TATE COHOMOLOGY AND PERIODIC LOCALIZATION
OF POLYNOMIAL FUNCTORS
NICHOLAS J. KUHN
Abstract. In this paper, we show that Goodwillie calculus, as applied
to functors from stable homotopy to itself, interacts in striking ways
with chromatic aspects of the stable category.
Localized at a fixed prime p, let T(n) be the telescope of a vn self
map of a finite Smodule of type n. The Periodicity Theorem of Hopkins
and Smith implies that the Bousfield localization functor associated to
T(n)* is independent of choices.
Goodwillie's general theory says that to any homotopy functor F
from Smodules to Smodules, there is an associated tower under F,
{PdF}, such that F ! PdF is the universal arrow to a dexcisive func
tor.
Our first main theorem says that PdF ! Pd1F always admits a
homotopy section after localization with respect to T(n)* (and so also
after localization with respect to Morava Ktheory K(n)*). Thus, after
periodic localization, polynomial functors split as the product of their
homogeneous factors.
This theorem follows from our second main theorem which is equiva
lent to the following: for any finite group G, the Tate spectrum tG(T(n))
is weakly contractible. This strengthens and extends previous theorems
of GreenleesSadofsky, HoveySadofsky, and MahowaldShick. The Pe
riodicity Theorem is used in an essential way in our proof.
The connection between the two theorems is via a reformulation of a
result of McCarthy on dual calculus.
1.Introduction and main results
Over the past twenty years, beginning with the Nilpotence and Periodic
ity Theorems of E. Devanitz, M. Hopkins, and J. Smith [DHS , HopSm , R2],
there has been a steady deepening of our understanding of stable homotopy
as organized by the chromatic, or periodic, point of view. During this same
period, there have been many new results in homotopical algebra, many fol
lowing the conceptual model offered by T. Goodwillies's calculus of functors
[G1 , G2 , G3 ].
____________
Date: July, 2003.
2000 Mathematics Subject Classification. Primary 55P65; Secondary 55N22, 55P*
*60,
55P91.
This research was partially supported by a grant from the National Science F*
*oundation.
1
2 KUHN
Here, and in a previous paper [Ku2 ], I prove theorems illustrating a beau
tiful interaction between these two strands of homotopy theory. These re
sults say that certain homotopy functors, stratified via Goodwillie calculus,
decompose into their homogeneous strata, after periodic localization. The
first paper concerned a highly stuctured splitting of the important functor
1 1 . Ignoring the extra structure, one is left with an illustration of the
main result here: after Bousfield localization with respect to a periodic ho
mology theory, all polynomial endofunctors of stable homotopy split into a
product of their homogeneous components.
We now explain our main results in more detail.
The periodic homology theories we consider are K(n)*, the nth Morava
Ktheory at a fixed prime p and with n > 0, and the `telescopic' variants
T (n)*, where T (n) denotes the telescope of a vnself map of a finite complex
of type n. A consequence of the Periodicity Theorem is that the associated
Bousfield class is independent of the choice of both the complex and
self map. Also, we recall that T (n)*acyclics are K(n)*acyclic1; thus the
associated localization functors are related by LK(n) ' LK(n)LT(n).
Our use of concepts from Goodwillie calculus and localization theory re
quire that we work within a good model category with homotopy category
equivalent to the standard stable homotopy category. Thus we work within
the category S, the category of Smodules of [EKMM ].
Goodwillie's general theory then says that a homotopy functor F : S ! S
admits a universal tower of fibrations under F ,
..
.


fflffl
P2F:(X):
tt
tttt p2
e2 tttt fflffl
tttt jP1F4(X)4
ttt jjjj
tttjjje1jjjjjt p1
tjjjjjjtte0 fflffl
F (X) ________________//_P0F (X),
such that
(1) PdF is dexcisive, and
(2) ed : F ! PdF is the universal natural transformation to a dexcisive
functor.
____________
1The Telescope Conjecture asserts the converse, and, these days, is consider*
*ed unlikely
to hold for n 2.
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 3
Our splitting theorem then is as follows.
Theorem 1.1. Let F : S ! S be any homotopy functor. For all primes p,
n 1, and d 1, the natural map
pd(X) : PdF (X) ! Pd1F (X)
admits a natural homotopy section after applying LT(n).
The theorem can be reformulated as follows. Let DdF (X) be the fiber
of pd(X) : PdF (X) ! Pd1F (X). Then DdF is both dexcisive and homo
geneous: Pd1DdF ' *. The theorem is equivalent to the statement that
there is a natural weak equivalence of filtered spectra
Yd
LT(n)PdF (X) ' LT(n)DcF (X).
c=0
Example 1.2. Here is the simplest example illustrating our theorem. Let
p = 2. For k 2 Z, let RPk1 be the Thom spectrum of k copies of the
canonical line bundle over RP 1. [Ku1 , Ex.5.7] implies that the cofibration
sequence
(1.1) RP11! RP01 ! S0
splits after K(n)localization, for all n, even though the connecting map
ffi : S0 ! RP11is nonzero in mod 2 homology.
As was, in essence, observed in a 1983 paper by J.Jones and S.Wegmann
[JW ], (1.1) is the suspension of the special case X = S1 of a natural
cofibration sequence of functors
(1.2) (X ^ X)hZ=2 ! P2(X) ! X.
One can also construct this sequence using Goodwillie calculus: see x3.
Theorem 1.1 says that (1.2) splits after applying LT(n) for all n and X,
even though the connecting map
ffi : X ! (X ^ X)hZ=2
is often nontrivial before localization.
Remark 1.3. There are various sorts of polynomial functors studied in the lit
erature differing slightly from Goodwillie's dexcisive functors: R.McCarthy
has studied dadditive functors [McC ], and his student A.MauerOats [MO ]
has studied an infinite family interpolating between additive and excisive.
As will be explained more fully in x6, the analogue of Theorem 1.1 holds in
all these generalized settings.
Remark 1.4. Theorem 1.1 and Corollary 1.7 below also has consequences
for using the tower {PdF (X)} to understand E*n(F (X)), where En is the
usual pcomplete integral height n complex oriented commutative Salgebra.
4 KUHN
Since it is known [H ] that K(n)*(X) = 0 if and only if E*n(X) = 0, our the
orem says that the spectral sequence associated to the tower will collapse at
E1.
Theorem 1.1 is deduced from a rather different result in equivariant stable
homotopy theory that we now describe.
If G is a finite group, let GS denote the category of Smodules with G
action: the category of socalled `naive Gspectra'. Note that any Smodule
can be considered as an object in GS by giving it trivial Gaction.
For Y 2 GS, we let YhG and Y hG respectively denote associated homo
topy orbit and homotopy fixed point Smodules. There are various con
structions in the literature, more [GM ] or less [ACD , AK , Kl1, WW1 ] so
phisticated, of a natural `Norm' map
N(Y ) : YhG ! Y hG
satisfying the key property that N(Y ) is an equivalence if Y is a finite free
GCW spectrum. Let the Tate spectrum tG (Y ) be defined as the cofiber of
N(Y ). As recently observed by J.Klein [Kl2 ], up to weak equivalence, these
constructions are unique: see x2.
We prove the following vanishing theorem.
Theorem 1.5. For all finite groups G, primes p, and n 1,
LT(n)tG (LT(n)S) ' *.
This theorem will turn out to be equivalent to the following corollary.
Corollary 1.6. If T (n) is the telescope of any vnself map of a type n com
plex, then tG (T (n)) ' *.
Besides implying Theorem 1.1, Theorem 1.5 also leads to the following
splitting result.
Corollary 1.7. For any Y 2 GS, the fundamental cofibration sequence
N(Y ) hG
YhG ! Y ! tG (Y )
splits after applying LT(n) for any n.
One also immediately deduces results similar to [HSt , Cor.8.7].
Corollary 1.8. For all finite groups G, the norm map induces an isomor
phism
T (n)*(BG) ~!T (n)*(BG).
Similarly, LT(n)( 1 BG+ ) is self dual in the category of T (n)local spectra.
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 5
Our two theorems are supported by three propositions.
The first of these is a slight variant of results of R. McCarthy in [McC ],
and establishes the connection between our two theorems.
We need to recall Goodwillie's classification of homogeneous polynomial
functors [G3 ]. Let d denote the dthsymmetric group. If our original functor
F is finitary (terminology from [G3 ]), i.e. commutes with directed homotopy
colimits, then DdF (X) is weakly equivalent to a homotopy orbit spectrum
of the form
(CF (d) ^ X^d)h d,
where CF (d) 2 dS is determined naturally by F . Important to us is that,
even without the finitary hypothesis, there is a natural weak equivalence of
the form
DdF (X) ' ( dF (X))h d,
where dF is a functor determined naturally by F , taking values in the
category dS.
Proposition 1.9. Let F : S ! S be any homotopy functor. For all d 1,
there is a homotopy pullback diagram
PdF (X) ______//( dF (X))h d
pd 
fflffl fflffl
Pd1F (X) ____//_t d( dF (X)).
This diagram is natural in both X and F .
Our other two propositions together imply Theorem 1.5. The first is a
new very general observation about Tate spectra.
Proposition 1.10. Let R be a ring spectrum and E* a homology theory. If
tZ=p(R) is E*acyclic for all primes p, then so is tG (M) for all Rmodules
M and for all finite groups G.
We remark that, by standard arguments, tZ=p(R) ' *, and thus is cer
tainly E*acyclic, for all primes p such that R* is uniquely pdivisible. In
particular, to apply the proposition to the pair (R, E*) = (LT(n)S, T (n)*),
one need to only look at the single prime involved in the periodic theory.
It is in proving our last proposition that deep results in periodic stable
homotopy will be used.
Proposition 1.11. For all primes p and n 1, LT(n)tZ=p(LT(n)S) ' *.
At this point we need to comment on results like Theorem 1.5 in the
literature.
6 KUHN
The main theorem of the 1988 article by M.Mahowald and P.Shick [MS ]
can be restated as
(1.3) tZ=2(T (n)) ' *.
A proof along their lines can presumably be done at odd primes as well. We
will see that the generalization of their theorem to all primes is equivalent
to Proposition 1.11, yielding one possible proof of that result. We will offer
a rather different proof, using the telescopic functors of Bousfield and the
author [B1 , Ku1 , B2 ].
The main theorem of the 1996 article by J.Greenlees and H.Sadofsky [GS ]
reads
(1.4) tG (K(n)) ' *.
Their proof is elementary (in the sense that consequences of the Nilpotence
Theorem are not needed), but heavily uses two special facts about K(n): it is
complex oriented, and K(n)*(BZ=p) is a finitely generated K(n)*module.
Note that neither of these two facts is available when considering T (n)*.
For readers interested in the simplest proof of (1.4), it is hard to imagine
improving upon the clever argument given in [GS , Lemma 2.1] showing
that tZ=p(K(n)) ' * , but our Proposition 1.10 offers an alternative way to
proceed starting from this.
The most substantial part of the main theorem of [HSa ] says that
(1.5) LK(n)tG (LK(n)S) ' *.
Note that, were the Telescope Conjecture true, then (1.5) and Theorem 1.5
would be equivalent; at any rate, the latter implies the former. The authors
prove their theorem by starting from (1.4), and then using the Periodicity
Theorem, together with the technical heart of Hopkins and D. Ravenel's
proof [R2 ] that LE(n) is a smashing localization. Our proof of Theorem 1.5
bypasses the need for the HopkinsRavenel argument.
The rest of the paper is organized as follows. In x2, we review properties
of the norm map and tG , leading to a proof of Proposition 1.10. In x3,
supported by the appendix, we first discuss models for LE tZ=p(LE S) for
a general spectrum E, and then use telescopic functors to show that the
model is contractible when E = T (n). The results of the previous two
sections are combined in x4 yielding a proof of Theorem 1.5. Also in this
section is a discussion of the equivalence of Theorem 1.5 and Corollary 1.6,
with arguments similar in spirit to ones in [MS , HSa ]. In x5, we review
what we need to about dexcisive functors, and prove Proposition 1.9 with
arguments similar to those in [McC ]. In x6, we prove our splitting results,
Theorem 1.1 and Corollary 1.7.
As is already evident, if E is an Smodule, we let LE denote Bousfield
localization with respect to the associated homology theory E*. Through
out we also use the following conventions regarding functors taking values
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 7
in S. We write F f!~G if f(X) : F (X) ! G(X) is a weak equivalence for
all X. By a weak natural transformation f : F ! G we mean a pair of
natural tranformations of the form F g~H h!G or F h!H g~G. Finally,
we say that a diagram of weak natural transformations commutes if, after
evaluation on any object X, the associated diagram commutes in the stable
homotopy category.
Acknowledgements I would like to thank various people who have helped
me with this project. Randy McCarthy and Greg Arone have helped me
learn about Goodwillie towers. Obviously Randy's paper [McC ] has been
important to my thinking, and Greg suggested the compelling reformulation
of Randy's results given in Proposition 1.9. Neil Strickland alerted me to
the fact a conjecture of mine, that (1.5) was true, was already a theorem
in the literature, and Hal Sadofsky similarly told me about Mahowald and
Shick's theorem (1.3). Our main results have been reported on in various
seminars and conferences, e.g. at the A.M.S. meetings in January, 2003, and
in Oberwolfach in March, 2003.
2. Tate spectra and Proposition 1.10
2.1. Homotopy orbit and fixed point spectra. For G a fixed finite
group, and Y 2 GS, the Smodules YhG and Y hG are defined in the usual
way:
YhG = (EG+ ^ Y )=G, and Y hG = (Map S (EG+ , Y ))G .
Both of these functors take weak equivalences and cofibration sequences
in GS to weak equivalences and cofibration sequences in S. (See [GM , Part
I] for these sorts of facts.)
YhG has an important additional property not shared with Y hG: it com
mutes with filtered homotopy colimits.
We record the following well known facts, which are fundamental when
one considers the behavior of YhG and Y hG under Bousfield localization.
Lemma 2.1. If f : Y ! Z is a map in GS that is an E*isomorphism,
then fhG : YhG ! ZhG is also an E*isomorphism.
Lemma 2.2. If Y 2 GS is Elocal, so is Y hG.
2.2. A characterization of the norm map. A recent paper by Klein
[Kl2 ] exploring axioms for generalized FarrellTate cohomology leads to a
nice characterization of norm maps, and thus Tate spectra.
Proposition 2.3. Let NG (Y ), N0G(Y ) : YhG ! Y hG be natural transfor
mations such that both NG ( 1 G+ ) and N0G( 1 G+ ) are weak equivalences.
8 KUHN
Then there is a unique weak natural equivalence f(Y ) : YhG ~! YhG such
that the diagram
NG(Y )
YhG _____//Y hG
FF OO
FFF N0G(Y)
f(Y )F""FF
YhG
commutes. It follows that the cofibers of NG (Y ) and N0G(Y ) are naturally
weakly equivalent.
We sketch the proof, using the sorts of arguments in [Kl2 ].
Call a homotopy functor H : GS ! S homological if it preserves homo
topy pushout squares and filtered homotopy colimits. Then Klein, in the
spirit of [WW2 ], observes that any homotopy functor F : GS ! S admits
a universal left approximation by a homological functor, i.e. there exists
homological functor F hom, and a natural transformation F hom(Y ) ! F (Y )
satisfying the expected universal property.
Applying this to the case F (Y ) = Y hG, and observing that YhG is homo
logical, shows that there is a unique weak natural transformation g : YhG !
Y hG,homyielding a commutative diagram of weak natural transformations
NG(Y )
YhG J______//YOhGO
JJ 
JJJ 
g(Y )JJ%%J
Y hG,hom.
The right upward map is certainly an equivalence for Y = 1 G+ , and, by
assumption, so is the top map. Thus g is a weak natural transformation
between homological functors that is an equivalence when Y = 1 G+ . It
follows that g is weak equivalence.
Applying this same argument to N0Gyields the proposition.
2.3. Tate spectra. We refer to any natural transformation NG as in the
last proposition as a norm map. The cofiber of NG (Y ) is the associated
Tate spectrum, denoted tG (Y ). Both NG and tG are unique in the sense of
Proposition 2.3; their existence is shown in the various references cited in
the introduction.
It is immediate that tG preserves weak equivalences and cofibration se
quences.
From [GM , Prop. I.3.5], we deduce
Lemma 2.4. If R is a (homotopy) ring spectrum with trivial G action, and
M is an Rmodule, then tG (R) is a ring spectrum, and tG (M) is a tG (R)
module. Furthermore, RhG ! tG (R) is a map of Ralgebra spectra.
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 9
Fix Y 2 GS. For each subgroup H of G, Y can be regarded as being in
HS by restriction. From [GM , pp.2829], one deduces
Lemma 2.5. The assignment G=H 7 ! tH (Y ) defines a Mackey functor
to the stable homotopy category. Furthermore, Y hH ! tH (Y ) is a map of
Mackey functors.
In x5, we will use the following familiar property of the norm map. In
the literature, this explicitly appears, with a short axiomatic proof, as [AK ,
Prop.2.10].
Lemma 2.6. If K is a finite free GCW complex, then for all Y 2 GS,
tG (MapS (K, Y )) ' *.
2.4. Proof of Proposition 1.10. Recall that R is a ring spectrum, and
we are assuming that tZ=p(R) is E*acyclic. We wish to show that tG (M) is
also E* acyclic, for all Rmodules M, and for all G.
We first note that we can assume M = R. For tG (M) is a tG (R)module,
and thus the former will be E*acyclic if the latter is.
Next we show that we can reduce to the case when G is a pgroup. For
each prime p dividing the order of G, let Gp < G be a pSylow subgroup.
Then we have
Lemma 2.7. Given Y 2 GS and E* a generalized homology theory, tG (Y )
will be E*acyclic if tGp(Y ) is E*acyclic for all p dividing the order of G.
Proof.We recall that the completion of the Burnside ring A(H) is denoted
bA(H). The assignment G=H 7 ! Y hH is then an bAmodule Mackey functor
in the sense of [MM ]. Thus so is G=H 7 ! tH (Y ), and then also G=H 7 !
E*(tH (Y )). Now [MM , Cor.4] implies the lemma.
Having reduced Proposition 1.10 to the case when G is a pgroup, and is
thus solvable, the next lemma implies the proposition.
Lemma 2.8. Let K be a normal subgroup of G, Q = G=K, R a ring
spectrum, and E* a homology theory. If tK (R) and tQ (R) are both E*
acyclic, so is tG (R).
Proof.For Y 2 GS, consider the composite
NK (Y )hQ hK NQ(Y hK) hK hQ hG
YhG ' (YhK )hQ ! (Y )hQ ! (Y ) ' Y .
We will know that this composite can be considered a norm map if we check
that each of these maps is an equivalence when Y = 1 G+ .
10 KUHN
As there is an equivalence of Smodules with Kaction
`
1 G+ ' 1 K+ ,
gK2Q
it follows that NK ( 1 G+ ), and thus NK ( 1 G+ )hQ , is an equivalence.
As there are equivalences of Smodules with Qaction
NK ( 1 G+) 1 1
( 1 G+ )hK ~ ( G+ )hK !~ Q+ ,
it follows that NQ (( 1 G+ )hK ) is an equivalence.
We conclude from this discussion that if both NK (R)hQ and NQ (RhK )
are E*isomorphisms, then NG (R) will also be an E*isomorphism, and thus
tG (R) will be E*acyclic.
By assumption, tK (R) is E*acyclic. Thus NK (R) is an E*isomorphism.
By Lemma 2.1, NK (R)hQ is also.
By assumption, tQ (R) is E*acyclic. As tQ (RhK ) is a tQ (R)module,
we conclude that tQ (RhK ) is also E*acyclic, so that NQ (RhK ) is an E*
isomorphism.
3. Telescopic functors and Proposition 1.11
The goal of this section is to prove that LT(n)tZ=pLT(n)S ' *. We will
prove this by establishing that the localized unit map
LT(n)S ! LT(n)tZ=pLT(n)S
is null.
In outline our argument showing this is as follows. It is well known that
tZ=pS can be written as certain inverse limit of Thom spectra. Starting from
this, we will show that the unit map S ! tZ=pS factors though an inverse
limit of `connecting maps' associated to the Goodwillie tower of the functor
1 1 applied to spheres in negative dimensions. We warn the reader of
technical complications: odd primes are less pleasant than p = 2, we use the
theorems of W.H.Lin and J.Gunawardena establishing the Segal conjecture
for Z=p, and a key homological calculation is deferred to an appendix.
It will follow that the localized unit will factor through the inverse limit *
*of
the localized connecting maps. That this inverse limit is null will then be an
easy consequence of constructions of Bousfield and the author [B1 , Ku1 , B2]
showing that LT(n) factors through 1 . These `telescopic' constructions
heavily use the Periodicity Theorem of Hopkins and Smith [HopSm ], and
thus are also heavily dependent on the Nilpotence Theorem of [DHS ].
3.1. Models for LE tZ=pLE Y and LE t pLE Y . If ff is an orthogonal real
representation of a finite group G, we let S(ff) and Sffrespectively denote
the associated unit sphere and one point compactified sphere. Thus S(ff)
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 11
has an unbased Gaction while the Gaction on Sffis based, and there is a
cofibration sequence of based Gspaces
S(ff)+ ! S0 ! Sff.
Fix a prime p, and let æ denote p acting on Rp= (R) in the usual way.
The action of Z=p < p on S(æ) is free, and one concludes that the infinite
join S(1æ) is a model for EZ=p. This quickly leads to the following well
known description of tZ=p.
Lemma 3.1. (Compare with [GM , Thm.16.1].) For Y 2 GS, there is a
natural weak equivalence
tZ=pY ' holim MapS (Skj, Y )hZ=p.
k
We need a generalization of this.
Lemma 3.2. For Y 2 GS, there is a natural weak equivalence
LE tZ=pLE Y ' holim LE (MapS (Skj, Y )hZ=p).
k
If (p  1)! acts invertibly on E*, e.g. if E is plocal, there is a natural weak
equivalence
LE t pLE Y ' holim LE (MapS (Skj, Y )h p).
k
These equivalences are also natural with respect to the partially ordered set
of Bousfield classes , and there are commutative diagrams
LE t pLE Y___~__//holimk LE (MapS (Skj, Y )h p)
 
 
fflffl fflffl
LE tZ=pLE Y _~__//_holimk LE (MapS (Skj, Y )hZ=p).
Proof.By definition, LE tZ=pLE Y is the cofiber of
LE NZ=p(LE Y ) : LE (LE Y )hZ=p ! LE (LE Y )hZ=p.
The domain of this map can be simplified:
LE YhZ=p ! LE (LE Y )hZ=p
12 KUHN
is an equivalence. Meanwhile, the range of this map rewritten via the fol
lowing chain of natural weak equivalences:
LE (LE Y )hZ=p~(LE Y )hZ=p
~!Map hZ=p
S (S(1æ)+ , LE Y )
~!holimMap (S(kæ) , L Y )hZ=p
k S + E
~!holimL Map (S(kæ) , L Y )hZ=p
k E S + E
~ holimL Map (S(kæ) , L Y )
k E S + E hZ=p
~ holimL Map (S(kæ) , Y ) .
k E S + hZ=p
The crucial second to last equivalence here is induced by norm maps which
are equivalences since Z=p acts freely on S(kæ).
Thus LE tZ=pLE Y has been identified:
LE tZ=pLE Y~!holim cofiber{LE YhZ=p ! LE MapS (S(kæ)+ , Y )hZ=p}
k
~!holim L Map (Skj, Y ) .
k E S hZ=p
The proof of the statements for t p are similar, noting that, under the
hypothesis that (p  1)! acts invertibly on E*, the norm maps
MapS (S(kæ)+ , LE Y )h p ! MapS (S(kæ)+ , LE Y )h p
will still be equivalences.
For r 0, and X an Smodule, we let DrX = (X^r)h r, and we recall
that there are natural transformations DrX ! Dr X. Specializing to
r = p, a quick check of definitions verifies the next lemma.
Lemma 3.3. There is a natural weak equivalence
kDp k X ' MapS (Skj, X^p)h p,
and thus there is a plocal equivalence
t pS ' holim k+1DpSk .
k
Define dk : S ! k+1DpSk to be be the composite
S unit!t pS ! k+1DpSk .
As the restriction map t pS ! tZ=pS is unital, our various observations
combine to yield the following proposition.
Proposition 3.4. If E is plocal, LE tZ=pLE S ' * if and only if
holim LE dk : LE S ! holimLE k+1DpSk
k k
is null.
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 13
3.2. The Goodwillie tower of 1 1 . Recall that 1 Z denotes the sus
pension spectrum of a space Z, and that 1 has right adjoint 1 , where
1 X is the zerothspace of a spectrum X.
Let Pr(X) denote the rth functor in the Goodwillie tower of the functor
1 1 : S ! S.
Thus this Goodwillie tower has the form
..
.


fflffl
P3(X)99
sss
ssss p3(X)
e3(X)ssss fflffl
ssss ii4P2(X)4
sse2(X)iiiiiiss
sss iiiiii p2(X)
ssiiiiise1(X) fflffl
1 1 X _________________//P1(X).
This tower has the following fundamental properties.
(1) If X is 0connected, then 1 1 X ! holimrPr(X) is an equivalence.
(2) The fiber of pr(X) : Pr(X) ! Pr1(X) is naturally weakly equivalent to
Dr(X).
(3) There are equivalences D1X ' P1X ' X, and via the second of these,
e1(X) : 1 1 X ! P1X can be identified the with evaluation map ffl(X) :
1 1 X ! X.
All of these properties can be deduced from Goodwillie's general theory.
For an explicit discussion of these (and more) see [AK ] or [Ku2 ].
3.3. Telescopic functors. Bousfield and the author have deduced the the
following consequence of the Periodicity Theorem.
Theorem 3.5. There exists a functor n : Spaces ! Smodules and a
natural weak equivalence n 1 X ' LT(n)X.
With the result stated at the level of homotopy categories, and with
K(n) replacing T (n), this is the main theorem of [Ku1 ]. However the sorts
of constructions given there, and in [B1 ] (for n = 1), yield the theorem as
stated: see [B2 ].
This has the following immediate corollary [Ku1 , Ku2 ].
14 KUHN
Corollary 3.6. There is a natural factorization by weak Smodule maps
LT(n) 1 1 X
''n(X)oo77oo OOOLT(n)ffl(X)OO
oooo OOO
ooo OO''O
LT(n)X _________________________LT(n)X.
To use this, we recall an observation about reduced homotopy functors,
functors F : S ! S such that F (X) is contractible whenever X is. Good
willie observes that then there is an induced weak natural transformation
F (X) ! F ( X).
The naturality is with respect to both X and F . For example, if F = Dr,
this natural transformation agrees with the one discussed previously.
In particular, we can apply this construction to both the domain and
range of the natural transformation
LT(n)Pp ! LT(n)P1,
evaluated on k X for all k 0. Recalling that LT(n) commutes with
suspension and P1(X) ' X, we obtain maps
holim kLT(n)Pp( k X) ! LT(n)X.
k
Theorem 3.7. holim kLT(n)Pp( k X) ! LT(n)X admits a homotopy sec
k
tion.
Proof.A section is given by holim k(LT(n)ep( k X) O jn( k X)).
k
3.4. Specialization to odd spheres. Standard homology calculations as
in [CLM , BMMS ] imply the next lemma.
Lemma 3.8. Localized at an odd prime p, DrSk ' * for odd k 2 Z, and for
2 r p  1. Thus (for all primes p) the natural map
holim kPp1(Sk ) ! S
k
is a plocal equivalence.
Continuing the cofibration sequence DpX ! Pp(X) ! Pp1(X) one step
to the right defines a natural transformation
ffi(X) : Pp1(X) ! DpX.
Localized at p, define ffik : S ! k+1DpSk to be the composite
kffi(Sk)k+1 k
S ~ kPp1(Sk ) ! DpS .
Proposition 3.9. holim LT(n)ffik : LT(n)S ! holimLT(n) k+1DpSk is
k k
null.
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 15
Proof.Localized at p, there is a cofibration sequence
holimLT(n) kPpSk ! LT(n)S ! holimLT(n) k+1DpSk .
k k
Theorem 3.7 says that the first map has a section. Thus the second map is
null.
3.5. Proof of Proposition 1.11. A comparison of Proposition 3.4 with
Proposition 3.9 shows that we will have proved Proposition 1.11 once we
check the following lemma.
Lemma 3.10. holim dk : S ! holim k+1DpSk factors through
k k
holimffik : S ! holim k+1DpSk .
k k
Proof.W.H.Lin's theorem [L ], when p = 2, and J.Gunawardena's theorem
[Gun , AGM ], when p is odd, can be stated in the following way:
holim dk : S ! holim k+1DpSk
k k
is padic completion. It follows that we need to check that holim ffik 2
k
ß0(holim k+1DpSk ) ' Zp is a topological generator. As topological gen
k
erators of Zp are detected mod p, the next lemma, whose proof is deferred
to the appendix, completes our argument.
Lemma 3.11. ffi(S1 ) : Pp1(S1 ) ! DpS1 is nonzero in mod p homol
ogy.
4. The proofs of Theorem 1.5 and Corollary 1.6
We begin this section by noting how Proposition 1.11 and Proposition 1.10
together imply Theorem 1.5. Proposition 1.11 can be restated as saying
that tZ=pLT(n)S is T (n)*acyclic. Recalling that the localization of a ring
spectrum (e.g. S) is again a ring spectrum, Proposition 1.10 can then be
applied to the pair (R, E*) = (LT(n)S, T (n)*), to conclude that tG LT(n)S is
T (n)*acyclic for all G. This is a restatement of Theorem 1.5.
Now we turn to showing how Corollary 1.6 can be deduced from Theo
rem 1.5, and vice versa.
We need to review some of the fine points of the Periodicity Theorem.
(A good reference for this is [R2 ].) We fix a prime p, and work with plocal
spectra. A finite spectrum F is of type n if K(n)*(F ) 6= 0, but K(i)*(F ) = 0
for i < n. Let Cn = {finiteF  F has type at leastn}. Then every F 2 Cn
admits a vn self map: a map f : dF ! F such that K(n)*(f) is an
isomorphism, but K(i)*(f) = 0 for all i 6= n. If n > 0, then d will necessarily
be positive. In all cases, f is unique and natural up to iteration. Thus there
is a well defined functor from Cn to spectra sending F to v1nF , the telescope
of any vn self map of F . We note that vn preserves both cofibration sequences
and retracts.
16 KUHN
The Thick Subcategory Theorem says that any thick subcategory of the
category of plocal spectra, i.e. any collection of plocal finite spectra clos*
*ed
under cofibration sequences and retracts, is Cn for some n 0.
We recall that LT(n) denotes Lv1nFfor any F of type n. From the facts
stated above, it is easily verified that this is independent of choice of F , a*
*nd
that for all F 2 Cn, LT(n)(F ) = v1nF . Finally we note that if F has type n
and F 0has type i 6= n, then v1nF ^ v1iF 0' *.
Lemma 4.1. Fix a finite group G. The following conditions are equivalent.
(1) tG (LT(n)S) is T (n)*acyclic.
(2) For all F 2 Cn, tG (v1nF ) ' *.
(3) For all type n complexes F , tG (v1nF ) ' *.
(4) There exists a type n complex F such that tG (v1nF ) ' *.
Note that statement (1) is the conclusion of Theorem 1.5 and (3) is the
conclusion of Corollary 1.6.
Clearly (2) implies (3), which in turn implies (4). To see that (4) implies
(2), note that the collection of F 2 Cn such that tG (v1nF ) ' * forms a thick
subcategory contained in Cn. Such a thick subcategory will equal all of Cn
if it contains any type n finite. (This type of reasoning appears in [MS ].)
Now suppose (1) holds. Since v1nF ' LT(n)F , it is an LT(n)Smodule,
and we see that tG (v1nF ) is (v1nF )*acyclic for all finite F of type n. It*
* is
easy to find a type n finite F that is a ring spectrum; thus so is R = v1nF .
But then tG (R) will be an R*acyclic Rmodule, and thus contractible, i.e.
statement (4) holds.
It remains to show that (2) implies (1). We reason as in [HSa ].
Define finite spectra F (0), . .,.F (n) by first setting F (0) = S, and then
recursively defining F (i + 1) to be the cofiber of a vi self map of F (i).
Ravenel [R1 ] observes that if f : dX ! X is a self map with cofiber C
and telescope T , then = . Applying this n times leads to an
equality of Bousfield classes
n1`
= .
i=0
Smashing this with tG (LT(n)S), and noting that
tG (LT(n)S) ^ F (n) ' tG (LT(n)F (n)) ' tG (v1nF (n)),
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 17
leads to
n1`
= .
i=0
Smashing this with T (n), and noting that T (n) ^ v1iF (i) ' * if i < n,
leads to
= .
If (2) holds, then the right side of this last equation is the Bousfield class *
*of
a contactible spectrum. Thus so is the left, i.e. (1) holds.
5. Polynomial functors and Tate cohomology
In this section we sketch a proof of Proposition 1.9. As I hope will be
clear, this proposition is just a variant of [McC , Prop.4], and our proof uses
precisely the same ideas that McCarthy does.
5.1. Review of Goodwillie calculus. In the series of papers [G1 , G2 ,
G3 ], Tom Goodwillie has developed his theory of polynomial resolutions of
homotopy functors. We need to summarize some aspects of Goodwillie's
work as they apply to functors from Smodules to Smodules. Throughout
we cite the version of [G3 ] of June, 2002.
In [G2 ], Goodwillie begins by defining and studying the total homotopy
fiber of a cubical diagram. For example the total homotopy fiber of a square
X0 _____//_X1
 
 
fflffl fflffl
X2 ____//_X12
is the homotopy fiber of the evident map from X0 to the homotopy pullback
of the square with X0 omitted. A cubical diagram is then homotopy cartesian
if its total fiber is weakly contractible. Dual constructions similarly define
total homotopy cofibers and homotopy cocartesion cubes. We note that in
a stable model category like S, a cubical diagram is homotopy cartesian
exactly when it is homotopy cocartesion.
A cubical diagram is strongly homotopy cocartesion if each of its 2 dimen
sional faces is homotopy cocartesion. A functor is then said to be dexcisive
if it takes strongly homotopy cocartesion (d + 1)cubical diagrams to homo
topy cartesian cubical diagrams.
In [G3 ], given a functor F , Goodwillie proves the existence of a tower
{PdF } under F so that F ! PdF is the universal arrow to a dexcisive
functor, up to weak equivalence.
For functors with range in a stable model category, Goodwillie [G3 ] gives
a description of how DdF (X), the fiber of PdF (X) ! Pd1F (X), can be
computed by means of cross effects. We describe how this goes in our setting.
18 KUHN
Let F : S ! S be a functor. Let d = {1, 2, . .,.d}. In [G3 , x3], crdF , the
dth cross effect of F , is defined to the the functor of d variables given as t*
*he
total homotopy fiber
`
(crdF )(X1, . .,.Xd) = TotFib F ( Xi).
T d i2dT
A dvariable homotopy functor H : Sd ! S is reduced if H(X1, . .,.Xd)
is contractible whenever any of the Xi are. Given such a functor, its multi
linearization L(H) : Sd ! S is defined by the formula
(5.1) L(H)(X1, . .,.Xd) = hocolimn n1+...+ndH( n1X1, . .,. ndXd).
i!1
This will be 1excisive in each variable.
Now define dF : S ! dS by the formula
dF (X) = L(crdF )(X, . .,.X).
Then [G3 , Theorems 3.5, 6.1] says that there is a natural weak equivalence
(5.2) DdF (X) ' ( dF )(X)h d.
We need to explain some of the ideas behind this formula.
Firstly, d(F ) ! d(PdF ) is always an equivalence, and it follows that
one can assume the original functor F is dexcisive.
If F is dexcisive then crdF is already 1excisive in each variable [G3 ,
Prop.3.3], and so dF (X) can be identified with (crdF )(X, . .,.X). In this
case, the natural map
DdF (X) ! PdF (X)
identifies with the natural transformation
ffd(X) : ( dF )(X)h d ! F (X)
defined to be the composite
`d
( dF )(X)h d ! F ( X)h d ! F (X).
i=1
W d
Here the second map is induced by the fold map i=1X ! X.
Goodwillie proves (5.2) by verifying that crd(ffd) is an equivalence, so that
Dd(ffd) is an equivalence. Enroute to this, he shows that there is a natural
equivariant weak equivalence
crd( dF ) ' d+ ^ crdF.
5.2. Dual constructions. In [McC ], McCarthy investigates `dual calculus'.
In this spirit, replacing wedges by products, fibers by cofibers, etc., leads to
constructions dual to the above. In particular, given F : S ! S, we define
crdF : Sd ! S by the formula
Y
(crdF )(X1, . .,.Xd) = TotCofibF ( Xi),
T d i2T
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 19
and then we define dF : S ! dS by
dF (X) = L(crdF )(X, . .,.X).
Because both the domain and range of F is a stable model category, one
sees that each of the natural transormations
crdF ! crdF
and
dF ! dF
are weak equivalences.
If F is dexcisive then dF (X) can be identified with (crdF )(X, . .,.X).
In this case, we define the weak natural transformation
ffd(X) : F (X) ! ( dF )(X)h d
to be the zigzag composite
F (X) ! F (Xd)h d ! ( dF )(X)h d ~ ( dF )(X)h d.
Here the first map is induced by the diagonal X ! Xd.
Arguments dual to Goodwillie's show that the next lemma holds.
Lemma 5.1. (Compare with [McC , Lemmas 3.7,3.8].) Let F : S ! S be
dexcisive.
(1) crd(ffd), and thus Dd(ffd), is an equivalence.
(2) There is a natural equivariant weak equivalence
crd( dF ) ' MapS ( + , crdF ).
5.3. Proof of Proposition 1.9. Proposition 1.9 is a formal consequence
of Lemma 5.1. First of all, we observe the following.
Lemma 5.2. (Compare with [McC , proof of Prop.4].) Let F be dexcisive.
Then t d( dF ) is (d  1)excisive. Thus the cofibration sequence
Dd(( dF )h d) ! Pd( dF )h d ! Pd1(( dF )h d)
identifies with the norm sequence
( dF )h d ! ( dF )h d ! t d( dF ).
Proof.For the first statement, we check that crd(t d( dF )) ' *:
crd(t d( dF )) ' t d(crd( dF )) ' t d(MapS ( + , crdF )) ' *.
Here we have used Lemma 5.1(2) and Lemma 2.6.
As ( dF )h d is dexcisive and homogeneous, the second statement fol
lows.
20 KUHN
Now we turn to the proof of Proposition 1.9. We can assume that F
is dexcisive. Assuming this, the last lemma implies that the weak natu
ral tranformation ffd(X) : F (X) ! ( dF (X))h d induces a commutative
diagram of weak natural transformations
DdF (X) _____//_( dF (X))h d
 
 
fflffl fflffl
PdF (X) ______//( dF (X))h d
 
 
fflffl fflffl
Pd1F (X) ____//_t d( dF (X)).
In this diagram each of the vertical columns is a homotopy fibration sequence
of Smodules. The top map is a weak equivalence thanks to Lemma 5.1(1).
Thus the bottom square is a homotopy pullback diagram.
5.4. Polynomial functor variants. McCarthy and his student Mauer
Oats [MO ] have explored various different notions of what it might mean to
say a functor F : A ! B is polynomial of degree at most d, with dexcisive
and dadditive as two special cases. In these variants B should surely be a
reasonable model category, but A can often be a category with much less
structure. As a hint of why this might be true, note that the definition of
cross effects only uses the existence of finite coproducts in A.
If B is any stable model category admitting norm maps, and A is also
appropriately stable, then the evident analogue of Proposition 1.9 still holds.
The discussion above goes through with one little change: the formula (5.1)
for the (multi)linearization process L needs to be adjusted to reflect the
notion of degree 1 functor at hand. Note that our proof of Proposition 1.9
didn't use this formula (nor did McCarthy's arguments in [McC ]).
Of relevance to the next section, we note that these variants of L are still
homotopy colimits, and thus preserve E*isomorphisms.
6.Localization and the proofs of Theorem 1.1 and
Corollary 1.7
In this section, we show how our vanishing Tate cohomology result, The
orem 1.5, leads to the splitting results Theorem 1.1 and Corollary 1.7. To
simplify notation, we let L = LT(n).
Proof of Corollary 1.7.Let Y be an Smodule with G action. We wish to
show that the norm sequence
N(Y ) hG
YhG ! Y ! tG (Y )
splits after applying L. Thus we need to construct a left homotopy inverse
to L(N(Y )).
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 21
The localization map Y ! LY induces a commutative diagram
N(Y )
YhG ______________//_Y hG
 
 
fflfflN(LY ) fflffl
(LY )hG ___________//(LY )hG .
Applying L to this yields the diagram
L(N(Y ))
L(YhG )______________//_L(Y hG)
o L(''hG)
fflfflL(N(LY )) fflffl
L((LY )hG )____~_____//_L((LY )hG ).
Here the left vertical map is an equivalence, as homology isomorphisms
are preserved by taking homotopy orbits (Lemma 2.1). The lower map,
L(N(LY )), is an equivalence by Theorem 1.5: its cofiber, L(tG (LY )), is a
module over L(tG (LS)), and is thus contractible.
Our desired left homotopy inverse is now obtained by composing the right
vertical map of the diagram with the inverses of the two indicated equiva
lences.
Proof of Theorem 1.1.We are given a functor F : S ! S and wish to prove
that
DdF (X) ! PdF (X) ! Pd1F (X)
splits after applying L. Thus we need to construct a left homotopy inverse
to LDdF (X) ! LPdF (X).
We need a lemma that plays the role that Lemma 2.1 played in the pre
vious proof. Call a natural transformation F ! G an E*isomorphism, if
F (X) ! G(X) is an E*isomorphism for all X.
Formula (5.2) says that Dd is the composition of constructions each of
which preserve E*isomorphisms, and thus we have
Lemma 6.1. If F ! G is an E*isomorphism, then so is DdF ! DdG.
Remark 6.2. This lemma holds for the variants on the notion of dexcisive,
as discussed above in x5.4.
Armed with this lemma, Theorem 1.1 is proved as follows.
The localization natural transformation F ! LF , together with Proposi
tion 1.9, induce a commutative diagram
DdF _____//Dd(LF )_~__//_ d(LF )h d
  
  
fflffl fflffl fflffl
PdF _____//_Pd(LF_)___// d(LF )h d.
22 KUHN
Applying L to this, gives the diagram
LDdF _~__//_LDd(LF )_~__//L( d(LF )h d)
  
  o
fflffl fflffl fflffl
LPdF _____//_LPd(LF )____//L( d(LF )h d).
Here the top left natural transformation is an equivalence by the lemma
just stated. The right vertical natural transformation is an equivalence by
Theorem 1.5, as its cofiber, L(t d( d(LF )), is an L(tG (LS))module, when
evaluated on any X. (Though not necessarily local, due to the hocolimit
construction L, d(LF )(X) is nevertheless an LSmodule.)
Our desired left homotopy inverse is now obtained by composing the nat
ural transformation along the bottom of this diagram with the inverses of
the three indicated equivalences.
Appendix A. Proof of Lemma 3.11
We begin with some needed notation.
Recall that Pr denotes the rth Goodwillie approximation to the functor
1 1 . We let
ffi(X) : Pr1(X) ! DrX
denote the connecting map for the cofibration sequence
DrX ! Pr(X) ! Pr1(X).
Given any reduced homotopy functor F : S ! S, we let
(X) : F (X) ! F ( X)
denote the canonical natural map.
Fixing a prime p, all homology will be with Z=p coefficients. The Steenrod
operations act on H*(X) as operations lowering dimensions. To unify the
`even' prime and odd prime cases, we let P1 = Sq2, when p = 2. Thus, for
all primes p, P1 lowers degree by 2p  2.
The goal of this appendix is to prove Lemma 3.11, which we restate more
precisely.
Lemma A.1. ffi* : H1(Pp1(S1 )) ! H1( Dp(S1 )) is an isomorphism
of one dimensional Z=pmodules.
Recall that H*(DrX) is a known functor of H*(X), both additively, and
as a module over the Steenrod algebra. Furthermore, the behavior of * :
H*( DrX) ! H*(Dr X) is known. See [CLM , BMMS ].
TATE COHOMOLOGY AND PERIODIC LOCALIZATION 23
Naturality implies that there is a commutative diagram:
ffi* 1
H1(Pp1(S1 )) ________________//_H1( Dp(S ))
o o
fflffl ffi* fflffl
H1(Pp1(O1HZ))O _____________//_H1( Dp(O1HZ))O
P1* oP1*
 ffi* 
H2p3(Pp1( 1HZ)) ____________//H2p3( Dp( 1HZ))
* o*
fflffl 2ffi* fflffl
H2p3( 2Pp1( HZ)) __________//_H2p3( 1Dp( HZ)),
where the top vertical maps are induced by the inclusion S1 ! 1HZ.
The top square is a square of homology groups of lowest degree. That
the indicated maps are isomorphisms, all between one dimensional vector
spaces, is an easy consequence of facts from [CLM , BMMS ]. For example,
the middle right map is an isomorphism due to the Nishida relation
P1*fiQ1x = fiQ0x 2 H2(Dp( 1HZ)),
for x 2 H1( 1HZ).
Using this diagram, to show that the top map is nonzero, and thus an
isomorphism, it suffices to show that the lower left map and the bottom map
are each isomorphisms. We state each of these as a separate lemma (one in
dual form).
Lemma A.2. * : H2p3(Pp1( 1HZ)) ! H2p3( 2Pp1( HZ)) is an
isomorphism of one dimensional Z=pmodules.
Proof.When p = 2, is an equivalence, and so * is an isomorphism.
When p is odd, the situation is more complicated, and we proceed as
follows. We have a commutative diagram
* 2
H2p3(Pp1( 1HZ)) __________//_H2p3( Pp1( HZ))

o 
fflffl 
H2p3(P2( 1HZ)) o

 
 
fflffl * fflffl
H2p3( 1HZ) ________~________//_H2p3( 1HZ)
with indicated isomorphisms. Thus, to show the top map is an isomorphism,
we need to check that the lower left map is an isomorphism. Equivalently,
we need to check that
ffi* : H2p3( 1HZ) ! H2p3( D2 1HZ)
24 KUHN
is zero. The map ffi : 1HZ ! D2 1HZ factors through
: 2D2 2HZ ! D2 1HZ,
and this map is zero on H2p3: the range is one dimensional, spanned by the
suspension of a *decomposable of the form x * y, with x 2 H1( 1HZ)
and y 2 H2p3( 1HZ). But nonzero *decomposables are never in the
image of * : H*( D2(X)) ! H*(D2( X)).
With our final lemma, we have reached the heart of the matter.
Lemma A.3. ffi* : H2p1( Dp( HZ)) ! H2p1(Pp1( HZ)) is an iso
morphism of one dimensional Z=pmodules.
Proof.Since HZ is 0connected, the Goodwillie tower Pr( HZ) converges
strongly to 1 1 ( HZ) = 1 S1. Thus the associated 2nd quadrant
spectral sequence converges strongly to H*(S1). For this to happen, P1(x)
must be in the image of ffi*, where x 2 H1(Pp1( HZ)) is a nonzero element,
for otherwise P1(x) 6= 0 2 H2p1(S1).
Thus ffi* is nonzero, and is thus an isomorphism.
Remark A.4. In work in progress, the author is studying the spectral se
quence converging to H*( 1 X) with Er,*+r1= H*(DrX). The sort of
argument just given generalizes to show that the first interesting differential
is dp1 : H*1(DpX) ! H*(X). This differential is determined by H*(X)
as a module over the Steenrod algebra, and has image imposing the unstable
condition on H*(X).
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Department of Mathematics, University of Virginia, Charlottesville, VA
22903
Email address: njk4x@virginia.edu