Explicit fibrant replacement for discrete G-spectra

0.2 MB·

by  Daniel G. Davis

EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE
G-SPECTRA
8
DANIELG.DAVIS
0
0
2 Abstract. If C isthe model category of simplicialpresheaves ona sitewith
enough points, with fibrations equal to the global fibrations, then it is well-
n
known that the fibrant objects are, in general, mysterious. Thus, it is not
a
surprisingthat, when G isaprofinitegroup, the fibrant objects inthemodel
J
category of discrete G-spectra are alsodifficult to get a handle on. However,
2
withsimplicialpresheaves,itispossibletoconstructanexplicitfibrantmodel
foranobjectinC,undercertainfinitenessconditions. Similarly,inthispaper,
T] weshowthatifGhasfinitevirtualcohomologicaldimensionandXisadiscrete
G-spectrum,thenthereisanexplicitfibrantmodelforX. Also,wegiveseveral
A applications ofthisconcretemodelrelatedtoclosedsubgroupsofG.
.
h
t
a
m
1. Introduction
[

In this paper, G always denotes a profinite group and, by “spectrum,” we mean
1
v a Bousfield-Friedlander spectrum of simplicial sets. In particular, a discrete G-
2 spectrum is a G-spectrum such that each simplicial set Xk is a simplicial object in
3 the category of discrete G-sets (thus, the action map on the l-simplices,
3
0 G×(Xk)l →(Xk)l,
.
1 iscontinuouswhen(X ) isregardedasadiscretespace,foralll ≥0). Thecategory
k l
0
of discrete G-spectra, with morphisms being the G-equivariant maps of spectra, is
8
denoted by Spt .
0 G
: Asshownin[3,Section3],SptGisasimplicialmodelcategory,whereamorphism
v
f in Spt is a weak equivalence (cofibration) if and only if f is a weak equivalence
i G
X (cofibration)inSpt,the simplicialmodelcategoryofspectra. GivenX ∈Spt ,the
G
r homotopyfixedpointspectrumXhGisthetotalrightderivedfunctoroffixedpoints:
a XhG = (X )G, where X → X is a trivial cofibration and X is fibrant, all
f,G f,G f,G
inSpt . This definitiongeneralizesthe classicaldefinitionofhomotopyfixedpoint
G
spectrum, in the case when G is a finite group (see [3, pg. 337]).
Notice that we can loosen up the requirements on X . If X → Xf is a weak
f,G
equivalence, with Xf fibrant, all in Spt , then, by the right lifting property of a
G
fibrant object, there is a weak equivalence X →Xf in Spt , so that
f,G G
XhG =(X )G →(Xf)G
f,G
is a weak equivalence. Thus, we can identify XhG and (Xf)G, and, hence, we only
need a fibrant replacement Xf to form XhG. Henceforth, we relabel any such Xf
as X and refer to it as a globally fibrant model for X. (Thus, from now on,
f,G
X →X does not have to be a cofibration.)
f,G
The preceding discussion shows that a globally fibrant model X is an impor-
f,G
tant object. Of course, the model category axioms guarantee that X always
f,G
1
2 DANIELG.DAVIS
exists. But it is reasonableto ask for more. For example, in Spt, there is a functor
Q: Spt→Spt, Z 7→Q(Z)=Zf,
where Zf is a fibrant spectrum,
(Zf)k =colimΩn(Ex∞(Zk+n)),
n
and there is a natural weak equivalence Z → Zf (see, for example, [13, pg. 524]).
Hence, for the model category Spt, there is always an explicit model for fibrant
replacement. Similarly, it is natural to wonder if an explicit model for X is
f,G
available.
Butthereisadifficultywiththis. LetG−Sets betheGrothendiecksiteoffinite
df
discrete G-sets (e.g., see [9, Section 6.2]). There is an equivalence between Spt
G
and the category of sheaves of spectra on G−Sets (the discrete G-spectrum X
df
correspondsto thesheafofspectraHom (−,X);see[3,Section3]fordetails), and
G
it is well-knownthat, in general, for categories of simplicial presheaves,presheaves
of spectra, and sheaves of spectra, there is no known explicit model for a globally
fibrant object. In fact, the situation is such that [7, pg. 1049] says that “[t]he
fibrant objects in all of these theories continue to be really quite mysterious” (a
similar statement appears in [10, between Corollary 19 and Definition 20]).
Nevertheless, under certain hypotheses, explicit models for globally fibrant ob-
jects are available in the cases of simplicial presheaves and presheaves of spectra.
Suchresults arebasedonJardine’sresultin[11, Proposition3.3],whichconstructs
anexplicitgloballyfibrantmodelforasimplicialpresheafP onthesite´et| ,where
S
P and the scheme S must satisfy certain finiteness conditions (and other hypothe-
ses). Forexample,undersimilarfinitenessconditions,[12,Proposition3.20]follows
theproofofJardine’sresulttoobtainaconcretegloballyfibrantmodelforapresheaf
of spectra on a site with enough points.
Now suppose that G has finite virtual cohomological dimension (see Definition
4.1) and that X is a discrete G-spectrum. In this paper, we show that there is an
explicit model for X , by expressing the homotopy limit for diagrams in Spt
f,G G
in terms of the homotopy limit for diagrams in Spt (see Theorem 2.3) and by
modifying the proofof [3, Theorem7.4](which applies the two results cited above,
[11,Proposition3.3]and[12,Proposition3.20]). WereferthereadertoTheorem4.2
for the precise statement of our main result; its formulationdepends ondefinitions
that are given in Section 3.
Let H be a closedsubgroupofG. If Y →Z is a weak equivalence inSpt , such
H
that Y is a globally fibrant model for X in Spt and Z is a globally fibrant model
G
for X in Spt , then we label the map Y →Z as rG. Note that the map
H H
X →(X ) ,
f,G f,G f,H
a weak equivalence in Spt , can be labelled as rG, so that rG always exists. In
H H H
Corollary4.7, we show that the explicit globally fibrantmodel constructedin The-
orem 4.2 yields an explicit model for rG (where, as before, we assume that G has
H
finite virtual cohomologicaldimension).
Section 5 explains that, when H is a closed normal subgroup of G and X is
a discrete G-spectrum, there are cases when XhH, unlike XH, is not known to
be a G/H-spectrum. In Corollary 5.4, we point out that, if G has finite virtual
cohomologicaldimension,thenTheorem4.2implies thatXhH canalwaysbe taken
to be a G/H-spectrum.
EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA 3
Throughout this paper, U < G means that U is an open subgroup of G.
o
Acknowledgements. I thank Mark Hovey for a helpful conversation.
2. Homotopy limits in the category of discrete G-spectra
To explicitly construct the desired fibrant discrete G-spectrum, we first need to
understand homotopy limits in the category of discrete G-spectra. Thus, we begin
this sectionby followingthe presentationin [8] to givethe generaldefinition of the
homotopylimitofaC-diagramX(−)inM,whereMisasimplicialmodelcategory
and X(−) is a diagram in M indexed by a small category C.
Recallthat, fora smallcategoryC,the classifying space ofC is the simplicialset
BC, with l-simplices (BC) equal to the set of compositions
l
c −σ→0 c −σ→1 ···σ−l→−1 c
0 1 l
in C (see, for example, [8, Definition 14.1.1] for the definition of the face and de-
generacy maps).
Definition 2.1 ([8, Definition 18.1.8]). As above, let M be a simplicial model
categoryand let C be a small category. Also, let X =X(−) be a C-diagramin M;
that is, X is a functor C → M, so that, for example, if c → d is a morphism in
C, then X(c) → X(d) is a morphism in M. Then the homotopy limit of X in M,
holimMX, is defined to be the equalizer of the diagram
C
α //
Q X(c)B(C↓c) // Q X(d)B(C↓c) ,
c∈C σ:c→d
β
where the secondproduct is indexed overallthe morphisms in C. Here, the map α
is defined as follows: the projection of α onto the factor indexed by σ: c → c is
0 1
equal to composing the projection
Q X(c)B(C↓c) →X(c )B(C↓c0)
c∈C 0
with the canonical map X(c )B(C↓c0) → X(c )B(C↓c0). The map β is defined by
0 1
letting the projectionof β onto the factor indexed by σ be givenby composing the
projection
Q X(c)B(C↓c) →X(c )B(C↓c1)
c∈C 1
with the canonical map X(c )B(C↓c1) → X(c )B(C↓c0) that is induced by the map
1 1
B(C↓c )→B(C↓c ).
0 1
Remark 2.2. Note that the homotopy limit is an equalizer of a diagram involv-
ing products and cotensors, and, given a simplicial set K, the cotensor functor
(−)K: M → M is a right adjoint. Hence, the homotopy limit holimM(−) com-
C
mutes with limits in M.
If X and Z are C-diagrams in Spt and Spt, respectively, then we use the
G
less cumbersome holimGX and holim Z to denote holimSptGX and holimSptZ,
C C C C
respectively. As in Definition 2.1, given c ∈ C, X(c) is the object in M that is
indexed by c in the diagram X.
Theorem 2.3. If X is a C-diagram in Spt , then there is an isomorphism
G
holimGX ∼=colim(holimX)N.
C N⊳oG C
4 DANIELG.DAVIS
Proof. Foreachc∈C,letB denotethe simplicialsetB(C↓c). GivenadiscreteG-
c
spectrumY andasimplicialsetK,let (YK) andYK denotethe cotensorobjects
G
in Spt and Spt, respectively. Also, let QG and Q denote products in Spt and
G G
Spt,respectively. Then,byDefinition2.1,holimGX istheequalizerofthediagram
C
α //
QGc∈C(X(c)Bc)G // QGσ:c→d(X(d)Bc)G
β
in Spt .
G
Togofurther,wenotehowlimitsandcotensorsinSpt areformed. Recallfrom
G
[3,Remark4.2]that,if{Y } isanydiagraminSpt ,thenthelimitofthisdiagram
α α G
in the categorySptG is colimN⊳oG(limαYα)N, where the limit in this expressionis
taken in the category of spectra. The colimit in this expression, and others like it,
canbe takeninthe categoryofspectra,sincethe forgetfulfunctor Spt →Sptis a
G
left adjoint, by [3, Corollary 3.8]. Given a discrete G-spectrum Y and a simplicial
set K, the spectrum YK can be regarded as a G-spectrum (but not necessarily a
discrete G-spectrum), by using only the G-action on Y. Then
(YK) =colim(YK)N
G
N⊳oG
(e.g., see [5, (1.2.2)] and [11, pg. 42]). We apply these observations as follows.
First of all, note that holimGX is the equalizer in Spt of the diagram
C G
α //
Nco⊳liomG(Qc∈C(X(c)Bc)G)N β // Nco⊳liomG(Qσ:c→d(X(d)Bc)G)N .
Furthermore, let S be a G-set (but not necessarily a discrete G-set) and let U be
an open normal subgroup of G. Then it is clear that (S SN)U ⊂ SU. Since
N⊳oG
U ∈ {N|N ⊳ G}, SU ⊂ S SN, and, hence, SU ⊂ (S SN)U. Thus, we
o N⊳oG N⊳oG
can conclude that
SU =(S SN)U.
N⊳oG
Similarly, if Y is a G-spectrum,
YU ∼=(colimYN)U.
N⊳oG
Therefore,
(Qc∈C(X(c)Bc)G)U =Qc∈C(Nco⊳liomG(X(c)Bc)N)U ∼=Qc∈C(X(c)Bc)U,
and, similarly,
(Qσ:c→d(X(d)Bc)G)U ∼=Qσ:c→d(X(d)Bc)U.
TheprecedingtwoisomorphismsimplythatholimGX isisomorphictotheequal-
C
izer in Spt of the diagram
G
α //
colim(Q X(c)Bc)N // colim(Q X(d)Bc)N.
N⊳oG c∈C β N⊳oG σ:c→d
Thus, holimGC X ∼=colimN⊳oGEN, where E is the equalizer in Spt of the diagram
α′ //
colimN⊳oG(Qc∈CX(c)Bc β′ // Qσ:c→dX(d)Bc)N,
where α′ and β′ are the maps in the equalizer diagram for holim X.
C
EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA 5
Since filteredcolimits andfinite limits commute, E ∼=colimN⊳oG(E′)N,whereE′
is the equalizer in Spt of the diagram
α′ //
Q X(c)Bc // Q X(d)Bc.
c∈C β′ σ:c→d
Notice that E′ =holim X.
C
If U is anopen normalsubgroupofG, then (holim X)U is a G/U-spectrum,so
C
thatcolimN⊳oG(holimCX)N isadiscreteG-spectrum. Also,givenY ∈SptG,there
isanisomorphismcolimN⊳oGYN ∼=Y. Therefore,puttingourvariousobservations
together, we obtain that
holimGX ∼=colimEN ∼=colim(colim(E′)N′)N
C N⊳oG N⊳oG N′⊳oG
∼= colim(E′)N′ =colim(holimX)N.
N′⊳oG N⊳oG C
(cid:3)
IfX isaC-diagramoffibrantdiscreteG-spectra(thatis,X(c)isfibrantinSpt ,
G
forallc∈C),thenholimGX isafibrantdiscreteG-spectrum,by[8,Theorem18.5.2
C
(2)], so that Theorem 2.3 gives the following result.
Corollary2.4. IfX isaC-diagram offibrantdiscreteG-spectra, thenthespectrum
colimN⊳oG(holimCX)N is a fibrant discrete G-spectrum.
We conclude this section with some observations about Corollary 2.4.
Definition 2.5. IfP isaC-diagramofpresheavesofspectraonthesiteG−Sets ,
df
then there is a presheaf of spectra holim P, defined by
C
(holimP)(S)=holimP(S),
C C
for each S ∈G−Sets .
df
LetX beaC-diagraminSpt . Thenitisnaturaltoformthepresheafofspectra
G
holim Hom (−,X). Also, let
C G
F =Hom (−,colim(holimX)N),
G
N⊳oG C
the canonical sheaf of spectra on the site G−Sets associated to the spectrum
df
colimN⊳oG(holimCX)N that is considered in Corollary 2.4.
The following lemma says that the presheaf holim Hom (−,X) is actually a
C G
sheaf of spectra, since it is isomorphic to F.
Lemma 2.6. If X is a C-diagram of discrete G-spectra, then the presheaves of
spectra F and holim Hom (−,X) on the site G−Sets are isomorphic.
C G df
Proof. Let S be a finite discrete G-set: S can be identified with a disjoint union
`m G/U , where each U is an open subgroup of G. Notice that the collection
j=1 j j
{N}N⊳oG of open normal subgroups of G is a cofinal subcollection of the collec-
tion {U} of open subgroups of G, so that, if Y is a G-spectrum, there is an
U<oG
6 DANIELG.DAVIS
isomorphism colimN⊳oGYN ∼=colimU<oGYU of G-spectra. Hence,
F(S)∼=Qm (colim(holimX)N)Uj
j=1 N⊳oG C
∼=Qm (colim(holimX)U)Uj
j=1 U<oG C
∼=Qm (holimX)Uj
j=1
C
∼=holimHom (S,X),
G
C
where the third isomorphism is due to the fact that U ∈ {U|U < G} (as in the
j o
proofof Theorem 2.3) and the lastisomorphismapplies Remark 2.2. This chain of
isomorphismsshowsthatthereisanisomorphismF(S)∼=holimCHomG(S,X)that
is natural for S ∈ G−Sets , so that F and holim Hom (−,X) are isomorphic
df C G
presheaves of spectra. (cid:3)
Remark 2.7. LetX be aC-diagramoffibrantdiscreteG-spectra. Thentheasser-
tion of Corollary 2.4 that colimN⊳oG(holimCX)N is a fibrant discrete G-spectrum
is equivalentto claiming thatF is a globallyfibrantpresheafofspectra (see [3, pg.
333]). Also, by Lemma 2.6, to show that F is globally fibrant, it suffices to show
thatholim Hom (−,X)isagloballyfibrantpresheaf. Thiscanbedonebyadapt-
C G
ing [11, Proposition 3.3] and [3, Lemma 7.3], since Hom (−,X) is a C-diagram
G
of globally fibrant presheaves of spectra. This gives a somewhat different way of
obtaining Corollary 2.4.
3. The explicit construction of a fibrant discrete G-spectrum
In this section, we use Corollary 2.4 to construct the fibrant object in Spt
G
that is our primary object of interest. We begin with several definitions that are
standard in the theory of discrete G-modules and discrete G-spectra.
Definition3.1. IfAisanabeliangroupwiththediscretetopology,letMap (G,A)
c
be the abelian group of continuous maps from G to A. If Z is a spectrum,
one can also define the discrete G-spectrum Map (G,Z), where the l-simplices
c
(Map (G,Z) ) of the kth simplicial set of the spectrum Map (G,Z) are given by
c k l c
Map (G,(Z ) ), the set of continuous maps from G to (Z ) . Here, (Z ) is given
c k l k l k l
the discrete topology and the G-action on Map (G,Z) is induced on the level of
c
sets by (g·f)(g′)=f(g′g), for g,g′ ∈G and f ∈Map (G,(Z ) ), for each k,l≥0.
c k l
This action also makes Map (G,A) a discrete G-module.
c
Definition 3.2. Consider the functor
Γ : Spt →Spt , X 7→Γ (X)=Map (G,X),
G G G G c
whereΓ (X)hastheG-actiongivenbyDefinition3.1. Asexplainedin[3,Definition
G
7.1], the functor Γ forms a triple and there is a cosimplicial discrete G-spectrum
G
Γ•X, where, for all n≥0,
G
(Γ•X)n ∼=Map (Gn+1,X).
G c
Here, the spectrum Map (Gn+1,X) is defined as in Definition 3.1, since the carte-
c
sian product Gn+1 is a profinite group, and its discrete G-action is given by the
G-action on the constituent sets that is given by
(g·f)(g ,g ,g ,...,g )=f(g g,g ,g ,...,g ).
1 2 3 n+1 1 2 3 n+1
EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA 7
The next definition restates Definition 3.2 in the context of discrete G-modules.
Definition 3.3. LetDMod(G) bethe categoryofdiscreteG-modules. Then,asin
Definition 3.2, there is a functor
Γ : DMod(G)→DMod(G), M 7→Γ (M)=Map (G,M),
G G c
and,givenadiscreteG-module M,thereis acosimplicialdiscrete G-module Γ•M.
G
Definition 3.4 ([3, Remark 7.5]). Given a discrete G-spectrum X, let
X =colim(XN)f.
b N⊳oG
Notice that X is a discrete G-spectrum, since functorialfibrantreplacementin Spt
b
(see the Introduction) implies that each (XN)f is a G/N-spectrum. Also, X is
b
fibrant as a spectrum and there is a weak equivalence
ψ: X ∼=colimXN →colim(XN)f =X
N⊳oG N⊳oG b
that is G-equivariant.
We define some useful terminology. If X• is a cosimplicial object in Spt , then
G
X•isacosimplicialdiscreteG-spectrum. IfX•isacosimplicialdiscreteG-spectrum
suchthatXnisafibrantdiscreteG-spectrum,foralln≥0,thenX•isacosimplicial
fibrant discrete G-spectrum.
The following result defines the explicit globally fibrant object that is of partic-
ular interest to us.
Theorem 3.5. Let G be a profinite group, and let H be a closed subgroup of G. If
X is a discrete G-spectrum, then the discrete H-spectrum
colim(holimΓ•X)K
K⊳oH ∆ G b
is fibrant in the model category of discrete H-spectra. In particular, the discrete
G-spectrum
colim(holimΓ•X)N
N⊳oG ∆ G b
is fibrant in Spt .
G
Proof. By Corollary 2.4, we only need to show that Γ•X is a cosimplicial fibrant
G b
discrete H-spectrum. By [14, Proposition 1.3.4 (c)], there is a homeomorphism
h: H×G/H →GthatisH-equivariant,whereH actsonthesourcebyactingonly
on the factor H and G/H is the profinite space G/H ∼=limN⊳oGG/NH.
Given a spectrum Z and a profinite space W = lim W , where each W is a
α α α
finite discrete space, as in Definition 3.1, we can form the spectrum Map (W,Z),
c
where (Map (W,Z) ) =Map (W,(Z )), and there is an isomorphism
c k l c k l
Map (W,Z)∼=colimQ Z.
c α w∈Wα
Thus,
(3.6) Map (G/H,X)∼=colimQ X.
c b N⊳oG G/NH b
Since filtered colimits commute with finite limits,
Q X ∼=Q colim(X)N′ ∼= colim(Q X)N′,
G/NH b G/NHN′⊳oG b N′⊳oG G/NH b
8 DANIELG.DAVIS
andit follows that Map (G/H,X)is a discrete G-spectrum, with G acting only on
c b
X. Therefore, by applying the homeomorphism h, there is an isomorphism
b
(3.7) Map (G,X)∼=Map (H,Map (G/H,X))
c b c c b
of discrete H-spectra.
Recall from [3, Corollary 3.8] that if Z is a fibrant spectrum, then Map (H,Z)
c
is a fibrant discrete H-spectrum. Also, since X is a fibrant spectrum, the product
b
Q X is also a fibrant spectrum, so that, by (3.6), Map (G/H,X) is fibrant
G/NH b c b
in Spt. Then, by applying these observations to (3.7), we obtain that Map (G,X)
c b
is a fibrant discrete H-spectrum. Hence, Map (G,X) is a fibrant spectrum, by [3,
c b
Lemma 3.10], so that Map (G,Map (G,X)) is a fibrant discrete H-spectrum, by
c c b
applyingthepreviousargumentagain. Thus,iterationofthisargumentshowsthat
Γ•X is a cosimplicial fibrant discrete H-spectrum. (cid:3)
G b
4. Completing the proof of the main result
In this section, we finish the proof of the main result. Also, we discuss several
consequencesof having a concrete model for X , givena discrete G-spectrum X.
f,G
Definition 4.1. Let H∗(G;M) denote the continuous cohomology of G with co-
c
efficients in the discrete G-module M. Then a profinite group G has finite virtual
cohomological dimension (or finite vcd) if there exists an open subgroup H of G
andanon-negativeintegerm, suchthat Hs(H;M)=0,for alldiscrete H-modules
c
M and all s≥m.
Many of the profinite groups that one works with, in practice, have finite vcd.
For example, if G is a compact p-adic analytic group, G has finite vcd (see the
discussion in [3, pg. 330]).
Let X be a discrete G-spectrum. Then there is a G-equivariant monomorphism
i: X →Map (G,X) that is defined, on the level of sets, by i(x)(g)=g·x. Then i
c
induces a map X →Γ•X of cosimplicial discrete G-spectra, where, here, X is the
G
constant diagram. Thus, the composition
X →ψ X →∼= limX →holimX →holimΓ•X
b b b G b
∆ ∆ ∆
of canonical maps defines the G-equivariant map
ψ: X →holimΓ•X
b G b
∆
(the canonical map lim X →holim X is defined in [8, Example 18.3.8 (2)]).
∆ b ∆ b
Note that there is a homotopy spectral sequence
Es,t =πs(π (Γ•X))⇒π (holimΓ•X),
2 t G b t−s G b
∆
whereE0,t ∼=π (X)andEs,t =0,whens>0,by[3,Section7]. Thus,the spectral
2 t 2
sequence collapses, so that the map ψ is a weak equivalence.
b
NowletH be aclosedsubgroupofG. ThenX isadiscreteH-spectrum,sothat
X ∼=colimK⊳oHXK. Composing this isomorphism with the map colimK⊳oH(ψb)K
gives the H-equivariant map
Ψ: X → colim(holimΓ•X)K.
K⊳oH ∆ G b
EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA 9
NowweshowthatifGhasfinitevcd,thenΨisaweakequivalence. Asmentioned
inthe Introduction, the proof(below) closely followsthe proofof[3, Theorem7.4],
so that our proof will be somewhat abbreviated. Also, we should mention that the
proof of [3, Theorem 7.4] follows the arguments given in [11, proof of Proposition
3.3] and [12, Proposition 3.20].
Theorem 4.2. Let G have finite vcd, let X be a discrete G-spectrum, and let H
be a closed subgroup of G. Then the map
Ψ: X → colim(holimΓ•X)K
K⊳oH ∆ G b
is a weak equivalence in the category of discrete H-spectra, such that the target is
a fibrant discrete H-spectrum.
Proof. Because of the earlier Theorem 3.5, we only have to prove that Ψ is a weak
equivalence of spectra.
Since H is closed in G, H also has finite vcd. Hence, H has a collection {U} of
open normal subgroups such that (a) {U} is a cofinal subcollection of {K}K⊳oH
(so, for example, H ∼=lim H/U) and (b) for all U, Hs(U;M)= 0, for all s ≥m,
U c
where m is some natural number that is independent of U, and for all discrete
U-modules M. Thus,
(4.3) colim(holimΓ•X)K ∼=colim(holimΓ•X)U,
K⊳oH ∆ G b U ∆ G b
so that, to show that Ψ is a weak equivalence, it suffices to show that the map
Ψ: X →colim(holimΓ•X)U ∼=colimholim(Γ•X)U,
b G b G b
U ∆ U ∆
induced by Ψ and (4.3), is a weak equivalence.
Notice that each U is a closed subgroup of G. Then, for each U, (Γ•X)U is a
G b
cosimplicialfibrantspectrum,sothatthereisaconditionallyconvergenthomotopy
spectral sequence
(4.4) Es,t(U)=πsπ ((Γ•X)U)⇒π (holim(Γ•X)U),
2 t G b t−s G b
∆
with
Es,t(U)∼=Hs(U;π (X))
2 c t
(these assertions are verified in the proof of [3, Lemma 7.12]).
SinceEs,∗(U)=0whenevers≥m,theE -termsE∗,∗(U)areuniformlybounded
2 2 2
on the right. Therefore, by [12, Proposition 3.3], taking a colimit over {U} of the
spectral sequences in (4.4) gives the spectral sequence
(4.5) Es,t =colimHs(U;π (X))⇒π (colimholim(Γ•X)U).
2 c t t−s G b
U U ∆
Notice that
E∗,t ∼=H∗(limU;π (X))∼=H∗({e};π (X)),
2 c t c t
U
whichisisomorphictoπ (X),concentratedindegreezero. Thus,spectralsequence
t
(4.5) collapses, so that, for all t, π (colim holim (Γ•X)U) ∼= π (X), and, hence,
t U ∆ G b t
Ψ is a weak equivalence. (cid:3)
b
Let X be a C-diagramof discrete G-spectra, where C is a small category. Then,
by Theorem 2.3, there is a canonical map
φ(X,G): holimGX ∼=colim(holimX)N →holimX
C N⊳oG C C
10 DANIELG.DAVIS
that is G-equivariant.
Corollary 4.6. Let G have finite vcd, let X be a discrete G-spectrum, and let H
be a closed subgroup of G. Then the H-equivariant map
φ(Γ•X,H): holimH Γ•X →holimΓ•X
G b G b G b
∆ ∆
is a weak equivalence in Spt.
Proof. Notice that ψ = φ(Γ•X,H)◦Ψ. Then the desired conclusion follows from
b G b
thefactthatψ andΨareweakequivalences,wherethe latterfactisfromTheorem
b
4.2. (cid:3)
In the Introduction, we pointed out that a weak equivalence rG: X →X
H f,G f,H
in Spt always exists. The following result uses Theorem 4.2 to give a concrete
H
model for rG.
H
Corollary 4.7. Let G have finite vcd, let X be a discrete G-spectrum, and let H
be a closed subgroup of G. Then there is a weak equivalence
rG: colim(holimΓ•X)N → colim(holimΓ•X)K
H N⊳oG ∆ G b K⊳oH ∆ G b
in Spt , where the source of this map is a fibrant discrete G-spectrum and the
H
target is a fibrant discrete H-spectrum.
Proof. Let N be an open normal subgroup of G. Then N ∩H is an open normal
subgroup of H and, hence, there is a canonical map
(holimΓ•X)N ֒→(holimΓ•X)N∩H → colim(holimΓ•X)K.
∆ G b ∆ G b K⊳oH ∆ G b
These maps, as N varies, induce the desired map, which is easily seen to be H-
equivariant. InSptH,theweakequivalenceX →colimK⊳oH(holim∆Γ•GXb)K isthe
composition of the weak equivalence X → colimN⊳oG(holim∆Γ•GXb)N and rHG, so
that rG is a weak equivalence. (cid:3)
H
The following result is a special case of the fact that, if H is open in G, then a
fibrantdiscreteG-spectrumisalsofibrantasadiscreteH-spectrum(see[4,Lemma
3.1] and [9, Remark 6.26]).
Corollary 4.8. Let G have finite vcd and let X be a discrete G-spectrum. If H
is an open subgroup of G, then colimN⊳oG(holim∆(Γ•GXb))N, a fibrant discrete G-
spectrum, is also a fibrant discrete H-spectrum.
Proof. ByTheorem4.2,thespectrumcolimK⊳oH(holim∆(Γ•GXb))K isafibrantdis-
creteH-spectrum. Thus,toverifythecorollary,itsufficestoshowthatthis fibrant
discrete H-spectrum is isomorphic to colimN⊳oG(holim∆(Γ•GXb))N in SptH.
Note that if U is an opensubgroupof H, then U is also anopen subgroupof G,
so that
{H∩V |V < G}={U|U < H}.
o o