OHKAWA'S THEOREM: THERE IS A SET OF BOUSFIELD CLASSES WILLIAM G. DWYER AND JOHN H. PALMIERI Introduction Bousfield classes are a very useful tool in stable homotopy theory. Ten years* * ago, Ohkawa proved in [5] that there is a set of Bousfield classes, with cardinality* * at most 22@0. This result has some interesting consequences_see [3], for example_but it does not seem to be as well known as it should; we are trying to remedy this. In this short note, we give a simple proof of Ohkawa's theorem, and we discuss some natural questions that arise from the proof. In particular, the proof lea* *ds us to consider "Ohkawa classes", which provide a weaker equivalence relation th* *an Bousfield classes do. Unfortunately, Ohkawa classes seem unwieldy, and we have not found any uses for them aside from proving Ohkawa's theorem. We define Ohkawa classes and prove Ohkawa's theorem (Theorem 1.2) in the Section 1; to prove the result,@we show that there is a set O of Ohkawa classes, of cardinality at most 22 , and0that this set maps onto the collection of Bous* *field classes. This map is in fact a map of partially order sets; in Section 2, we ex* *amine the partial ordering on O. It turns out to be rather different from the orderin* *g on the set B of Bousfield classes; we are able to conclude from this that the surj* *ection O -! B is not@one-to-one. In Section 3, we show that the cardinality of O is exactly 22 , 0but we are unable to deduce anything useful about the cardinalit* *y of B from this. Throughout, we ask a number of questions; knowing the answers would shed some light on O and B and the relationship between the two. The authors would like to thank Dan Christensen and Neil Strickland for inter- esting discussions about this material. 1. Ohkawa's theorem The following definition is due to Bousfield [1, 2]. Definition 1.1.The Bousfield class of a spectrum E is defined to be the collection of E-acyclic spectra: = {X | E ^ X = 0}: Two spectra E and F are Bousfield equivalent if = . Let B be the collec* *tion of Bousfield classes. Our goal for this paper is to provide a simple proof of the following result,* * and to discuss some related results. We write @0 for the cardinality of the integer* *s. ____________ Date: 13 May 1999. 1991 Mathematics Subject Classification. 55P42, 55P60, 55U35. Partially supported by the National Science Foundation, Grant DMS98-02386. 1 2 WILLIAM G. DWYER AND JOHN H. PALMIERI Theorem 1.2 (Ohkawa [5]).B is a set, with cardinality at most 22@0. Our proof is a reformulation of Neil Strickland's reformulation (private comm* *u- nication) of Ohkawa's original proof. As we hope will be clear, this result hol* *ds in a fair amount of generality: Lemma 1.4 contains the main facts we need. __ Definition 1.3.Let F be the homotopy category of finite spectra, and let F be the set of isomorphism classes of objects of F. We need two facts about finite spectra. __ Lemma 1.4. (a) F has cardinality @0, and for any two finite spectra A and B, [A; B] is at most countable. (b) Every spectrum X can be written as a colimit of finite spectra. Proof.This is standard. Part (b) requires Brown representability of homology * *__ functors; see [4, Theorem 4.2.4] for a general statement. |* *__| These are the main requirements for the proof of Theorem 1.2, so that theorem will hold essentially whenever these conditions are satisfied. For example, in * *any stable homotopy category (as defined in [4]) which satisfies Brown representabi* *lity for homology functors, the collection B of Bousfield classes is a set. Definition 1.5.A left ideal I in the category F is a set of maps between finite spectra which is closed under left composition: if f :A ! B is in I, then so is g O f :A ! C for any map g :B ! C. If every map in I has the same domain A, we say that I is based at A. Let E be a spectrum. Given a finite spectrum A and a homology class x 2 E*A, define the annihilator ideal of x to be __ annA(x) = {f 2 [A; B] | [B] 2 F; (E*f)(x) = 0}: This is a left ideal based at A. Define the Ohkawa class of E, written <>, t* *o be __ <> = {annA (x) | [A] 2 F; x 2 E*A}: Write O for the collection of all Ohkawa classes. Lemma 1.6. O is a set, with cardinality at most 22@0. (In Theorem 3.1, we show that O has cardinality at least 22@0.) Proof.By Lemma 1.4(a), the set of annihilator ideals has cardinality 2@0. Since Ohkawa classes are sets of annihilator@ideals, then the collection O of all of * *them is a set, of cardinality at most 22 . 0 |__* *_| In order to show that the Bousfield classes form a set, we will show that the function sending <> to defines a surjection O i B. In fact, this is not * *just a map of sets: both O and B have partial orderings, and we will show that this map is a map of posets. The partial ordering on B is defined as follows: say that if E ^ X = 0 =) F ^ X = 0. Equivalently, viewing Bousfield classes as collections of acycl* *ics, as in Definition 1.1, the ordering is by reverse inclusion. The partial orderin* *g on O is by inclusion: <> <> if for all annihilator ideals annA(x) 2 <>* *, then annA(x) = annA(y) for some y 2 E*A. Lemma 1.7. If <> <>, then . OHKAWA'S THEOREM 3 Proof.Suppose that <> <> and that E ^ X = 0; we want to show that F ^ X = 0. Write X = colimXff. It suffices to show that for all ff and all x 2 F*(Xff), then x 7! 0 in F*X. Given such an x, then because <> <>, we have annXff(x) = annXff(y) for some y 2 E*(Xff). Since E ^ X = 0, then ifffi:Xff! Xfiis in annXff(y) for all large fi; hence it is also in annXff(x)._* *Thus x goes to zero in F*X = colimF*Xff, and thus F ^ X = 0. |__| Corollary 1.8.The map f :O ! B defined by f<> = is well-defined, sur- jective, and order-preserving. Theorem 1.2 follows immediately. By the way, we point out in Corollary 2.7 th* *at O -! B is not one-to-one. 2.The partial ordering on O In this section, we examine the partial ordering on O. There is a minimal ele* *ment and a maximal element, and we also find that the partial orderings on O and B are rather different: many pairs of Ohkawa classes are incomparable, while the corresponding Bousfield classes are not. We use this observation to deduce that* * the map O -! B is not one-to-one. Given a left ideal I in F based at a finite spectrum A, we write dom(I) for A* *, the domain of the maps in I. For any finite A, we let (1)A denote the ideal consist* *ing of every map with domain A. Lemma 2.1. <> = {I \ J | I 2 <>; J 2 <>; dom (I)W= dom(J)}. More generally, for any set of spectra {Eff}, the Ohkawa class of ffEffconsists of* * all finite intersections of elements of the Ohkawa classes of the Eff: _ << Eff>> = {Iff1\ . .\.Iffn| Iffj2 <>; dom (Iffj) = dom(Iffk)}: ff Proof.This is clear. |___| Corollary 2.2.<> _ <> = <> is well-defined,Wand is at least as l* *arge as both <> and <>; the analogous statement holds for ff<>. Proof.To see that <>_<> <>, given an ideal annA(x) 2 <>, consider* * the class x0 2 E*AF*A. This class has annihilator ideal equal to annA(x)\(1)A_= annA(x). |__| Question 2.3. Is <> the least upper bound of <> and <>? The answer is probably "no"; in fact, <> _ <> will be strictly larger t* *han <> whenever <> is not closed under intersections of ideals. Lemma 2.4. Write 0 for the trivial spectrum. Then <<0>> is the minimal element in the poset O. Proof.<<0>> contains only the ideals (1)A for each finite A. * *|___| This is somewhat heartening, because <0> is the minimal element in B. Things rapidly turn sour, though. Lemma 2.5. <> 6 <>. In contrast, in the Bousfield lattice B, is the unique maximal element, * *hence is at least as large as any Bousfield class, while is a minimal nonzero el* *ement. 4 WILLIAM G. DWYER AND JOHN H. PALMIERI Proof.Consider the unit j :S0 ! HQ, viewed as an element in HQ*S0. Then the annihilator ideal of j contains every map from S0 to every finite torsion spect* *rum. On the other hand, given any nonzero element y 2 ss*S0, for some prime p and for every n 0, y is not pn-divisible, so the composite jn O y is non-trivial, * *where jn :S0 ! M(pn) is the inclusion of the bottom cell. Hence annS0(j) is not_an element of <>. |__| A similar argument should show that <> 6 <> for all n 0, where * *S0(p) denotes the p-local sphere. (And as with , each is a minimal nonzero Bousfield class.) We conclude that the partial ordering on O does not bear much relation to the partial ordering on B. Corollary 2.6.<> < <>. Since = , we have the following. Corollary 2.7.The surjection f :O ! B is not one-to-one. Corollary 2.8.<> is not the largest element in O. _ Note that O does have a largest element: <>. <>2O Question 2.9. Is this the Ohkawa class of some familiar spectrum? Remark 2.10. Dan Christensen has suggested several modifications to the defini- tion of Ohkawa class, in order to make the partial orderingWbetter behaved. If * *one defines the modified Ohkawa class of E to be [[E]] = << 1n=1E>>, then this has* * the effect of closing <> under finite intersections, and hence making _ the leas* *t upper bound. Another more involved modification changes the partial ordering so that, for example, the class of the sphere is larger than that of HQ. 3.Cardinality of O In this section, we determine the cardinality of the set O of Ohkawa classes. Theorem 3.1. There are at least 22@0Ohkawa classes. Hence by Lemma 1.6, O has cardinality 22@0. In order to prove the theorem, we need some notation and some definitions. Definition 3.2.Given a spectrum E and a finite spectrum A, we write <>A for the annihilator ideals in <> based at A. Let P = {2; 3; 5; : :}:be the set o* *f positive prime numbers. If I is an ideal based at S0, let P(I) be this set of prime numb* *ers: P(I) = {p 2 P | S0 ! S0 [p e1 2 I}: Given two sets of primes S and T , we say_that S and T are commensurable if (S - T ) [ (T - S) is finite. We write S for the commensurability class of S. Proof of Theorem 3.1.Given an infinite set S = {p1; p2; : :}:of prime numbers, define a spectrum X(S) by X(S) = colim(S0 p1-!S0 p2-!: :):: We want to examine the Ohkawa class of X(S) in terms of sets of primes. First, we note that ss0X(S) = {a_b2 Q | b = pn1. .p.nk; pni2 S; n1 < . .<.nk}; OHKAWA'S THEOREM 5 while ssiX(S) = ssiS0 if i 6= 0. For each ideal I in the Ohkawa class of X(S) b* *ased at S0, we can then determine the possible commensurability class of P(I): ____ _ __ __ {P(I)| I 2 <>S0} = {;; S ; P}: For instance, given x = a_b2 ss0X(S) with b = pn1: :p:nk, then P(annS0(x)) will contain all primes in S except the ni; hence P(annS0(x)) and S are commensurabl* *e. Now consider a set of subsets of P, indexed by some indexing set J: {Sff| ff * *2 J}. Then by Lemma 2.1, ____ _ {P(I)| I 2 << X(Sff)>>S0} ff2J _ __ ___ consists of ;, P, Sfffor each ff, and the commensurability classes of finite in* *tersec- tions of the Sff. Lemma 3.3 below shows that there is a set T of subsets of P, * *with cardinality 2@0, so that no member of T is commensurable with any subset of any other member. Given any set {Sff| Sff2 T} of elements of T, one can recover the commensurability classes of the Sfffrom_the set of all commensurability classes* * of finiteWintersections of them_the Sffare the maximal@elements. Hence the spectra 0 __ J X(Sff), with the Sffchosen from T, provide 22 different Ohkawa classes. |_* *_| Lemma 3.3. There is a set T of subsets of P, so that T has cardinality 2@0, and so that no element of T is commensurable with a subset of any other element of * *T. Proof.First, note that any countably infinite set N has 2@0 subsets, none of wh* *ich is a subset of any other. For example, partition N into countably many sets N1, N2, N3, : :,:where each Nihas cardinality at least 2. Then the subsets of N of * *the form {xi| i = 1; 2; : :;:xi2 Ni} do the job. Now, partition P into infinitely many infinite sets S1, S2, S3, : :.: The set N = {1; 2; 3; : :}:has 2@0 subsets, none of which is a subset of any other; for* * each such subset Tff N, let [ Sff= Sn: n2Tff Since no Tffis a subset of any other, then this gives 2@0 subsets Sffof P, none* *_of which is commensurable with a subset of any other. |__| Note that each X(S) constructed in the proof of Theorem 3.1 is Bousfield equi* *v- alent to the sphere, so this result gives little insight into the cardinality o* *f the set of Bousfield classes. This also gives a dramatic example of the failure of O ! * *B to be one-to-one. By way of comparison, we have this result, which was pointed out by Neil Strickland. Lemma 3.4. There are at least 2@0 Bousfield classes. Proof.For_each subset S of the set {0; 1; 2; : :}:[{1}, we have a distinct Bous* *field class: < K(n)>. |___| n2S Question 3.5. What is the cardinality of B? 6 WILLIAM G. DWYER AND JOHN H. PALMIERI References [1]A. K. Bousfield, The Boolean algebra of spectra, Comment. Math. Helv. 54 (19* *79), no. 3, 368-377. [2]____ , The localization of spectra with respect to homology, Topology 18 (19* *79), no. 4, 257- 281. [3]M. Hovey and J. H. Palmieri, The structure of the Bousfield lattice, preprin* *t. [4]M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy th* *eory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. [5]T. Ohkawa, The injective hull of homotopy types with respect to generalized * *homology functors, Hiroshima Math. J. 19 (1989), no. 3, 631-639. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: dwyer.1@nd.edu Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: palmieri@member.ams.org