, then the abelian gr*
*oup
sst(En ^ MI ^ X) is finite.
Corollary 2.14 ([14, pg. 116]). If X is a finite spectrum, then sst(En ^ X) ~=
limIsst(En ^ MI ^ X).
We recall the definition of a second version of continuous cohomology [20].
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 9
Definition 2.15. Let CG be the category of discrete G-modules and tow(CG ) the
category of towers in CG . Then Hscont(G; {Mi}), the continuous cohomology of G
with coefficients in the tower {Mi}, is the sth right derived functor of the le*
*ft exact
functor limi(-)G :tow(CG ) ! Ab , which sends {Mi} to limiMGi. By [20, Theorem
2.2], if the tower of abelian groups {Mi} satisfies the Mittag-Leffler conditio*
*n, then
Hscont(G; {Mi}) ~=Hscts(G; limiMi), for s 0, where Hscts(G; M) is the cohomol*
*ogy
of continuous cochains with coefficients in the topological G-module M (see [40,
x2]).
Theorem 2.16 ([20, (2.1)]). Let {Mi}i 0 be a tower of discrete G-modules sat-
isfying the Mittag-Leffler condition. Then, for each s 0, there is a short ex*
*act
sequence
0 ! lim1iHs-1c(G; Mi) ! Hscont(G; {Mi}) ! limiHsc(G; Mi) ! 0,
where H-1c(G; -) = 0.
Definition 2.17. Let G be a closed subgroup of Gn, let X be a finite spectrum,
and let In = (p, u1, ..., un-1) En*. Then, by [5, Rk. 1.3],
sst(En ^ X) ~=limksst(En ^ X)=Iknsst(En ^ X)
is a profinite continuous Zp[[G]]-module (since it is the inverse limit of fini*
*te discrete
G-modules), and the definition of Hsc(G; sst(En ^ X)) is given by
Hsc(G; sst(En ^ X)) = limkHsc(G; sst(En ^ X)=Iknsst(En ^ X)).
By [5, Rk. 1.3], for s 0, there are isomorphisms
Hsc(G; sst(En^X)) ~=Hscts(G; sst(En^X)) ~=Hscont(G; {sst(En^X)=Iknsst(En ^ X)}).
3.The model category of discrete G-spectra
In this section, we begin explaining the theory of homotopy fixed points for
discrete G-spectra. We note that much of this theory (in this section and in Se*
*ctions
5, 7, and 8, through Theorem 8.5) is already known, in some form, especially in
the work of Jardine mentioned above, in the excellent article [29], by Mitchell*
* (see
also the opening remark of [31, x5]), and in Goerss's paper [11]. However, sin*
*ce
the above theory has not been explained in detail before, using the language of
homotopy fixed points for discrete G-spectra, we give a presentation of it.
A pointed simplicial discrete G-set is a pointed simplicial set that is a sim*
*plicial
discrete G-set, such that the G-action fixes the basepoint.
Definition 3.1. A discrete G-spectrum X is a spectrum of pointed simplicial sets
Xk, for k 0, such that each simplicial set Xk is a pointed simplicial discrete
G-set, and each bonding map S1 ^ Xk ! Xk+1 is G-equivariant (S1 has trivial G-
action). Let SptG denote the category of discrete G-spectra, where the morphisms
are G-equivariant maps of spectra.
As with discrete G-sets, if X 2 SptG, there is a G-equivariant isomorphism
X ~=colimNXN . Also, a discrete G-spectrum X is a continuous G-spectrum since,
for all k, l 0, the set Xk,lis a continuous G-space with the discrete topolog*
*y, and
all the face and degeneracy maps are (trivially) continuous.
10 DANIEL G. DAVIS
Definition 3.2. As in [23, x6.2], let G - Setsdfbe the canonical site of finite
discrete G-sets. The pretopology of G-Setsdf is given by covering families of t*
*he
form {fff:Sff! S}, a finite`set of G-equivariant functions in G-Setsdffor a fix*
*ed
S 2 G-Setsdf, such that ffSff! S is a surjection.
Let Shv be the Grothendieck topos consisting of sheaves of sets on the site
G-Setsdf. The topos Shv has a unique point u: Sets! Shv. The left and right
adjoints, respectively, of the topos morphism u are
u*: Shv ! Sets, F 7! colimNF(G=N), and
u*: Sets! Shv, X 7! Hom G(-, Mapc(G, X))
[23, Rk. 6.25]. The G-action on the discrete G-set Map c(G, X) is defined by
(g . f)(g0) = f(g0g), for g, g0 in G, and f a continuous map G ! X, where X is
given the discrete topology.
Recall that the topos Shv is equivalent to TG , the category of discrete G-
sets (see [23, Prop. 6.20] or [26, III-9, Thm. 1], for example). The functor
Map c(G, -): Sets! TG prolongs to the functorQMap c(G, -): Spt! SptG. Thus, if
X is a spectrum, then Map c(G, X) ~=colimN G=NX is the discrete G-spectrum
with (Map c(G, X))k = Map c(G, Xk), where Map c(G, Xk) is a pointed simplicial
set, with l-simplices Map c(G, Xk,l) and basepoint G ! *, where Xk,lis regarded
as a discrete set. The G-action on Map c(G, X) is defined by the G-action on the
sets Map c(G, Xk,l). It is not hard to see that Map c(G, -) is right adjoint to*
* the
forgetful functor U :SptG ! Spt.
Note that if X 2 SptG, then Hom G(-, X): (G-Setsdf)op ! Sptis a presheaf,
such that, for S 2 G - Setsdf, Hom G (S, X) 2 Spt satisfies Hom G (S, X)k,l=
Hom G(S, Xk,l), a pointed set with basepoint S ! *.
Let ShvSpt be the category of sheaves of spectra on the site G-Setsdf. A
sheaf of spectra F is a presheaf F :(G-Setsdf)op ! Spt, such that, for any S 2
G-Setsdfand any covering family {fff:Sff! S}, the usual diagram (of spectra) is
an equalizer. Equivalently, a sheaf of spectra F consists of pointed simplicial*
* sheaves
Fn, together with pointed maps of simplicial presheaves oe :S1 ^ Fn ! Fn+1, for
n 0, where S1 is the constant simplicial presheaf. A morphism between sheaves
of spectra is a natural transformation between the underlying presheaves.
We equip the category PreSpt of presheaves of spectra on the site G-Setsdf
with the stable model category structure (see [22], [23, x2.3]). Recall that, i*
*n this
model category structure, a map h: F ! G of presheaves of spectra is a weak
equivalence if and only if the associated map of stalks u*(F) ! u*(G) is a weak
equivalence of spectra, where u*(F) = colimNF(G=N), by prolongation.
Definition 3.3. In the stable model category structure, fibrant presheaves are *
*often
referred to as globally fibrant, and if F ! G is a weak equivalence of presheav*
*es,
with G globally fibrant, then G is a globally fibrant model for F.
We recall the following fact, which is especially useful when S = *.
Lemma 3.4. Let S 2 G-Setsdf. The S-sections functor PreSpt ! Spt, defined
by F 7! F(S), preserves fibrations, trivial fibrations, and weak equivalences b*
*etween
fibrant objects.
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 11
Proof.The S-sections functor has a left adjoint, obtained by left Kan extension,
that preserves cofibrations and weak equivalences. See [23, pg. 60] and [29, Co*
*r.
3.16] for the details.
Let L2 denote the sheafification functor for presheaves of sets, simplicial p*
*re-
sheaves, and presheaves of spectra, so that L2F ~=Hom G (-, u*(F)), by [23, Cor.
6.22]. Then i: ShvSpt ! PreSpt , the inclusion functor, is right adjoint to L2.
By [10, Rk. 3.11], ShvSpt has the following model category structure. A map
h: F ! G of sheaves of spectra is a weak equivalence (fibration) if and only if*
* i(f)
is a weak equivalence (fibration) of presheaves. Also, h is a cofibration of sh*
*eaves
of spectra if the following holds:
(1) the map h0: F0 ! G0 is a cofibration of simplicial presheaves; and
(2) for each n 0, the canonical map L2((S1 ^ Gn) [S1^Fn Fn+1) ! Gn+1 is
a cofibration of simplicial presheaves.
Since i preserves weak equivalences and fibrations, (L2, i) is a Quillen pair*
* for
(PreSpt , ShvSpt). Thus, for F 2 PreSpt , F ! L2F is a weak equivalence, and
Ho(PreSpt ) ~=Ho(ShvSpt ) is a Quillen equivalence.
Since Shv ~=TG , prolongation gives an equivalence of categories ShvSpt ~=
SptG, via the functors u*: ShvSpt ! SptG, and R: SptG ! ShvSpt , where
R(X) = Hom G(-, X).
Definition 3.5. For the remainder of this paper, if X is a discrete G-set, a si*
*m-
plicial discrete G-set, or a discrete G-spectrum, we let u*X = colimNXN .
Exploiting the above equivalence, we make SptG a model category in the follow-
ing way. Define a map f of discrete G-spectra to be a weak equivalence (fibrati*
*on)
if and only if Hom G(-, f) is a weak equivalence (fibration) of sheaves of spec*
*tra.
Also, define f to be a cofibration if and only if f has the left lifting proper*
*ty with
respect to all trivial fibrations. Thus, f is a cofibration if and only if Hom *
*G(-, f) is
a cofibration in ShvSpt . Using this, it is immediate that SptG is a model cate*
*gory,
and there is a Quillen equivalence Ho(ShvSpt ) ~=Ho(SptG).
In the theorem below, we define the model category structure of SptG without
reference to sheaves of spectra. This extends the model category structure on t*
*he
category SG (the category of simplicial objects in TG ), that is given in [11, *
*Thm.
1.12], to SptG.
Theorem 3.6. Let f :X ! Y be a map in SptG. Then f is a weak equivalence
(cofibration) in SptG if and only if f is a weak equivalence (cofibration) in S*
*pt.
Proof.For weak equivalences, the statement is clearly true. Assume that f is
a cofibration in SptG. Since Hom G(-, X0) ! Hom G (-, Y0) is a cofibration of
simplicial presheaves, evaluation at G=N implies that XN0 ! Y0N is a cofibration
in S. Thus, X0 ~=u*X0 ! u*Y0 ~=Y0 is a cofibration in S.
Since colimits commute with pushouts,
Hom G(-, (S1 ^ Yn) [S1^Xn Xn+1) ! Hom G(-, Yn+1)
is a cofibration of simplicial presheaves, and hence, the map of simplicial sets
u*((S1 ^ Yn) [S1^Xn Xn+1) ! Yn+1 is a cofibration.
Let W be a simplicial pointed discrete G-set. Then S1^W ~=colimN(S1^W N),
so that S1^W is also a simplicial pointed discrete G-set. Since the forgetful f*
*unctor
12 DANIEL G. DAVIS
U :TG ! Setsis a left adjoint, pushouts in TG are formed in Sets, and thus, the*
*re
is an isomorphism
u*((S1 ^ Yn) [S1^Xn Xn+1) ~=(S1 ^ Yn) [S1^Xn Xn+1
of simplicial discrete G-sets. Hence, (S1^Yn)[S1^Xn Xn+1 ! Yn+1 is a cofibration
in S, and f is a cofibration in Spt.
The converse follows from the fact that if j is an injection of simplicial di*
*screte
G-sets, then Hom G(-, j) is a cofibration of simplicial presheaves.
The preceding theorem implies the following two corollaries.
Corollary 3.7. If f :X ! Y is a weak equivalence (cofibration) in SptG, then, f*
*or
any K 0 and A is any discrete abelian
group [45, Lemma 9.4.5]. Also, recall that Xf = Xf,G(Definition 5.2).
Theorem 7.4. Let G be a profinite group with vcd(G) m, and let X be a discrete
G-spectrum. Then there are weak equivalences
Hom G (-, X) -'!Hom G (-, Xf) -'!holimHom G (-, oXf),
and holim Hom G(-, oXf) is a globally fibrant model for Hom G(-, X). Thus,
evaluation at * 2 G-Setsdf gives a weak equivalence XhG ! holim ( oXf)G .
Remark 7.5. The weak equivalence X ~= colimNXN ! colimN(XN )f, whose
target is a discrete G-spectrum that is fibrant in Spt, induces a weak equivale*
*nce
Xf,G! (colimN(XN )f)f,G. Thus, there are weak equivalences
Ho(*, Xf) ! Ho(*, (colimN(XN )f)f,G) Ho(*, colimN(XN )f),
so that Ho(*, colimN(XN )f) is a model for XhG that does not require the model
category SptG for its construction.
Proof of Theorem 7.4.Since Xf is fibrant in Spt, Xf is fibrant in SptG by Coro*
*l-
lary 3.8. By iteration, Hom G(-, oXf) is a cosimplicial globally fibrant presh*
*eaf,
so that holim Hom G (-, oXf) is globally fibrant, by Lemma 7.3. It only remains
to show that ~: Hom G(-, X) ! holim Hom G(-, oXf) is a weak equivalence.
By hypothesis, G contains an open subgroup H with cd(H) m. Then by
[45, Lem. 0.3.2], H contains a subgroup K that is an open normal subgroup of
G. Let {N} be the collection of open normal subgroups of G. Let N0 = N \ K.
Observe that {N0} is a cofinal subcollection of open normal subgroups of G so t*
*hat
G ~=limN0G=N0 . Since N0 m + 1, whenever M is a discrete N0-module. Henceforth, we drop the 0from
N0 to ease the notation: N is really N \ K.
Any presheaf of sets F has stalk colimNF(G=N), so that ~ is a weak equivalence
if ~u: X ~= colimNXN ! colimNholim ( oXf)N is a weak equivalence. Since
Hom G(-, oXf) is a cosimplicial globally fibrant spectrum, the diagram ( oXf)N
is a cosimplicial fibrant spectrum. Then, for each N, there is a conditionally *
*con-
vergent spectral sequence
(7.6) Es,t2(N) = ssssst(( oXf)N ) ) sst-s(holim( oXf)N ).
Because sst(X) is a discrete G-module, we have
Q
sst(Map c(G, Xf)N ) ~=sst(Map c(G=N, Xf)) ~= G=N sst(X) ~=Map c(G, sst(X))N
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 19
Q
and sst(Map c(G, Xf)) ~= sst(colimN G=NXf) ~= Map c(G, sst(X)). By iterating
such manipulations, we obtain ssssst(( oXf)N ) ~= Hs(( *sst(X))N ). The cochain
complex 0 ! sst(X) ! *sst(X) of discrete N-modules is exact (see e.g. [11, pp.
210-211]), and, for k 1 and s > 0,
Hsc(N; ksst(X)) ~=Hsc(N; Mapc(G, k-1sst(X))) = 0.
Thus, the above cochain complex is a resolution of sst(X) by (-)N -acyclic modu*
*les,
so that Es,t2(N) ~= Hsc(N; sst(X)). Taking a colimit over {N} of (7.6) gives the
spectral sequence
(7.7) Es,t2= colimNHsc(N; sst(X)) ) sst-s(colimNholim( oXf)N ).
Since Es,*2(N) = 0 whenever s > m+1, the E2-terms E2(N) are uniformly bounded
on the right. Therefore, by [29, Prop. 3.3], the colimit of the spectral sequen*
*ces
does converge to the colimit of the abutments, as asserted in (7.7).
Finally,
E*,t2~=H*c(limNN; sst(X))~=H*({e}; sst(X)),
which is isomorphic to sst(X), concentrated in degree zero. Thus, (7.7) collaps*
*es and
for all t, sst(colimN holim ( oXf)N ) ~=sst(X), and hence, ~u is a weak equival*
*ence.
Remark 7.8. Because of Theorem 7.4, if vcd(G) < 1 and X is a discrete G-
spectrum, we make the identification
XhG = holim( oGXf,G)G = Ho(*, Xf,G).
In [43, x2.14], an expression that is basically equivalent to holim ( oGXf,G)G
is defined to be the homotopy fixed point spectrum XhG , even if vcd(G) = 1.
This approach has the disadvantage that (-)hG need not always be the total right
derived functor of (-)G . Thus, we only make the identification of Remark 7.8 w*
*hen
vcd(G) < 1.
Now it is easy to construct the descent spectral sequence. Note that if X is a
discrete G-spectrum, the proof of Theorem 7.4 shows that
ssssst(( oGX)G ) ~=sss(( oGsst(X))G ) ~=Hsc(G; sst(X)).
Theorem 7.9. If vcd(G) < 1 and X is a discrete G-spectrum, then there is a
conditionally convergent descent spectral sequence
(7.10) Es,t2= Hsc(G; sst(X)) ) sst-s(XhG ).
Proof.As in Theorem 7.4, ( oXf)G is a cosimplicial fibrant spectrum. Thus, we
can form the homotopy spectral sequence for ss*(holim ( oXf)G ).
Remark 7.11. Spectral sequence (7.10) has been constructed in other contexts:
for simplicial presheaves, presheaves of spectra, and SG , see [21, Cor. 3.6],*
* [23,
x6.1], and [11, xx4, 5], respectively. In several of these examples, a Postnik*
*ov
tower provides an alternative to the hypercohomology spectrum that we use. In
all of these constructions of the descent spectral sequence, some kind of finit*
*eness
assumption is required in order to identify the homotopy groups of the abutment
as being those of the homotopy fixed point spectrum.
20 DANIEL G. DAVIS
Let X be a discrete G-spectrum. We now develop a second model for XhK ,
where K is a closed subgroup of G, that is functorial in K.
The map X ! Xf,G in SptK gives a weak equivalence XhK ! (Xf,G)hK .
Composition with the weak equivalence (Xf,G)hK ! holim ( oK((Xf,G)f,K))K
gives a weak equivalence XhK ! holim ( oK((Xf,G)f,K))K between fibrant spec-
tra. The inclusion K ! G induces a morphism G (Xf,G) ! K (Xf,G), giving a
map oG(Xf,G) ! oK(Xf,G) of cosimplicial discrete K-spectra.
Lemma 7.12. There is a weak equivalence
ae: holim( oG(Xf,G))K ! holim( oK(Xf,G))K ! holim( oK((Xf,G)f,K))K .
Proof.Recall the conditionally convergent spectral sequence
Hsc(K; sst(X)) ~=Hsc(K; sst((Xf,G)f,K)) ) sst-s(holim( oK((Xf,G)f,K))K ).
We compare this spectral sequence with the homotopy spectral sequence for
holim ( oG(Xf,G))K . Note that if Y isQa discrete G-spectrum that is fibrant as*
* a
spectrum, then Map c(G, Y ) ~=colimN G=NY and
Q
Map c(G, Y )K ~=Map c(G=K, Y ) ~=colimN G=(NK)Y
are fibrant spectra. Thus, ( oG(Xf,G))K is a cosimplicial fibrant spectrum, a*
*nd
there is a conditionally convergent spectral sequence
Es,t2= Hs(( *Gsst(Xf,G))K ) ) sst-s(holim( oG(Xf,G))K ).
As in the proof of Theorem 7.4, 0 ! sst(Xf,G) ! *G(sst(Xf,G)) is a (-)K -acycl*
*ic
resolution of sst(Xf,G), and thus, we have Es,t2~=Hsc(K; sst(Xf,G)) ~=Hsc(K; ss*
*t(X)).
Since ae is compatible with the isomorphism between the two E2-terms, the spe*
*c-
tral sequences are isomorphic and ae is a weak equivalence.
Remark 7.13. Lemma 7.12 gives the following weak equivalences between fibrant
spectra:
XhK = (Xf,K)K ! holim( oK((Xf,G)f,K))K holim( oG(Xf,G))K .
Thus, if vcd(G) < 1, X is a discrete G-spectrum, and K is a closed subgroup of
G, then holim ( oG(Xf,G))K is a model for XhK , so that
XhK = holim( oG(Xf,G))K
is another definition of the homotopy fixed points.
This discussion yields the following result.
Theorem 7.14. If X is a discrete G-spectrum, with vcd(G) < 1, then there is a
presheaf of spectra P (X): (OG )op ! Spt, defined by
P (X)(G=K) = holim( oG(Xf,G))K = XhK .
Proof.If Y is a discrete G-set, any morphism f :G=H ! G=K, in OG , induces a
map
Map c(G, Y )K ~=Map c(G=K, Y ) ! Map c(G=H, Y ) ~=Map c(G, Y )H .
Thus, if Y 2 SptG, f induces a map Map c(G, Y )K ! Map c(G, Y )H , so that ther*
*e is
a map P (X)(f): holim ( oG(Xf,G))K ! holim ( oG(Xf,G))H . It is easy to check
that P (X) is actually a functor.
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 21
We conclude this section by pointing out a useful fact: smashing with a finite
spectrum, with trivial G-action, commutes with taking homotopy fixed points. To
state this precisely, we first define the relevant map.
Let X be a discrete G-spectrum and let Y be any spectrum with trivial G-actio*
*n.
Then there is a map
(holim( oGXf)G ) ^ Y ! holim(( oGXf)G ^ Y ) ! holim(( oGXf) ^ Y )G .
Also, there is a natural G-equivariant map Map c(G, X) ^ Y ! Map c(G, X ^ Y )
that is defined by the composition
Q Q Q
(colimN G=N X) ^ Y ~=colimN(( G=N X) ^ Y ) ! colimN G=N (X ^ Y ),
Q
by using the isomorphism Map c(G, X) ~=colimN G=N X. This gives a natural G-
equivariant map ( G G X)^Y ! G (( G X)^Y ) ! G G (X^Y ). Thus, iteration
gives a G-equivariant map ( oX) ^ Y ! o(X ^ Y ) of cosimplicial spectra. Hence,
if vcd(G) < 1, XhG ^ Y ! (Xf ^ Y )hG is a canonical map that is defined by
composing XhG ^ Y ! holim (( oXf) ^ Y )G , from above, with the map
holim(( oXf) ^ Y )G ! holim( o(Xf ^ Y ))G ! holim( o(Xf ^ Y )f)G .
Lemma 7.15 ([29, Prop. 3.10]). If vcd(G) < 1, X 2 SptG, and Y is a finite
spectrum with trivial G-action, then XhG ^Y ! (Xf,G^Y )hG is a weak equivalence.
Remark 7.16. By Lemma 7.15, when Y is a finite spectrum, there is a zigzag of
natural weak equivalences XhG ^ Y ! (Xf,G^ Y )hG (X ^ Y )hG. We refer to
this zigzag by writing XhG ^ Y ~=(X ^ Y )hG.
8. Homotopy fixed points for towers in SptG
In this section, {Zi} is always in tow(SptG) (except in Definition 8.7). For *
*{Zi} a
tower of fibrant spectra, we define the homotopy fixed point spectrum (holimiZi*
*)hG
and construct its descent0spectral sequence. Also, recall from x5 that, if G is*
* finite
and X 2 SptG, then Xh G = holimGXf, where X ! Xf is a weak equivalence that
is G-equivariant, with Xf fibrant in Spt.
Definition 8.1. If {Zi} in tow (SptG) is a tower of fibrant spectra, we define
Z = holimiZi, a continuous G-spectrum. The homotopy fixed point spectrum ZhG
is defined to be holimiZhGi, a fibrant spectrum.
We make some comments about Definition 8.1. Let H be a closed subgroup
of G. Then the map holimi((Zi)f)H ! holimiholim ( oH(Zi)f)H and the map
holimiholim ( oG(Zi)f,G)H ! holimiholim ( oH((Zi)f,G)f,H)H are weak equiva-
lences. Thus, in Definition 8.1, each of our three definitions for homotopy fi*
*xed
points (Definition 5.2, Remarks 7.8, 7.13) can be used for ZhHi.
In Definition 8.1, suppose that not all the Zi are fibrant in Spt. Then the m*
*ap
Z = holimiZi ! holimi(Zi)f,{e}= holimiZh{e}i= Zh{e} need not be a weak
equivalence. Thus, for an arbitrary tower in SptG, Definition 8.1 can fail to h*
*ave
the desired property that Z ! Zh{e}is a weak equivalence.
Below,0Lemmas 8.2 and 8.3, and Remark 8.4, show that when G is a finite group,
ZhG ' Zh G, and, for any G, ZhG can be obtained by using a total right derived
functor that comes from fixed points. Thus, Definition 8.1 generalizes the noti*
*on
of homotopy fixed points to towers of discrete G-spectra.
22 DANIEL G. DAVIS
Lemma 8.2. Let G be a finite group and let {Zi} in tow (SptG)0be a tower of
fibrant spectra. Then there is a weak equivalence ZhG ! Zh G.
Proof.It is not hard to see that the map ZhG ! Zh0G can be defined to be
holimilimG(Zi)f ! holimiholimG(Zi)f ~=holimGholimi(Zi)f,
which is easily seen to be a weak equivalence.
In the lemma below, whose elementary proof is omitted, the functor
R(limi(-)G ): Ho(tow (SptG)) ! Ho(Spt)
is the total right derived functor of the functor limi(-)G :tow(SptG) ! Spt.
Lemma 8.3. If {Zi} is an arbitrary tower in SptG, then
holimi((Zi)f)G -'! holimi((Zi)0f)G -' limi((Zi)0f)G = R(limi(-)G )({Zi}).
Remark 8.4. By Lemma 5.3, if X 2 SptG, then XhG = (R(-)G )(X). Also, by
Lemma 8.3, if {Zi} in tow(SptG) is a tower of fibrant spectra, then
ZhG = holimiZhGi= holimi((Zi)f)G ~=R(limi(-)G )({Zi}).
Thus, the homotopy fixed point spectrum ZhG is again given by the total right
derived functor of an appropriately defined functor involving G-fixed points.
Given any tower in SptG of fibrant spectra, there is a descent spectral seque*
*nce
whose E2-term is a version of continuous cohomology.
Theorem 8.5. If vcd(G) < 1 and {Zi} in tow(SptG) is a tower of fibrant spectra,
then there is a conditionally convergent descent spectral sequence
(8.6) Hscont(G; {sst(Zi)}) ) sst-s(ZhG ).
We omit the proof of Theorem 8.5, since it is a special case of [8, Prop. 3.1*
*.2],
and also because (8.6) is not our focus of interest. However, we point out that
spectral sequence (8.6), whose construction goes back to the `-adic descent spe*
*ctral
sequence of algebraic K-theory ([41], [29]), is the homotopy spectral sequence
Es,t2= lims sst(( oG((Zi)f,G))G ) ) sst-s(holim( oG((Zi)f,G))G ).
x{i} x{i}
For our applications, instead of spectral sequence (8.6), we are more interes*
*ted in
descent spectral sequence (8.9) below. Spectral sequence (8.9), a homotopy spec*
*tral
sequence for a particular cosimplicial spectrum, is more suitable for compariso*
*n with
the K(n)-local En-Adams spectral sequence (see [5, Prop. A.5]), when (8.9) has
abutment ss*((En ^ X)hG), where X is a finite spectrum.
Definition 8.7. If {Zi} is a tower of spectra such that {sst(Zi)} satisfies the*
* Mittag-
Leffler condition for every t 2 Z, then {Zi} is a Mittag-Leffler tower of spect*
*ra.
Theorem 8.8. If vcd(G) < 1 and {Zi} in tow(SptG) is a tower of fibrant spectra,
then there is a conditionally convergent descent spectral sequence
(8.9) Es,t2= ssssst(holimi( oG(Zi)f)G ) ) sst-s(ZhG ).
If {Zi} is a Mittag-Leffler tower, then Es,t2~=Hscont(G; {sst(Zi)}).
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 23
Remark 8.10. In Theorem 8.8, when {Zi} is a Mittag-Leffler tower, spectral
sequence (8.9) is identical to (8.6). However, in general, spectral sequences (*
*8.6)
and (8.9) are different. For example, if G = {e}, then in (8.6), E0,t2= limisst*
*(Zi),
whereas in (8.9), E0,t2= sst(holimiZi).
Proof of Theorem 8.8.Note that ZhG ~= holim holimi( oG(Zi)f)G , and the dia-
gram holimi( oG(Zi)f)G is a cosimplicial fibrant spectrum.
Let {Zi} be a Mittag-Leffler tower. For k 0, Lemma 2.10 and Remark 2.11
imply that lim1iMapc(Gk, sst+1(Zi)) = 0. Therefore,
sst(holimi(Map c(Gk+1, (Zi)f))G ) ~=limiMapc(Gk+1, sst(Zi))G ,
and hence, sst(holimi( o(Zi)f)G ) ~=limi( osst(Zi))G . Thus,
Es,t2~=sss(limi( osst(Zi))G ) ~=Hs(limi(-)G { *sst(Zi)}i).
Consider the exact sequence {0} ! {sst(Zi)} ! { *sst(Zi)} in tow (CG ). Note
that, for s, k > 0, by Theorem 2.16,
Hscont(G; { ksst(Zi)}) ~=limiHsc(G; ksst(Zi)) = 0,
since the tower { ksst(Zi)} satisfies the Mittag-Leffler condition, and, for ea*
*ch i,
ksst(Zi) ~=Map c(G, k-1sst(Zi)) is (-)G -acyclic. Thus, the above exact seque*
*nce
is a (limi(-)G )-acyclic resolution of {sst(Zi)}, so that we obtain the isomorp*
*hism
Es,t2~=Hscont(G; {sst(Zi)}).
By Remark 8.4, we can rewrite spectral sequence (8.9), when {Zi} is a Mittag-
Leffler tower, in a more conceptual way:
Rs(limi(-)G ){sst(Zi)} ) sst-s(R(limi(-)G )({Zi})).
Spectral sequence (8.6) can always be written in this way.
9.Homotopy fixed point spectra for E_(X)
Recall that E_(X) = ^L(En ^ X). In this section, for any spectrum X and for
G 0, and equals Map c(Gj-1n, sst(En)), when s = 0. Thus,
ss*((^L(E(j+1)n))hGn) ~=Map c(Gj-1n, ss*(En)) ~=ss*(^L(E(j)n)),
as abelian groups. Therefore, for j 1, there is an isomorphism
(^L(E(j+1)n))hGn ~=^L(E(j)n).
The techniques described in this paper do not allow us to handle the j = 0 case,
which would say that EhGnnand ^L(S0) ' EdhGnnare isomorphic.
HOMOTOPY FIXED POINTS FOR LK(n)(En ^ X) 27
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Purdue University, Department of Mathematics, 150 N. University Street, West
Lafayette, IN 47907-2067, USA
E-mail address: dgdavis@math.purdue.edu