THE STRUCTURE OF THE BOUSFIELD LATTICE MARK HOVEY AND JOHN H. PALMIERI Abstract.Using Ohkawa's theorem that the collection B of Bousfield class* *es is a set, we perform a number of constructions with Bousfield classes. * *In particular, we describe a greatest lower bound operator; we also note th* *at a certain subset DL of B is a frame, and we examine some consequences of t* *his observation. We make several conjectures about the structure of B and DL. 1.Introduction In [Bou79a ] and [Bou79b ], Bousfield introduced an equivalence relation on s* *pec- tra that has turned out to be extremely important. Given a spectrum E, we define the Bousfield class of E to be the collection of E-acyclic spectra X, where* * X is E-acyclic if and only if E ^ X = 0. Then we say that E and F are Bousfield equivalent if and only if = . The notion of Bousfield equivalence, and * *hence Bousfield class, plays a major role in much of modern stable homotopy theory. We can order the collection of Bousfield classes using reverse inclusion. We then have a partially ordered class associated to the stable homotopy category,* * and Bousfield and others have investigated properties of this partially ordered cla* *ss. The nilpotence theorem of Devinatz, Hopkins, and Smith [DHS88 ], for example, is equivalent to the classification of Bousfield classes of finite spectra [HS * *]. We recently learned that Ohkawa has proved the surprising result that there is onl* *y a set of Bousfield classes [Ohk89 ]; see also [Str97]. He proves there are at mos* *t i2 Bousfield classes, where ii= 2ii-1and i0 = @0. In light of this result, the aut* *hors decided to re-examine the structure of the partially ordered set of Bousfield c* *lasses. The goal of this paper is to provide some kind of global understanding of the partially ordered set B of Bousfield classes. Using Ohkawa's result, we are ab* *le to perform certain constructions in B, such as a greatest lower bound operation. We also bring to bear many methods and results from lattice theory; for instanc* *e, the sub-partially ordered set DL of B, which consists of all Bousfield classes* * for which = , is a very nice sort of distributive lattice known as a frame. This has some nice consequences, and it also leads to some interesting questions. Much of our understanding of the Bousfield lattice is only conjectur* *al; we hope that the conjectures and their implications are interesting enough to prom* *pt further study of this material. There are several questions we have not address* *ed. In particular, a frame such as DL has an associated topological space. It wou* *ld be interesting to understand something about this space, even conjecturally. Ja* *ck Morava has asked whether this space has a structure sheaf, probably of stable homotopy categories, associated to it. The stalk at K(n), for example, might be ____________ Date: January 21, 1998. 1991 Mathematics Subject Classification. 55P42, 55P60, 06D10. Research partially supported by a National Science Foundation grant. Research partially supported by National Science Foundation grant DMS-940745* *9. 1 2 MARK HOVEY AND JOHN H. PALMIERI the K(n)-local category. There are also many frame-theoretic properties that DL may or may not have, such as coherence. Here is one of the conjectures that we do discuss. Call a Bousfield class strange if < . For instance, the Brown-Comenetz dual of the p-local sphere has such a Bousfield class. By general lattice theory, the inclusion DL * *,! B has a "right adjoint" r :B -! DL which is a retraction onto DL . One can see that r sends every strange Bousfield class to <0>, and also that r induces a map r0: B=(strange) -!DL , where B=(strange) is the quotient lattice of B by the id* *eal of strange Bousfield classes. Conjecture 3.12 states that r0is an isomorphism; * *this implies, for example, that = for all spectra E. Our other m* *ain conjectures are Conjectures 5.1, 6.12, 7.4-7.6, and 9.1. Here is the structure of the paper. In Section 2, we define Bousfield classes* * and the basic operations one can perform on them: join, smash, meet, and complemen- tation. Next we examine DL and its relation to B; in particular, we note that * *DL is a frame, and we construct a retraction from B to DL . We also give the conje* *c- tured description of this retraction in terms of strange Bousfield classes. We * *discuss more basic structure in Section 4: we discuss minimal and complemented Bousfield classes, and we recall some properties of BA , the set of complemented Bousfield classes. For example, we recall Bousfield's observation that BA is a Boolean * *al- gebra. In Section 5 we examine spectra X for which there is a finite spectrum F with X ^ F = 0; we give a conjectured classification of the Bousfield classes of such X. This provides some information about BA . In Section 6, we return to the fact that DL is a frame; this allows us to construct a complete Boolean algebra cBA DL which (properly) contains BA , and we give a conjectured description of cBA . Then in the next section, we examine Bousfield classes of spectra X for which X ^ F 6= 0 for all finite F . This leads to a discussion of some properti* *es of I, the Brown-Comenetz dual of the p-local sphere, as well as several conjectures about spectra with no finite acyclics. We show that these conjectures are all e* *quiv- alent, and we discuss some of their consequences. Much of the paper to this poi* *nt suggests that the set of strange Bousfield classes, those classes of p-local sp* *ectra X with < , is interesting; in Section 8 we examine some examples of such spectra. We end the paper in Section 9 with a discussion of the partially order* *ed class of localizing subcategories_recall that a subcategory is called localizin* *g if it is thick and is closed under coproducts; the main conjecture is that every loca* *lizing subcategory is equal to the class of E-acyclics for some spectrum E. This conje* *cture has several equivalent formulations, and some deep structural consequences. We work p-locally throughout the paper, except for Section 9, in which we work globally. As in all discussions of Bousfield classes of spectra, we work in the* * stable homotopy category of spectra, as described for example in [HPS97 ]. The authors would like to thank Dan Christensen and Neil Strickland for many helpful discussions about Bousfield classes. 2. Basic structure of the Bousfield lattice: _, ^, f, and a In this section we discuss the basic structure of the Bousfield lattice, incl* *uding the wedge (a.k.a. the join) _, the smash product ^, the meet f, and the comple- mentation operator a. We start with the definition of Bousfield equivalence and related ideas, due * *to Bousfield in [Bou79a ] and [Bou79b ]. THE STRUCTURE OF THE BOUSFIELD LATTICE 3 Definition 2.1.Let E, F , X, and Z be spectra. (a) Z is E-acyclic if and only if E ^ Z = 0. (b) The Bousfield class of E, written , is the collection of E-acyclic spec* *tra. (c) The spectra E and F are Bousfield equivalent if and only if = . (d) The Bousfield classes are partially ordered by reverse inclusion: we write if and only if E ^ Z = 0 ) F ^ Z = 0. (e) The wedge of and is defined to be _ = . The wedge of an arbitrary set of Bousfield classes is defined the same way. (f) Similarly, the smash product of and is defined to be ^ = . (g) X is E-local if and only if [Z; X] = 0 for all E-acyclic spectra Z. We denote the p-local sphere by S; then is the largest Bousfield class in* * this ordering, and <0> is the smallest.WIt is clear that _ is the least upper* * bound, or join, of and ; indeed, is the join of the set {}. Now we recall Ohkawa's result. Theorem 2.2 ([Ohk89 ]).The class of Bousfield classes forms a set. We use B to denote the set of Bousfield classes. B is a partially ordered set in which every subset has a least upper bound (i* *.e., B is a complete join semilattice). Since there is a smallest element, then eve* *ry subset also has a greatest lower bound, or meet, obtained by taking the join of* * all the lower bounds (we are using the fact there is a (nonempty) set of these lower bounds, so we can in fact take the join). Since B has both finite joins and fin* *ite meets, it is a lattice; since it has arbitrary joins, it is a complete lattice.* * We denote the meet of and by f . Unfortunately, this meet is not easily described. In particular, we do not know whether B is distributive: in other wo* *rds, is _ f = _ f _ ? The meet certainly does not distribute over infinite joins; see Example 7.3. In contrast, the smash product of Bousfield classes distributes over infinite* * joins. ^ is a lower bound for and , but it need not be the greatest lower bound; for example, if I is the Brown-Comenetz dual of the sphere, then I ^ I = 0 (see [Bou79a , Lemma 2.5] and Lemma 3.8 below). In general, then, we have ^ f . In any complete lattice with an operation that distributes over infinite join* *s, we can define a complementation operator a: we define a to be the join of all <* *Y > such that ^ = <0>. Here are some of the basic properties of a, most of which are due to Bousfield [Bou79b ]. Lemma 2.3. Let a be the complementation operator on the Bousfield lattice B. Then a has the following properties. (a) a if and only if E ^ X = 0. (b) a is order-reversing: if and only if a a. (c) a2 = . (d) f = a a _ a. (e) More generally, a converts arbitrary joins to meets and arbitrary meets to joins. Proof.Part (a) holds since the smash product distributes over infinite joins, so a ^ = 0. For the next part, suppose . Since a ^ = 0, 4 MARK HOVEY AND JOHN H. PALMIERI then a ^ = 0. Hence a a, so a is order-reversing. The other half of part (b) follows from part (c). For part (c), it is formal to verify that a2. Since a is order-revers* *ing, it follows that a a3. Thus a = a3. Now suppose X ^ Z = 0. Then a = a a2 , so a2 ^ = 0. Thus a2, completing the proof of part (c). Parts (d) and (e) are formal consequences of the_other_p* *arts, given Ohkawa's theorem. |__| Note that not all of these properties would hold if we tried to define a usin* *g the meet instead of the smash. Bousfield's work predates Ohkawa's, so he had to work harder to construct the operator a. In particular, he constructs an operator at (or closer to) the spec* *trum level, and shows that it descends to give an operator on Bousfield classes. For any spectrum E, Bousfield shows in [Bou79b , Lemma 1.13] that the localizing subcategory of E-acyclic spectra is generated by a single spectrum aE. So for instance, a spectrum X is E-local if and only if [aE; X]* = 0. The spectrum aE is not well-defined, but any other choice generates the same localizing subcate* *gory, so in particular has the same Bousfield class. Thus is well-defined. In fa* *ct, = a, since if Z ^ E = 0, then Z is in the localizing subcategory genera* *ted by aE, so . As with the meet, it is rather difficult to compute the effect of the operato* *r a. We will discuss it further and give some examples in Section 4. Dan Christensen has pointed out that, just as one can define the meet operati* *on f by f = a a _ a, one can define an operation g by g = a a ^ a: Then g _ , and this inequality may be strict; for example, a g a = , even though a 6= . 3.The retraction onto DL The Bousfield lattice B is a complete lattice, but may not be distributive; we show in Example 7.3 that the meet does not distribute over infinite joins. In any case, the smash and the meet certainly do not coincide. To get around this problem, Bousfield introduced the sub-partially ordered set DL of B in [Bou79a* * ]: DL consists of the Bousfield classes satisfying = ^ . The goal * *of this section is to study DL and its relationship to B. In particular, we point* * out that there is a retraction B -!DL , and we make some conjectures about it. Example 3.1. Bousfield observes in [Bou79a ] that if E is a ring spectrum or a finite spectrum, then is in DL . On the other hand, I, the Brown-Comenetz dual of the p-local sphere is not: since I ^ I = 0, then > ^ . We mention the following in passing. Question 3.2. Let E be a spectrum. Must the sequence ^2 ^3 : : : stabilize? See Proposition 3.13(e) for a conjectured answer to this question. THE STRUCTURE OF THE BOUSFIELD LATTICE 5 Lemma 3.3. Suppose 2 DL , , and . Then ^ . Proof.We have = ^ ^ . |___| A frameWisWa complete lattice in which the meet distributes over infinite joi* *ns: a ^ bi = (a ^ bi). For example, a topology on a space X has the structure of a frame, in which the open subsets of X are ordered by inclusion. Frames are al* *so called locales, complete Heyting algebras, or complete Brouwerian lattices. Th* *ey are used in categorical topology [Joh86], where a locale is viewed as a general* *ized topological space, lattice theory [Bir79], and logic [FS90 ]. W Proposition 3.4.DL is a frame. In DL , the join of {} is , and the meet of and is ^ . The inclusion i: DL -!B preserves arbitrary joins but does not preserve meets. Proof.Much of this is due to Bousfield [Bou79a ], eitherWexplicitly or implicit* *ly. We leave to the reader the straightforward check that and ^W are in * *DL if all , , and are in DL . It follows from this that is the * *join of {}, that DL is a complete lattice, and that the inclusion i: DL -!B preser* *ves joins. Lemma 3.3 implies that the meet in DL is the smash product, and, since the smash product distributes over infinite joins, thatWDL is a frame. To see * *that i does not preserve meets, note that both and n are in DL . Their smash product, and hence their meet in DL , is 0, but their meet in B is at_lea* *st by Proposition 7.2. |__| We can think of a complete lattice, or indeed any partially ordered set, as a* * cat- egory with a unique map from x to y if and only if x y. A complete lattice is * *just a partially ordered set that is complete and cocomplete as a category; the coli* *mit of a functor to a lattice is the join of all of the objects in the image, and dual* *ly for the limit. From this point of view, an order-preserving map of partially ordered se* *ts corresponds to a functor on the associated categories. A functor between comple* *te lattices preserves colimits if and only if it preserves arbitrary joins. Obvio* *usly a left adjoint must have this property, and, for complete lattices, the converse * *is true as well. Note that, for maps of partially ordered sets f and g, g is right adjo* *int to f if and only if fx y is equivalent to x gy. Lemma 3.5. Suppose f :C -! D is an order-preserving map between complete lattices. Then f has a right adjoint if and only if f preservesWarbitrary joins* *. In this case, the right adjoint of f is the map g defined by gy = {x | fx y}. Proof.One can easily verify that g is order-preservingWand fx y implies x gy. Conversely, if f preserves colimits, then fgy = {fx | fx y} y, so x_ gy implies fx fgy y. |__| Johnstone proves in [Joh86, Theorem I.4.2] the (equivalent) statement that a functor between complete lattices has a left adjoint if and only if it preserve* *s ar- bitrary meets. Applying Lemma 3.5 to DL , we get the following corollary, first pointed out to us by Neil Strickland. Corollary 3.6.The inclusionWfunctor DL -!B has a right adjoint r :B -! DL defined by r = { 2 DL | }. The functor r preserves arbitrary meets, r for all X, and r = if 2 DL . 6 MARK HOVEY AND JOHN H. PALMIERI In fact r preserves the smash product as well. Lemma 3.7. The functor r :B -!DL preserves the smash product: r ^ = r ^ r. Proof.Since ^ is a lower bound for and , r ^ is a low* *er bound for r and r, so r ^ r ^ r. Conversely, r ^ r ^ and r ^ r 2 DL , so r ^ r r ^ . |* *___| We would like to understand this map r more explicitly. We begin by pointing out that r does kill some Bousfield classes. Lemma 3.8. If < and , then E ^ F = 0. In particular, ^ = 0, so r = 0. Proof.We must have E ^HFp = 0, since otherwise . Hence E ^F = 0. In particular ^ = 0. Since r 2 DL , Lemma 3.3 implies that r_ ^ = 0, so r = 0. |__| By the argument in [Rav84 , 2.6] (see also Lemma 7.1(c) below), I does have < , so there are nontrivial examples of such spectra. Definition 3.9.We define a spectrum E to be strange if < . Hence every strange spectrum is in the kernel of r. We will study some more examples of strange spectra in Section 8. A subset J of a complete lattice C is called a complete ideal if it is closed* * under arbitrary joins, and if x 2 J and y x, then y 2 J. Every complete ideal in a complete lattice is principal; if we let m be the join of all the elements of J* *, then y 2 J if and only if y m. For the complete ideal of strange spectra, we can identify the maximal element m "explicitly." Lemma 3.10. Let D = aHFp _ HFp. Then the collection of strange Bousfield classes is the principal ideal generated by a = a f . Proof.Note that a if and only if E ^ HFp = 0 and E ^ aHFp = 0. This second condition holds if and only if a2 = . Hence a if * *__ and only if and E ^ HFp = 0, that is, if and only if < . |* *__| Given a (complete) ideal J in a (complete) lattice C, we can define a b (mod J) if there is some x 2 J such that a _ x = b _ x. If J is principal, the* *n a b (mod J) if a _ m = b _ m, where m is the largest element in J. The equivalence classes under this congruence relation define a complete lattice C=J (see [Bir7* *9, II.4], and note that a complete join semilattice is a complete lattice). The ob* *vious epimorphism C -!C=J preserves arbitrary joins, and has kernel J. There are often other epimorphisms with kernel J; hence given a poset map C -! D with kernel containing J, there may not be an induced map C=J -! D. Proposition 3.11.Let J be the principal ideal of strange Bousfield classes. If (mod J), then r = r. Proof.As before, we let D = aHFp_ HFp. Since J is the principal ideal generated by a, we have (mod J) if and only if _ a = _ a. It therefore suffices to show that r _ a= r. So suppose 2 DL with _a. Then Lemma 3.3 implies that = ^ _ ^ a. THE STRUCTURE OF THE BOUSFIELD LATTICE 7 Now, if Z ^ HFp = 0, then Z ^ aD = 0, and so = ^ . On the other hand, if Z ^ HFp is nonzero, then (X _ aD) ^ HFp is nonzero, so X ^ HFp is nonzero. Hence ^ a , so in this case as well. Thus r _ a= r as required. |___| It follows from Proposition 3.11 that the epimorphism r :B -! DL factors through an epimorphism r0: B=J -! DL , where J is the ideal of strange spectra. Conjecture 3.12. The epimorphism r0: B=J -! DL is an isomorphism. This conjecture has two parts: that J is the kernel of r, and (since epimor- phisms of lattices are not determined by their kernels) that the induced map is* * an isomorphism. The conjecture has several consequences. Proposition 3.13.Suppose Conjecture 3.12 holds. Then the following properties hold. (a) r = 0 if and only if E is strange. (b) r preserves arbitrary joins. (c) If E ^ HFp 6= 0, then 2 DL . (d) r = ^ . (e) Hence ^n = ^(n+1)when n 2. Proof.The first two parts are immediate. For part (c), note that Conjecture 3.12 implies that r (mod J), so that _ a = r _ a, where D = aHFp_ HFp as usual. If E ^ HFp 6= 0, then > a, so _ a = . Similarly, r > a, so r _ a = r. Thus = r, and so 2 DL . Part (d) is proved similarly. We can assume that E ^ HFp = 0. Then ^ = _ a^ _ a = r _ a^ r _ a = r ^ r = r: Part (e) follows immediately. |___| Note that, if r preserves arbitrary joins, it must have a right adjoint r*: D* *L -! B. This right adjoint must be defined by r* = _ a, where D = aHFp _ HFp. We can define this map without knowing Conjecture 3.12, of course, but we do not know that it preserves arbitrary meets without Conjecture 3.12. Another corollary of Conjecture 3.12 would be some understanding of the diffe* *r- ence between the meet and the smash in B. In particular, the meet and the smash are equivalent, modulo strange spectra. Proposition 3.14.Suppose Conjecture 3.12 holds. Let D = aHFp_ HFp, so that a is the maximum strange Bousfield class. Then if and are arbitrary Bousfield classes, we have f _ a = ^ _ a: Proof.Since r preserves both meets and the smash product, we have r f = r ^ . Conjecture 3.12 completes the proof. |___| 8 MARK HOVEY AND JOHN H. PALMIERI 4.More structure of B: minimal and complemented classes In this section we discuss minimal, maximal, and complemented Bousfield class* *es. We say that a nonzero Bousfield class is minimal if there is no nonzero B* *ous- field class strictly less than . Maximal Bousfield classes are defined simil* *arly. Example 4.1. For n 0, the nth Morava K-theory spectrum K(n) has a minimal Bousfield class_see Section 5. We conjecture below (Conjecture 5.1) that is minimal when n 2, where A(n) is a spectrum that measures the failure of the telescope conjecture; we also conjecture (see Lemma 7.8) that is minimal, w* *here I is the Brown-Comenetz dual of the sphere. It is natural to wonder whether a given Bousfield class can be written as the least upper bound of minimal ones, or dually, whether a class is the greatest l* *ower bound of maximal ones. Since the least upper bound has a much more convenient description, we will focus on minimal Bousfield classes. Using the complementat* *ion operator a, one can easily check that is minimal if and only if a is max* *imal. Although we have referred to a as the complementation operator, it is not al- ways the case that a _ = ; when this happens, we say that is complemented. One can easily check that if there is a Bousfield class so t* *hat _ = and ^ = <0>, then = a. This is the reason for the term "complemented." We also define to be f-complemented if there is a Bousfield class so that f = <0> and _ = . Now we note that we should only have made one definition. Proposition 4.2. is f-complemented if and only if is complemented. If these conditions hold, then the f-complement of is a. Proof.Since ^ f , we see that if is f-complemented, then is complemented, with the same complement. Conversely, suppose that a_ = . Then 2 a f = a a _ a = a = <0>; so a is the f-complement of . |___| The collection of all complemented Bousfield classes is denoted BA . Here are some of the basic properties of BA ; these are all due to Bousfield [Bou79a ]. Lemma 4.3. Suppose that and are in BA , and is an arbitrary Bous- field class. Then: (a) = ^ _ ^ a. (b) if and only if = ^ . (c) f = ^ . (d) Hence BA DL . (e) ^ is in BA , and a ^ = a _ a. (f) _ is in BA , and a _ = a ^ a. (g) BA is a Boolean algebra. (Recall that a Boolean algebra is a distributive lattice in which every eleme* *nt has a complement.) Proof.For the first part, use the identity = ^ = ^ _ a. The second part then follows immediately. For part (c), suppose and THE STRUCTURE OF THE BOUSFIELD LATTICE 9 . Then = ^ = ^ ^ . Hence ^ . Part (d) is clear. For part (e), note that a ^ = a f = a _ a. Furthermore, = a _ = a _ ^ _ ^ a ^ _ a _ a: Thus ^ is complemented, as required. The proof of part (f) is similar,* *_and part (g) follows immediately from the preceding parts. |__| Example 4.4. Bousfield shows in [Bou79a ] that if F is a finite spectrum, then is in BA . He also notes that is not in BA ; in particular, the inclu* *sion BA DL is proper. We show in Section 5 that and are in BA . The structure theory of infinite Boolean algebras is considerably more compli- cated than the structure theory of finite Boolean algebras. In particular, BA * *is not closed under infinite joins (see Corollary 7.10), and so is certainly not isomo* *rphic to the complete Boolean algebra of all subsets of some infinite set. The simple* *st infinite Boolean algebra that is not complete is the Boolean algebra of all fin* *ite and cofinite subsets of an infinite set. We have noted that every finite spectrum is complemented; some other exam- ples of complemented spectra are provided by smashing localizations. Recall that every spectrum E determines a Bousfield localization functor LE , as described * *in [Bou79b ]. If E and F are Bousfield equivalent, then the functors LE and LF are equal_Bousfield localization only depends on the Bousfield class of the spectru* *m. We say that a Bousfield class is smashing if the natural map LE S ^ X -! LE* * X is an equivalence. Ravenel proves the following in [Rav84 , 1.31]. Proposition 4.5.Every smashing Bousfield class is complemented, with com- plement given by the fiber AE S of S -! LE S. Proof.For a general Bousfield localization functor LE , we have = ___ . Because LE is smashing, we have LE S ^ AE S = LE AE S = 0. |__| 5. Bousfield classes with finite acyclics In this section we give a brief summary of what is known about Bousfield clas* *ses which contain finite spectra; this leads to information about the Boolean algeb* *ra BA . Details can be found in [Hov95a ]. As above, we denote the (p-local) sphere by S; we write M(p) for the mod p Moore spectrum. A generic finite spectrum of type n will be denoted by F (n); then any choice for F (n) generates the same thick subcategory Cn, by the thick subcategory theorem of Hopkins-Smith [HS , Rav92a]. In particular, the Bousfield class of F (n) is well-defined. Any F (n) has an essentially unique vn-self map* * whose cofiber is an F (n + 1) and whose telescope we will denote by T (n). The Bousfi* *eld class of T (n) is also well-defined. By repeated use of [Rav84 , 1.34], we have a Bousfield class decomposition = _ _ . ._. _ : Furthermore, T (i) ^ T (j) = 0 unless i = j, and T (i) ^ F (n) = 0 for i < n. 10 MARK HOVEY AND JOHN H. PALMIERI It follows that localization with respect to T (0) _ T (1) _ . ._.T (n - 1), * *writ- ten Lfn-1, is smashing and that its kernel is precisely the localizing subcateg* *ory generated by F (n)_see [Mil92]. By the above decomposition (see also Proposi- tion 4.5), is complemented with complement ; in other words, we have = _ , and F (n) ^ Lfn-1S = 0. Given a spectrum E, we say that E has a finite acyclic if there is a nontrivi* *al finite spectrum X such that E ^X = 0. In this case, the thick subcategory theor* *em says that the collection of finite E-acyclics is Cn for some finite n, and we h* *ave . The Morava K-theory spectra K(n) play an important role here. They are known to be field spectra, so that K(n) ^ E is a wedge of suspensions of K(n) f* *or any E. The telescope conjecture, recently proved to be false for n = 2 by Raven* *el, asserts that = . If this were true, then for any E with a finite * *acyclic, we would have _ _ = = : n {n | E^K(n)6=0} The failure of the telescope conjecture is measured by the fiber A(n) of the natural map T (n) -!LK(n)T (n). Once again, A(n) is well-defined up to Bousfield class. With a little work, we have _ = ; clearly A(n)^K(n) = 0. It follows easily from this that and are both complemented, as of course is . Since K(n) is a complemented field spectrum, then * *is minimal, by [HPS97 , 3.7.3]. The spectrum A(n) is rather odd, as for example ^ = , yet BP ^ A(n) = 0. So, for instance, A(n) is not (Bousfield equivalent to) a nonzero ring spectrum. As far as detecting finite spectra goes, A(n) behaves as K(n) and T (n) do: ( ^ = if i n; 0 if i >:n Other than this, very little is known about A(n). Since the telescope conjectu* *re fails when n = 2, it seems likely that it fails for all n 2, in which case A(n* *) is nonzero when n 2. We make the following conjectures. The first is a replacemen* *t, of sorts, for the telescope conjecture; it says that, although the telescope co* *njecture is false, the spectra A(n) that measure its failure behave as well as possible. Conjecture 5.1. If n 2, is a minimal nonzero Bousfield class. Hence, if E has a finite acyclic, then E is Bousfield equivalent to a finite wedge of * *spectra K(n) and A(n); in particular, _ _ = _ : {n | E^K(n)6=0} {n | E^A(n)6=0} Note that each of the wedges here is finite. This would mean that there are o* *nly countably many Bousfield classes with a finite acyclic. We also have the follow* *ing proposition, whose proof is immediate. Proposition 5.2.Suppose Conjecture 5.1 holds. Then every Bousfield class with a finite acyclic is complemented. THE STRUCTURE OF THE BOUSFIELD LATTICE 11 6.The complete Boolean algebra of spectra We have seen that the sublattice DL of the Bousfield lattice is a frame, and that the retraction map r :B -! DL preserves arbitrary meets. We have conjec- tured that r preserves arbitrary joins. We have not discussed how r behaves with respect to complements, however, and we do so in this section. We also explore * *the relationship between DL and its sub-poset BA . Definition 6.1.Define the complement operation A: DL -!DL by DL -A-!DL ; 7! r(a): Then we have the following straightforward lemma, whose proof we leave to the reader. Lemma 6.2. (a) If and are inWDL , then A if and only if Y ^ X = 0. In other words, A = { 2 DL | ^ = 0}. (b) A is order-reversing: if in DL , then A A. (c) If 2 DL , then A2 and A = A3. W (d) A converts arbitrary joins to meets: if is in DL for all i, then A * * is the meet of the A. Note that this lemma actually holds in any frame, and the complement operator is well-known in the theory. See [Bir79, V.11], for example. We will recall som* *e of this theory in the results below for the reader's convenience. W Also note that A does not convert meets to joins. For example, let X = n K(n) and let Y = HFp. Then X and Y are both in DL , and X ^ Y = 0, and thus A ^ = . On the other hand, by the computations in Example 7.3, we have A _ A a _ a = a f a < : Of course, we do have A ^ A _ A for any and in DL . This argument also implies that A2 is not the identity_indeed, if A2 were the identity, one can check that A would have to convert meets to joins. However, we do not know a specific spectrum X in DL for which A2 6= . Given Conjecture 3.12, a is in DL by Proposition 3.13(c), and A a = r = 0, * *so A2 a = . Definition 6.3.A Bousfield class is closed if 2 DL and A2 = . The sub-partially ordered set of DL consisting of the closed elements is denot* *ed cBA . Note that every Bousfield class of the form A is closed, by Lemma 6.2(c). We have the following theorem, which again holds in considerably more generality than we state it; see [Bir79, V.10-11] for the general approach. Theorem 6.4. The sub-poset cBA of DL is closed under arbitrary meets,Wand therefore is a complete lattice. The join in cBA of {} is A2 . Every element in cBA is complemented, so cBA is in fact a complete Boolean algebra. The inclusion cBA -! DL preserves arbitrary meets, and its left adjoint is giv* *en by A2: DL -! cBA . We will write the join in cBA as _cBA . 12 MARK HOVEY AND JOHN H. PALMIERI V Proof.Note that A2 is order-preserving.VThus,VifVwe denote by i the meet in DL of {}, we haveV i A2 i iA2. In particular, if each is closed, so is i. So cBA is closed under arbitrary meets, and henc* *e is a complete lattice, withWthe join defined to be the meet of all upper bounds. Now, certainly A2 i is closed and is an upper bound for {}.WIf is closed and an upper bound for {}, we have = A2 A2< iXi>, so the join in cBA is as claimed. One can easily check that A2 is the left adjoin* *t to the inclusion. It remains to show that an arbitrary element of cBA is complemented in cBA . To see this, note that ^ A = 0, and 2 A2 _ A = A A ^ A = A(0) = ; since A converts joins to meets. Thus A is the complement of in cBA_,_so cBA is a complete Boolean algebra. |__| This theorem explains our choice of symbol cBA . Note that a complete Boolean algebra need not be isomorphic to the lattice of subsets of a set. Note that, if is already complemented in the Bousfield lattice, so that <* *X> 2 BA , then certainly A2 = , so BA is a subBoolean algebra of cBA . Of course, the inclusion BA cBA is proper, because cBA is complete and BA is not. Also, the lattice cBA is not a sublattice of the Bousfield lattice: the m* *eets and joins are different in the two sets. We now investigate how A and A2 behave on meets. The following lemma appears in [Bir79, V.11]; we reproduce its proof for the reader's convenience. Lemma 6.5. Suppose and are in DL . Then (a) A ^ = A A2 ^ A2. (b) A converts meets to joins in cBA : that is, A ^ = A2 A _ A . (c) A2 preserves finite meets: that is, A2 ^ = A2 ^ A2. Proof.Certainly A ^ A A2 ^ A2. Conversely, suppose A ^ , so that Z ^ X ^ Y = 0. It suffices to show that = ^ A2 ^ A2 = 0 as well. To see this, note that ^ ^ = 0, so ^ A. On* * the other hand, ^ A2 by definition. Thus ^ A^A2 = 0. Hence A. Since A2 by definition, we have A ^ A2 = 0. Part (b) follows from part (a), since A converts joins to meets, so that 2 2 A2 A _ = A A ^ A : Similarly, part (c) follows from part (b), since 3 2 2 __ A2 ^ = A A _ A = A A _ A = A ^ A : |__| This lemma allows us to understand the map A2: DL -! cBA . Definition 6.6.A Bousfield class is said to be dense if 2 DL and A2 = . The following theorem is a special case of Theorem V.26 of [Bir79], where it * *is attributed to Glivenko. THE STRUCTURE OF THE BOUSFIELD LATTICE 13 Theorem 6.7. For and in DL , A2 = A2 if and only if there is a dense Bousfield class such that ^ = ^ . Proof.First suppose there is a dense such that ^ = ^ . Then A2 ^ = A2 ^ . But since A2 preserves finite meets, this means that A2 ^ A2 = A2 ^ A2. Since A2 = , this means A2 = A2. Conversely, suppose A2 = A2. Let = _ A^ A _ . Then one can easily check that ^ = ^ , so it remains to prove t* *hat is dense. To see this, note that A2: DL -! cBA preserves joins, so 2 2 3 2 2 A2 _ A = A A _cBA A = A A _cBA A= : as required. |___| Theorem 6.7 leads us to consider the dense Bousfield classes. Lemma 6.8. Let = a _ . If Z is in DL and , then is dense. Conversely, if Conjecture 3.12 holds, then an arbitrary Bousfield cl* *ass 2 B is dense if and only if . Proof.If , then A = ra ra = 0, since a is the maximum strange Bousfield class. Hence A2 = , so is dense. If Conjecture 3.12 holds, then any Z with is automatically in DL , so we can drop that hypothesis. Furthermore, if is dense, then A = A3 = A = 0, so ra = 0. Given Conjecture 3.12, we can conclude that a is strange,_and so a a. Thus . |__| The following corollary is an immediate consequence of Lemma 6.8 and Theo- rem 6.7. Corollary 6.9.Let = a _ . Suppose Conjecture 3.12 holds. Then for any and in DL , A2 = A2 if and only if ^ = ^ <* *D>. This corollary suggests that a characterization of cBA can be obtained from = a _ . Let LD B denote the sub-partially ordered set of B con- sisting of all elements of the form ^ . Then LD B is closed under arbitr* *ary joins, and so is a complete lattice. The inclusion LD B -! B preserves those ar* *bi- trary joins, so has a right adjoint B -!LD B; this right adjoint takes to _ { 2 LD B | }: If Conjecture 3.12 holds, then D 2 DL , so ^ implies that ^ ^ . Thus, assuming Conjecture 3.12, the right adjoint B -! LD B is just given by smashing with . Smashing with D preserves arbitrary joins, so has a right adjoint LD B -!B as well. This right adjoint takes 2 LD B to the larg* *est such that ^ = . Lemma 6.10. Suppose Conjecture 3.12 holds. Then LD B DL . Proof.By Conjecture 3.12, we have _ a = r _ a. Thus ^ = ^ _ a = ^ r _ a= ^ r: Furthermore, we have r ^ = r ^ r = ^ r, since Con- jecture 3.12 also implies that D is in DL . Thus r ^ = ^ ,_so_ ^ is in DL for all . |__| 14 MARK HOVEY AND JOHN H. PALMIERI Theorem 6.11. Suppose Conjecture 3.12 holds. Then A2: DL -! cBA fac- tors through the epimorphism ^ (-): DL -! LD B to define an isomorphism F :LD B -!cBA . Proof.We define F ^ = A2. By Corollary 6.9, F is well-defined, injective, and order-preserving. On the other hand, F is obviously surjective_s* *ince A2 is. |__| Naturally, we would like a better description of LD B, in light of Theorem 6.* *11. See Conjecture 5.1 for a related result. Conjecture 6.12. We have _ _ = _ _ : n0 n2 Note that a and a for all n, so the half of the equality in Conjecture 6.12 holds. By the definition of D and the computations in Section 5, the conjecture is equivalent to the following: _ = a _ = _ : n0 The following proposition completes our conjectural identification of cBA up* * to isomorphism. Proposition 6.13.Suppose Conjectures 3.12, 5.1 and 6.12 hold. Then cBA is isomorphic to the complete Boolean algebra generated by the atoms for n 0, for n 2, and . This isomorphism is given by applying A2, so to actually identify cBA we need to understand the behavior of A2. PropositionW6.14.SupposeWConjecture 3.12, 5.1 and 6.12 hold. Then every sub- wedge of n0 _ n2 is closed. However, A2 6= . Proof.Let denote an arbitrary subwedge of such that ^ = <0>. Let denote the complementary subwedge of . We will show that A = , so that is closed. It is clear that A, since ^ = <0>* * and 2 DL . On the other hand, , so A A . Since A 2 DL , it follows that A = A ^ and so A is a subwedge of . This subwedge cannot contain any term in , so we must have A = . In particular, it follows that _ A = ; n0 and hence 0 1 _ A2 = A @ A: n0 THE STRUCTURE OF THE BOUSFIELD LATTICE 15 We now prove that this is strictly larger that , using [Rav84 , Theorem 2.* *10]. Let J = (pi0; vi11; : :;:vinn; : :):be an infinite regular sequence in BP*. Th* *en we can form a spectrum BPJ with BPJ* = BP*=J in various ways; Ravenel uses the Bass-Sullivan construction. By [Rav84 , Corollary 2.14], BPJ is a ring spectrum and hence is in DL . Since BPJ is built from BP , we have BPJ ^ A(n) = 0 for all n. On the other hand, one can easily see that BPJ ^ K(n) = 0 for all n, since a power of vn is invariant modulo (pi0; vi11; : :;:vin-1n-1)iand this power has t* *o act both W j invertibly and nilpotently on K(n)*BPJ. Hence A n0 for all infinite regular sequences J. On the other hand, Theorem 2.10 of [Rav84 ] impli* *es__ that, for almost all such infinite regular sequences J, we have > . * * |__| In light of these results, we would like to understand A2. Given a regul* *ar sequence J as in the proof of Proposition 6.14, we can form a spectrum S=J by taking the sequential colimit of the partial quotients S=Jn. This spectrum may not be well-defined even up to Bousfield class, though each S=Jn is. The obvious conjecture is that A2 should be the wedge of the over all infinite r* *egular sequences J and all representatives S=J. 7.Bousfield classes without finite acyclics We have been discussing Bousfield classes with finite acyclics; in this secti* *on, we examine the rest of the Bousfield classes. No spectrum can have both a nonzero finite acyclic and a nonzero finite local; we conjecture that every spectrum ha* *s one or the other. In any case, we pay some attention to spectra with finite locals,* * and we discuss Brown-Comenetz duality and its relation to such spectra. We also show that a number of conjectures related to Bousfield classes without finite acycli* *cs are equivalent. Brown-Comenetz duality [BC76 ] is the main source of counterexamples in the theory of Bousfield classes. Given a spectrum X, we denote by IX its Brown- Comenetz dual, obtained by applying Brown representability to the cohomology theory Y 7! Hom (ss0(X ^ Y ); Q=Z(p)). Let I denote the Brown-Comenetz dual of the sphere. Note that IX is the same as the function spectrum F (X; I), and the* *re is a natural map X -! I2X which is an isomorphism when the homotopy groups of X are finite. Also note that IX = 0 if and only if X = 0, since Q=Z(p)is an injective cogenerator of the category of p-local abelian groups. The spectrum I* * is the central example of this paper. Recall the spectra X(n) from [Rav84 , Section 3], which interpolate between t* *he Bousfield classes of S = X(0) and BP = X(1): = > > . .>. = : Some of the basic properties of I are as follows. Lemma 7.1. (a) I is in the localizing subcategory generated by HFp; hence . (b) X(1) ^ I = 0; hence X(n) ^ I = 0 for all n 1, and BP ^ I = 0. (c) HFp ^ I = 0; hence > , and I ^ I = 0. (d) T (n) ^ I = 0 for all n. (e) ^ = = for all n. (f) The mod p Moore spectrum M(p) (and every finite-dimensional torsion spec- trum) is I-local. 16 MARK HOVEY AND JOHN H. PALMIERI Proof.Part (a) follows immediately from the fact that the homotopy of I is boun* *ded- above and torsion, as in [Rav84 , 2.6]. Part (b) follows from [Rav84 , Lemma 3.2], where it is shown that [X(1); M(p)] = 0. Using the isomorphism M(p) = F (IM(p) ; I) and adjointness, we find that I(X(1) ^ IM(p) )= 0, so that X(1) ^ IM(p) = 0. Since the homotopy groups of I are torsion, one can readily verify t* *hat = , so that X(1) ^ I = 0. Since = , then part (c) follows from (a) and (b). Part (d) follows from part (a) and the well-* *known fact that HFp ^ T (n) = 0 (because a vn-self map must have positive Adams filtr* *a- tion). Part (e) follows from (d) and the Bousfield class decomposition of Secti* *on 5. It is proved in [HS97 , Corollary B.13] that M(p) is I-local, using the isomorp* *hism M(p) = I2M(p) . It follows from [HS97 , Theorem B.6] that every finite-dimensio* *nal_ (defined in [HS97 ]) torsion spectrum is I-local. |* *__| Another useful property of I is that it detects when a spectrum has a finite local. We have already discussed spectra with a finite acyclic; similarly, we s* *ay that a spectrum E has a finite local if there is a nonzero finite spectrum X which is E-local. Note that no spectrum can have both a nonzero finite local and a nonze* *ro finite acyclic: if M is a finite E-local and W is a finite E-acyclic, then M ^ * *W is both local and acyclic, and nonzero if both M and W are. In [Hov95a , Lemma 3.7* *], the first author shows that if E has a finite local, then every finite torsion * *spectrum is E-local. This was extended in [HS97 , Theorem B.6] to all finite-dimensional torsion spectra. Proposition 7.2.The following are equivalent for a spectrum E: (a) M(p) is E-local. (b) E has a finite local. (c) aE ^ I = 0. (d) . Proof.We have already noted that (a) and (b) are equivalent. To see that (c) and (d) are equivalent, note that aE ^ I = 0 if and only if a a. This holds * *if and only if . Since M(p) is I-local, it follows that (d))(a). To see that (a))(c), suppose * *that M(p) is E-local. Then [aE; M(p)]* = 0. Using the isomorphism M(p) = I2M(p) and adjointess, we find that I(aE ^ IM(p) )= 0. Thus aE ^ IM(p) = 0. We have already seen in the proof of Lemma 7.1 that = , completing_the proof. |__| Note that this proposition implies for example that every dissonant spectrum * *is I-acyclic, since finite spectra are harmonic. W ExampleW7.3. Since finite spectra are harmonic, < n . In particular, f n = . But for each n, f = <0>, since is minim* *al and K(n) ^ I = 0. Thus the meet does not distribute over infinite joins in the Bousfield lattice. We now consider three conjectures, which we will prove are equivalent. Note t* *hat for X finite, X ^ I = F (DX; I) = IDX , where DX is the Spanier-Whitehead dual of X. In particular, X ^ I 6= 0 for every finite X. This, combined with Lemma 7* *.1, suggests the following conjecture, first made in [HS97 , Appendix B]. Conjecture 7.4. If E ^ I 6= 0, then for some n. THE STRUCTURE OF THE BOUSFIELD LATTICE 17 Note that the converse to Conjecture 7.4 is immediate from part (d) of Lemma * *7.1. The following conjecture appeared in [Hov95a , Conjecture 3.10]. Conjecture 7.5 (The Dichotomy Conjecture).Every spectrum has either a finite local or a finite acyclic. It was pointed out in [Hov95b ] that the Dichotomy Conjecture is equivalent to the following conjecture. Conjecture 7.6. If E has no finite acyclics, then . The converse to Conjecture 7.6 follows from Lemma 7.1(e). Theorem 7.7. The following are equivalent: (a) Conjecture 7.4. (b) The Dichotomy Conjecture 7.5. (c) Conjecture 7.6. Proof.We will prove that (a),(b) and (b),(c). To see that (a))(b), suppose that E has no finite locals. Then aE ^ I 6= 0, by Proposition 7.2. Hence, by part (a), for some n. It follows that = , and so E has a finite acyclic. To see that (b))(a), suppose E ^ I 6= 0. Then a2E ^ I 6= 0, so aE has no finite locals, again using Proposition 7.2. Hence aE must have a finite acyclic* *, by part (b), and so for some n. It follows that for so* *me n. Proposition 7.2 shows that (b))(c). To see that (c))(b), suppose that E has no finite acyclics. Then part (c) implies . Since M(p) is I-local by part_* *(e) of Lemma 7.1, it is also E-local. |__| The Dichotomy Conjecture has a few interesting consequences. The most obvious one is that it implies that is minimal. Lemma 7.8. If E is a nontrivial spectrum with < , then E has no finite locals or finite acyclics. Hence, if the Dichotomy Conjecture holds, there are* * no such E, and is a minimal Bousfield class. Proof.Proposition 7.2 implies that E has no finite locals. Since < , E ^ Lfn-1S = 0 for all n. Thus E can have no finite acyclics either, by the Bousfie* *ld_ class decomposition of Section 5. |__| The Dichotomy Conjecture also gives us a partial classification of complement* *ed Bousfield classes, when combined with the following lemma. Lemma 7.9. (a) Suppose that K is a field spectrum. Then for any E, either or . (b) At least one of E and aE has a finite local. (c) If E is complemented and has a finite local, then E ^ I 6= 0. Proof.(a): If E ^ K 6= 0, then = , since E ^ K is a nontrivial wedge of suspensions of K. If E ^ K = 0, then by definition of . (b): Apply part (a) to HFp. 18 MARK HOVEY AND JOHN H. PALMIERI (c): By Proposition 7.2, since E has a finite local, aE ^ I = 0. Since E is complemented, aE must be its complement and E_aE must detect every spectrum._ Thus E ^ I 6= 0. |__| CorollaryW7.10.NoneWof the following spectra is complemented: X(n), BP , HFp, n K(n), nT (n), and I. Furthermore, if the Dichotomy Conjecture holds and E is complemented, then either for some n or for some n. Proof.This follows from Lemma 7.1. |___| We have already seen that K(n) is complemented for all n. Hence Corollary 7.10 shows that BA is not closed under infinite joins. By Proposition 5.2, Conjecture 5.1 implies the converse to the second half of* * the corollary: every E with or is complemented. We can restate this as the following corollary. Corollary 7.11.Suppose both the Dichotomy Conjecture and Conjecture 5.1 hold. Then the atoms of BA are and, for n 2, . Every element of BA can either be written as a finite join of atoms or the complement of a finite j* *oin of atoms, in a unique way. In particular, BA is isomorphic to the Boolean algebra* * of finite and cofinite subsets of a countable set. 8. Strange Bousfield classes In this section, we investigate some strange Bousfield classes. We start with* * the following problem. As above, we write IX for the Brown-Comenetz dual of X. Problem 8.1. Classify the strange Bousfield classes. For instance, is every str* *ange spectrum Bousfield equivalent to IA for some connective A with finitely generat* *ed homotopy groups? Or to IR for some connective ring spectrum R? Note that for any connective spectrum A with finitely generated homotopy groups, since then IA will have homotopy groups bounded-above and torsion, so will be in the localizing subcategory generated by HFp. While the set of strange Bousfield classes may be more complicated than the guesses given in Problem 8.1, these guesses at least give us a starting place f* *or the study of strange Bousfield classes. We find that when A is as above, IA is very much like I. Lemma 8.2. Suppose A is a connective spectrum with finitely generated homotopy groups. Then the following are equivalent for a spectrum E. (a) A ^ M(p) is E-local. (b) A ^ X is E-local for some finite torsion spectrum X. (c) aE ^ IA = 0. (d) . The proof of this lemma is very similar to that of Proposition 7.2. We require A to have finitely generated homotopy groups so that A ^ X = I2(A ^ X) for all finite torsion X. We require that A be connective as well so that . This guarantees that IA ^ T (n) = 0 for all n, and thus that = * *for all finite X. We leave the rest of the proof to the reader. Similarly, we have the following analogue of Theorem 7.7. THE STRUCTURE OF THE BOUSFIELD LATTICE 19 Theorem 8.3. Suppose A is connective and has finitely generated homotopy group* *s. Then the following are equivalent. (a) If E ^ IA 6= 0, then ^ for some n. (b) For every E, either A ^ M(p) is E-local, or E ^ A ^ F (n) = 0 for some n. (c) If E ^ A has no finite acyclics, then . Again we leave the proof to the reader. The converses of parts (a) and (c) al* *ways hold, the key point being that X ^ IX is never zero unless X is. Indeed, there * *is a map X ^ IX -! I adjoint to the identity map of IX, and hence nontrivial. We now examine some specific strange spectra. We introduced the spectra X(n) in Section 7. Theorem 8.4. We have = < < . .<. < < . .<. < : Proof.We first show that . By Lemma 8.2, this is equivalent to showing that X(n) ^ M(p) is IX(n + 1)-local. Because X(n + 1) ^ M(p) = I2(X(n + 1) ^ M(p)), we can use the same argument as in the proof of Lemma 7.1(* *e) to find that X(n + 1) ^ M(p) is IX(n + 1)-local. It therefore suffices to show * *that X(n)^M(p) is in the colocalizing subcategory generated by X(n+1)^M(p) (recall that a colocalizing subcategory is a thick subcategory_closed_under products). * *We use the X(n + 1)-based Adams tower. That_is,_we let X(n + 1) be the fiber of ^s the unit map of X(n + 1), we let Xs = X(n + 1) ^ X(n) ^ M(p), and we let Ks = X(n + 1) ^ Xs. There are then cofiber sequences Xs+1 -!Xs -!Ks -!Xs+1; and the homotopy inverse limit holim(Xs) is trivial for connectivity reasons. We turn this around by letting Xs be the cofiber of the map Xs -!X0 = X(n)^M(p). Then we have cofiber sequences Xs+1 -!Xs -!Ks -!Xs+1; and the homotopy inverse limit of Xs is X(n) ^ M(p). It therefore suffices to s* *how that each Ks is in the colocalizing subcategory generated by X(n + 1) ^ M(p). To see this, we use [DHS88 , Proposition 2.3], which shows that X(n + 1)*X(k) is a free module over X(n + 1)* for k n + 1. It follows that X(n + 1) ^ X(n + * *1) and X(n + 1) ^ X(n) are wedges of suspensions of X(n + 1). Then one can easily check that Ks is a wedge of suspensions of X(n + 1) ^ M(p), and since everything is connective and locally finite, this wedge is also a product. Hence Ks is in * *the colocalizing subcategory generated by X(n + 1) ^ M(p), and so X(n) ^ M(p) is as well. A similar proof, using the fact that BP*X(n) is a free BP*-module, shows that X(n) ^ M(p) is in the colocalizing subcategory generated by BP ^ M(p). Thus we have . We have already seen that for any connective X with finitely generated homotopy groups. It remains to show that all of the inequalities above are strict. For this we* * recall the method used by Ravenel in [Rav84 , Sections 2 and 3]. He shows that there a* *re no maps from X(n+1) to X(n)^M(p), and that this is equivalent to the statement that X(n + 1) ^ I(X(n) ^ M(p))= 0: 20 MARK HOVEY AND JOHN H. PALMIERI We have already seen that = . Hence X(n+1)^IX(n) = 0, but X(n + 1) ^ IX(n + 1) is nonzero. Thus < . Similarly, Ravenel's proof that there are no maps from BP to X(n) ^ M(p) shows that BP ^ IX(n) = 0. Since BP ^ IBP is nonzero, this shows that < . Final* *ly, since there are no maps from HFp to BP , then HFp ^ IBP = 0, and so < . |__| There are probably more strange Bousfield classes than the ones described in Theorem 8.4. For example, Ravenel discusses spectra BPJ for infinite invariant regular sequences J in BP* in [Rav84 , Section 2]. We have already met these spectra in the proof of Proposition 6.14. He shows that > > f* *or J 6= (p; v1; : :):. Presumably the Brown-Comenetz duals of these spectra give o* *ther strange spectra. In addition, at p = 2, we have MSp as well. Ravenel sketched an argument to the first author once that > , and presumably one would also have < . We do, however, make the following conjecture. Conjecture 8.5. The spectra X(n) and X(n + 1) are adjacent in the Bousfield lattice. That is, if > , then . Note that if Conjecture 8.5 holds, then a = . Indeed, since X(1)^I =* * 0, we have a . Similarly, we have seen above that X(n+1)^IX(n) = 0, so a . But X(n)^IX(n) is nonzero, so we must have a = if Conjecture 8.5 holds. Thus Conjecture 8.5 also implies that IX(n) and IX(n + 1) are adjacent in the Bousfield lattice. Conjecture 8.5 also implies the following result. Conjecture 8.6. ^ = for all n and k. Hopkins has proved Conjecture 8.6, but the authors have not seen a proof. To * *see that Conjecture 8.5 implies Conjecture 8.6, proceed by induction on n. We will * *only indicate the proof for n = 1. Conjecture 8.5 implies that _ = . By smashing with T (k), we find that = ^ , as required. We mention that Hopkins has proved the following, though again the authors do not know the proof. Conjecture 8.7. = . 9. Localizing and colocalizing subcategories In this last section, we make a few remarks about general localizing and colo- calizing subcategories. The outstanding question here is whether every localizi* *ng subcategory is the collection of E-acyclics for some E. As pointed out by Neil Strickland, Ohkawa's result [Ohk89 ] is relevant here. Recall that a subcategory of the stable homotopy category is called localizin* *g if it is thick and is closed under coproducts. The basic conjecture here is the follo* *wing. Conjecture 9.1. Every localizing subcategory is the collection of E-acyclics for some E (and is therefore principal). There are several equivalent formulations of this conjecture. First we need s* *ome notation. Given a spectrum X, let loc(X) denote the localizing subcategory gen- erated by X. THE STRUCTURE OF THE BOUSFIELD LATTICE 21 Proposition 9.2.The following are equivalent. (a) Conjecture 9.1 holds. (b) Every principal localizing subcategory loc(X) is the collection of E-acycl* *ics for some E. (c) For each X, loc(X) is the collection of aX-acyclics. (d) if and only if X 2 loc(Y ). Proof.It is clear that (a))(b). To see that (b))(c), suppose loc(X) is the E- acyclics for some E. Then E ^ X = 0 so . On the other hand, if E ^ Z = 0, then Z 2 loc(X), so Z ^ aX = 0. Thus = , so loc(X) is also the collection of aX-acyclics. To see that (c))(d), note that X 2 loc(Y ) implies that . Conversely, if , then . In particular, X is an aY -acyclic. Thus, from part (c), we have X 2 loc(Y ). It remains to show that (d))(a). We will first show (d))(c). Indeed, suppose Y is aX-acyclic. Then = . By part (d), we have Y 2 loc(X). Hence loc(X) is the collection of aX-acyclics, as required. It is clear that (c))(b),* * so it remains to show that (b))(a). We will do so by showing that, given part (b), ev* *ery localizing subcategory is principal. Given a localizing subcategory C, there is* * only a set of Bousfield classes represented by objects of C by [Ohk89 ]. Since (b))(* *d), this means there is only a set of principal localizing subcategories of C. Cho* *ose a representative for each such principal localizing subcategory, and let X be t* *he __ wedge of all of those representatives. Then loc(X) = C, so C is principal. * * |__| Note that Conjecture 9.1, together with Ohkawa's result, would imply that the* *re is only a set of localizing subcategories. It would also imply that the cohomol* *ogical localizations studied in [Hov95b ] always exist, and are in fact homological lo* *caliza- tions. We would like a similar understanding of colocalizing subcategories (thick ca* *t- egories which are closed under products), but such an understanding has eluded us. The obvious conjecture is that there is a one-to-one correspondence between localizing subcategories and colocalizing subcategories, so that every colocali* *zing subcategory would be the collection of E-locals for some E, given Conjecture 9.* *1. One could also ask whether every colocalizing subcategory is principal. We do n* *ot know the answer, but we do have the following intriguing result. Recall that a coideal is a thick subcategory C with the additional property t* *hat if X 2 C and Y is arbitrary, then F (Y; X) 2 C. Proposition 9.3.The colocalizing subcategory generated by I is the entire stable homotopy category, as is the coideal generated by I. Proof.We use the results of [CS ]. Recall that they call a spectrum X injective if there are no phantom maps to it. They show in [CS , Proposition 3.9] that IX is injective for all X. They show in [CS , Proposition 4.15] that any X fits in* *to a cofiber sequence X -! I2X -! K, where K is injective. It follows from [CS , Lem* *ma 4.14] that K is a retract of I2K. Now, IY = F (Y; I) is in the coideal generate* *d_by I for any Y , so both I2X and K are as well. Hence X is too. |_* *_| 22 MARK HOVEY AND JOHN H. PALMIERI References [Bir79]Garrett Birkhoff, Lattice theory, corrected reprint of the 1967 third ed* *., American Math- ematical Society Colloquium Publications, vol. 25, American Mathematical* * Society, Providence, R. I., 1979. [BC76] E. H. Brown and M. Comenetz, Pontrjagin duality for generalized homology* * and coho- mology theories, Amer. J. Math. 98 (1976), 1-27. [Bou79a]A. K. Bousfield, The Boolean algebra of spectra, Comment. Math. Helv. 5* *4 (1979), 368-377. [Bou79b]A. K. Bousfield, The localization of spectra with respect to homology, * *Topology 18 (1979), 257-281. [CS] J. D. Christensen and N. P. Strickland, Phantom maps and homology theori* *es, Topology, to appear. [DHS88]E. S. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable ho* *motopy theory, Ann. of Math. (2) 128 (1988), 207-241. [FS90] Peter J. Freyd and Andre Scedrov, Categories, allegories, North-Holland * *Mathematical Library, vol. 39, North-Holland Publishing Co., Amsterdam, 1990. [HS] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II,* * Ann. of Math. (2), to appear. [Hov95a]M. Hovey, Bousfield localization functors and Hopkins' chromatic splitt* *ing conjecture, The Cech Centennial (Providence, RI) (M. Cenkl and H. Miller, eds.), Con* *temporary Mathematics, no. 181, Amer. Math. Soc., 1995, pp. 225-250. [Hov95b]M. Hovey, Cohomological Bousfield classes, J. Pure Appl. Algebra 103 (1* *995), 45-59. [HPS97]M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotop* *y theory, vol. 128, Mem. Amer. Math. Soc., no. 610, American Mathematical Society, 1997. [HS97] M. Hovey and N. P. Strickland, Morava K-theories and localization, submi* *tted to Mem- oirs of the Amer. Math. Soc., 1997. [Joh86]Peter T. Johnstone, Stone spaces, reprint of the 1982 ed., Cambridge Stu* *dies in Ad- vanced Mathematics, vol. 3, Cambridge University Press, Cambridge-New Yo* *rk, 1986. [Mil92]H. R. Miller, Finite localizations, Boletin de la Sociedad Matematica Me* *xicana 37 (1992), 383-390, special volume in memory of Jose Adem, in book form, ed* *ited by Enrique Ramirez de Arellano. [Ohk89]T. Ohkawa, The injective hull of homotopy types with respect to generali* *zed homology functors, Hiroshima Math. J. 19 (1989), 631-639. [Rav84]D. C. Ravenel, Localization with respect to certain periodic homology th* *eories, Amer. J. Math. 106 (1) (1984), 351-414. [Rav92a]D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, An* *nals of Mathe- matics Studies, vol. 128, Princeton University Press, 1992. [Str97]N. P. Strickland, Counting Bousfield classes, preprint, 1997. Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: palmieri@member.ams.org