=
and
= :
Proof. If there were a spectrum V (n - 1) with
BP*V (n - 1) = BP*=(p; v1; : :;:vn-1 );
it would be type n and we would have BP ^V (n-1) = P (n), so the result would
be obvious. In general, there are not such spectra, but there are appropriate
substitutes M(pi0; vi11; : :;:vin-1n-1) constructed by Devinatz in [Dev ]. The*
*se exist
for sufficiently large (i0; i1; : :;:in-1 ), they are finite of type n, and the*
*y have
the evident BP -homology. Furthermore, BP ^ M(pi0; vi11; : :;:vin-1n-1) can be
constructed from P (n) using cofibre sequences, in the same way that the mod
pn Moore space can be constructed from the mod p Moore space. Therefore

:
Note that there is a natural map of BP -module spectra
BP ^ M(pi0; vi11; : :;:vin-1n-1) ! P (n):
The unit map S0 ! P (n) of the ring spectrum P (n) factors through this map,
so by the proceeding lemma,

:
It can actually be shown using a variant of the Landweber exact functor theo-
rem and Lemma 2.13 of [Rav84 ] that P (n) is a module spectrum over BP ^
M(pi0; vi11; : :;:vin-1n-1), but we do not need this.
To see that = , we proceed similarly. A vn self map on
M(pi0; vi11; : :;:vin-1n-1) induces multiplication by a power of vn on BP -homo*
*logy,
so
BP ^ T el(M(pi0; vi11; : :;:vin-1n-1)) = v-1n(BP ^ M(pi0; vi11; : :;:vin-1*
*n-1)):
The maps that build BP ^ M(pi0; vi11; : :;:vin-1n-1) from P (n) by cofibre sequ*
*ences
can all be chosen to be BP module maps. Thus they will also build v-1n(BP ^
M(pi0; vi11; : :;:vin-1n-1)) from v-1nP (n). Thus
= :
The latter equality comes from Theorem 2.1 of [Rav84 ].
The unit map of v-1nP (n) factors through v-1n(BP ^ M(pi0; vi11; : :;:vin-1n*
*-1)),
so we also have . __|_ |
Corollary 1.10. BP ^ A(n) = 0, so that the natural map LfnX ! LnX is
a BP equivalence.
8 MARK HOVEY
Proof.
= = = 0:
__|_ |
Corollary 1.11. Every BP -module spectrum with finite acyclics is Bous-
field equivalent to a finite wedge of Morava K-theories.
Proof. Suppose E is a BP -module spectrum with FA (E) = Cn+1 . Since E
is a BP module spectrum, E is a retract of BP ^ E, so
= = _ . ._.:
But = . Since K(n) is a field spectrum, we have
that is either 0 or . __|_ |
A particularly good kind of BP -module spectrum is a Landweber exact spec-
trum [Land ]. Recall that E is Landweber exact if the natural map
BP*(X) BP* E* ! E*(X)
is an isomorphism. The most common examples are E(n) and elliptic cohomo-
logy. Call E vn-periodic if vn 2 BP* maps to a unit in E*=(p; v1; : :;:vn-1 ).
Corollary 1.12. If E is a vn-periodic Landweber exact spectrum, then
= = :
Proof. Recall that if E is vn-periodic and Landweber exact then vj is not a
zero-divisor mod (p; v1; : :;:vj-1) for j < n, and vn is a unit mod (p; v1; : :*
*;:vn-1 )
[Land ]. It suffices to show that E ^ K(j) 6= 0 for j n, and that E ^ F (n + *
*1) =
0. Since E is a BP -module spectrum, = , it suffi-
ces to show that E ^ v-1jM(pi0; vi11; : :;:vij-1n-1) 6= 0. But the homotopy *
*of
E ^ v-1jM(pi0; vi11; : :;:vij-1n-1) is v-1jE*=(pi0; vi11; : :;:vij-1j-1), which*
* is not 0 by
Landweber exactness.
Similarly, the homotopy of E ^ M(pi0; vi11; : :;:vinn) is E*=(pi0; vi11; : :*
*;:vinn).
We know that vn is a unit mod (p; v1; : :;:vn-1 ), and it follows that vn is al*
*so a
unit mod (pi0; vi11; : :;:vin-1n-1), so the homotopy is 0. __|_ |
2. Localizations with respect to finite spectra
In this section we consider what localization with respect to a finite spect*
*rum
looks like. We also determine the K(n)-localization of BP . All of the results *
*in
this section are known to Hopkins and possibly others. Special cases of some of
these results have appeared in [MS ].
CHROMATIC SPLITTING CONJECTURE 9
We have already used the M(pi0; vi11; : :;:vin-1n-1) in the previous section*
*. We
need them again here, and we need to know that they exist for sufficiently large
(i0; : :;:in). Furthermore, there are natural maps
M(pj0; vj11; : :;:vjn-1n-1) ! M(pi0; vi11; : :;:vin-1n-1)
for jk sufficiently large compared to ik, which induce the evident map on BP -
homology. Notice that these maps fix the bottom cell.
The following result says that localization with respect to F (n) is complet*
*ion
at p; v1; : :;:vn-1 .
Theorem 2.1. For arbitrary X, the map X ! lim-(X^M(pi0; vi11; : :;:vin-1n-*
*1))
induced by inclusion of the bottom cell is F (n)-localization.
Proof. First we verify that the right-hand side is F (n)-local. Suppose Z *
*is
F (n)-acyclic. Then
[Z; X ^ M(pi0; vi11; : :;:vin-1n-1)] = [Z ^ DM(pi0; vi11; : :;:vin-1n-1)*
*; X];
where DY denotes the Spanier-Whitehead dual of Y . Since
K(i)*(DY ) = K(i)*(Y ) = Hom K(i)*(K(i)*(Y ); K(i)*);
DM(pi0; vi11; : :;:vin-1n-1) also has type n. Thus
Z ^ DM(pi0; vi11; : :;:vin-1n-1) = 0;
so X ^ M(pi0; vi11; : :;:vin-1n-1) is F (n)-local. Then the inverse limit of *
*X ^
M(pi0; vi11; : :;:vin-1n-1) is also F (n)-local.
Now we must check that the map is an F (n)-isomorphism. By Spanier-
Whitehead duality, it suffices to show that
[F (n); X] ! [F (n); lim-(X ^ M(pi0; vi11; : :;:vin-1n-1))]
is an isomorphism. We have an exact sequence
lim-1[F (n); X ^ M(pi0; vi11; : :;:vin-1n-1)] ,! [F (n); lim-(X ^ M(pi0; vi11; *
*: :;:vin-1n-1))]
! lim-[F (n); X ^ M(pi0; vi11; : :;:vin-1n-1)] ! 0:
There is a dimension shift on the lim-1term, but we will show it is 0 so that w*
*ill
not matter.
So we need to investigate [F (n); X ^ M(pi0; vi11; : :;:vin-1n-1)], or equiv*
*alently,
[F (n) ^ DM(pi0; vi11; : :;:vin-1n-1); X]. Note that DM(pi0; vi11; : :;:vin-1n*
*-1) is just
a desuspension of M(pi0; vi11; : :;:vin-1n-1), so that the top cell is in degre*
*e 0.
(This is easy to see from the construction of the M(pi0; vi11; : :;:vin-1n-1).)*
* Also
note that if X is type n, X ^ M(pi0; vi11; : :;:vin-1n-1) is a wedge of copies *
*of
X, for large enough indices (i0; : :;:in-1 ). Indeed, at each stage of the con-
struction of M(pi0; vi11; : :;:vin-1n-1), one takes the cofiber of a vj self ma*
*p on
10 MARK HOVEY
M(pi0; vi11; : :;:vij-1n-1). Since X is type n, that vj self map must be nilpot*
*ent on
X ^ M(pi0; vi11; : :;:vij-1n-1), so that if we take large enough indices, it wi*
*ll be null.
Thus [F (n); X ^ M(pi0; vi11; : :;:vin-1n-1)] is a direct sum of copies of [*
*F (n); X]
in dimensions corresponding to the cells of DM(pi0; vi11; : :;:vin-1n-1). The *
*maps
in the inverse system are all multiplication by a vj to some power, except on
the top cell, which is fixed. So they are nilpotent, and for large enough indic*
*es
will be 0. Hence lim-[F (n); X ^ M(pi0; vi11; : :;:vin-1n-1)] = [F (n); X] as *
*required.
Furthermore, the system is Mittag-Leffler, so the lim-1term vanishes as well. *
*__|_ |
Corollary 2.2.
LF(n)^E X = LF(n)LE X:
Proof. The map X ! LF(n)LE X is an F (n) ^ E-isomorphism, so it suffices
to show that LF(n)LE X is F (n) ^ E-local. Since
LF(n)LE X = lim-(LE X ^ M(pi0; vi11; : :;:vin-1n-1));
it will suffice to show that LE X^M(pi0; vi11; : :;:vin-1n-1) is F (n)^E-local.*
* Suppose
Z is F (n) ^ E-acyclic, and consider
[Z; LE X ^ M(pi0; vi11; : :;:vin-1n-1)] = [Z ^ DM(pi0; vi11; : :;:vin-1n-1)*
*; LE X]:
DM(pi0; vi11; : :;:vin-1n-1) is type n, so Z ^ DM(pi0; vi11; : :;:vin-1n-1) is *
*E-acyclic,
since Z is F (n) ^ E-acyclic. Thus this group is 0 as required. __|_ |
Note that if X is finite,
LF(n)LE X = lim-(LE X ^ M(pi0; vi11; : :;:vin-1n-1))
= lim-(X ^ LE M(pi0; vi11; : :;:vin-1n-1)):
In particular, recalling from [MS ] that
LTel(n)M(pi0; vi11; : :;:vin-1n-1) = T el(M(pi0; vi11; : :;:vin-1n-1)*
*);
and taking E = T el(0) _ . ._.T el(n), we recover their result that
LTel(n)S0 = lim-(T el(M(pi0; vi11; : :;:vin-1n-1))):
We can use a similar argument to calculate LK(n)BP .
Lemma 2.3.
LK(n)BP = LF(n)(v-1nBP ) = lim-(v-1nBP ^ M(pi0; vi11; : :;:vin-1n-1)):
Proof. Note that v-1nBP is Landweber exact and vn periodic, so has Bous-
field class : As a ring spectrum, it is self-local, so
Ln(v-1nBP ) = v-1nBP:
Thus
LK(n)(v-1nBP ) = LF(n)Ln(v-1nBP ) = LF(n)(v-1nBP ):
CHROMATIC SPLITTING CONJECTURE 11
So it suffices to show that BP ! v-1nBP is a K(n)-isomorphism, or equiva-
lently that BP xvn!BP is a K(n)-isomorphism. (We have left out the evident
suspension). Since K(n) is a field spectrum and so has a Kunneth isomorphism,
it will suffice to show that
BP ^ M(pi0; vi11; : :;:vin-1n-1) xvn!BP ^ M(pi0; vi11; : :;:vin-1n-1)
is a K(n)-isomorphism.
Note that xvn induces multiplication by jR vn on BP*BP or K(n)*BP . Here
jR is the right unit, discussed in [Rav86 ], where it is shown that
jR vn vn mod (p; v1; : :;:vn-1 ):
Thus, xvn is an isomorphism on K(n)*P (n). But BP ^ M(pi0; vi11; : :;:vin-1n-1)
can be built from P (n) using cofiber sequences where the maps are BP -module
maps. Thus xvn is also an isomorphism on
K(n)*(BP ^ M(pi0; vi11; : :;:vin-1n-1)):
__|_ |
The homotopy of LK(n)BP is then easily calculated to be (v-1nBP*)In, the
completion of v-1nBP* at the ideal In = (p; v1; : :;:vn-1 ). Note that vn is no*
*t a
unit in LnBP , but becomes one upon completion at In. In particular, one sees
that LK(n)BP is Landweber exact, so we have
Corollary 2.4.
=
Proof. , sice LK(n)S0 is an LnS0 module spectrum.
Since Ln is smashing, = . On the other hand,
L K(n)BP is an LK(n)S0-module spectrum, and since LK(n)BP is Landweber
exact,
= :
__|_ |
3. Ring spectra without finite acyclics
In this section we prove our BP -version of the chromatic splitting conjectu*
*re
and use it to deduce that finite torsion spectra are local with respect to any
infinite wedge of Morava K-theories. A corollary of this is that localization w*
*ith
respect to a ring spectrum that has no finite acyclics must be the identity fun*
*ctor
or p-completion on finite complexes.
Recall that all spectra are p-local, and Xp denotes the p-completion of X.
Throughout this section (ni) will be an infinite increasing sequence of nonnega-
tive integers.
12 MARK HOVEY
Theorem 3.1. The natural map
1Y
BPp ! LK(ni)BPp
i=1
is the inclusion of a wedge summand.
To prove this theorem, we use Brown-Comanetz duality. Recall that the
Brown-Comanetz dual of a spectrum X is the spectrum IX which represents the
functor Y ! Hom (ss0(X ^ Y ); Q=Z): In particular, if X has finitely generated
homotopy groups, then I2X = Xp: Recall as well that a map Y ! X is called
f-phantom if, for all finite Z and maps Z ! Y , the composite Z ! Y ! X is
null. Recall the following lemma, on page 66 of [Mar ].
Lemma 3.2. For any spectrum X, any f-phantom map into IX is null.
Q 1
Let F be the fibre of BPp ! i=1LK(ni)BPp. Since BPp = I(I(BP )), we will
have proved the theorem if we can show that the map F ! BPp is f-phantom.
First we remove the p-completion.
Lemma 3.3. Let F 0be the fibre of
1Y
BP ! LK(ni)BP:
i=1
If F 0! BP is f-phantom, then F ! BPp is null.
Proof of lemma. Let C be the fiber of BP ! BPp. Then C is a rational
space, so LK(n)C = * , and LK(n)BP = LK(n)(BPp) for n > 0: Consider the
following diagram.
C1 - ---! C - ---! C2
?? ? ?
y ?y ?y
Q
F 0- ---! BP - ---! LK(ni)BP
?? ? ?
y ?y ?y
Q
F - ---! BPp - ---! LK(ni)BPp
We consider two cases. If LK(0) appears in the product, then
C2 = fiber(LK(0)BP ! LK(0)(BPp)) = C;
so that F = F 0. Then if F 0! BP is f-phantom, so is F ! BPp, and so it is
null.
On the other hand, if LK(0) does not appear in the product, then C2 = 0, and
we have a cofiber sequence
F 0! F ! C:
CHROMATIC SPLITTING CONJECTURE 13
If F 0! BP is f-phantom, then F 0! BP ! BPp is null, so F ! BPp factors
through C. But C is M(p)-acyclic and BPp is M(p)-local, so the map must be
null. __|_ |
So to complete the proof of the theorem, it will suffice to prove:
Lemma 3.4. If X is finite, the map BP *(X) ! (LK(n)BP )*(X) is injective
for large n.
Proof. By using Spanier-Whitehead duality, it suffices to prove the lemma
in homology rather than cohomology. Recall from the preceeding sections that
LK(n)BP* = (v-1nBP*)In, where In = (p; v1; : :v:n-1) as usual. Note that
LK(n)BP clearly satisfies the hypotheses of the Landweber exact functor theo-
rem, so that
(LK(n)BP )*(X) = BP*(X) BP* LK(n)BP*:
The Landweber filtration theorem [Land ] says that BP*(X) has a finite filtrat*
*ion
by BP*BP subcomodules Mi for i = 1; : :;:m, such that the quotient Mi+1=Mi
is isomorphic to BP*=Imi for some mi. Choose n larger than all the mi. Then
BP*=Imi injects into BP*=Imi BP* (v-1nBP*)In. The proof of the Landweber
exact functor theorem actually shows that (v-1nBP*)In is flat in the category of
BP*BP comodules which are finitely generated over BP*. Now an easy induction
on the Mi using the 5-lemma completes the proof. __|_ |
W
Corollary 3.5. BPp is local with respect to E = K(ni) for any infinite
sequence (ni) of integers. BP is E-local if and only if the sequence contains 0.
Proof. LK(ni)BPp is certainly E-local, and any product of local spectra is
local. Thus BPp is a retract of a local space, so is local. We have the cofib*
*er
sequence C ! BP ! BPp, where C is rational. Thus, BP is E-local if and only
if C is E-local if and only if HQ is E-local. This is true if and only if 0 is *
*in the
sequence. __|_ |
It is natural to ask if the analogue of chromatic convergence holds. Define
Xj = LK(n0)_:::_K(nj)BPp: One would then ask if BPp is the inverse limit X of
the Xj. I don'tQknow the answer to this question. Note though that the map
from BPp ! LK(ni)BPp factors through X, so that BPp is a retract of X.
Theorem 3.6. Suppose R is a ring spectrum with no finite acyclics. If HQ ^
R 6= *, then LR X = X for all finite X. If HQ ^ R = *, then LR X = Xp for all
finite X.
First we show
Lemma 3.7. Suppose E is any spectrum such that LE X = X for some finite
X. Then if If HQ ^ E 6= *, then LE X = X for all finite X. If HQ ^ E = *,
then LE X = Xp for all finite X.
14 MARK HOVEY
Proof. Consider the class C of all finite X that are local with respect to*
* E.
It is easy to see that C is closed under retracts, suspensions, and cofibration*
*s.
It is nonempty by hypothesis, so it must be a Cn for some n. Suppose n > 1,
and let X be a space of type n - 1. Then X has a vn-1 -self map f, which must
be of positive degree d. In the cofiber sequence dX ! X ! Y , Y has type n
so is E-local. Thus, if Z is E-acyclic, any map Z !g X factors through dX.
Repeating this process, we find that g factors through the inverse limit of the
kdY , which is null. Thus X is E-local, which is a contradiction.
Thus C C1. In particular, the Moore space M(p) is E-local. Consider the
cofiber sequence S0pxp!S0p! M(p): Again, if Z is E-acyclic, any map Z ! S0p
factors through the inverse limit of the times p map on the p-complete sphere,
which is null. So S0pis E-local.
Now consider the cofibre sequence
F ! S0 ! S0p:
F is a rational space, so it is either E-acyclic or E-local according to whether
E ^ HQ is trivial or not. Localizing the cofibre sequence at E completes the
proof of the lemma. __|_ |
Proof of theorem. Thus to prove the theorem, we only need to show that
some finite X is R-local. A corollary of the nilpotence theorem [Hop ] tells *
*us
that any ring spectrum must be detected by one of the K(n), for 0 n 1: If
R is detected by K(1) = HFp then the Bousfield class of R is at least as big as
that of HFp. Since the Moore space M(p) is HFp-local, it is also R-local, and
we are done.
So suppose that R ^ HFp is null. I claim that R ^ K(n) must then be nonzero
for infinitely many n < 1. Indeed, for all n, there is a ring spectrum Yn of ty*
*pe
n. (see [Dev ] for specific examples). Then R ^ Yn is also a ring spectrum, wh*
*ich
is nonzero since R has no finite acyclics. It is not detected by any K(i) with
i < n or i = 1, so it must be detected by some K(i) for i n.
This means by [Rav84 , Thm 2.1] that the Bousfield class of R is as least as
big as that of some infinite wedge of Morava K-theories. Thus it will suffice to
show that M(p) is local with respect to such a wedge, for then it will be R-loc*
*al
as well. To do this we follow the argument of [Rav84 , Thm 4.4]. We know
already that BPp is local. It follows that any locally finite wedge of suspensi*
*ons
of BP ^ M(p) is local. We then use the Adams tower based on BP homology_
to write_M(p) as an inverse limit of spaces Ks of the form BP_^_BP ^s ^ M(p).
Here BP is the fiber of the unit map S0 ! BP . Since BP ^ BP is a locally
finite wedge of suspensions of BP , each Ks is local. Then M(p), as the inverse
limit of local spectra, is also local. __|_ |
Corollary 3.5 and Theorem 3.6 can be used to show that some mapping groups
are 0. For example, they imply that [BP ; BPp] = 0 and [BP ; Xp] = 0,
where X is finite. Indeed, BP has no K(i) homology for i > n.
CHROMATIC SPLITTING CONJECTURE 15
Recall the problem of Bousfield, mentioned in Section 1, which asks for a
classification of smashing localization functors.
Corollary 3.8. If the localization functor LE is smashing and E has no
finite acyclics, then LE is the identity functor.
Proof. If LE is smashing, then = , which is a ring spectrum.
Since E has no finite acyclics, neither does LE S0. So the proceeding theorem
tells us that LE S0, which is LLE S0S0, is either S0 or S0p. But S0phas the same
Bousfield class as the sphere itself. Indeed, suppose S0p^ X is zero. Then, usi*
*ng
the cofibre sequence
F ! S0 ! S0p
we find that X is a rational space. But S0p^ HQ is not zero, so S0p^ X can't be
either. Thus = = , as required. __|_ |
This also proves the following conjecture in the case that E is a ring spect*
*rum
with no finite acyclics. Hopkins and possibly others have made this conjecture
independently.
Conjecture 3.9. If E is arbitrary, then LE S0 is smashing.
This brings us to the question of localization with respect to an arbitrary
spectrum with no finite acyclics. I make the following conjecture.
Conjecture 3.10. If E has no finite acyclics, then LE S0 is either the sph*
*ere
itself or S0p.
Our method above relied on showing that BPp is E-local. This will certainly
not be true in general. There are E with no finite acyclics such that BP ^ E is
zero. An example of such a spectrum is IS0, the Brown-Comanetz dual of the
sphere. It is a consequence of sections 2 and 3 of [Rav84 ] that BP ^ IS0 = 0.
However, torsion finite spectra are local with respect to IS0. In fact I2X is
always IX-local, since
[Z; I2X] = [Z; F (IX; IS0)] = [Z ^ IX; IS0]:
So S0p= I2S0 is local with respect to IS0.
4. The chromatic splitting conjecture
In this section, we describe Hopkins' chromatic splitting conjecture and de-
duce some corollaries of it. The conjecture is concerned with the fibre of the
map LnS0 ! LK(n)S0. The following lemma is a generalization of a lemma of
Hopkins.
Lemma 4.1. Suppose E; F are spectra such that F ^ LE S0 is null. Then
for arbitrary X, the fibre of the natural map LE_F X ! LF X is the function
spectrum F (LE S0; LE_F X).
16 MARK HOVEY
Proof. Let Y denote the fibre. Then Y is E _ F local and F acyclic. We
claim that Y is therefore E local. Consider the map Y ! LE Y . This is an
E isomorphism, and F*Y = 0. Now LE Y is an LE S0 module spectrum, so
since F ^ LE S0 is null, so is F ^ LE Y . Thus the map Y ! LE Y is an E _ F
isomorphism. Since both sides are E _ F local, it is therefore an equivalence, *
*so
Y is E local.
To show that Y is F (LE S0; LE_F X), it will suffice to show that Y has the
same universal property, i.e. that
[Z; Y ] = [Z ^ LE S0; LE_F X]:
Since Y is E local, and the natural map Z ! Z ^ LE S0 is an E isomorphism, we
have [Z; Y ] = [Z ^ LE S0; Y ]: Since LE S0 is F acyclic, so is Z ^ LE S0. Appl*
*ying
[Z; ] to the cofibre sequence
Y ! LE_F X ! LF X
we see that [Z ^ LE S0; Y ] = [Z ^ LE S0; LE_F X], as required. __|_ |
The main example we are interested in here is the cofibre sequence
F (Ln-1 S0; LnX) ! LnX ! LK(n)X:
To describe the chromatic splitting conjecture, I must briefly describe some
work of Hopkins-Ravenel and Hopkins-Miller based on Morava's philosophy. Un-
fortunately, little of this work has appeared. The idea is this: the Morava sta-
bilizer group Sn is essentially the group of automorphisms of the formal group
law over K(n)*. This is not quite true: it is actually the automorphisms of the
same formal group law, but considered over the ring Fpn[u; u-1 ]. Here u has
degree -2 and is a -(pn - 1)-fold root of vn. It is technically advantageous to
use u instead of vn. The work of Lubin and Tate gives an action of Sn on a
complete ring whose residue field is Fpn[u; u-1 ]. We take this ring to be the *
*flat
E(n)*-module
En* = W (Fpn)[[u1; : :;:un-1 ]][u; u-1 ]:
Here the ui have degree 0, u has degree -2, and W (Fpn) is the Witt vectors of
the field with pn elements. The map
E(n)* = Z(p)[v1; : :;:vn; v-1n] ! En*
takes vi to uiu1-pi and vn to u1-pn : The residue field of the complete local r*
*ing
En* is then Fpn[u; u-1 ]: Now given an element of Sn, it lifts to an isomorphism
from the formal group F over En* to a possibly different formal group F 0. The
work of Lubin-Tate [LT ] shows that there is a well-defined automorphism of
the ring En* taking F 0to a formal group law which is *-isomorphic to F; i.e.
isomorphic by an isomorphism which reduces to the identity on the residue field
Fpn[u; u-1 ]: This gives a (continuous) action of Sn on En*.
CHROMATIC SPLITTING CONJECTURE 17
Now, En* is actually the homotopy of a spectrum En. In fact, En* is a flat
E(n)*-module, so one can simply tensor with it. In [HM ] it is shown that Sn
actually acts on the spectrum En, in fact by E1 maps. They show that the
homotopy fixed point spectrum of this action is LK(n)S0. (Actually, one has
to cope with the Galois group Z=n of the extension W (Fpn) over Zp as well.)
There is then a homotopy fixed point set spectral sequence
E2 = H*;*(Sn; En*)Z=n =) ss*(LK(n)S0):
The group cohomology here must be taken to be continuous cohomology. This
spectral sequence was known before the work of [HM ]: I believe it is due to
Hopkins-Ravenel, and a brief description of it appears in [HMS ]. It collaps*
*es
and there are no extensions when the prime p is large with respect to n.
Now, consider the inclusion
W (Fpn) ! En;0:
This is a map of Sn-modules, where Sn acts trivially on W (Fpn) , so induces
H*(Sn; W (Fpn)) ! H*(Sn; En;0):
(Here and below I always mean the Z=n invariants of cohomology groups.) Com-
putations suggest that H*(Sn; W (Fpn)) is, or at least contains, an exterior al*
*ge-
bra on n generators x1; : :;:xn. Of these, x1 is most familiar: it is usually c*
*alled
in. It arises from the determinant map Sn ! Zp, where we are thinking of Zp
as a subgroup of its own group of units. This is a crossed homomorphism with
respect to the trivial action of Sn on Zp W (Fpn) , so gives rise to a class in
H1(Sn; W (Fpn)):
I will describe Hopkins' chromatic splitting conjecture first in the case n *
*= 2
where it is simpler and also known to be true, at least for p > 3. This is all *
*due to
Hopkins, and uses essentially the calculations of Shimomura-Yabe in [Sh-Y ]. In
that case, H*(S2; W (Fpn)) is an exterior algebra on classes traditionally deno*
*ted
i and ae, in bidegrees (1; 0) and (3; 0). Both of these classes survive to homo*
*topy
classes
i : S-1p! LK(2)S0 ae : S-3p! LK(2)S0:
Multiplication also gives us a class
iae : S-4p! LK(2)S0:
Compose these maps with the map
LK(2)S0 ! F (L1S0; L2S0p):
This is L1-local, so we get maps i; ae; andiae from L1S0pto F (L1S0; L2S0p) of *
*de-
grees -2; -4; -5: But, strangely enough, ae and iae actually factor further thr*
*ough
L0S0p. Thus we get a map
-2 L1S0p_ -4 L0S0p_ -5 L0S0p! F (L1S0; L2S0p):
18 MARK HOVEY
This map is in fact a homotopy equivalence. This is the chromatic splitting
conjecture for n = 2. The general case is more complicated and is stated below.
Conjecture 4.2 (Hopkins' chromatic splitting conjecture). Fix an
integer n 1.
(i)H*(Sn; W (Fpn)) contains the exterior algebra E(x1; : :;:xn):
(ii)Each class of nonzero degree xi1. .x.ijin the exterior algebra survives
to a homotopy class
xi1. .x.ij: S-2(ik)+jp ! LK(n)S0:
(iii)The composite
xi1...xij
S-2(ik)+jp ! LK(n)S0 ! F (Ln-1 S0; LnS0p)
factors through
Ln-max ikS-2(ik)+jp:
(iv)The maps above split F (Ln-1 S0; LnS0p) into 2n - 1 summands.
(v) The cofibre sequence
F (Ln-1 S0; LnS0p) ! Ln-1 S0p! Ln-1 LK(n)S0
splits, so that
Ln-1 LK(n)S0 ' Ln-1 S0p_ F (Ln-1 S0; LnS0p):
As mentioned above, this conjecture is known to be true for n = 1 and for
n = 2; p > 3. The only other thing known about this conjecture is that x1 = in
always survives to give a homotopy class [HM ]. One would expect that part
1 should be possible to do, and that part 2 may be approachable using the
techniques of [HM ]. To this author at least, part 3 is a complete mystery. *
* It
seems to be suggesting that there is some interesting relationship between the
different Sn. Note that part 5 of the chromatic splitting conjecture has not re*
*ally
been tested yet. The summand -2 Ln-1 S0pof F (Ln-1 S0; LnS0p) corresponding
to in always maps trivially to Ln-1 S0pby construction. For n = 2, the maps
from the other 2 summands to L1S0pare trivial for dimensional reasons. For
n = 3 there are possible maps from the other summands to L2S0p. It is also part
5 from which the striking corollaries below can be derived.
Theorem 4.3. If the chromatic splitting conjecture is true, and if f : X !*
* Y
is a map between two finite spectra such that LK(n)f : LK(n)X ! LK(n)Y is null
for infinitely many n, then f is null.
Proof. It suffices to show that f : X ! Yp is null. Note that LK(n)Yp =
LK(n)Y if n > 0. We have the diagram
CHROMATIC SPLITTING CONJECTURE 19
X ----! Yp ----! LK(n)Yp
?? ?
y ?y
Ln-1 Yp ----! Ln-1 LK(n)Yp
By the preceeding result, Ln-1 Yp is a summand of Ln-1 LK(n)Yp. Thus if
X ! LK(n)Yp is null, so is X ! Ln-1 Yp. The chromatic convergence theorem
says that the tower Ln-1 Y is pro-isomorphic to the constant tower. It is easy
to see that Ln-1 Yp is also pro-isomorphic to the constant tower. Thus, since
X ! Ln-1 Yp is null for a cofinal sequence of n's, X ! Yp is null. __|_ |
We can use the results in the previous section to prove that such a map must
at least be null upon smashing with BP .
Proposition 4.4. If f : X ! Y is a map between two finite spectra such that
LK(n)f is null for infinitely many n, then the composite
X ! Y ! BP ^ Y
is null. In particular, if E is a BP -module spectrum, E*(f) : E*(X) ! E*(Y )
is zero.
Proof. First note that infinite products commute with smashing with fi-
niteQspectra, by Spanier-Whitehead duality. Thus, BPp ^ Y is a retract of
LK(ni)BPp ^ Y; for any infinite sequence (ni): Since the map X ! Y !
BPp ^ Y becomes null on localizaing with respect to K(n) for infinitely many n,
it is null. It follows from general facts about p-completions of spectra of fi*
*nite
type that the map
X ! Y ! BP ^ Y
is null (see Chapter 9 of [Mar ]). Smashing with E, we find that the composite
E ^ X ! E ^ Y ! E ^ BP ^ Y
is null. But if E is a BP -module spectrum, then E is a wedge summand of
E ^ BP , so in fact E ^ X ! E ^ Y is null. __|_ |
For several years, Hopkins has been saying that one does not need to reas-
semble the monochromatic parts of X to recover the homotopy theory of finite
spectra. The following corollary indicates a precise sense in which this is tru*
*e.
CorollaryQ4.5. If the chromatic splitting conjecture is true, then the nat*
*ural
map Xp ! LK(ni)Xp is the inclusion of a summand. In particular, if Y is
arbitrary, and Y ! X is a map such that the composite Y ! X ! LK(n)X is
null for infinitely many n, then Y ! X ! Xp is null.
20 MARK HOVEY
Q
Proof. By the preceeding theorem, the map Xp ! iLK(ni)Xp is injective
on maps from finite complexes. Thus , if F denotes the fibre, the map F ! Xp
is f-phantom. Since there are no f-phantom maps to Xp, it is null. __|_ |
5. Appendix: The p-completion
In this appendix, we investigate the consequences that the chromatic splitti*
*ng
conjecture would have on the structure of ss*LnS0. In particular, we show how
to determine the divisible summands, and show that except for those summands
and the free one in dimension 0, ss*LnS0 is a direct sum of cyclic groups which
have bounded torsion in each dimension.
The idea is that the adjoint properties of function spectra together with the
fact that Ln is smashing allow us to deduce the structure of F (LiS0; LnS0p) fr*
*om
just knowing it for i = n - 1. Indeed, we have
F (Li-1S0; LnS0p) = F (Li-1(LiS0); LnS0p) = F (Li-1S0 ^ LiS0; LnS0p)
= F (Li-1S0; F (LiS0; LnS0p)):
We illustrate the technique for n = 2, where we have
F (L1S0; L2S0p) = -2 L1S0p_ -4 L0S0p_ -5 L0S0p
Thus,
F (L0S0; L2S0p) = -4 L0S0p_ -4 L0S0p_ -5 L0S0p:
As we will see below, this splitting reflects the three Q=Zp summands in ss*L2S0
as calculated by Shimomura-Yabe in [Sh-Y ].
Proposition 5.1. Suppose the chromatic splitting conjecture is true. Then
F (L0S0; LnS0p) splits into a wedge of 3n-1 copies of HQp, in dimensions ranging
from -2n to -n2 - 1, with 2n-1 copies in dimension -2n.
It is of course possible to explicitly calculate the locations of these summ*
*ands
for any specific n. For example, for n = 3, there are 4 summands in dimension
-6, 3 in dimension -7, and 1 each in dimensions -9 and -10.
Proof. We proceed by induction on n. Suppose that we know the result is
true for all k n - 1. We have
F (L0S0; LnS0p) = F (L0S0; F (Ln-1 S0; LnS0p))
as above. We know by the chromatic splitting conjecture that F (Ln-1 S0; LnS0p)
splits into 2n - 1 summands, with LkS0poccuring 2n-k-1 times. The highest
dimensional occurence of LkS0 is in dimension -2(n - k) (corresponding to
xn-k 2 H2k-1 (Sn; W (Fpn))) and the lowest dimensional occurence of LkS0 is
in -(n - k)2 - 1 (corresponding to xn-k xn-k-1 . .x.1).
Thus the total number of summands in F (L0S0; LnS0p) is
3n-2 + 2 x 3n-3 + : :+:2n-2 + 2n-1 = 3n-1 :
CHROMATIC SPLITTING CONJECTURE 21
The only summands that can arise in dimension -2n arise from the highest
dimensional occurence of LkS0. Thus there are
2n-2 + 2n-3 + : :+:1 = 2n-1
of them. The lowest dimensional summand arises from choosing k = 0 in the
above paragraph, and occurs in dimension -n2 - 1: __|_ |
This proposition also explains why we need to complete the sphere in the chr*
*o-
matic splitting conjecture. If we did not, there would also be maps -1 HQ !
LnS0 coming from the fiber of the natural map LnS0 ! LnS0p.
One certainly expects that F (HQ; LnS0p) should correspond to the divisible
summands in ss*LnS0. The rest of this section is devoted to proving that. Lemma
4.1 tells us that this function spectrum is the fibre of the map
LnS0p! LK(1)_..._K(n)S0p:
Since
LK(1)_..._K(n)S0p= LM(p)LnS0p;
we have a cofibre sequence
F (L0S0; LnS0p) ! LnS0p! (LnS0p)p:
Now there are two things we need to do. First, we need to know something
about how the homotopy groups of Xp are related to the homotopy groups of
X. One might like them to be the p-completions of the homotopy groups of X.
This is false in general, but the following proposition says that they are clos*
*e to
being p-complete. That something like this proposition might be true was first
suggested to me by Hal Sadofsky. T
For an abelian group G, let p1 G = pnG:
Proposition 5.2. For arbitrary X; Y , [Y; Xp] is a module over Zp, has no
divisible summands, and [Y; Xp]=p1 [Y; Xp] is the p-completion of [Y; Xp].
Proof. We can assume X = Xp and Y = Yp: Then Y is a module spectrum
over S0p, so maps out of it are a module over ss0S0p= Zp: We will first show
that [Y; X] has no divisible summands. Consider the system of cofibre sequences
whose nth and n - 1st terms are displayed below.
n
X --xp--! X ----! X ^ M(pn)
? ? ?
xp?y =?y ?y
n-1
X -xp---! X ----! X ^ M(pn-1 )
If f 2 [Y; X] generates a divisible summand, there are maps fn 2 [Y; X] for
all n, such that pfn = fn-1 , where f0 = f: These will define a map into the
inverse limit Z = lim-(xp : X ! X) of the left column in the above diagram.
Now inverse limits do not behave very well in general, but the inverse limit of
22 MARK HOVEY
cofibre sequences is still a cofibre sequence, as we will prove below. Thus we *
*get
a cofibre sequence
Z ! X ! lim-(X ^ M(pn)) = LM(p)X:
Since X is already M(p)-local, Z must be null. Since f factors through Z, f is
null too.
Now we will show that the map
[Y; X] ! [Y; X]p = lim-[Y; X]=pn[Y; X]
is surjective. The proposition will then follow, since the kernel of the map
A ! Ap for abelian groups A is always p1 A: Suppose (fn) 2 [Y; X]p, so
fn 2 [Y; X]=pn[Y; X]. Let A = [Y; X] and B = [Y; X], and denote the ele-
ments of B killed by xpn by B(pn). Then, using the cofibre sequence
n i
X xp! X ! X ^ M(pn);
we get a diagram of short exact sequences
A=pnA ---i-! [Y; X ^ M(pn)] ----! B(pn)
?? ? ?
y ?y xp?y
A=pn-1 A ---i-! [Y; X ^ M(pn-1 )] ----! B(pn-1 )
Since (fn) is a compatible sequence, so is (i(fn)), se we get an element of
lim-[Y; X ^ M(pn)]: The map
[Y; X] = [Y; lim-(X ^ M(pn))] ! lim-[Y; X ^ M(pn)]
is not an isomorphism in general, but it is always surjective. So we get a map
f 2 [Y; X], and it is easy to see that f maps to (fn) 2 [Y; X]p: __|_ |
This completes the proof of the proposition modulo the following lemma,
which I learned from Hal Sadofsky.
Lemma 5.3. The inverse limit of cofibre sequences is a cofibre sequence.
Proof. It is easy to see that products of cofibre sequences are cofibre se-
quences. Thus, given cofibre sequences
An ----! Bn ----! Cn
?? ? ?
y ?y ?y
An-1 ----! Bn-1 ----! Cn-1
we get a diagram of cofibre sequences
Q Q Q
An ----! Bn ----! Cn
?? ? ?
y ?y ?y
Q Q Q
An ----! Bn ----! Cn
CHROMATIC SPLITTING CONJECTURE 23
where the vertical arrows are the maps whose fibres are the inverse limits. Now
it is not always true that the fibres in suchQa situationQform a cofibre sequen*
*ce,
butQit isQtrue in this case since the map Cn ! Cn is induced by the map
Bn ! Bn: __|_ |
Corollary 5.4. The kernel of the map
[Y; X] ! [Y; Xp]
is precisely the divisible summands in [Y; X].
Proof. Any divisible summand in [Y; X] must map to 0 in [Y; Xp], by the
proposition. To see the converse, note that we showed that the fibre of X ! Xp
is the rational spectrum Z = lim-(xp : X ! X). So [Y; Z] is a divisible group,
and thus its image in [Y; X] is also divisible. __|_ |
Now the second thing we need to do is to get some kind of control over the
homotopy of LnS0.
Lemma 5.5. ssiLnS0 is a countable abelian group.
Proof. We will show this using the Adams-Novikov spectral sequence
Es;t2= Exts;s+iBP*BP(BP*; BP*(LnS0)) =) ssiLnS0
This spectral sequence converges in a very strong sense, in that Es;s+i1is 0 for
large enough s (and fixed i) [Rav92 ]. Thus, if Es;s+i1is countable for all s*
*; t,
so is ssiLnS0. However, Es;t1is a subquotient of Es;s+i2, so it will suffice to*
* show
that E2 is countable in each bidegree.
One way to calculate the Ext groups of a BP*BP comodule M is to use the
cobar complex, made up out of
s(M) = M BP* BP*BP BP* . .B.P*BP*BP
where there are s factors of BP*BP . Note that BP*BP is countable in each
degree. I claim that if M; N are countable in each degree their tensor product
willLbe too. Indeed, the degree t part of their tensor product is a quotient of
Mk Nt-k. The tensor product of two countable abelian groups is countable,
as is the countable direct limit (or sum) of countable abelian groups. Thus, if
M is countable in each degree, so is sM, and thus also ExtsBP*BP (BP*; M):
Thus it will suffice to show that BP*(LnS0) is countable in each degree.
Since Ln is smashing, BP*(LnS0) = ss*(LnBP ). This is calculated by Ravenel
in [Rav84 ]. His result is that
ss*(LnBP ) = BP* -n Nn+1
for n 1, where Nn+1 is defined inductively by N0 = BP* and the short exact
sequence
0 ! Nk ! v-1kNk ! Nk+1 ! 0:
24 MARK HOVEY
If Nk is countable in each degree, so is v-1kNk , as it is a direct limit of co*
*untable
groups. So by induction, Nn+1 is countable in each degree, so is BP*(LnS0) and
we are done. __|_ |
Note that there is a sense in which countable torsion groups A are completely
classified (Ulm's Theorem [Kap ]). This classification is complicated, howeve*
*r,
because p1 A may not be 0. One certainly hopes that this complication does
not arise in LnS0. We will see below that it does not if the chromatic splitting
conjecture is true.
For the purposes of the theorem below, let nk denote the number of summands
in F (L0S0; LnS0p) in dimension k.
Theorem 5.6. Suppose the chromatic splitting conjecture is true. Then for
all k, sskLnS0 = Dk Tk, where D0 = Z, Dk is a direct sum of nk copies of
Q=Z(p), and Tk is a bounded torsion group which is a countable direct sum of
cyclic groups.
Proof. It is clear that D0 = Z, since the composite ss0S0 ! ss0LnS0 !
ss0L1S0 is the identity. The rest of ssiLnS0 is all torsion. It suffices to pro*
*ve the
theorem for X = LnS0p, which differs from LnS0 only in that D0 = Zp instead
of Z.
So we have the cofibre sequence
Y ! X !f Xp;
where Y = F (L0S0; LnS0p) is a finite wedge of suspensions of HQp: We investi-
gate the image of f on homotopy. We have a short exact sequence
0 ! A ! G ! H ! 0
where A = ssk+1 X=(divisible summands ) = Tk+1 is a countable torsion group,
G = ssk+1 Xp is a Zp module with no divisible summands whose p-completion is
G=p1 G; and H = im f is torsion-free. This means that A = Tor(G), and that
this is necessarily a short exact sequence of Zp modules. There is an induced
short exact sequence of Zp modules
0 ! A=p1 A ! G=p1 G ! H0:
The induced map of Zp modules H ! H0 is surjective, so since H is a Zp
submodule of a finite direct sum of Qp's, H0 can have at most only countable
torsion.
Now, B = A=p1 A is a countable torsion group, and p1 B = 0. Thus, by
Theorem 11 of [Kap ], it must be a direct sum of cyclics
M
B = Z=pni:
i
Further, it is sitting inside the p-complete group G=p1 G. Therefore, its p-
completion Bp is also inside G=p1 G. If B is unbounded torsion, one can see
CHROMATIC SPLITTING CONJECTURE 25
from the direct sum decomposition of B that Bp=B H is uncountable, and
in fact even the torsion of Bp=B is uncountable. This is impossible, given the
possibilities for H0. Thus B must be bounded torsion, say pN B = 0. But in
that case, we have pN A p1 A, and so we can deduce that the times p map
from pN A to itself is surjective. This means pN A is divisible, and since A h*
*as
no divisible summands, it must be 0. Therefore A = Tk+1 has bounded torsion,
and is therefore a direct sum of cyclics, proving half of the proposition. But
A = Tor(G), and a torsion subgroup which is bounded torsion always splits off.
Thus G = A H. Thus H = im f can have no divisible summands. H is
therefore a free Zp module.
If the rank of H is less than nk, then there is a Qp summand in sskLnS0. This
summand survives to sskL0S0, which is impossible. It follows that the image of
ssk(Y ) in sskLnS0 is a direct sum of nk copies of Q=Z(p), as required. __|_ |
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University of Kentucky, Lexington, KY
E-mail address: hovey@ms.uky.edu