ITERATED HOMOTOPY FIXED POINTS FOR THE
LUBIN-TATE SPECTRUM
DANIEL G. DAVIS1
Abstract. When G is a profinite group and H and K are closed subgroups,
with H normal in K, it is not always possible to form the iterated homotopy
fixed point spectrum (Z hH )hK/H , where Z is a continuous G-spectrum. However, we show that, if G = Gn , the extended Morava stabilizer group, and
b n ∧ X), where L
b is Bousfield localization with respect to Morava
Z = L(E
K-theory, En is the Lubin-Tate spectrum, and X is any spectrum with trivial
Gn -action, then the iterated homotopy fixed point spectrum can always be
hH )hK/H is just E hK , extending a result
constructed. Also, we show that (En
n
of Devinatz and Hopkins.
1. Introduction
Let G be a profinite group and let X be a continuous G-spectrum. Thus, X is
the homotopy limit of a tower of discrete G-spectra that are fibrant as spectra (see
Section 2 for a quick review of this notion that is developed in detail in [3]). Let
H and K be closed subgroups of G, with H normal in K. Note that H is closed in
K, and K, H, and the quotient K/H are all profinite groups.
It is easy to see that the fixed point spectrum X H is a K/H-spectrum and
(X H )K/H = X K .
Since, in this context, homotopy fixed points are a total right derived functor of
fixed points, in some sense (see [3, Remark 8.4]), it is reasonable to wonder (a) if
the H-homotopy fixed point spectrum X hH is a continuous K/H-spectrum; and
(b) assuming that (a) holds, if there is an isomorphism
(1.1)
(X hH )hK/H ∼
= X hK
in the stable homotopy category, where (X hH )hK/H is the iterated homotopy fixed
point spectrum. (The preceding discussion is partly taken from [5, Introduction].)
It is not hard to see that, if H is open in K, then the isomorphism of (1.1)
always holds; see Theorem 3.4 for the details. However, if H is not open in K, then
the situation is much more complicated. In this case, it is not always possible to
even construct X hH as a K/H-spectrum, and, whenever we can construct X hH as
a K/H-spectrum, the assertion that X hH is a continuous K/H-spectrum need not
be valid. Thus, when H is not open in K, additional hypotheses are needed just to
get past step (a) - see Sections 3 and 4 for a discussion of this point.
Now we consider the above issues in an example that is of interest in chromatic
stable homotopy theory. Let n ≥ 1, let p be a fixed prime, and let K(n) be the
nth Morava K-theory spectrum (so that K(n)∗ = Fp [vn±1 ], where |vn | = 2(pn − 1)).
1
The author was partially supported by an NSF VIGRE grant at Purdue University. Part of
this paper was written during a visit to the Institut Mittag-Leffler (Djursholm, Sweden).
1
2
DANIEL G. DAVIS
Then, let En be the Lubin-Tate spectrum: En is the K(n)-local Landweber exact
spectrum whose coefficients are given by En∗ = W (Fpn )[[u1 , ..., un−1 ]][u±1 ], where
W (Fpn ) is the ring of Witt vectors of the field Fpn , each ui has degree zero, and
the degree of u is −2.
Let Sn be the nth Morava stabilizer group (the automorphism group of the
Honda formal group law Γn of height n over Fpn ) and let
Gn = Sn o Gal(Fpn /Fp )
be the extended Morava stabilizer group, a profinite group. Then, given any closed
subgroup K of Gn , let EndhK be the construction, due to Devinatz and Hopkins,
that is denoted EnhK in [6]. This work ([6]) shows that the spectra EndhK behave
like homotopy fixed point spectra, though they are not constructed with respect to
a continuous K-action on En .
As in [3], let EnhK be the K-homotopy fixed point spectrum of En , with respect
to the continuous K-action on En . In [4], the author showed that
(1.2)
EnhK ∼
= EndhK ,
for all K closed in Gn . (Here, and elsewhere in this paper, we use “∼
=” to denote an
isomorphism in the stable homotopy category.) Now, as explained in [5, pg. 8], the
profinite group K/H acts on EndhH . Suppose that H is open in K, so that K/H
is a finite group. Then, Devinatz and Hopkins (see [6, Theorem 4 and Section 7])
showed that the canonical map
EndhK → holim EndhH = (EndhH )hK/H
K/H
is a weak equivalence. Thus, by applying (1.2), we obtain that (EnhH )hK/H ∼
= EnhK ,
and this is indeed the case, since it is a special case of Theorem 3.4. Therefore, it
is clear that, for En and finite quotients K/H, the iterated homotopy fixed point
spectrum behaves as desired.
However, when K/H is not finite, the papers [6, 5] do not construct (EndhH )hK/H ,
and hence, they are not able to consider the question of whether this spectrum is
just EndhK . Thus, in this more complicated situation, we show how to construct
b denotes the
(EndhH )hK/H . More precisely, we prove the following result, where L
Bousfield localization functor LK(n) .
Theorem 1.3. Let H and K be any closed subgroups of Gn , such that H is a
normal subgroup of K. Also, let X be any spectrum (with no K/H-action). Then
b ndhH ∧ X) is a continuous K/H-spectrum.
the spectrum L(E
The theory of [3] (reviewed in Section 2), applied to Theorem 1.3, automatically
yields
b dhH ∧ X))hK/H
(L(E
n
as the homotopy fixed points with respect to the continuous K/H-action. Running
through the argument for Theorem 1.3 again, by omitting the (− ∧ X) everywhere,
gives the desired object (EndhH )hK/H . Note that, if we apply (1.2), we immediately
obtain that the actual iterated homotopy fixed point spectrum (EnhH )hK/H can
always be formed, as desired.
b ndhH ∧ En ))hK/H ,
In Section 5, we use the descent spectral sequence for (L(E
where K/H is acting trivially on (the second) En , to prove the following result,
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
3
which says that the iterated homotopy fixed point spectrum for En behaves as
desired, if one smashes with En before taking the K/H-homotopy fixed points.
Theorem 1.4. If H and K are as in Theorem 1.3, then
b ndhH ∧ En ))hK/H ∼
b ndhK ∧ En ).
(L(E
= L(E
Finally, in Section 6, we use (1.2) to prove that the iterated homotopy fixed
point spectrum for En behaves as desired.
Theorem 1.5. Let H and K be as above. Then there is an equivalence
(EnhH )hK/H ∼
= EnhK .
To study the iterated homotopy fixed points of En , we consider the notion of
a hyperfibrant discrete G-spectrum, and we show that, for such a spectrum, the
iterated homotopy fixed points can always be formed (see Definition 4.1 and Construction 4.4). Also, Corollary 4.7 shows that if a discrete G-spectrum is a hyperfibrant discrete K-spectrum, then the iterated homotopy fixed points associated to
the pair (H, K) behave as desired. Also, we point out that, for a totally hyperfibrant discrete G-spectrum, the iterated homotopy fixed points are well-behaved,
for all pairs (H, K) (see Definition 4.9). However, in this paper, we do not delve
any further into developing a general theory of iterated homotopy fixed points.
We mention that in [2], in studying Galois extensions that satisfy certain hypotheses that are different from hyperfibrancy or total hyperfibrancy, Mark Behrens
and the author consider a situation where iterated homotopy fixed points work out
in the desired way.
The author tried hard (and consulted several experts) to find an example of a
discrete G-spectrum that is not hyperfibrant, but he was not successful. Though
he has concluded that there are discrete G-spectra that are not hyperfibrant, he
believes that a general theoretical understanding of this issue might be quite intricate.
Recall (see [3, Section 3]) that a fibrant discrete G-spectrum can be viewed
as a globally fibrant presheaf of spectra on a certain Grothendieck site. Thus,
one approach to a general theory of iterated homotopy fixed points might involve
a better understanding of globally fibrant objects (which are well-known to be
mysterious). Ben Wieland suggested to the author that one way of approaching
this matter might involve working with intermediate model category structures for
presheaves of spectra (e.g., see [16]). (For some of the issues that come up in this
approach, see [8, Introduction, Appendix A, Example A.10], [17, Section 2.10], [18,
Section 6.5.4].) Since the author’s primary interest was in obtaining well-behaved
iterated homotopy fixed points for the Lubin-Tate spectrum, rather than pursue
these other approaches, he has restricted himself to the considerations described in
the paragraph after Theorem 1.5.
At the end of this paper, we discuss how the results in this paper carry over
to other spectra (often referred to as versions of Morava E-theory), like En , that
replace the role of Fpn with certain other finite fields.
We make a comment about notation in this paper. If a limit or colimit is indexed
over {N }, as in limN G/N , then the (co)limit is indexed over all open normal
subgroups of G, unless stated otherwise.
4
DANIEL G. DAVIS
We point out that, independently of our work, in [10, Sections 10.1, 10.2], Halvard Fausk considers the iterated homotopy fixed point pro-spectrum for pro-Gorthogonal spectra, and the associated descent spectral sequence, by making general use of Postnikov towers. However, our approach to iterated homotopy fixed
points is different in technique from his: instead of Postnikov towers, we rely on
the notion of hyperfibrancy, which, in the case of En , makes use of technical results
about the Morava stabilizer group (see [6, the proof of Theorem 4.3]). Also, Mark
Behrens has thought substantially about iterated homotopy fixed points, especially
in the case of En (see [2]).
We conclude this Introduction by mentioning that in [9, Lemma 10.5], it is shown
that, if G is any discrete group, with K any normal subgroup, and X is a G-space,
then X hK is homotopy equivalent to (X hK )hG/K . This fact is described as being
a “transitivity property” and the author thinks that [9, Lemma 10.5] is the first
reference to iterated homotopy fixed points in the literature.
Acknowledgements. I am grateful to Mark Behrens for a series of profitable
conversations about iterated homotopy fixed points. I thank Tilman Bauer, Ethan
Devinatz, and Paul Goerss for helpful discussions about this work. Also, I thank
Halvard Fausk, Rick Jardine, and an early referee for their comments.
2. A summary of the theory of continuous G-spectra and their
homotopy fixed points
This section contains a quick review of material from [3] that is needed for our
work in this paper. We begin with some terminology.
All of our spectra are Bousfield-Friedlander spectra of simplicial sets. Let G be a
profinite group. A discrete G-set is a G-set S such that the action map G × S → S
is continuous, where S is regarded as a discrete space.
Definition 2.1. A discrete G-spectrum X is a G-spectrum such that each simplicial
set Xk is a simplicial object in the category of discrete G-sets. SptG is the category
of discrete G-spectra, where the morphisms are G-equivariant maps of spectra.
Theorem 2.2 ([3, Theorem 3.6]). SptG is a model category, where f in SptG is a
weak equivalence (cofibration) if and only if it is a weak equivalence (cofibration) of
spectra.
It is helpful to note that, by [3, Lemma 3.10], if X is fibrant in SptG , then X is
fibrant as a spectrum. In this paper, we often take the smash product (in spectra)
of a discrete G-spectrum X with a spectrum Y that has no G-action (and we never
take the smash product in the case where Y has a non-trivial G-action). Then
(following [3, Lemma 3.13]), since X N , where N is an open normal subgroup of G,
is a G/N -spectrum, the isomorphisms
X ∧Y ∼
= (colim X N ) ∧ Y ∼
= colim(X N ∧ Y )
N
N
show that X ∧ Y is also a discrete G-spectrum. (For the reader who is interested
in working with discrete G-spectra with a symmetric monoidal smash product, we
point out that this is being worked out in [2], where the authors use symmetric
spectra, instead of Bousfield-Friedlander spectra.)
Definition 2.3. If X ∈ SptG , then let X → Xf,G → ∗ be a trivial cofibration
followed by a fibration, all in SptG . Then X hG = (Xf,G )G .
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
5
When G is a finite group, this definition is equivalent to the classical definition
of X hG for a finite discrete group G (see [3, Section 5]).
Definition 2.4. A tower of discrete G-spectra {Xi } is a diagram
X0 ← X1 ← X2 ← · · ·
in SptG , such that each Xi is fibrant as a spectrum.
Definition 2.5. If {Xi } is a tower of discrete G-spectra, then holimi Xi is a continuous G-spectrum. Also, holimi (Xi )hG is the G-homotopy fixed point spectrum of
the continuous G-spectrum holimi Xi . Given a tower {Xi }, we write X = holimi Xi
and X hG = holimi (Xi )hG .
We call the above construction “homotopy fixed points” because: (1) if G is a
finite discrete group, then the above definition agrees with the classical definition
of X hG (see [3, Lemma 8.2]); and (2) it is the total right derived functor of fixed
points in the appropriate sense (see [3, Remark 8.4]).
Definition 2.6. We say that G has finite virtual cohomological dimension (finite
vcd), and we write vcd(G) < ∞, if there exists an open subgroup U in G such that,
for some positive integer m, Hcs (U ; M ) = 0, for all s > m, whenever M is a discrete
U -module.
It is useful to note that if G is a compact p-adic analytic group, then G has finite
vcd (see the explanation just before Lemma 2.10 in [3]).
Before stating the next result, we need some notation. When X is a discrete set,
let Mapc (G, X) denote the discrete G-set of continuous maps G → X, where the
G-action is given by (g · f )(g 0 ) = f (g 0 g), for g, g 0 ∈ G. Now consider the functor
ΓG : SptG → SptG ,
X 7→ ΓG (X) = Mapc (G, X),
where the action of G on Mapc (G, X) is induced on the level of sets by the G-action
on Mapc (G, (Xk )l ), the set of l-simplices of the pointed simplicial set Mapc (G, Xk ),
where k, l ≥ 0. As explained in [3, Definition 7.1], the functor ΓG forms a triple and
there is a cosimplicial discrete G-spectrum Γ•G X. Given a tower {Mi } of discrete Gs
(G; {Mi }) denote continuous cohomology in the sense of Jannsen
modules, let Hcont
(see [14]).
Theorem 2.7 ([3, Theorem 8.8]). If G has finite vcd and {Xi } is a tower of discrete
G-spectra, then there is a conditionally convergent descent spectral sequence
E2s,t ⇒ πt−s (holim(Xi )hG ) = πt−s (X hG ),
i
E2s,t
s
πt (holimi (Γ•G (Xi )f,G )G ).
= π
If the tower of abelian groups {πt (Xi )}
where
satisfies the Mittag-Leffler condition for each t ∈ Z, then
E s,t ∼
= H s (G; {πt (Xi )}).
cont
2
3. Iterated homotopy fixed points - a discussion
Let G be a profinite group, and let X be a continuous G-spectrum, so that
X = holimi Xi , where each Xi is a discrete G-spectrum that is fibrant as a spectrum.
(Every discrete G-spectrum Z is a continuous G-spectrum, in the sense that there
is a weak equivalence Z → holimi Zf,G that is given by the composition
'
∼
=
'
Z −→ Zf,G −→ lim Zf,G −→ holim Zf,G ,
i
i
6
DANIEL G. DAVIS
which is G-equivariant, where holimi Zf,G is a continuous G-spectrum and the limit
and holim are each applied to a constant diagram.) In this section, we consider
more carefully the iterated homotopy fixed point spectrum (X hH )hK/H and its
relationship with X hK .
The author thanks Mark Behrens and Paul Goerss for help with the next lemma
and its proof.
Lemma 3.1. Let G be a profinite group and let H be an open subgroup of G. If X
is a fibrant discrete G-spectrum, then X is a fibrant discrete H-spectrum.
Remark 3.2. If H is normal in G, then Lemma 3.1 is a consequence of [15, Remark
6.26], using the fact that the presheaf of spectra HomG (−, X) is globally fibrant in
the model category of presheaves of spectra on the site G−Setsdf (for an explanation
of this fact, see [3, Section 3]).
Proof of Lemma 3.1. Note that the forgetful functor U : SptG → SptH has a left
H
adjoint IndH
G : Spt
WH → SptG , where, as a spectrum, IndG (Z) can be identified with
the finite wedge G/H Z of copies of Z indexed by G/H.
Let X be a fibrant discrete G-spectrum. To show that X is fibrant in SptH ,
we will show that the forgetful functor U preserves all fibrations. To do this, it
suffices to show that IndH
G preserves all weak equivalences and cofibrations. Thus,
we only have to show that if f : Y → Z is a map of discrete H-spectra that is a
weak equivalence (cofibration) of spectra, then IndH
G (f ) is also a weak equivalence
(cofibration) of spectra.
If f is a weak equivalence, then, as a map of spectra, π∗ (IndH
G (f )) is equivalent
to ⊕G/H π∗ (f ), which is an isomorphism, so that IndH
G (f ) is a weak equivalence.
Now let f be a cofibration of spectra. Given n ≥ 0, let, for example, Yn be
the nth pointed simplicial set of Y (a Bousfield-Friedlander spectrum of simplicial
sets). Then f0 : Y0 → Z0 and each induced map (S 1 ∧ Zn ) ∪(S 1 ∧Yn ) Yn+1 → Zn+1
are cofibrations of simplicial sets. It is easy to see that
_
_
Y0 →
Z0
(IndH
G (f ))0 :
G/H
G/H
is a cofibration of simplicial sets. Also, the composition
_
_
_
_
((S 1 ∧ Zn ) ∪(S 1 ∧Yn ) Yn+1 ) →
Zn+1
Yn+1 ) ∼
(S 1 ∧
Zn ) ∪(S 1 ∧WG/H Yn ) (
=
G/H
G/H
is a cofibration of simplicial sets. Thus,
G/H
IndH
G (f )
G/H
is a cofibration of spectra.
Corollary 3.3. Let G and H be as in Lemma 3.1. If X is a discrete G-spectrum,
then X hH = (Xf,G )H .
Proof. Since X → Xf,G is a trivial cofibration in SptG , it is also a trivial cofibration
in SptH , by [3, Corollary 3.7].
For the rest of this section, let H and K be closed subgroups of G, with H normal
in K. The following result shows that, if H is open in K, then the isomorphism of
(1.1) always holds.
Theorem 3.4. Let G be a profinite group with closed subgroups H and K, with H
normal in K. Given {Xi }, a tower of discrete G-spectra, let X = holimi Xi be a
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
7
continuous G-spectrum. If H is open in K, then there is a weak equivalence
X hK → (X hH )hK/H .
Proof. Since each Xi is a discrete H- and K-spectrum, X is a continuous H- and
K-spectrum. By Corollary 3.3,
X hH = holim(Xi )hH = holim((Xi )f,K )H .
i
i
Since HomK (−, (Xi )f,K ) is a fibrant presheaf of spectra on the site K−Setsdf , the
canonical map
∼ holim((Xi )f,K )H
(Xi )hK = HomK (∗, (Xi )f,K ) → holim HomK (K/H, (Xi )f,K ) =
K/H
K/H
is a weak equivalence, by [15, Proposition 6.39]. By Corollary 3.3, there is the
equality holimK/H ((Xi )f,K )H = holimK/H (Xi )hH , so that the previous sentence
yields a weak equivalence
(Xi )hK → holim(Xi )hH .
K/H
Note that both the source and target of this weak equivalence are fibrant spectra,
since, for any profinite group L, the functor (−)L : SptL → Spt preserves fibrant
objects (see [3, Corollary 3.9]).
Thus, there is a weak equivalence
X hK = holim(Xi )hK → holim holim(Xi )hH ∼
= holim X hH .
i
i
K/H
K/H
Since the identity map X hH → X hH , where X hH = holimi ((Xi )f,K )H is, of course,
a weak equivalence and a K/H-equivariant map, with X hH fibrant, we can make
the identification holimK/H X hH = (X hH )hK/H , completing the proof.
Now we consider the more difficult case, when H is not open in K. Let us
simplify the situation by assuming that the continuous G-spectrum X is a discrete
G-spectrum.
First of all, in this case, consider the statement that Xf,K is a fibrant object in
SptH . When H is open in K, this statement is Lemma 3.1, which we proved by
showing that the forgetful functor U : SptK → SptH preserves fibrations, and we
did this by using the fact that U is a right adjoint. However, when H is not open
in K, the forgetful functor U does not preserve limits, so that it cannot be a right
adjoint (see [2, Section 3.6]). Thus, when H is not open in K, there is no reason
to expect that Xf,K is fibrant in SptH , and, without this fact, the above proof of
Theorem 3.4 fails to work.
Another problem in this case is that the notation (X hH )hK/H might not even
make sense. For example, X hH = (Xf,H )H need not even be a K/H-spectrum,
since Xf,H , in general, has no K-action. The spectrum (Xf,K )H certainly has
a K/H-action, but, as mentioned previously, we cannot necessarily identify this
spectrum with (Xf,H )H .
Suppose that K has finite vcd. Then, by [3, Remark 7.13], we can make the
identification
X hH = holim(Γ• (Xf,K ))H ∼
= (holim Γ• (Xf,K ))H .
∆
K
∆
K
Thus, since holim∆ Γ•K (Xf,K ) is a K-spectrum, X hH can be identified with the
K/H-spectrum (holim∆ Γ•K (Xf,K ))H .
8
DANIEL G. DAVIS
Now we need to show that the K/H-spectrum (holim∆ Γ•K (Xf,K ))H is actually
a continuous K/H-spectrum, so that we can take its K/H-homotopy fixed points.
Towards this end, note that
(holim Γ•K (Xf,K ))H ∼
= holim colim(Γ•K (Xf,K ))N H ,
∆
∆
N
where the colimit is over all open normal subgroups of K. If the homotopy limit
and the colimit were to commute, then
(holim Γ•K (Xf,K ))H ∼
= colim(holim Γ•K (Xf,K ))N H
∆
N
∆
is a discrete K-spectrum, since (holim∆ Γ•K (Xf,K ))N H is a K/(N H)-spectrum and
K/(N H) is a finite group. However, the homotopy limit and the colimit do not
necessarily commute, so that, in general, (X hH )hK/H can not even be formed.
We conclude that, when H is not open in K, unless various hypotheses are
satisfied, the expression (X hH )hK/H does not have any meaning.
4. Hyperfibrant and totally hyperfibrant discrete G-spectra
In this section, we consider a situation in which the iterated homotopy fixed
point spectrum can always be formed. As before, throughout this section, we let
H and K be closed subgroups of the profinite group G, with H normal in K.
Let X be a discrete G-spectrum. Then there is a map
Xf,G → (Xf,G )f,K ,
which gives a map
ψ : colim X hN K = colim(Xf,G )N K ∼
= (Xf,G )K → ((Xf,G )f,K )K = X hK ,
N
N
where the last equality is due to the fact that X → Xf,G → (Xf,G )f,K is a trivial
cofibration in SptK and (Xf,G )f,K is fibrant in SptK .
It is easy to see that ψ is a weak equivalence for all K open in G, since Xf,G is
always fibrant in SptK , for such K. But if Xf,G is fibrant in SptK for all closed
K, then ψ is a weak equivalence for all closed K. This motivates the following
definition.
Definition 4.1. Let X be a discrete G-spectrum. If ψ is a weak equivalence for
all K closed in G, then X is a hyperfibrant discrete G-spectrum.
It is not hard to see that there are profinite groups G such that not every discrete
G-spectrum is hyperfibrant. For example, let G be an infinite profinite group with
a nontrivial finite subgroup K (which is automatically closed in G). As explained
in [3, Section 5], since Xf,G is a fibrant spectrum, X hK = holimK Xf,G . Then we
have the canonical map
colim X hN K = colim(Xf,G )N K ∼
= (Xf,G )K ∼
= lim Xf,G → holim Xf,G = X hK ,
N
N
K
K
which need not always be a weak equivalence. Thus, there exists G with some
discrete G-spectrum X that is not hyperfibrant.
For the example below and the paragraph after it, we assume that G has finite
vcd.
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
9
Example 4.2. Let X be any spectrum (with no G-action), so that Mapc (G, X),
as defined in Section 2, is a discrete G-spectrum. The descent spectral sequence for
Mapc (G, X)hK has the form
Hcs (K; Mapc (G, πt (X))) ⇒ πt−s (Mapc (G, X)hK ).
Since Hcs (K; Mapc (G, πt (X))) = 0, for s > 0,
π∗ (Mapc (G, X)hK ) ∼
= Hc0 (K; Mapc (G, π∗ (X))) = Mapc (G, π∗ (X))K
∼
= Map (G/K, π∗ (X)) ∼
= π∗ (Map (G, X)K ),
c
hK
and thus, Mapc (G, X)
have:
c
K
' Mapc (G, X) , for any K closed in G. Therefore, we
Mapc (G, X)hK ' Mapc (G, X)K ∼
= colim Mapc (G, X)N K
N
hN K
' colim Mapc (G, X)
N
.
Thus, for any spectrum X, Mapc (G, X) is a hyperfibrant discrete G-spectrum.
For any discrete G-spectrum X,
∼ holim colim(Γ• (Xf,G ))N K ,
X hK =
∆
G
N
and similarly,
colim X hN K = colim holim(Γ•G (Xf,G ))N K .
N
N
∆
Thus, if ψ is a weak equivalence, for some K, then the canonical map
φ : colim holim(Γ•G (Xf,G ))N K → holim colim(Γ•G (Xf,G ))N K
N
∆
∆
N
is a weak equivalence. Therefore, if X is hyperfibrant, then, for all K, the colimit
and the holim involved in defining φ always commute, which is not true, in general.
Lemma 4.3. If X is a discrete G-spectrum and K is normal in L, where L is a
closed subgroup of G, then colimN X hN K is a discrete L/K-spectrum.
Proof. Note that the spectrum X hN K = (Xf,G )N K is an N L/N K-spectrum, where
N L/N K is a finite group. The continuous map L/K → N L/N K, given by
lK 7→ lN K, makes (Xf,G )N K a discrete L/K-spectrum. Since the forgetful functor
SptL/K → Spt is a left adjoint, colimits in SptL/K are formed in the category of
spectra, so that colimN (Xf,G )N K is a discrete L/K-spectrum.
If X is hyperfibrant, then we can identify X hH with the discrete K/H-spectrum
colimN X hN H , giving the following.
Construction 4.4. If X is a hyperfibrant discrete G-spectrum, then the iterated
homotopy fixed point spectrum can always be formed:
(X hH )hK/H = (colim X hN H )hK/H .
N
It is natural to wonder if (1.1) holds given any hyperfibrant discrete G-spectrum
X. However, it is not hard to see that hyperfibrancy need not imply (1.1). For
example, note that
(X hH )hK/H = (colim(Xf,G )N H )hK/H ' (((Xf,G )H )f,K/H )K/H ,
N
10
DANIEL G. DAVIS
and
X hK ' colim(Xf,G )N K ∼
= (Xf,G )K = ((Xf,G )H )K/H ,
N
Therefore, (X hH )hK/H ∼
= X hK if and only if the canonical map
((Xf,G )H )K/H → (((Xf,G )H )f,K/H )K/H
(4.5)
is a weak equivalence, which need not be the case. (To see this, for example, note
that if (Xf,G )H were fibrant in SptK/H , then, in SptK/H , the map
(Xf,G )H → ((Xf,G )H )f,K/H
is a weak equivalence between fibrant objects, so that the map in (4.5) is a weak
equivalence (see the comment just before Corollary 3.9 in [3]). However, (Xf,G )H
need not be fibrant in SptK/H .)
We have seen that if X is a hyperfibrant discrete G-spectrum, then the iterated
homotopy fixed point spectrum, while it can be constructed, is not necessarily as
well-behaved as we would like, since (1.1) need not be true. Thus, we make the
following considerations.
Let X be any discrete K-spectrum. As in the proof of Lemma 4.3, colimN X N H ,
where the colimit is over all open normal subgroups of K, is a discrete K/Hspectrum, so that the isomorphism X H ∼
= colimN X N H implies that X H is also a
discrete K/H-spectrum. Hence, there is a functor (−)H : SptK → SptK/H .
The author thanks Mark Behrens for help with the next lemma, which is basically
a version of [15, Lemma 6.35], which is for simplicial presheaves.
Lemma 4.6. The functor (−)H : SptK → SptK/H preserves fibrant objects.
Proof. It is easy to see that the functor (−)H has a left adjoint t : SptK/H → SptK
that sends a discrete K/H-spectrum X to X, where now X is regarded as a discrete
K-spectrum via the canonical map K → K/H. It suffices to show that t preserves
all weak equivalences and cofibrations, since this implies that (−)H preserves all
fibrations.
If f is a weak equivalence (cofibration) in SptK/H , then f is a K/H-equivariant
map that is a weak equivalence (cofibration) of spectra. Since, as a map of spectra,
t(f ) = f , t(f ) is a weak equivalence (cofibration) of spectra. Because t(f ) is also
K-equivariant, it is a weak equivalence (cofibration) in SptK .
Corollary 4.7. Let X be a discrete G-spectrum and let K be a closed subgroup
of G. If X is a hyperfibrant discrete K-spectrum, then there is a weak equivalence
(X hH )hK/H ' X hK , where H is any closed subgroup of G that is also normal in
K.
Proof. Since X is hyperfibrant in SptK , ψ : colimN X hN H → X hH is a weak equivalence, where the colimit is over all open normal subgroups of K. Thus, by Construction 4.4,
(X hH )hK/H = (colim X hN H )hK/H = (colim(Xf,K )N H )hK/H ' ((Xf,K )H )hK/H .
N
N
Since Xf,K is fibrant in SptK , Lemma 4.6 implies that (Xf,K )H is fibrant in SptK/H .
Also, the identity map (Xf,K )H → (Xf,K )H is a trivial cofibration in SptK/H .
Thus, we can let (Xf,K )H = ((Xf,K )H )f,K/H , and hence,
((Xf,K )H )hK/H = (((Xf,K )H )f,K/H )K/H = ((Xf,K )H )K/H = (Xf,K )K = X hK .
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
11
Corollary 4.8. Let X be a discrete G-spectrum and suppose that X is a hyperfibrant discrete L-spectrum, for every subgroup L that is closed in G. Then, if H and
K are closed subgroups of G, with H normal in K, then there is a weak equivalence
(X hH )hK/H ' X hK .
The preceding corollary says that if the discrete G-spectrum X is hyperfibrant
as a discrete K-spectrum, for all K closed in G, then not only can the iterated
homotopy fixed point spectrum always be constructed, but (1.1) is always true.
Also, note that if X is a hyperfibrant discrete G-spectrum, then X does not have
to be a hyperfibrant discrete K-spectrum, for all K closed in G, since this would
mean that (1.1) is always true, which we saw above is not necessarily the case.
These considerations motivate the following definition.
Definition 4.9. Let X be a discrete G-spectrum. If X is a hyperfibrant discrete Kspectrum, for all K closed in G, then X is a totally hyperfibrant discrete G-spectrum.
Therefore, if X is a totally hyperfibrant discrete G-spectrum, then (X hH )hK/H '
X hK , for any H and K closed in G, with H normal in K.
Example 4.10. For any spectrum X, Mapc (G, X) is a totally hyperfibrant discrete
G-spectrum. To see this, let K be any closed subgroup of G, and let L be any closed
subgroup of K. Since L is closed in G, Mapc (G, X)hL is defined. Also, if N is an
open normal subgroup of K, then N L is closed in K, since K is a profinite group
and N L is an open subgroup of K. Thus, N L is a closed subgroup of G, so that
Mapc (G, X)hN L is defined. Then, by Example 4.2 and by taking the colimit over
all open normal subgroups N of K, we have:
colim Map (G, X)hN L ' colim Map (G, X)N L ∼
= Map (G, X)L ' Map (G, X)hL .
N
c
N
c
c
c
5. Iterated homotopy fixed points for En
In this section, we show that it is always possible to construct the spectrum
(EndhH )hK/H . As mentioned in the Introduction, applying (1.2) allows us to identify
EndhH with EnhH , yielding the iterated homotopy fixed point spectrum (EnhH )hK/H .
We begin with some notation.
We use (−)f to denote functorial fibrant replacement in the category of spectra.
Let E(n) be the Johnson-Wilson spectrum, with E(n)∗ = Z(p) [v1 , ..., vn−1 ][vn±1 ],
where the degree of each vi is 2(pi − 1). Let MI0 ← MI1 ← MI2 ← · · · be a
b 0) ∼
tower of generalized Moore spectra such that L(S
= holimi (Ln MIi )f , where Ln
is Bousfield localization with respect to E(n). This tower of finite type n spectra
exists by [13, Proposition
T 4.22]. Let {Uj }j≥0 be a descending chain of open normal
subgroups of Gn , with j Uj = {e} (see [6, (1.4)]), so that Gn ∼
= limj Gn /Uj . Also,
let H and K be closed subgroups of Gn , with H normal in K.
We recall the following useful result from [12, Section 2].
Lemma 5.1. If Z is an E(n)-local spectrum, then, in the stable category, there is
an isomorphism
b ∼
LZ
= holim(Z ∧ MIi )f .
i
Theorem 5.2 ([6]). For each i,
EndhH
dhUj H
∧ MIi ' colimj (En
∧ MIi ).
12
DANIEL G. DAVIS
dhUj H
b
Proof. By [6, Definition 1.5], EndhH ∼
). Thus, we have:
= L(colim
j En
b
EndhH ∧ MIi ∼
EndhUj H ) ∧ MIi
= L(colim
j
∼
= Ln (colim EndhUj H ) ∧ MIi
j
∼
= colim(EndhUj H ∧ MIi ),
j
where the second equivalence applies [13, Lemma 7.2] and the third equivalence is
dhU H
due to the fact that each En j is E(n)-local (since each is K(n)-local).
The next definition, which we recall from [3], defines a useful spectrum.
dhUj
Definition 5.3. Let Fn = colimj En
Gn -spectrum.
. By [3, Lemma 3.13], Fn is a discrete
Remark 5.4. Using [3, Corollary 9.8] and (1.2), Theorem 5.2 implies that
(Fn ∧ MIi )hH ∼
= EndhH ∧ MIi ∼
= colim(EndhUj H ∧ MIi )
j
∼
= colim(Fn ∧ MIi )hUj H ∼
= colim(Fn ∧ MIi )hN H ,
j
N
where the last colimit is over all open normal subgroups of Gn . Therefore, since
Theorem 5.2 is valid if H is replaced with any closed subgroup of Gn , Fn ∧ MIi is
a hyperfibrant discrete Gn -spectrum.
For the rest of this section, we let X be any spectrum, with no K/H-action.
dhUj H
Lemma 5.5. The spectrum colimj (En
∧MIi ∧X)f is a discrete K/H-spectrum.
dhU H
dhU H
Proof. Note that En j is a Uj K/Uj H-spectrum. Thus, En j ∧ MIi ∧ X is a
dhU H
Uj K/Uj H-spectrum. By functoriality, (En j ∧ MIi ∧ X)f is also a Uj K/Uj Hspectrum. The argument is completed as in the proof of Lemma 4.3.
Corollary 5.6. The spectrum
b ndhH ∧ X) ∼
L(E
= holim(colim(EndhUj H ∧ MIi ∧ X)f )
i
j
is a continuous K/H-spectrum. In particular,
EndhH ∼
= holim(colim(EndhUj H ∧ MIi )f )
i
j
is a continuous K/H-spectrum.
Proof. By Lemma 5.1,
b ndhH ∧ X) ∼
L(E
= holim(EndhH ∧ MIi ∧ X)f
i
∼
= holim(colim(EndhUj H ∧ MIi ∧ X)f ).
i
dhU H
j
dhU H
Since (En j ∧ MIi ∧ X)f is fibrant, so is colimj (En j ∧ MIi ∧ X)f , so that the
homotopy limit is of a diagram of fibrant spectra, as required.
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
13
Remark 5.7. We can just as well phrase Corollary 5.6 as saying that
b dhH ∧ X) ∼
L(E
= holim ((colim E dhUj H ) ∧ MI ∧ X)f,K/H .
n
i
n
j
i
dhU H
In particular, note that EndhH ∼
= holimi ((colimj En j ) ∧ MIi )f,K/H . Now recall
dhU H
that En ∼
= holimi (Fn ∧ MIi )f,Gn . Thus, colimj En j is playing the same role as
Fn and is the analogue of Fn .
Construction 5.8. Let X be any spectrum with trivial K/H-action. Then
b ndhH ∧ X))hK/H = (holim(colim(EndhUj H ∧ MI ∧ X)f ))hK/H
(L(E
i
j
i
= holim((colim(EndhUj H ∧ MIi ∧ X)f )hK/H ).
i
j
6. The descent spectral sequence
Let H and K be closed subgroups of Gn , with H normal in K. Note that a
compact p-adic analytic group is simply a p-adic analytic group that is compact.
By [7, Theorem 9.6], if G is a p-adic analytic group, U a closed subgroup of G, and
N a closed normal subgroup of G, then U and G/N , with the quotient topology,
are p-adic analytic groups. Thus, we obtain the following useful result.
Lemma 6.1. The profinite group K/H is a compact p-adic analytic group and
vcd(K/H) < ∞.
b dhH ∧ X) is a continuous K/H-spectrum, for any
Since, by Corollary 5.6, L(E
n
spectrum X with trivial K/H-action, Theorem 2.7 gives a descent spectral sequence
b dhH ∧ X))hK/H ),
(6.2)
E s,t ⇒ πt−s ((L(E
n
2
where
E2s,t = π s πt (holim(Γ•K/H (colim(EndhUj H ∧ MIi ∧ X)f ))K/H ).
i
j
s
(G; M ) denote the continuous cohomology of continuous cochains of G,
Let Hcts
with coefficients in the topological G-module M (see [20, Section 2]). If the tower
of abelian groups {πt (EndhH ∧ MIi ∧ X)}i satisfies the Mittag-Leffler condition for
each t ∈ Z, then
s
s
b ndhH ∧ MI ∧ X))}) ∼
b ndhH ∧ X))),
E s,t ∼
(K/H; {πt (L(E
(K/H; πt (L(E
= Hcont
= Hcts
2
i
where the second isomorphism applies [14, Theorem 2.2]. In particular, when X is
b ndhH ∧ X) ' EndhH ∧ X, and [6, Remark 1.3] and [5, pg. 9]
a finite spectrum, L(E
imply that spectral sequence (6.2) has the form
(6.3)
Hcs (K/H; πt (EndhH ∧ X)) ⇒ πt−s ((EndhH ∧ X)hK/H ),
where the E2 -term is continuous cohomology for profinite Zp [[K/H]]-modules (see
[6, Remark 1.3]).
Recall from [5] that there is a Lyndon-Hochschild-Serre spectral sequence
(6.4)
Hcs (K/H; πt (EndhH ∧ X)) ⇒ πt−s (EndhK ∧ X),
where X is a finite spectrum. The fact that spectral sequences (6.3) and (6.4) have
identical E2 -terms suggests that there is an equivalence (EndhH )hK/H ∼
= EndhK . In
the next result, we use descent spectral sequence (6.2), when X = En , to show that
this equivalence holds after smashing with En , before taking K/H-homotopy fixed
points. First of all, we define the relevant map.
14
DANIEL G. DAVIS
Let
Ci = colim(EndhUj H ∧ MIi ∧ En )f .
j
Then there is a map
b ndhK ∧ En ) → (L(E
b ndhH ∧ En ))hK/H ,
θ : L(E
which is defined by making the identifications
b dhK ∧ En ) = holim(colim(E dhUj K ∧ MI ∧ En )f )
L(E
n
n
i
i
j
and
b ndhH ∧ En ))hK/H = holim(((Ci )f,K/H )K/H ),
(L(E
i
and by taking the composite of the canonical map
holim(colim(EndhUj K ∧ MIi ∧ En )f ) → lim (holim Ci ),
i
j
K/H
the map
∼
=
i
∼
=
lim (holim Ci ) −→ holim lim Ci −→ holim((Ci )K/H ),
i
K/H
i
K/H
i
and the map
holim((Ci )K/H ) → holim(((Ci )f,K/H )K/H ).
i
i
b ndhH ∧ En ))hK/H ,
To avoid confusion, we point out that, in the expression (L(E
the lone En has trivial K/H-action, as stated just after Lemma 6.1.
Theorem 6.5. The map
'
b dhK ∧ En ) −→
b dhH ∧ En ))hK/H
θ : L(E
(L(E
n
n
(6.6)
is a weak equivalence.
Proof. By [6, Proposition 6.3],
πt (EndhH ∧ En ∧ MIi ) ∼
= Mapc (Gn /H, πt (En ∧ MIi )).
Thus, since {πt (En ∧ MIi )} satisfies the Mittag-Leffler condition for every integer
t, the tower {πt (EndhH ∧ En ∧ MIi )} does too, since the functor
Mapc (Gn /H, −) : Abdis → Abdis ,
where Abdis is the category of discrete abelian groups, is exact and additive (see
[3, Lemma 2.10, Remark 2.11]). Therefore, descent spectral sequence (6.2) takes
the form
s
b ndhH ∧ En ))hK/H ).
E s,t = Hcts
(K/H; Mapc (Gn /H, πt (En ))) ⇒ πt−s ((L(E
2
By [19, Exercise 2.2.3], the continuous epimorphism π : Gn /H → Gn /K has a
continuous section σ : Gn /K → Gn /H, such that σ(eK) = eH. Thus, by [5, proof
of Lemma 3.15] (or, e.g., [21, Proposition 1.3.4 (c)]), there is a homeomorphism
K/H × Gn /K → Gn /H,
(kH, gK) 7→ kH · σ(gK).
Notice that this homeomorphism is K/H-equivariant, with K/H acting on the
source by acting only on K/H, since
k 0 H · (kH, gK) = (k 0 kH, gK) 7→ k 0 kH · σ(gK) = k 0 H · (kH · σ(gK)).
Thus,
Mapc (Gn /H, πt (En )) ∼
= Mapc (K/H, Mapc (Gn /K, πt (En )))
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
15
is an isomorphism of topological K/H-modules. Therefore,
E2s,t ∼
= lim i Hcs (K/H; Mapc (K/H, Mapc (Gn /K, πt (En ∧ MIi )))),
which is 0, for s > 0, and equals Mapc (Gn /K, πt (En )), when s = 0. Thus,
b ndhH ∧ En ))hK/H ) ∼
b ndhK ∧ En )).
π∗ ((L(E
= Mapc (Gn /K, π∗ (En )) ∼
= π∗ (L(E
Remark 6.7. In Theorem 7.3, by using (1.2), we will show that (EnhH )hK/H ∼
=
EnhK . Therefore, using (1.2) again, (6.6) implies that taking K/H-homotopy fixed
points commutes with smashing with En :
b nhH ∧ En ))hK/H ∼
b nhK ∧ En ).
(L(E
= L(E
This is interesting because such a commutation need not hold in general, unless
one is smashing with a finite spectrum. However, (6.6) is not surprising, because it
is known that, for all H,
b ndhH ∧ En ) ∼
b n ∧ En ))hH ,
L(E
= (L(E
where on the right-hand side, the second En has the trivial H-action. This last
equivalence follows from the fact that
b ndhH ∧ En )) ∼
b n ∧ En ))hH ),
π∗ (L(E
= Mapc (Gn , En∗ )H ∼
= π∗ ((L(E
where the second isomorphism is obtained from the descent spectral sequence for
b n ∧ En ))hH and the fact that π∗ (L(E
b n ∧ En )) ∼
(L(E
= Mapc (Gn , En∗ ) (see the last
paragraph of [3]).
7. A proof of (1.1) in the case of En
In [4], the author showed that, as stated in (1.2), EnhK can be identified with
for all closed subgroups K of Gn . In this section, we use this result to show
that (1.1) holds when X equals En .
EndhK ,
Lemma 7.1. Let K be any closed subgroup of Gn and let L be any closed subgroup
of K. Then the canonical map colimN (EndhN L ∧ MIi ) → EndhL ∧ MIi , where the
colimit is over all open normal subgroups N of K, is a weak equivalence.
Proof. As in Example 4.10, both L and N L are closed subgroups of Gn , so that
EndhL and EndhN L are defined. Since both the source and target of the map are
K(n)-local, it suffices to show that the associated map
colim(EndhN L ∧ En ∧ MIi ) → EndhL ∧ En ∧ MIi
N
is a weak equivalence. For any integer t, by using [6, Proposition 6.3] and the
fact that Mapc (Gn , πt (En ∧ MIi )) is a discrete K-module (since it is a discrete
Gn -module), we have:
πt (colim(EndhN L ∧ En ∧ MIi )) ∼
= colim Mapc (Gn , πt (En ∧ MIi ))N L
N
N
∼
= Mapc (Gn , πt (En ∧ MIi ))L
∼
= πt (E dhL ∧ En ∧ MI ),
n
completing the proof.
i
16
DANIEL G. DAVIS
Corollary 7.2. The spectrum Fn ∧ MIi is a totally hyperfibrant discrete Gn spectrum.
Proof. Let K be any closed subgroup of Gn and let L be any closed subgroup of
K. Then, by [3, Corollary 9.8] and Lemma 7.1,
(Fn ∧ MIi )hL ∼
= EnhL ∧ MIi ∼
= EndhL ∧ MIi ∼
= colim(EndhN L ∧ MIi )
N
∼
= colim(EnhN L ∧ MIi ) ∼
= colim(Fn ∧ MIi )hN L ,
N
N
where the colimit is over all open normal subgroups N of K.
Since EnhH ∼
= EndhH , we identify these two spectra, so that the construction
dhH hK/H
of (En )
automatically yields the iterated homotopy fixed point spectrum
(EnhH )hK/H .
Theorem 7.3. For any H and K, there is an isomorphism
(EnhH )hK/H ∼
= EnhK
in the stable homotopy category.
Proof. By applying Corollary 4.8 to Corollary 7.2, we have that
((Fn ∧ MIi )hH )hK/H ∼
= (Fn ∧ MIi )hK .
Thus, we obtain:
(EnhH )hK/H = (holim(Fn ∧ MIi )hH )hK/H = holim((Fn ∧ MIi )hH )hK/H
i
i
∼
= holim(Fn ∧ MIi )hK = EnhK .
i
Remark 7.4. By Theorem 7.3, the strongly convergent Lyndon-Hochschild-Serre
spectral sequence of (6.4) for π∗ (EndhK ), constructed in [5], can be identified with
the descent spectral sequence of (6.3) for π∗ ((EnhH )hK/H ).
We conclude this paper by pointing out that the results in this paper also apply
to other spectra that are closely related to En .
Let k be any finite field that contains Fpn . Then [11, Section 7] shows that there
is a commutative S-algebra E(k, Γ) associated to the field k and a height n formal
group law Γ over k, with
E(k, Γ)∗ = W(k)[[u1 , ..., un−1 ]][u±1 ],
where E(k, Γ)∗ is graded in the same way that En∗ is.
Ethan Devinatz has informed the author that all the results of [6] and [5] go
through as is, with En replaced with E(k, Γ) and Gn replaced with the profinite
group
G(k) = Sn o Gal(k/Fp ).
(This fact, in reference to [6], is also stated in [1, Section 5.2].) Also, G(k) is a
compact p-adic analytic group with finite vcd (see the two paragraphs following
[3, Lemma 2.9]). Thus, for example, [6] gives the following results: (a) G(k) acts
on the spectrum E(k, Γ) by maps of commutative S-algebras; (b) for any closed
subgroup K of G(k), there is a commutative S-algebra E(k, Γ)dhK that behaves
ITERATED HOMOTOPY FIXED POINTS FOR THE LUBIN-TATE SPECTRUM
17
like a homotopy fixed point spectrum with respect to a continuous K-action; (c)
there is an isomorphism
b
π∗ (L(E(k,
Γ)dhK ∧ E(k, Γ))) ∼
= Mapc (G(k), π∗ (E(k, Γ)))K ;
dhUj K
b
and (d) E(k, Γ)dhK ∼
), where {Uj } is a nested sequence of
= L(colim
j E(k, Γ)
open normal subgroups of G(k), as before.
Part (d) above implies that
E(k, Γ) ∼
= holim((colim E(k, Γ)dhUj ) ∧ MI ),
i
j
i
so that E(k, Γ) is a continuous G(k)-spectrum, and, for any closed subgroup K of
G(k), there is a homotopy fixed point spectrum E(k, Γ)hK . Though [4] does not
consider E(k, Γ)hK , the argument given there - by letting E(k, Γ) and G(k) play
the role of En and Gn , respectively - shows that
E(k, Γ)hK ∼
= E(k, Γ)dhK .
Furthermore, statements (c) and (d) above imply, as before, that the spectrum
(colimj E(k, Γ)dhUj ) ∧ MIi is a totally hyperfibrant discrete G(k)-spectrum. Thus,
all the assertions in this paper are valid when En and Gn are replaced with E(k, Γ)
and G(k), respectively, so that, for example, we obtain that
(E(k, Γ)hH )hK/H ∼
= E(k, Γ)hK .
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