ContemporaryVMathematicsolume 00, 0000 Bousfield Localization Functors and Hopkins' Chromatic Splitting Conjecture MARK HOVEY July 1993 Introduction This paper arose from attempting to understand Bousfield localization func- tors in stable homotopy theory. All spectra will be p-local for a prime p throu* *g- hout this paper. Recall that if E is a spectrum, a spectrum X is E-acyclic if E ^ X is null. A spectrum is E-local if every map from an E-acyclic spectrum to it is null. A map X ! Y is an E-equivalence if it induces an isomorphism on E*, or equivalently, if the fibre is E-acyclic. In [Bou79 ], Bousfield sho* *ws that there is a functor called E-localization, which takes a spectrum X to an E-local spectrum LE X, and a natural transformation X ! LE X which is an E-isomorphism. Studying LE X is studying that part of homotopy theory which E sees. These localization functors have been very important in homotopy theory. Ravenel [Rav84 ] showed, among other things, thatWfinite spectra are local with respect to the wedge of all the Morava K-theories n<1 K(n): This gave a con- ceptual proof of the fact that there are no non-trivial maps from the Eilenberg- MacLane spectrum HFp to a finite spectrum X. Hopkins and Ravenel later extended this to the chromatic convergence theo- rem [Rav92 ]. If we denote, as usual, the localization with respect to the fi* *rst n + 1 Morava K-theories K(0) _ . ._.K(n) by Ln, the chromatic convergence theorem says that for finite X, the tower : :s:siLnX ! ssiLn-1 X : :!:ssiL0X is pro-isomorphic to the constant tower {ssiX}: In particular, X is the inverse limit of the LnX. ______________ 1991 Mathematics Subject Classification. 55P42, 55P60, 55N22, 55N20. This paper is in final form and no version of it will be submitted elsewhere. cO00000American0Mathematical0So* *ciety0-0000/00 $1.00 + $.25 per page 1 2 MARK HOVEY The major result of this paper is that finite torsionWspectra are local with respect to any infinite wedge of Morava K-theories i<1 K(ni): This has several interesting corollaries. For example, it implies that there are no maps from the Johnson-Wilson spectra BP to a finite spectrum. It also implies that if E is a ring spectrum which detects all finite spectra, so that E*(X) 6= 0 if X finit* *e, then LE X is either X or Xp, the p-completion of X, for finite X. This in turn implies that the only smashing localization which detects all finite complexes * *is the identity functor. W In order to prove that finiteQtorsion spectra are iK(ni) local, I show that BPp is a wedge summand of iLK(ni)BPp. This is saying that one does not have to reassemble the chromatic pieces of BPp into an inverse limit to recover the homotopy theory of BPp. This result is a BP analogue of the chromatic splitting conjecture of Hopkins. I will describe this conjecture in Section 4, * *but for now suffice it to say that the conjecture is that Ln-1 Xp is a wedge summand in Ln-1 LK(n)Xp. The chromatic splitting conjecture is actually stronger than that, for it also explains how this splitting occurs, but most of the corollari* *es I draw from the chromatic splitting conjecture only need the splitting itself. One corollary of the chromaticQsplitting conjecture would be that, for finite X, Xp* * is a wedge summand of iLK(ni)Xp, explaining how my result is a BP analogue of the chromatic splitting conjecture. I do not know if Ln-1 BPp is a wedge summand of Ln-1 LK(n)BPp: In the first two sections of this paper, I describe some other results about Bousfield localization functors, this time with respect to spectra E which kill* * a finite spectrum. The pedigree of these results is somewhat confusing. Almost all of the results in Sections 1 and 2 have been known to Hopkins for some time. Others may have known them as well, but they have not appeared in print before. I feel that they warrant a larger audience. In addition, I discov* *ered many of these results independently, and there are a couple of new results as well. For example, I show that LK(n) is a minimal localization functor. That is, if the natural transformation X ! LE X factors through LK(n)X, then LE X is either the zero functor or is LK(n)X itself. I also provide some new examples of smashing localizations. The last section of the paper discusses the consequences of the chromatic splitting conjecture on the homotopy groups of LnS0. We show that, given the chromatic splitting conjecture, the divisible summands in ss*LnS0 for n 1 can be determined. There are 3n-1 of them, with 2n-1 of them occuring in dimension -2n, and the others spread out from dimension -2n - 1 to dimension -n2 - 1: This therefore explains part of the Shimomura-Yabe calculation of ss*L2S0 for p > 3 in [Sh-Y ]. This paper is written in the homotopy category of p-local spectra. In parti- cular, `=' is equality in the homotopy category, namely homotopy equivalence. Similarly, I will often write that a map or spectrum is 0, by which I mean that it is null-homotopic or contractible. CHROMATIC SPLITTING CONJECTURE 3 I would like to thank Mike Hopkins for sharing his ideas so freely. I thank Hal Sadofsky for hundreds of discussions on matters related to this paper. I also thank David Johnson for helpful discussions, and Paul Eakin and Avinash Sathaye for convincing me that my original ideas about infinite abelian groups were too naive. 1. Spectra with finite acyclics Before describing the results of this section, I need to recall the definiti* *on of the Bousfield class of a spectrum [Bou79 ]. Definition 1.1. Two spectra E and F are Bousfield equivalent if, given any spectrum X, E ^ X = 0 if and only ifF ^ X = 0: Denote the equivalence class of E by . Define if and only if F ^ X = 0 implies E ^ X = 0: Define ^ = and _ = : There is a minimal Bousfield class <*>, which we will often denote by 0, and a maximal Bousfield class . I remind the reader that it is perfectly possib* *le to have while nonetheless = 0: In this section we investigate Bousfield classes of spectra E which have fin* *ite acyclics, i.e. there is some finite X with E ^ X = 0. Highlights of this section include the minimality of the Bousfield class of K(n) (Corollary 1.7) and the n* *ew examples of smashing localizations (Proposition 1.5). We also show that every BP -module spectrum with finite acyclics has the Bousfield class of a wedge of Morava K-theories, and that a vn-periodic Landweber exact spectrum has the same Bousfield class as E(n). First, we need to recall some corollaries of the nilpotence theorem [DHS ,* * HS ]. Recall that a finite spectrum X has type n if K(i)*(X) = 0 for i < n but K(n)*(X) 6= 0: Every finite spectrum is of some finite type, and the periodicity theorem of J. Smith, written up in [Rav92 ], says that there is a spectrum of * *type n for all n. Let Cn denote the class of all finite spectra of type at least n. * *Then [HS ] any nonempty collection of finite spectra that is closed under cofibrati* *ons and retracts is some Cn. Lemma 1.2 (Hopkins-Smith). All finite spectra of type n have the same Bousfield class, which we denote F (n). Proof. This is an easy application of the nilpotence theorem. Given an X of type n, let C consist of all finite spectra Y such that . It is eas* *y to see that C is closed under retracts, cofibrations, and suspensions, so must be a 4 MARK HOVEY Ck for some k. Since X 2 C, C Cn. In particular, if Y is type n, . Interchanging X and Y completes the proof. __|_ | A spectrum X in Cn has a vn self-map, that is, a map inducing an isomorphism on K(n)*(X) [HS ]. Any two such become equal upon iterating them enough times, so that there is a well-defined telescope T eln(X). T elnis actually an exact functor on the category of finite spectra with vn self maps. This follows from the fact that a map between two such finite spectra will commute with large enough iterates of the vn self maps. By following a similar line of proof* * as in the above lemma, we get Lemma 1.3. The telescopes of finite spectra of type n all have the same Bo* *us- field class, which we denote T el(n). This lemma was also known to Hopkins and Smith, and it appears in [MS ] as well. Recall the lemma of [Rav84 ]: if f is a self-map of X and T el(X) is its telescope and Y its cofibre, then = _. Applying this repeatedly using vn self maps, we get (1) = _ . ._._ : This decomposition is the key to most of our results in this section. Note that* * it is orthogonal, in the sense that T el(m) ^ T el(n) = T el(m) ^ F (n) = 0 if m <* * n. This is proven in [MS ]. Given any spectrum E, let FA(E) = {X|X is finite andE ^ X = 0}: In this section, we will discuss spectra which have finite acyclics, so that we assume FA (E) 6= {*}: It is easy to see that FA (E) is closed under cofibration* *s and retracts, so it must be Cn+1 for some n. We then have the following observation. Lemma 1.4. If FA (E) = Cn+1 , then = _ . ._.: In particular, is the largest Bousfield class with fini* *te acy- clics Cn+1 , and therefore localization with respect to it, denoted Lfn, is sma* *shing. Proof. Just smash equation (1) with E. To see Lfnis smashing, note that for any spectrum E, FA (LE S0) = FA (E): Thus, : This implies by Prop. 1.27 of [Rav84 ] that Lfnis smashing. __|_ | CHROMATIC SPLITTING CONJECTURE 5 Lfnhas been investigated by many authors [Bou92 , MS , Mil , Rav92a ]. All of them noticed that it is smashing, though I think this is the most transparent proof. The telescope conjecture is usually stated as saying that if X is type n then LK(n)X = T el(X): This is equivalent to = , and also to Lfn= Ln. (For details, see [MS ] or one of the other cited papers above.) This conjecture is now known to be false for n = 2 [Rav92b , MRS ], and is presu* *med to be false for larger n as well. As an amusing example of what the failure of the telescope conjecture means, we include the following proposition. Proposition 1.5. Localization with respect to T el(0) _ . ._.T el(m) _ K(m + 1) _ . ._.K(n) is smashing. Proof. Call this localization functor Lm;n . We have that : We need to show that = if m < i n. Note that = = : But F (i) is T el(0) _ . ._.T el(m) acyclic, so Lm;n F (i) = LnF (i). Since Ln* * is smashing [Rav92 ], = , and the result follows. __|* *_ | It is an old problem of Bousfield's to classify all smashing localization fu* *nctors. We address another part of this problem in Section 3. To measure the extent to which the telescope conjecture fails, note that the* *re is a natural map LfnX ! LnX. Let AnX be the fibre of this map. Note that if X is type n, this is also the fiber of the map T el(X) ! LK(n)(X), for then Lfn(X) = T el(X) ([MS ]), and LnX = LK(n)(X). Proposition 1.6. If X is finite and type n, the Bousfield class of AnX does not depend on X. We denote it A(n). = _ , and A(n) ^ K(m) = 0 for all m. Proof. First note that because Lfnand Ln are both smashing (see [Rav92 ] for Ln), so is An. That is, AnX = AnS0 ^ X for all X. In particular, if X and Y are Bousfield equivalent, so are AnX and AnY . This shows that is well-defined. The map LfnX ! LnX is an isomorphism on K(m) homology for all m, (and also on BP homology as we will see below), so A(n) ^ K(m) = 0 for all m. If X is type n, = and = , and it follows that = _ : __|_ | A(n) behaves very much like MnX, the nth monochromatic layer, which is the fiber of LnX ! Ln-1 X. In particular, we have that A(n)^A(n) is homotopy equivalent to A(n), and LA(n)A(n) is homotopy equivalent to LA(n)S0. 6 MARK HOVEY Corollary 1.7. is a minimal Bousfield class. That is, if < , then E is null. Proof. Suppose . Then = 0 if n 6= m: Similarly, = 0. Thus, from equation (1), we have that = . But, also = 0; so by the preceeding proposition, we have = . Since K(n) is a field spectrum, E ^ K(n) is a wedge of suspensions of K(n), so there are only two possibilities for , 0 or . __|_ | Note that the corresponding result is not true for the other field spectrum, HFp. Indeed, in the proof of Theorem 2.2 of [Rav84 ], Ravenel shows that < , where Y denotes the Brown-Comanetz dual of BP ^ M(p). A similar argument to the above shows that if is less than or equal to some finite wedge of Morava K-theories, then E must be Bousfield equivalent to a finite wedge of Morava K-theories. This says in particular that the chromatic tower is unrefineable. There is no localization functor LE that fits between Ln and Ln-1 . In the light of this result and the failure of the telescope conjecture, one* * might ask if A(n) is also a minimal Bousfield class. This would say that the telescope conjecture is not so badly wrong. I think this is likely to be true, but since* * I have no data, I will not be so bold as to conjecture it. The following theorem will show that the telescope conjecture is true after smashing with BP . This has been known to Hopkins, Ravenel, and probably others, though it has not appeared before. First we need a lemma. Lemma 1.8. Suppose R is a ring spectrum and the unit map S0 !j R factors through some spectrum E. Then . Proof. Since R is a ring spectrum, the composite R = S0 ^ R j^1!R ^ R ! R is the identity. Since j factors through E, the identity map on R factors throu* *gh E ^ R. So if E ^ Z is null, so is R ^ Z. __|_ | Recall that P (n) is a BP -module spectrum whose homotopy is ss*P (n) = BP*=(p; v1; : :;:vn-1 ): The first part of the following theorem is Ravenel's theorem 2.1(g) in [Rav84 * *]. We reprove it so as to make the second part clearer. CHROMATIC SPLITTING CONJECTURE 7 Theorem 1.9. =

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. __|_ | References [Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257-281. [Bou92] A. K. Bousfield, Localization and periodicity in unstable homo* *topy theory, preprint (1992). [Dev] E. S. Devinatz, Small ring spectra, Jour. Pure Appl. Alg., 81 (1992) 11-16. [DHS] E. Devinatz, M. Hopkins, and J. Smith, Nilpotence and stable homotopy theory I, Ann. Math. 128 (1988) 207-241. [Hop] M. Hopkins, Global methods in homotopy theory, Proceedings of the 1985 LMS Symposium on Homotopy Theory, (J. D. S. Jones, E. Rees eds.), 1987, 73-96. [HMS] M. Hopkins, M. Mahowald and H. Sadofsky, Constructions of elements in Picard groups, preprint (1992). [HM] M. Hopkins and H. Miller, personal communication. [HS] M. Hopkins and J. Smith, Nilpotence and stable homotopy theory II, preprint. [Kap] I. Kaplansky, Infinite Abelian Groups, Univ. of Michigan Press (1954). [Land] P. Landweber, Homological properties of comodules over MU *(MU ) and BP *(BP ), Amer. J. Math. 98 (1976) 591-610. [LT] J. Lubin and J. Tate, Formal moduli for one-parameter Lie groups, Bull. Soc. Math. France 94 (1966) 49-60. [MRS] M. Mahowald, D. Ravenel, and P. Shick, The v2-periodic homotopy of a certain Thom complex, preprint (1992). [MS] M. Mahowald and H. Sadofsky, vn telescopes and the Adams spectral sequence, preprint (1992). [Mar] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland (1983). [Mil] H. R. Miller, Finite localizations, preprint (1992). [Rav84] D. Ravenel, Localization with respect to certain periodic homology the* *o- ries, Amer. J. Math. 106, (1984) 351-414. 26 MARK HOVEY [Rav86] D. Ravenel, Complex cobordism and Stable Homotopy Groups of Sphe- res, Academic Press (1986). [Rav92] D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Princeton (1992). [Rav92a] D. Ravenel, Life after the telescope conjecture, preprint (1992). [Rav92b] D. Ravenel, A counterexample to the telescope conjecture, preprint (1992). [Sh-Y] K. Shimomura and A. Yabe, The homotopy groups ss*(L2S0), preprint (1992). University of Kentucky, Lexington, KY E-mail address: hovey@ms.uky.edu