Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action

0.39 MB·English

by  Daniel G. Davis

HOMOTOPY FIXED POINTS FOR L (E ∧X) USING THE
K(n) n
CONTINUOUS ACTION
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DANIELG.DAVIS∗
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6 Abstract. LetGbeaclosedsubgroupofGn,theextendedMoravastabilizer
2 group. Let En be the Lubin-Tate spectrum, X an arbitrary spectrum with
trivial G-action, and let Lˆ =LK(n). We prove that Lˆ(En∧X) is a continu-
T] ousG-spectrumwithhomotopyfixedpointspectrum(Lˆ(En∧X))hG,defined
with respect to the continuous action. Also, we construct a descent spectral
A sequencewhoseabutmentisπ∗((Lˆ(En∧X))hG).Weshowthatthehomotopy
h. fixed points of Lˆ(En∧X) come from the K(n)-localization of the homotopy
t fixedpointsofthespectrum(Fn∧X).
a
m
1. Introduction
[

1 Let En be the Lubin-Tate spectrum with En∗ = W[[u1,...,un−1]][u±1], where
v the degree of u is −2, and the complete power series ring over the Witt vectors
4 W = W(F ) is in degree zero. Let S denote the nth Morava stabilizer group,
pn n
7 the automorphism group of the Honda formal group law Γ of height n over F .
n pn
4
Then let G = S ⋊Gal, where Gal is the Galois group Gal(F /F ), and let G
1 n n pn p
be a closed subgroup of G . Morava’s change of rings theorem yields a spectral
0 n
5 sequence
0
h/ (1.1) Hc∗(Gn;π∗(En∧X))⇒π∗Lˆ(X),
t where the E -term is continuous cohomology, X is a finite spectrum, and Lˆ is
a 2
m Bousfield localization with respect to K(n) = F [v ,v−1], where K(n) is Morava
∗ p n n
K-theory(see[15,Prop. 7.4],[32]). UsingtheG -actiononE bymapsofcommu-
: n n
v tative S-algebras(work ofGoerss andHopkins ([12], [13]), andHopkins and Miller
Xi [34]), Devinatz and Hopkins [6] construct spectra EhG with strongly convergent
n
spectral sequences
r
a
(1.2) Hs(G;π (E ∧X))⇒π (EhG∧X).
c t n t−s n
Also, they show that EhGn ≃Lˆ(S0), so that EhGn ∧X ≃Lˆ(X).
n n
We recall from [6, Rk. 1.3] the definition of the continuous cohomology that
appears above (see also Lemma 2.16). Let I = (p,u ,...,u ) ⊂ E . The iso-
n 1 n−1 n∗
morphismπ (E ∧X)=lim π (E ∧X)/Ikπ (E ∧X)presentsπ (E ∧X)asthe
t n k t n n t n t n
inverse limit of finite discrete G-modules. Then the above continuous cohomology
is defined by
Hs(G;π (E ∧X))=lim Hs(G;π (E ∧X)/Ikπ (E ∧X)).
c t n k c t n n t n
Date:January12,2005.
∗
Theauthor wassupportedbyNSFgrantDMS-9983601.
1
2 DANIELG.DAVIS
We compare the spectrum EhG and spectral sequence (1.2) with constructions
n
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 YhK =
Map (EK ,Y), whereEK is a free contractible K-space. Also, there is a descent
K +
spectral sequence
Es,t =Hs(K;π (Y))⇒π (YhK),
2 t t−s
where the E -term is group cohomology [29, §1.1].
2
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 ×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 Y is a simplicial discrete K-set (that is, for each l ≥ 0, Y is a
k k,l
discrete K-set, and all the face and degeneracy maps are K-equivariant). Then,
due to work of Jardine and Thomason, as explained in Sections 5 and 7, there is a
homotopy fixed point spectrum YhK defined with respect to the continuous action
of K, and, in nice situations, a descent spectral sequence
Hs(K;π (Y))⇒π (YhK),
c t t−s
where the E -term is the continuous cohomology of K with coefficients in the
2
discrete K-module π (Y).
t
Notice that we use the notation EhG for the construction of Devinatz and Hop-
n
kins, and (−)hK for homotopy fixed points with respect to a continuous action,
althoughhenceforth,whenK isfiniteandY isaK-spectrumoftopologicalspaces,
we write Yh′K for holim Y, which is an equivalent definition of the homotopy
K
fixed point spectrum Map (EK ,Y).
K +
After comparing the spectral sequence for EhG ∧X with the descent spectral
n
sequenceforYhK,E ∧X appearstobeacontinuousG -spectrumwith“descent”
n n
spectralsequencesfor“homotopyfixedpointspectra”EhG∧X.Indeed,weapply[6]
n
toshowthatE ∧X is acontinuousG -spectrum; thatis,E ∧X isthe homotopy
n n n
limit of a tower of fibrant discrete G -spectra. Using this continuous action, we
n
define the homotopy fixed point spectrum (E ∧X)hG and construct its descent
n
spectral sequence.
Inmoredetail,the K(n)-localspectrumE hasanactionbythe profinitegroup
n
G through maps of commutative S-algebras. The spectrum EhG, a K(n)-local
n n
commutativeS-algebra,isreferredtoasa“homotopyfixedpointspectrum”because
it has the following desired properties: (a) for any finite spectrum X, there exists
spectral sequence (1.2), which has the form of a descent spectral sequence; (b)
whenGisfinite, thereisaweakequivalenceEhG →Eh′G,andthedescentspectral
n n
sequenceforEh′G isisomorphictospectralsequence(1.2)(whenX =S0)[6,Thm.
n
3]; and(c) EhG is an N(G)/G-spectrum, where N(G) is the normalizerof G in G
n n
[6, pg. 5]. Thus, their constructions strongly suggest that G acts on E in a
n n
continuous sense.
However,in[6],the actionofG onE is notproventobe continuous,andEhG
n n n
is not defined with respect to a continuous G-action. Also, when G is profinite,
homotopy fixed points should always be the total right derived functor of fixed
points, in some sense, and, in [6], it is not shown that the “homotopy fixed point
spectrum” EhG can be obtained through such a total right derived functor.
n
After introducing some notation, we state the main results of this paper. Let
BP be the Brown-Petersonspectrum with BP =Z [v ,v ,...], where the degree
∗ (p) 1 2
HOMOTOPY FIXED POINTS FOR LK(n)(En∧X) 3
of vi is 2(pi − 1). The ideal (pi0,v1i1,...,vnin−−11) ⊂ BP∗ is denoted by I; MI is
the corresponding generalized Moore spectrum M(pi0,v1i1,...,vnin−−11), a trivial Gn-
spectrum. Given an ideal I, M need not exist; however, enough exist for our
I
constructions. The spectrum M is a finite type n spectrum with BP (M ) ∼=
I ∗ I
BP /I. The set {i ,...,i } of superscripts varies so that there is a family of
∗ 0 n−1
ideals {I}. ([4, §4], [19, §4], and [27, Prop. 3.7] provide details for our statements
about the spectra M .) The map r: BP → E - defined by r(v ) = u u1−pi,
I ∗ n∗ i i
where u = 1 and u = 0, when i > n - makes E a BP -module. Then, by the
n i n∗ ∗
Landweber exact functor theorem for BP, π (E ∧M )∼=E /I.
∗ n I n∗
The collection {I} contains a descending chain of ideals {I ⊃ I ⊃ I ⊃ ···},
0 1 2
such that there exists a correspondingtower of generalizedMoore spectra {M ←
I0
M ← M ← ···}. In what follows, the functors lim and holim are always
I1 I2 I I
taken over the tower of ideals {I }, so that lim and holim are really lim and
i I I Ii
holim , respectively. Also, in this paper, the homotopy limit of spectra, holim, is
Ii
constructedlevelwise in S, the categoryof simplicial sets, as defined in [3] and [41,
5.6].
Asin[6,(1.4)],letG =U (cid:13)U (cid:13)···(cid:13)U (cid:13)··· beadescendingchainofopen
n 0 1 i
normalsubgroups,suchthatT U ={e}andthecanonicalmapG →lim G /U
i i n i n i
is a homeomorphism. We define
F =colim EhUi.
n i n
Then the key to getting our work started is knowing that
E ∧M ≃F ∧M ,
n I n I
and thus, E ∧M has the homotopy type of the discrete G -spectrum F ∧M .
n I n n I
This result (Corollary 6.5) is not difficult, thanks to the work of Devinatz and
Hopkins.
Given a tower {Z } of discrete G -spectra, there is a tower {(Z ) }, with G -
I n I f n
equivariant maps Z → (Z ) that are weak equivalences, and (Z ) is a fibrant
I I f I f
discrete G -spectrum (see Def. 4.1). For the remainder of this section, X is any
n
spectrumwith trivialG -action. We use ∼= to denote anisomorphisminthe stable
n
homotopy category.
Theorem 1.3. As the homotopy limit of a tower of fibrant discrete G -spectra,
n
E ∼=holim (F ∧M ) is a continuous G -spectrum. Also, for any spectrum X,
n I n I f n
Lˆ(E ∧X)∼=holim (F ∧M ∧X) is a continuous G -spectrum.
n I n I f n
Using the hypercohomology spectra of Thomason [41], and the theory of pre-
sheaves of spectra on a site and globally fibrant models, developed by Jardine
(e.g. [21], [22], [23], [24]), we present the theory of homotopy fixed points for
discrete G-spectra by considering the site of finite discrete G-sets. Using the fact
that G has finite virtual cohomologicaldimension, Thomason’s hypercohomology
n
spectrumgivesaconcretemodelforthehomotopyfixedpointsthatmakesbuilding
the descent spectral sequence easy. We apply this theory to define homotopy fixed
points for towers of discrete G-spectra.
We point out that much of the theory described above (in Sections 3, 5, 7, and
8, through Remark 8.8) 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, §5]), and in Goerss’s paper [11]. However, since the above
4 DANIELG.DAVIS
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.
After defining homotopy fixed points for towers of discrete G-spectra, we show
that these homotopy fixed points are the total right derivedfunctor of fixed points
intheappropriatesense,andweconstructtheassociateddescentspectralsequence.
This enables us to define the homotopy fixed point spectrum (Lˆ(E ∧X))hG, us-
n
ing the continuous G -action, and construct its descent spectral sequence. More
n
specifically, we have the following results.
Definition 1.4. GivenaprofinitegroupG,letO betheorbit category ofG. The
G
objectsofO arethe continuousleftG-spacesG/K,forallK closedinG, andthe
G
morphismsare the continuousG-equivariantmaps. Note thateachobjectin O is
G
a profinite space.
We use Sp to denote the model category (spectra)stable of Bousfield-Friedlander
spectra.
Theorem 1.5. There is a functor
P: (O )op →Sp, P(G /G)=EhG,
Gn n n
where G is any closed subgroup of G .
n
We also show that the G-homotopy fixed points of Lˆ(E ∧X) can be obtained
n
by taking the K(n)-localization of the G-homotopy fixed points of the discrete
G-spectrum (F ∧X). This result shows that the spectrum F is an interesting
n n
spectrum that is worth further study.
Theorem 1.6. For any closed subgroup G and any spectrum X, there is an iso-
morphism
(Lˆ(E ∧X))hG ∼=Lˆ((F ∧X)hG)
n n
in the stable homotopy category. In particular, EhG ∼=Lˆ(FhG).
n n
Theorem 1.7. Let G be a closed subgroup of G and let X be any spectrum. Then
n
there is a conditionally convergent descent spectral sequence
(1.8) Es,t ⇒π ((Lˆ(E ∧X))hG).
2 t−s n
If the tower of abelian groups {π (E ∧M ∧X)} satisfies the Mittag-Leffler con-
t n I I
dition, for each t∈Z, then
Es,t ∼=Hs (G;π (Lˆ(E ∧X))),
2 cts t n
the cohomology of continuous cochains. If X is a finite spectrum, then (1.8) has
the form
(1.9) Hs(G;π (E ∧X))⇒π ((E ∧X)hG),
c t n t−s n
where the E -term is the continuous cohomology of (1.2).
2
Also,Theorem9.9showsthat, whenX is finite,(E ∧X)hG ∼=EhG∧X,sothat
n n
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 other.
Also, the spectra EhG and EhG should be the same. We plan to say more about
n n
the relationship between EhG and EhG and their associated spectral sequences in
n n
future work.
HOMOTOPY FIXED POINTS FOR LK(n)(En∧X) 5
It is important to note that the model given here for E as a continuous G -
n n
spectrumisnotcompletelysatisfactory,sincethecontinuousactionisbymorphisms
thatarejustmapsofspectra. BecausetherearemodelsforE wheretheG -action
n n
isbyA -andE -mapsofringspectra,wewouldliketoknowthatsuchstructured
∞ ∞
actions are actually continuous.
We outline the contents of this paper. In Section 2, after establishing some
notation and terminology, we provide some background material and recall useful
facts. In Section 3, we study the model category of discrete G-spectra. In Section
4, we study towers of discrete G-spectra and we give a definition of continuous
G-spectrum. In Section 5, we define homotopy fixed points for discrete G-spectra
andstate somebasicfacts aboutthis concept. Section6showsthatE is acontin-
n
uous G -spectrum, proving the first half of Theorem 1.3. Section 7 constructs two
n
useful models of the G-homotopy fixed point spectrum, when G has finite virtual
cohomologicaldimension. Section8defineshomotopyfixedpointsfortowersofdis-
crete G-spectra, builds a descent spectral sequence in this setting, and shows that
these homotopy fixed points are a total right derived functor, in the appropriate
sense. Section 9 completes the proof of Theorem 1.3, studies (Lˆ(E ∧X))hG, and
n
proves Theorems 1.5 and 1.6. Section 10 considers the descent spectral sequence
for (Lˆ(E ∧X))hG and proves Theorem 1.7.
n
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 [6] with Mike Hopkins. I am
grateful to Halvard Fausk, Christian Haesemeyer, Rick Jardine, and Charles Rezk
for useful conversations.
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 ∼= lim G/N, the inverse limit over the open normal subgroups. The
N
notation H < G means that H is a closed subgroup of G. We use G to denote
c
arbitrary profinite groups and, specifically, closed subgroups of G .
n
Let C be a category. A tower {C } of objects in C is a diagram in C of the form
i
···→ C →C → ···→ C → C . We always use Bousfield-Friedlander spectra
i i−1 1 0
[2], except when another category of spectra is specified. If C is a model category,
then Ho(C) is its homotopy category. The phrase “stable category” always refers
to Ho(Sp).
In S, the category of simplicial sets, Sn = ∆n/∂∆n is the n-sphere. Given a
spectrum X, X(0) =S0, and for j ≥1, X(j) =X∧X∧···∧X, with j factors. L
n
denotes Bousfield localization with respect to E(n) =Z [v ,...,v ][v ,v−1].
∗ (p) 1 n−1 n n
Definition 2.1. [17, Def. 1.3.1] Let C and D be model categories. The functor
F: C →D is a left Quillen functor if F is a left adjoint that preserves cofibrations
and trivial cofibrations. The functor P: D →C is a right Quillen functor if P is a
right adjoint that preserves fibrations and trivial fibrations. Also, if F and P are
an adjoint pair and left and right Quillen functors, respectively, then (F,P) is a
Quillen pair for the model categories (C,D).
6 DANIELG.DAVIS
Recall [17, Lemma 1.3.10] that a Quillen pair (F,P) yields total left and right
derived functors LF and RP, respectively, which give an adjunction between the
homotopy categories Ho(C) and Ho(D).
WeuseMap (G,A)=Γ (A)todenotethesetofcontinuousmapsfromGtothe
c G
topologicalspaceA, whereAis oftenaset, equippedwith the discretetopology,or
a discrete abelian group. Instead of Γ (A), sometimes we write just Γ(A), when
G
the G is understood from context. Let (Γ )k(A) denote (Γ Γ ···Γ )(A), the
G G G G
application of Γ to A, iteratively, k+1 times, where k ≥0. Let Gk be the k-fold
G
productofGandletG0 =∗. Then,ifAisadiscretesetoradiscreteabeliangroup,
thereisaG-equivariantisomorphism(Γ )k(A)∼=Map (Gk+1,A)ofdiscreteG-sets
G c
(modules), where Map (Gk+1,M) has G-action defined by
c
(g′·f)(g ,...,g )=f(g g′,g ,g ,...,g ).
1 k+1 1 2 3 k+1
Also, we often write Γk(A), or ΓkA, for Map (Gk,A).
G c
LetAbeadiscreteabeliangroup. ThenMapℓ(Gk,A)isthediscreteG -module
c n n
of continuous maps Gk →A with action defined by
n
(g′·f)(g ,...,g )=f((g′)−1g ,g ,g ,...,g ).
1 k 1 2 3 k
It is helpful to note that there is a G -equivariant isomorphism of discrete G -
n n
modules
p: Mapℓ(Gk,A)→Map (Gk,A), p(f)(g ,g ,...,g )=f(g−1,g ,...,g ).
c n c n 1 2 k 1 2 k
Mapℓ(Gk,A) is also defined when A is an inverse limit of discrete abelian groups.
c n
Byatopological G-module,wemeananabelianHausdorfftopologicalgroupthat
isaG-module,withacontinuousG-action. NotethatifM =lim M istheinverse
i i
limit of a tower {M } of discrete G-modules, then M is a topological G-module.
i
Fortheremainderofthissection,werecallsomefrequentlyusedfactsanddiscuss
backgroundmaterial, to help get our work started.
In [6], Devinatz and Hopkins, using work by Goerss and Hopkins ([12], [13]),
and Hopkins and Miller [34], show that the action of G on E is by maps of
n n
commutative S-algebras. Previously, Hopkins and Miller had shown that G acts
n
on E by maps of A -ring spectra. However, the continuous action presented
n ∞
hereis not structured. As alreadymentioned, the startingpointfor the continuous
action is the spectrum F ∧M , which is not known to be an A -ring object in
n I ∞
the category of discrete G -spectra. Thus, we work in the unstructured category
n
Sp of Bousfield-Friedlander spectra of simplicial sets, and the continuous action is
simply by maps of spectra.
As mentioned above, [6] is written using E , the category of commutative S-
∞
algebras, and M , the category of S-modules (see [8]). However, [18, §4.2], [28,
S
§14, §19], and [36, pp. 529-530] show that M and Sp are Quillen equivalent
S
model categories [17, §1.3.3]. Thus, we can import the results of Devinatz and
Hopkins from M into Sp. For example, [6, Thm. 1] implies the following result,
S
where R+ is the category whose objects are finite discrete left G -sets and G
Gn n n
itself (a continuous profinite left G -space), and whose morphisms are continuous
n
G -equivariant maps.
n
Theorem 2.2 (Devinatz, Hopkins). There is a presheaf of spectra
F: (R+ )op →Sp,
Gn
HOMOTOPY FIXED POINTS FOR LK(n)(En∧X) 7
such that (a) for each S ∈ R+ , F(S) is K(n)-local; (b) F(G ) = E ; (c) for U
Gn n n
an open subgroup of Gn, EnhU :=F(Gn/U); and (d) F(∗)=EnhGn ≃LˆS0.
Now we define a spectrum that is essential to our constructions.
Definition 2.3. Let Fn = colimiEnhUi, where the direct limit is in Sp. Because
HomGn(Gn/Ui,Gn/Ui)∼=Gn/Ui, F makes EnhUi a Gn/Ui-spectrum. Thus, Fn is a
G -spectrum, and the canonical map η: F →E is G -equivariant.
n n n n
Note that Fn is the stalk of the presheaf of spectra F|(G−Setsdf)op, at the unique
point of the Grothendieck topos (see §3). The following useful fact is stated in [6,
pg. 9] (see also [38, Lemma 14]).
Theorem 2.4. For j ≥ 0, let X be a finite spectrum and regard Lˆ(E(j+1) ∧X)
n
as a G -spectrum, where G acts only on the leftmost factor of the smash product.
n n
Then there is a G -equivariant isomorphism
n
π (Lˆ(E(j+1)∧X))∼=Mapℓ(Gj,π (E ∧X)).
∗ n c n ∗ n
WereviewsomefrequentlyusedfactsaboutthefunctorL andhomotopylimits
n
of spectra. First, L is smashing,e.g. L X ≃X∧L S0, for any spectrum X, and
n n n
E(n)-localization commutes with homotopy direct limits [33, Thms. 7.5.6, 8.2.2].
Note that this implies that F is E(n)-local.
n
Definition 2.5. If ···→X →X →···→X →X isatowerofspectrasuch
i i−1 1 0
that each X is fibrant in Sp, then {X } is a tower of fibrant spectra.
i i
If {X } is a tower of fibrant spectra, then there is a short exact sequence
i
0→lim1 π (X )→π (holim X )→lim π (X )→0.
i m+1 i m i i i m i
Also,ifeachmapinthetowerisafibration,themaplim X →holim X isaweak
i i i i
equivalence. If J is a small category and the functor P: J → Sp is a diagram of
spectra,suchthatP isfibrantforeachj ∈J,thenholim P isafibrantspectrum.
j j j
Definition 2.6. There is a functor (−)f: Sp → Sp, such that, given Y in Sp, Yf
is a fibrant spectrum, and there is a natural transformation idSp → (−)f, such
that, for any Y, the map Y → Yf is a trivial cofibration. For example, if Y is a
G-spectrum, then Yf is also a G-spectrum, and the map Y →Yf is G-equivariant.
The following statement says that smashing with a finite spectrum commutes
with homotopy limits.
Lemma 2.7 ([42, pg. 96]). Let J be a small category, {Z } a J-shaped diagram of
j
fibrant spectra, and let Y be a finite spectrum. Then the composition
(holimjZj)∧Y →holimj(Zj ∧Y)→holimj(Zj ∧Y)f
is a weak equivalence.
We recall the result that is used to build towers of discrete G-spectra.
Theorem 2.8 ([16, §2], [5, Remark 3.6]). If X is an E(n)-local spectrum, then, in
the stable category, there is an isomorphism
LˆX ∼=holimI(X ∧MI)f.
Lemma 2.9 ([19, Lemma 7.2]). If X is any spectrum, and Y is a finite spectrum
of type n, then Lˆ(X ∧Y)≃Lˆ(X)∧Y ≃L (X)∧Y.
n
8 DANIELG.DAVIS
We recall some useful facts about compact p-adic analytic groups. Since S is
n
compact p-adic analytic, and G is an extension of S by the Galois group, G is
n n n
a compact p-adic analytic group [37, Cor. of Thm. 2]. Any closed subgroup of a
compact p-adic analytic group is also compact p-adic analytic [7, Thm. 9.6]. Also,
since the subgroup in S of strict automorphisms is finitely generated and pro-p,
n
[35, pp. 76, 124] implies that all subgroups in G of finite index are open.
n
Let the profinite groupG be a compact p-adic analytic group. Then G contains
an open subgroup H, such that H is a pro-p group with finite cohomological p-
dimension; that is, cd (H) = m, for some non-negative integer m (see [25, 2.4.9]
p
or the exposition in [39]). Since H is pro-p, cd (H) = 0, whenever q is a prime
q
different from p [45, Prop. 11.1.4]. Also, if M is a discrete H-module, then, for
s ≥ 1, Hs(H;M) is a torsion abelian group [35, Cor. 6.7.4]. These facts imply
c
that, for any discrete H-module M, Hs(H;M) = 0, whenever s > m+1. We
c
expressthisconclusionbysayingthatGhasfinite virtualcohomologicaldimension
and we write vcd(G)≤m. Also, if K is a closed subgroup of G, H∩K is an open
pro-p subgroupof K with cd (H∩K)≤m, so that vcd(K)≤m, and thus, m is a
p
uniform bound independent of K.
Now we state various results related to towers of abelian groups and continuous
cohomology. The lemmabelowfollowsfromthe factthatatowerofabeliangroups
satisfies the Mittag-Leffler condition if and only if the tower is pro-isomorphicto a
tower of epimorphisms [20, (1.14)].
Lemma 2.10. Let F: Ab → Ab be an exact additive functor. If {A } is a
i i≥0
tower of abelian groups that satisfies the Mittag-Leffler condition, then so does the
tower {F(A )}.
i
Remark 2.11. Let G be a profinite group. Then the functor
Map (G,−): Ab→Ab, A7→Map (G,A),
c c
is defined by giving A the discrete topology. The isomorphism Map (G,A) ∼=
c
colim Q A shows that Map (G,−) is an exact additive functor. Later, we
N G/N c
will use Lemma 2.10 with this functor.
ThenextlemmaisaconsequenceofthefactthatlimitsinAbandintopological
spaces are created in Sets.
Lemma 2.12. Let M = lim M be an inverse limit of discrete abelian groups,
α α
so that M is an abelian topological group. Let H be any profinite group. Then
Map (H,M)→lim Map (H,M ) is an isomorphism of abelian groups.
c α c α
Lemma 2.13. If X is a finite spectrum, G < G , and t any integer, then the
c n
abelian group π (E ∧M ∧X) is finite.
t n I
Proof. The starting point is the fact that π (E ∧M )∼=π (E )/I is finite. In the
t n I t n
stable homotopycategoryofCW-spectra, since X is a finite spectrum, there exists
some m such that X = Σl−mX whenever l ≥ m, and X is a finite complex.
l m m
Since π (E ∧M ∧X)∼=π (E ∧M ∧X )andX canbe built outofa finite
t n I t−m n I m m
number of cofiber sequences, the result follows. (cid:3)
Corollary 2.14 ([15, pg. 116]). If X is a finite spectrum, then π (E ∧X) ∼=
t n
lim π (E ∧M ∧X).
I t n I
HOMOTOPY FIXED POINTS FOR LK(n)(En∧X) 9
We recallthe definition of a second versionof continuous cohomology. If M is a
topologicalG-module,thenHs (G;M)isthesthcohomologygroupofthecochain
cts
complex M → Map (G,M) → Map (G2,M) → ··· of continuous cochains for a
c c
profinitegroupGwithcoefficientsinM (see[40, §2]). IfM isadiscreteG-module,
this is the usual continuous cohomology Hs(G;M). There is the following useful
c
relationship between these cohomology theories.
Theorem 2.15 ([20, (2.1), Thm. 2.2]). Let {M } be a tower of discrete G-
n n≥0
modules satisfying the Mittag-Leffler condition and let M =lim M as a topolog-
n n
ical G-module. Then, for each s≥0, there is a short exact sequence
0→lim1 Hs−1(G;M )→Hs (G;M)→lim Hs(G;M )→0,
n c n cts n c n
where H−1(G;−)=0.
c
Thenextresultisfrom[6,Rk. 1.3]andisduetothefactthat,whenX isafinite
spectrum, π (E ∧X) ∼= lim π (E ∧X)/Ikπ (E ∧X) is a profinite continuous
t n k t n n t n
Z [[G]]-module.
p
Lemma 2.16. If G is closed in G and X is a finite spectrum, then, for s≥0,
n
Hs (G;π (E ∧X))∼=lim Hs(G;π (E ∧X)/Ikπ (E ∧X))
cts t n k c t n n t n
=Hs(G;π (E ∧X)).
c t n
We make a few remarks about the functorial smash product in Sp, defined in
[23, Chps. 1, 2].
Definition 2.17. Given spectra X and Y, their smash product X∧Y is given by
(X∧Y) =X ∧Y and (X∧Y) =X ∧Y , where, for example, X ∈S ,
2k k k 2k+1 k k+1 k ∗
the category of pointed simplicial sets.
Since K ∧(−): S → S is a left adjoint, for any K in S , smashing with any
∗ ∗ ∗
spectrum in either variable commutes with colimits in Sp.
3. The model category of discrete G-spectra
A pointed simplicial discrete G-set is a pointed simplicial set that is a simplicial
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
X , for k ≥ 0, such that each simplicial set X is a pointed simplicial discrete
k k
G-set, and each bonding map S1∧X →X is G-equivariant (S1 has trivial G-
k k+1
action). Let Sp denote the category of discrete G-spectra, where the morphisms
G
are G-equivariant maps of spectra.
As with discrete G-sets, if X ∈ Sp , there is a G-equivariant isomorphism
G
X ∼=colim XN. Also,adiscreteG-spectrumX isacontinuousG-spectrumsince,
N
for allk,l≥0,the set X is a continuousG-space with the discrete topology,and
k,l
all the face and degeneracy maps are (trivially) continuous.
Definition 3.2. As in [23, §6.2], let G−Sets be the canonical site of finite
df
discrete G-sets. The pretopology of G−Sets is given by covering families of the
df
form{f : S →S},a finite set of G-equivariantfunctions in G−Sets for a fixed
α α df
S ∈G−Sets , such that ` S →S is a surjection.
df α α
10 DANIELG.DAVIS
We use Shv to denote the category of sheaves of sets on the site G−Sets .
df
Also, T signifies the category of discrete G-sets, and, as in [11], S is the cate-
G G
gory of simplicial objects in T . The Grothendieck topos Shv has a unique point
G
u: Sets→Shv. The left adjoint of the topos morphism u is
u∗: Shv →Sets, F 7→colim F(G/N),
N
with right adjoint
u : Sets→Shv, X 7→Hom (−,Map (G,X))
∗ G c
[23, Rk. 6.25]. The G-action on the discrete G-set Map (G,X) is defined by
c
(g ·f)(g′) = f(g′g), for g,g′ in G, and f a continuous map G → X, where X is
given the discrete topology.
The functor Map (G,−): Sets→T prolongs to the functor
c G
Map (G,−): Sp→Sp .
c G
Thus, if X is a spectrum, then Map (G,X) ∼= colim Q X is the discrete
c N G/N
G-spectrum with (Map (G,X)) =Map (G,X ), where Map (G,X ) is a pointed
c k c k c k
simplicial set, with l-simplices Map (G,X ) and basepoint G → ∗, where X is
c k,l k,l
regardedas a discreteset. The G-actiononMap (G,X) is definedby the G-action
c
on the sets Map (G,X ).
c k,l
It is not hard to see that Map (G,−) is right adjoint to the forgetful functor
c
U: Sp →Sp. NotethatifX isadiscreteG-spectrum,thenthereisacontravariant
G
functor(presheaf)Hom (−,X): (G−Sets )op →Sp,where,foranyS ∈G−Sets ,
G df df
Hom (S,X)isthespectrumwith(Hom (S,X)) =Hom (S,X ),apointedset
G G k,l G k,l
with basepoint S →∗.
LetShvSptbethecategoryofsheavesofspectraonthesiteG−Sets . Asheafof
df
spectraF isapresheafF: (G−Sets )op →Sp,suchthat,foranyS ∈G−Sets and
df df
any covering family {f : S →S}, the usual diagram (of spectra) is an equalizer.
α α
Equivalently,asheafofspectraF consistsofpointedsimplicialsheavesFn,together
with pointed maps of simplicial presheaves σ: S1∧Fn → Fn+1, for n ≥ 0, where
S1 is the constant simplicialpresheaf. A morphismbetween sheavesof spectra is a
natural transformation between the underlying presheaves.
The category PreSpt of presheaves of spectra on the site G−Sets has the
df
following “stable” model category structure ([22], [23, §2.3]). A map h: F → G
of presheaves of spectra is a weak equivalence if and only if the associated map of
stalkscolim F(G/N)→colim G(G/N) is a weak equivalence ofspectra. Recall
N N
that a map k of simplicial presheaves is a cofibration if, for each S ∈ G−Sets ,
df
k(S)is amonomorphismofsimplicialsets. Thenh is a cofibrationofpresheavesof
spectra if the following two conditions hold:
(1) the map h0: F0 →G0 is a cofibration of simplicial presheaves; and
(2) for each n ≥ 0, the canonical map (S1 ∧Gn)∪ Fn+1 → Gn+1 is a
S1∧Fn
cofibration of simplicial presheaves.
Fibrations are those maps with the right lifting property with respect to trivial
cofibrations.
Definition3.3. Inthestablemodelcategorystructure,fibrantpresheavesareoften
referred to as globally fibrant, and if F → G is a weak equivalence of presheaves,
with G globally fibrant, then G is a globally fibrant model for F. We often use GF
to denote such a globally fibrant model.