COHOMOLOGICAL BOUSFIELD CLASSES
MARK HOVEY
August 1993
Abstract. In this paper, we begin the study of Bousfield classes for coho*
*mology
theories defined on spectra. Our main result is that a map f : X ! Y indu*
*ces
an isomorphism on E(n)-cohomology if and only if it induces an isomorphis*
*m on
E(n)-homology. We also prove this for variants of E(n) such as elliptic c*
*ohomology
and real K-theory. We also show that there is a nontrivial map from a spe*
*ctrum Z
to the K(n)-local sphere if and only if K(n)*(Z) 6= 0.
Introduction
Homological localization functors, introduced by Bousfield in [2], are very i*
*mpor-
tant in stable homotopy theory. See, for example, [11] or [13]. It seems natu*
*ral to
ask whether cohomological localization functors might not also be interesting. *
*The
main reason these have not been considered very much is that they are not known
to exist: the proof of Bousfield and all other known proofs of the existence of*
* homo-
logical localization functors run into potential set-theoretic difficulties whe*
*n they are
applied to cohomology theories. See [10, Ch. 7] for a nice discussion of this p*
*oint.
Nonetheless, one can still consider cohomological Bousfield classes. The coho*
*molog-
ical Bousfield class of E, , is simply the class of spectra X such that E*(*
*X) = 0.
Here and throughout the paper we will assume that all spectra have been localiz*
*ed at
some prime p. It turns out that this is a generalization of the usual Bousfield*
* class.
That is, for every spectrum E there is a spectrum IE (the Brown-Comenetz dual)
such that
= :
We calculate the cohomological Bousfield classes of most standard spectra, such*
* as
finite spectra, BP , BP , E(n) , and KO. Recall the rule of thumb for calcul*
*ating
ordinary Bousfield classes: look at which vn are non-nilpotent and remember tha*
*t in-
verting vn kills all vi-periodic information for i > n: The corresponding rule *
*of thumb
for cohomological Bousfield classes is to look at which vn appear and remember *
*that
completing at vn kills all vn-adic information. For example, a connective spect*
*rum is
___________
1991 Mathematics Subject Classification. 55P42, 55P60, 55N22, 55N20.
1
2 MARK HOVEY
already complete with respect to the vi for i > 0 because they have positive de*
*gree.
So one does not expect the cohomological Bousfield class of a connective spectr*
*um
to be larger than that of ordinary homology HZ. We show that this is correct, *
*at
least for connective spectra of finite type, in section 3. Similarly, one expec*
*ts
= = ;
since E(n) is not complete with respect to any vi for i n. We show this in sec*
*tion
2. Note that this implies the theorem stated in the abstract: that a map is an *
*E(n)-
cohomology isomorphism if and only if it is an E(n)-homology isomorphism. On
the other hand, one expects that the version of E(n) preferred by Hopkins and h*
*is
co-authors, En, will have
=
since it is complete with respect to all of the vi for i < n. We show this in s*
*ection 3.
Note that all of the cohomological Bousfield classes we calculate in this pap*
*er turn
out in fact to be homological Bousfield classes, and thus have localization fun*
*ctors.
We conjecture that every cohomological Bousfield class is a homological Bousfie*
*ld
class.
The organization of this paper is as follows. In the first section, we define*
* coho-
mological Bousfield classes and point out some general facts. The second secti*
*on
contains our calculation of the cohomological Bousfield class of E(n), as well *
*as that
of KO. The last section discusses cohomological Bousfield classes of other spec*
*tra,
in particular that of connective spectra, En, and LK(n)S0. We also point out h*
*ow
the work in this paper shows that Conjecture 3.10 of [9] is an analog of the te*
*lescope
conjecture for spectra E such that E*(X) 6= 0 for all finite X.
There is considerable overlap between this paper and an unpublished paper of
Bousfield [3]. In particular, he also noticed Proposition 1.1, and used it to c*
*onstruct
cohomological localizations for many spectra. The author would like to thank Pe*
*te
Bousfield for sharing his work, John Greenlees for his interest, and the refere*
*e for
improving the proof of Theorem 3.1.
1.Generalities
We begin by reminding the reader about ordinary Bousfield classes. Given a sp*
*ec-
trum E, define a spectrum X to be E-acyclic if E*(X) = 0, or equivalently if E *
*^ X
is null. Define a spectrum X to be E-local if there are no nontrivial maps from*
* any
E-acyclic spectrum Z to X. Define a map f : X ! Y to be an E-equivalence if it
is an isomorphism on E-homology. It is easy to see that an E-equivalence between
E-local spectra is a homotopy equivalence.
The Bousfield class of E, , is defined as the class of E-acyclic spectra. *
*Bousfield
classes are ordered by reverse inclusion, so if and only if every E-ac*
*yclic
COHOMOLOGICAL BOUSFIELD CLASSES 3
spectrum is also F -acyclic. There is a join, defined by
_ = :
In general, Bousfield classes do not form a lattice, but we can define
^ =
as long as we are careful not to treat it as a meet operation.
We define cohomological Bousfield classes in a similar way. Given a spectrum *
*E,
define a spectrum X to be E*-acyclic if E*(X) = 0, or equivalently, if the func*
*tion
spectrum F (X; E) is null. Then define a spectrum X to be E*-local, if there a*
*re
no non-trivial maps from any E*-acyclic spectrum Z to X. A map f : X ! Y
is called an E*-equivalence if it induces an isomorphism on E-cohomology. Just *
*as
in the homological case, an E*-equivalence between E*-local spectra is a homoto*
*py
equivalence.
Define the cohomological Bousfield class of E, , as the class of all E*-a*
*cyclic
spectra. Together with the usual Bousfield classes , we get a partially ord*
*ered
class, where the ordering is defined by reverse inclusion. Thus if e*
*very
X*-acyclic spectrum is also Y *-acyclic. Similarly, if every X*-acyc*
*lic
spectrum is Y -acyclic. One can define the join of two cohomological Bousfield *
*classes
by
_ = <(X _ Y )*>;
and it is indeed a join operation. Note that infinite joins do exist, but they *
*are given
by the product, not the wedge. So we have
_ Y
= <( Xn)*>:
One can also define
^ = ;
but again this is not really a meet operation. I do not think it is possible to*
* reasonably
define ^ :
The first thing to point out is that cohomological Bousfield classes are a ge*
*neraliza-
tion of ordinary Bousfield classes. Recall that IX denotes the Brown-Comenetz d*
*ual
[4] of X, defined as the spectrum which represents the exact (contravariant) fu*
*nctor
Z ! Hom (X*(Z); Q=Z(p)):
Remember that all our spectra are assumed to be p-local: otherwise we would rep*
*lace
Q=Z(p)by Q=Z. The indexing works out so that
(IX)n(Z) = Hom (Xn(Z); Q=Z(p)):
Proposition 1.1. For any spectrum X,
= :
4 MARK HOVEY
Proof.Let I denote IS0. A fundamental fact about Brown-Comenetz duality is that
IX = F (X; I):
This is easily proved, as follows.
[Z; F (X; I)] = [Z ^ X; I] = I0(Z ^ X)
= Hom (ss0(Z ^ X); Q=Z) = Hom (X0(Z); Q=Z) = [Z; IX]:
It follows then that
I(Z ^ X) = F (Z ^ X; I) = F (Z; IX):
Note that for any (p-local) abelian group A, Hom (A; Q=Z(p)) = 0 if and only if*
* A = 0:
Thus IY is null if and only if Y is null. Hence Z ^ X is null if and only if F *
*(Z; IX)
is null, proving the proposition. __|_ |
We will calculate many cohomological Bousfield classes below. Every one of th*
*em
is a homological Bousfield class. This leads us to make the following conjectur*
*e.
Conjecture 1.2. For any spectrum X, there is a spectrum Z such that
= :
If X has no rational homology, we can take Z = IX.
The main evidence for this conjecture is Theorem 3.1 below, and it is also ex*
*plained
there why we need the assumption that X has no rational homology to deduce that*
* we
can take Z = IX: Note that this conjecture would obviate the need for construct*
*ing
cohomological localization functors. It would also provide a partial fix to the*
* failure of
Brown-Comenetz duality to be a true duality. In general, there is a map X ! I2X,
but it is not an equivalence. The conjecture would say that if X has no ration*
*al
homology, we would at least have
= :
We now give some simple lemmas on cohomological Bousfield classes.
Lemma 1.3. (1) Any spectrum E is E*-local.
(2) If
X ! Y ! Z
is a cofibre sequence, then
_ :
(3) If Y is a retract of X, then
:
(4) X is E*-local if and only if :
COHOMOLOGICAL BOUSFIELD CLASSES 5
The proof is simply unwinding definitions, so we leave it to the reader. But *
*note
that the fact that E is always E*-local gives an interesting interpretation of *
*the
conjecture above. The typical way to show that [X; Y ]* is zero is to find a sp*
*ectrum
Z such that X is Z-acyclic and Y is Z-local. Conjecture 1.2 says you can always*
* do
this. Indeed, if Y *(X) = 0, X is by definition Y *-acyclic, and Y is always Y *
**-local.
Conjecture 1.2 says there is a spectrum Z such that = , so X is Z-acyc*
*lic,
and Y is Z-local.
Lemma 1.4. Suppose R is a ring spectrum, and that M is an R-module spectrum.
Then .
Proof.Suppose R*(X) = 0, and f : X ! M is a map. The composite
f
X ! M ! R ^ M
with the unit of R is the same as
R^f
X ! R ^ X ! R ^ M
which is null. But M is a retract of R ^ M, so f itself must be null. __|_ |
Note that it is not in general true that when M is an R-module
spectrum. A counterexample will be provided in section 3.
The most interesting lemma is the analog of Ravenel's lemma about the Bousfie*
*ld
class of a telescope. Given a self-map f : nX ! X, form the (homotopy) inverse
limit
f n f
lim-(. .!. X ! X):
We will need this construction frequently, so we will call it the microscope of*
* f, as it
is the dual construction to the telescope of f, and denote it by Mic(f).
Lemma 1.5 (Microscope lemma). Suppose f : nX ! X is a self-map, and let
Y denote its cofibre. Then
= _ :
Proof.Suppose Z is X*-acyclic. The Milnor exact sequence for Mic(f)*(Z) involves
a lim-and lim-1term, but each group in the sequence is 0. So Mic(f)*(Z) = 0, and
clearly Y *(Z) = 0. Conversely, if Y *(Z) = 0, the map
[Z; nX] ! [Z; X]
induced by f is an isomorphism. Thus the tower is Mittag-Leffler, and so
[Z; Mic(f)] = [Z; X]: __|_ |
Smashing does not behave very well with respect to cohomology in general, but*
* we
do have
Lemma 1.6. If F is finite, then <(X ^ F )*>.
6 MARK HOVEY
Proof.Recall that if F is finite,
F (Z; X ^ F ) = F (Z; X) ^ F:
Indeed,
[W; F (Z; X ^ F )] = [W ^ Z; X ^ F ]
= [W ^ Z ^ DF; X] = [W ^ DF; F (Z; X)] = [W; F (Z; X) ^ F ]:
Thus, if Z is X*-acyclic, F (Z; X) is null, so F (Z; X) ^ F is also null. H*
*ence
F (Z; X ^ F ) is null. __|_ |
Again, it is in general false that <(X ^ Y )*>. A counterexample will a*
*ppear
in section 3.
We will also need to know the cohomological Bousfield class of a free module *
*over
a ring spectrum.
Lemma 1.7. Suppose E is a ring spectrum with = , and X is a wedge of
suspensions of E. Then = .
Proof.E is a retract of X, so
:
X is an E-module spectrum, so = : __|_ |
I believe some hypothesis in the last lemma is necessary, though I don't have*
* a
counterexample.
Finally, we point out that cohomological Bousfield classes are not so well be*
*haved
with respect to p-localization as homological Bousfield classes are. For the re*
*st of this
section, spectra are not assumed to be p-local, and E(p)denotes the p-localizat*
*ion.
We have = if and only if = for all primes p. This is fal*
*se
for cohomological Bousfield classes in general. Indeed, let X denote the p-comp*
*lete
sphere S0p. By the microscope lemma, we have
= _
But since X is p-complete, Mic(p) is null. (For any p-local X, Mic(p) is the fi*
*ber of
the map from X to LM(p)X, the p-completion of X [9].) Thus = . But
X(q), for q a prime not equal to p, is a nontrivial rational spectrum. So by Le*
*mma
1.7, = . But M(p)(q)is null, so
6= <(M(p)(q))*>:
But we do have the following lemma.
Lemma 1.8. If = for all primes p, then = .
COHOMOLOGICAL BOUSFIELD CLASSES 7
Proof.We have
[Z; X](p)= [Z(p); X(p)] = [Z; X(p)]
but these are not in general equal to [Z(p); X]. Thus if Z is X*-acyclic, we h*
*ave
[Z; Y ](p)= 0 for all p, so [Z; Y ] = 0. __|_ |
2. E(n)-Cohomology and E(n)-Homology
The goal of this section is to prove
Theorem 2.1.
= :
Here E(n) is the Landweber exact homology theory whose homotopy groups are
Z(p)[v1; : :;:vn; v-1n]. E(n) is a ring spectrum. This is not completely obviou*
*s, because
Landweber exactness only tells us that E(n)*(X) has a natural external product,*
* but
this subtlety is dealt with definitively in [6].
It then follows = , and we will also prove that = , and t*
*hat
= , where Ell denotes any version of nonconnective torsion-free elli*
*ptic
cohomology. We begin with the following trivial but crucial lemma.
Lemma 2.2. For 0 n 1,
= :
Proof.Use the duality isomorphism
K(n)*(X) = Hom K(n)*(K(n)*(X); K(n)*): __|_ |
Proof of theorem.By Lemma 1.4, we know that
:
We also know from [11] that
= _ . ._.:
So we must show that for 0 j n:
Consider the ring spectrum
E(n)j = E(n)=(p; v1; : :;:vj-1):
E(n)j is obtained from E(n) by taking iterated cofibres, so
:
For j = n, E(n)j is just K(n), so we can assume j < n. Now E(n)j has a self-map
given by multiplication by vj, so
:
8 MARK HOVEY
Thus it suffices to show that Mic(vj) is a (nontrivial) wedge of suspensions of*
* K(j),
for then we would have
= :
To show this, we first need to recognize that Mic(vj) is in fact the fiber of*
* a map
of ring spectra. To do this, consider the following diagram of cofibre sequence*
*s.
xvk+1j k+1
E(n)j - --! E(n)j - --! E(n)j=vj
?? ?? ??
xvj?y =?y ?y
xvkj
E(n)j - --! E(n)j - --! E(n)j=vkj
The inverse limit of cofibre sequences is still a cofibre sequence [9, Lemma *
*5.3].
Thus we get a cofibre sequence
Mic (vj) ! E(n)j ! lim-E(n)j=vkj:
This shows in particular that Mic(vj) is non-trivial, since ss*(E(n)j) is not v*
*j-complete
(Remember j < n). The right-hand inverse limit is in fact a localization of E(*
*n)j,
according to the following lemma.
Lemma 2.3. The map
E(n)j ! lim-E(n)j=vkj
is localization at a finite spectrum of type j + 1.
Proof.Recall from [9, Theorem 2.1] that the localization of any spectrum X at a
finite spectrum of type j + 1 is given by the map
X ! lim-(X ^ M(pi0; vi11; : :;:vijj))
induced by inclusion of the bottom cell. Here M(pi0; vi11; : :;:vijj) is a fini*
*te spectrum
with the evident BP -homology, which exists for a cofinal set of multi-indices.*
* Now,
if X = E(n)j, because the attaching maps used to build M(pi0; vi11; : :;:vijj) *
*induce
multiplication by the appropriate vikkon E(n)-homology, we get
_ ij
E(n)j ^ M(pi0; vi11; : :;:vijj) = E(n)j=vj :
There are 2j summands in this decomposition, corresponding to the cells of
M(pi0; vi11; : :;:vij-1j-1):
The maps that make up the inverse system take all the cells but the bottom one *
*to 0,
as each cell is multiplied by some power of a vi. Thus we are left with lim-E(n*
*)j=vkj,
as required. __|_ |
COHOMOLOGICAL BOUSFIELD CLASSES 9
It would be nice to conclude at this point that Mic(vj) is thus an E(n)j-modu*
*le
spectrum, as it is the fiber of a module map. The technology of [5] may in fact*
* allow
us to do that, but we do not actually need it.
Now we would like to invert vj. Consider the diagram of cofibre sequences bel*
*ow.
xvj
E(n)j ---! E(n)j ---! E(n)j+1
?? ?? ??
xvj?y xvj?y 0?y
xvj
E(n)j ---! E(n)j ---! E(n)j+1
By taking the inverse limit, we see that
xvj : Mic(vj) ! Mic(vj)
is a homotopy equivalence. Here we are calling the map xvj, but remember that
we do not have any actual module structure. Nonetheless, the following diagram *
*of
cofiber sequences commutes, where LF(j+1)denotes localization at a finite spect*
*rum
of type j + 1.
Mic(vj)- --! E(n)j ---! LF(j+1)E(n)j
?? ?? ??
xvj?y' xvj?y xvj?y
Mic(vj)- --! E(n)j ---! LF(j+1)E(n)j
Taking the direct limit, we get a cofibre sequence
Mic(vj) ! v-1jE(n)j ! v-1jLF(j+1)E(n)j
where the right-hand map is a map of ring spectra. In particular, though we can*
*not
conclude that Mic(vj) is a module spectrum over v-1jE(n)j because of associativ*
*ity
problems, we do have that Mic(vj) is a retract of v-1jE(n)j ^ Mic(vj):
But v-1jE(n)j is itself a module spectrum over B(j) = v-1jBP=(p; v1; : :;:vj-*
*1):
W"urgler shows in [14] that B(j) splits additively, but not multiplicatively, a*
*s a wedge
of suspensions of K(j). (It splits multiplicatively after appropriate completio*
*n.) Now
any retract of a wedge of suspensions of K(j) is itself a wedge of suspensions *
*of K(j),
by [7, Prop 1.9]. Thus v-1jE(n)j, which is a retract of B(j) ^ v-1jE(n)j, is a*
*lso a
wedge of suspensions of K(j). Hence Mic(vj), as a retract of v-1jE(n)j ^ Mic(vj*
*), is
also a (nontrivial) wedge of suspensions of K(j). __|_ |
Corollary 2.4.
= and =
where KT denotes self-conjugate K-theory.
10 MARK HOVEY
Proof.Certainly : Conversely, K = KO ^ RP 2, so by Lemma 1.6,
= = :
One can do a similar argument for KT , using the fact that KT ^RP 2= K _3K: __*
*|_ |
Corollary 2.5.
= :
Proof.The same proof as given above for E(n) works fine, except that it is not *
*so
obvious that . Here one can use Baker's results [1] that show that
Ell=(p; v1) is a wedge of suspensions of K(2). __|_ |
3. Other Cohomological Bousfield Classes
In this section, we calculate the cohomological Bousfield classes of spectra *
*of finite
type, En, and LK(n)S0. We also point out the connection between Conjecture 3.10*
* of
[9] and the telescope conjecture.
Theorem 3.1. If X is any spectrum of finite type,
= _ :
Proof.We first show that, for any X,
= _
where Xp denotes the p-completion of X. Indeed, applying the microscope lemma to
X, we have
= _ <(X ^ M(p))*>:
Now recall that Mic (p) is the fiber of the natural map X -! Xp. In particula*
*r,
applying the microscope lemma to Xp which is already p-complete, we find that
= <(Xp ^ M(p))*>= <(X ^ M(p))*>:
Now, because X is finite type, Xp = I(IX), the Brown-Comenetz double dual.
Thus = : Also Mic(p) is always rational, and is trivial if and only i*
*f X is
p-complete. Since X is finite type, this happens only when X has finite homotopy
groups. Thus = . Thus
= _ : __|_ |
This verifies Conjecture 1.2 for finite type spectra.
Corollary 3.2. If X is a connective spectrum of finite type, then :
(Here HZ really means HZ(p)since all spectra are p-local. )
Proof.In this case, IX has homotopy which is bounded above. Then, following [11,
Lemma 2.6], IX is the limit of its Postnikov sections. Each of these has Bousf*
*ield
class less than or equal to , so the limit will as well. __|_ |
COHOMOLOGICAL BOUSFIELD CLASSES 11
We now calculate several specific Bousfield classes. We begin with the finite*
* spec-
tra.
Corollary 3.3. Suppose X is a finite spectrum. If X has nontrivial rational hom*
*ol-
ogy, then
= _ *:
Otherwise,
= **:
( Recall I denotes IS0. )
Proof.First note that the class of all finite spectra Y such that is*
* closed
under cofibrations and retracts. The thick subcategory theorem of [7] then impl*
*ies
that finite spectra of same type have the same cohomological Bousifeld class. S*
*o it
suffices to prove the proposition for a single X of type n. (We will need the s*
*tandard
facts about finite spectra of type n, which can be found in [7].) When n = 0, t*
*ake
X = S0. Then apply Theorem 3.1 to get the required result. Now suppose X has
type n > 0. Then X has a vn self-map f, which is necessarily of positive degre*
*e.
Since X is connective, Mic(f) is null. Thus, if Y denotes the cofibre of f, we *
*have
= :
Since Y has type n + 1, this completes the proof. __|_ |
Note that the statement that a map induces an isomorphism in cohomotopy is
weaker than saying it induces an isomorphism in integral homology.
Corollary 3.4. (1)*>= : On the other hand < :
(2) = and = :
Proof. (1)Apply the microscope lemma consecutively to the self-maps given by
multiplication by vi for 1 i n: Since BP is connective, all of the
microscopes vanish and we are left with HZ. For , note that
= **;
and recall that Ravenel shows in [11] that
**< :
Ravenel actually uses duality based on R=Z, but the same proof applies.
(2) For ko, apply the microscope lemma to xff, where ff is the generator of s*
*s1ko.
The microscope is trivial, and the cofibre is a finite wedge of suspensio*
*ns of
BP <1>. For ell, apply the microscope lemma to xffi and xffl:
__
|_ |
12 MARK HOVEY
This corollary enables us to give counterexamples to some pleasant properties
that hold for ordinary Bousfield classes. In particular, is incomparable*
* with
, despite the fact that K(n) is a module over BP . Similarly, is *
*in-
comparable with , showing that cohomological Bousfield classes do *
*not
behave well with respect to the smash product.
Recall the version of E(n), called En, used by, for example, Hopkins and Mill*
*er in
[8]. En is a flat E(n)-algebra spectrum, whose coefficients are
En* = W (Fpn)[[u1; : :;:un-1]][u; u-1]
where W (Fpn)denotes the Witt vectors of Fpn, each uihas degree 0, and u has de*
*gree
-2. The spectrum En is defined by the Landweber exact functoritheorem, where th*
*en
E(n)*-algebra structure is defined by sending vito uiu1-p for i < n, and to u1-*
*p for
i = n.
We will need to know that En is a ring spectrum. This is not guaranteed by
the Landweber exact functor theorem. That theorem only shows that En is a ring
object in the category of homology theories, which is the quotient of the categ*
*ory of
spectra by the phantom maps [10]. (Note that the objects of this quotient categ*
*ory
lift uniquely, up to isomorphism, though the maps do not.) The methods of [6] do
not apply because En* is not countable. But we have the following lemma.
Lemma 3.5. There are no phantom maps to En, so En admits a unique ring spec-
trum structure compatible with the ring structure on the homology theory En*X.
Proof.There are several ways to prove this. The proof below I learned from Hal
Sadofsky. First, it is not very hard to see that
En = lim-(En ^ Yk
where Yk = M(pi0; vi11; : :;:vin-1n-1) is a type n spectrum with the evident BP*
* -homology
and we are of course taking the homotopy inverse limit. Each of the En^Yk has f*
*inite
homotopy groups, so for X finite, (En ^ Yk)*X is also finite. Thus, there is no*
* lim-1-
term, and E*nX is an inverse limit of finite groups when X is finite. In parti*
*cular,
it is compact. Now, if X is an arbitrary spectrum, write X as the homotopy dire*
*ct
limit of finite spectra Xff. Then lim-1E*n(Xff) = 0, since each group is compa*
*ct. It
follows that E*n(X) = lim-E*n(Xff), so that there are no phantom maps to En. _*
*_|_ |
Proposition 3.6. = :
Proof.Apply the microscope lemma to the multiplication by p and ui maps. Since
the coefficient ring is already complete with respect to p and ui, the microsco*
*pes are
all 0. Thus we are left with the iterated cofibre whose coefficients are Fpn[u*
*; u-1]:
This is a wedge of suspensions of K(n). __|_ |
COHOMOLOGICAL BOUSFIELD CLASSES 13
This proposition shows that
=
does not necessarily imply that
= :
Indeed, take X = E(n) and Y = En.
In general, any K(n)-local spectrum X will have . We conjecture
that there is always equality if X is non-trivial. We have been unable to prove*
* this
in general, but we do have the following theorem.
Theorem 3.7.
<(LK(n)S0)*>= :
Proof.Any E-local spectrum X has : On the other hand, let F be a finite
spectrum of type n. Then
<(LK(n)S0)*> <(LK(n)S0 ^ F )*>= :
But LK(n)F has a finite K(n)-Postnikov tower, by a result of Hopkins and Ravenel
that has apparently never been published. It is implied by the results of [13].*
* It is
then easy to see by induction up the Postnikov tower that the natural map LK(n)*
*F !
I2LK(n)F is an equivalence. Thus
= :
But ILK(n)F is built up out of IK(n) in the same way that LK(n)F is built from
K(n). Since IK(n) = K(n), we see that : But we showed in [9]
that is a minimal Bousfield class, so in fact
= :
__
|_ |
This theorem shows that it is not always true that :
We conclude by pointing out the connection between a conjecture of [9] and the
telescope conjecture that follows from the considerations of this paper. Recall*
* that
the finite acyclics of a spectrum E, FA (E), are defined to be the class of all*
* finite X
such that E*(X) = 0. It is easy to see that FA (E) is a thick subcategory in th*
*e sense
of [7], so that it must be one of the Cn. We say that E has no finite acyclics *
*if FA (E)
consists of only the trivial spectrum.
Conjecture 3.8 (Conjecture 3.10 of [9]). If E has no finite acyclics, then the*
* E-
local sphere is either S0 or S0 completed at p. Equivalently, M(p) ( and every *
*finite
torsion spectrum ) is E-local.
14 MARK HOVEY
We proved this for ring spectra E in [9]. Now, from Proposition 1.1 and Lemma
1.3, we have that = , and that M(p) is IE*-local if and only if
. We saw above that = **: Thus we find that the conjecture above *
*is
equivalent to
Conjecture 3.9. If E has no finite acyclics, then
**:
This would say that ** is a minimal Bousfield class, and the unique minimal
Bousfield class with no finite acyclics. The telescope conjecture is equivalent*
* to
Conjecture 3.10 (Ravenel's telescope conjecture). If FA (E) = Cn+1, then
:
That this conjecture is equivalent to the telescope conjecture follows from [*
*9]. Of
course, the telescope conjecture is now known to be wrong for n = 2, by [12]. I*
*n any
case, this indicates that Conjectures 3.8 and 3.9 should be thought of as a sor*
*t of
telescope conjecture at n = 1.
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*conjecture, preprint
(1993).
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*s, Amer. J. Math.
106 (1984) 351-414.
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* University
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*(-), Manuscripta
Math. 29 (1979) 93-111.
University of Kentucky, Lexington, KY
E-mail address: hovey@ms.uky.edu
*