PHANTOM MAPS AND CHROMATIC PHANTOM MAPS J. DANIEL CHRISTENSEN AND MARK HOVEY Abstract. In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V ^ X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type at least n. We define divisibility and finite type conditions which are suitable for studying n-phantom maps. We show that the duality functor Wn-1 defined by Mahowald and Rezk is the analog of Brown-Comenetz duality for chromatic phantom maps, and give conditions under which the natural map Y ! Wn2-1Y is an isomorphism. Contents 1. Introduction 1 2. Phantom maps 2 2.1. Divisibility 2 2.2. Rational homotopy 6 2.3. Preserving nothing 7 3. Chromatic phantom maps 9 3.1. Brown-Comenetz duality and universal n-phantom maps 9 3.2. Divisibility 14 References 18 1. Introduction This paper takes place in the stable homotopy category. A map X ! Y of spectra is said to be phantom if for every finite spectrum V and every map V ! X the composite V ! X ! Y is zero. We write Ph (X, Y ) for the group of all phantom maps from X to Y . The first half of the paper, Section 2, deals with some results of a general nature. The theme is to determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V ^ X _______________ Date: August 9, 1999. 1991 Mathematics Subject Classification. Primary 55P42. Key words and phrases. Phantom map, chromatic phantom map, n-phantom map, cohomotopy, stable homotopy, spectrum, n-finite type. Copyright cO1999 The Johns Hopkins University Press. This article first appeared in the American Journal of Mathematics, 1999. 1 2 J. DANIEL CHRISTENSEN AND MARK HOVEY for V finite. Our main tool is the observation that for Y of finite type, a map X ! Y is phantom if and only if it is a divisible element of the group [X, Y ]. We also make use of the connection between phantom maps and Brown-Comenetz duality. In the second half of the paper, Section 3, we work in the p-local stable homotopy cat- egory and introduce generalizations of phantom maps called chromatic phantom maps. A map X ! Y is n-phantom if for every finite spectrum V of type at least n and every map V ! X the composite V ! X ! Y is zero. The results of this section, which require more background of the reader, generalize some of the results of the first half of the paper. We define analogs of the divisibility and finite type conditions which are suitable for studying chromatic phantom maps. The significance of chromatic phantom maps is demonstrated by the fact that the kernel in homotopy of the important map Ln S0 ! LK(n) S0 is precisely the n-phantom homotopy classes. We show that the dual- ity functor Wn-1 defined by Mahowald and Rezk [11 ] is the analog of Brown-Comenetz duality. Our Theorem 3.13 gives conditions under which the natural map Y ! Wn2-1Y is an isomorphism, generalizing a result of Mahowald and Rezk. Special thanks are due to Neil Strickland for asking the question which led to Sec- tion 2.3 and to Sharon Hollander for asking the question which led to many of the results in the first half of the paper. We also appreciate helpful conversations with Amnon Nee- man. Correspondence with Haynes Miller led to the results in Section 2, and we thank him for sharing his insights, ideas and arguments. 2. Phantom maps 2.1. Divisibility. In this section we describe conditions on spectra X and Y which ensure that all maps from X to Y are phantom. This will turn out to have close connections to the divisibility of the group [X, Y ]. We write IY for the Brown-Comenetz dual of Y [2], which is characterized by the natural isomorphism [X, IY ] = Hom (ss0(X^Y ), Q=Z). We write I for IS0, and note that IY = F (Y, I), where F (-, -) denotes the function spectrum. I(-) is a contravariant functor on the stable homotopy category, and there is a natural map Y ! I2Y . As a special case of the defining property, we have sskIY = Hom (ss-k Y, Q=Z). We will make use of the following proposition which gives several characterizations of phantom maps. This is Proposition 4.12 from [3]. Proposition 2.1. The following conditions on a map X ! Y are equivalent: (i)For each finite spectrum V and each map V ! X, the composite V ! X ! Y is zero (i.e., X ! Y is phantom). (ii)For each homology theory h taking values in abelian groups, the map h(X) ! h(Y ) is zero. (iii)The composite X ! Y ! I2Y is zero. |___| Part (iii) of Proposition 2.1 implies that the fiber F ! Y of the map Y ! I2Y is phan- tom. Hence, there are no nonzero phantom maps to Y if and only if the map Y ! I2Y is PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 3 a split monomorphism. This observation gives us some information about the homotopy groups of such a spectrum Y . Indeed, if there are no nonzero phantom maps to Y , then each homotopy group ssn Y must split off its double dual Hom (Hom (ssn Y, Q=Z), Q=Z). Furthermore, each summand of ssn Y must split off its double dual. In particular, if Z is a summand of ssn Y , there must be a nonzero phantom map to Y . Indeed, if not, Z would be a summand of its double dual ^Z, the pro-finite completion of Z. But a summand of a pro-finite group is pro-finite, so this can't happen. Recall that a spectrum Y is of finite type if each homotopy group ssn Y is a finitely generated abelian group. A finite type spectrum Y is rationally trivial if and only if its homotopy groups are finite, i.e., if and only if the map Y ! I2Y is an isomorphism. Thus for Y of finite type, there are no nonzero phantom maps to Y if and only if Y is rationally trivial. Next we present a characterization of phantom maps, based on this observation. A map f : X ! Y is divisible if it is a divisible element of the group [X, Y ], i.e., if for each nonzero integer n, we have f = ng for some g. Theorem 2.2. Let Y be a spectrum of finite type. Then a map X ! Y is phantom if and only if it is divisible. In particular, all maps from X to Y are phantom if and only if [X, Y ] is divisible. Moreover, the group Ph (X, Y ) of phantom maps is divisible. That a phantom map to a finite type target is divisible is Proposition 4.17 of [3]. For the convenience of the reader, we include the proof here. Proof. Let f : X ! Y be a divisible map and let V ! X be a map from a finite. Then the composite V ! X ! Y is a divisible element of the group [V, Y ]. This group is finitely generated (since Y has finite type), and so the composite is zero. Thus f is phantom. Now suppose that f : X ! Y is phantom. By part (iii) of Proposition 2.1, the given map f factors through the fiber F of the natural map Y ! I2Y . Since Y has finite type, the fiber F is rational [12 , Lemma A3.13]. Thus f must be divisible. In fact, for any nonzero integer n, we can write f as ng, with g phantom, since the map F ! Y is phantom (by Proposition 2.1). This proves that Ph (X, Y ) is divisible. We have proved the first and third statements. From the first it is clear that all maps from X to Y are phantom if and only if [X, Y ] is divisible. |___| The proof shows that any phantom map X ! Y , with Y of finite type, factors through a rational spectrum, and therefore through the rationalization of X. This is a stable analogue of [13 , Theorem 5.1 (ii)]. Phantom maps are not always divisible. For example, there is a spectrum Y such that Ph (HFp, Y ) is nonzero [3, Proposition 4.18]. Of course, by the above, no such Y has finite type. Also, a divisible map is not always phantom. For example, the identity map of HQ is divisible but not phantom. 4 J. DANIEL CHRISTENSEN AND MARK HOVEY A natural question is the following. Suppose that all graded maps from X to Y are phantom, i.e., that Ph (X, Y )* = [X, Y ]*. Does it follow that all maps from V ^ X to Y are phantom for V finite? The answer to this question is no, as we will see after the following theorem. Theorem 2.3. Let Y be a spectrum of finite type. Then the following are equivalent. (i)Ph (V ^ X, Y )* = [V ^ X, Y ]* for all finite V . (ii)Ph (V ^ X, Y )* = [V ^ X, Y ]* for all V . (iii)X ^ IY = 0. (iv) [X, Y ]* is rational. Proof. (iii) =) (ii): Assume that X ^ IY = 0. Then V ^ X ^ IY = 0 for any V . So [V ^ X, I2Y ]* = Hom (ss-* (V ^ X ^ IY ), Q=Z) = 0. So every map from V ^ X to Y is phantom, by part (iii) of Proposition 2.1. (ii) =) (i): This is clear. (i) =) (iv): Assume that all graded maps V ^ X ! Y are phantom for finite V . Let p be a prime and write M (p) for the mod p Moore spectrum. By assumption, any graded map M (p) ^ X ! Y is phantom. By Theorem 2.2, any such map is divisible. But since M (p) is torsion, we conclude that any map M (p) ^ X ! Y is zero. Therefore, multiplication by p on [X, Y ]* is an isomorphism. This is true for all primes, so [X, Y ]* is rational. (iv) =) (iii): Assume that [X, Y ]* is rational. By the following lemma, F (X, I2Y ) is the pro-finite completion of F (X, Y ). By hypothesis, F (X, Y ) is rational, and so its pro-finite completion is trivial. Thus Hom (ss*(X ^ IY ), Q=Z) = 0. Since Q=Z is a cogenerator, it follows that ss*(X ^ IY ) = 0. Therefore X ^ IY = 0. |___| We write Z ! Z^ for the pro-finite completion of a spectrum Z. This is Bousfield localization with respect to the wedge over all primes of the mod p Moore spectra. Put another way, Z ! ^Zis the initial map to a spectrum which has no nonzero maps from a rational spectrum. We write A^ for the pro-finite completion of an abelian group A. The above proof used the following lemma. Lemma 2.4. For Y finite type and any X, the natural map F (X, Y ) ! F (X, I2Y ) is pro-finite completion. In particular, Y ! I2Y is pro-finite completion. Proof. The second statement is [12 , Theorem 9.11]. The general case follows. Indeed, for a general spectrum E, if Y is E-local, so is F(X, Y ) for any X. In particular, F(X, I2Y ) is local with respect to the wedge of all the M (p). Since the fiber F of Y -! I2Y is rational, so is F (X, F ). Thus F (X, F ) is acyclic with respect to the wedge of all the M (p). |___| We now find specific spectra X and Y where Y is of finite type and [X, Y ]* is divisible, but not rational. In view of Theorems 2.2 and 2.3, this implies that all maps from X to Y are phantom, but that there is a map M (p) ^ X -! Y which is not phantom. PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 5 Proposition 2.5. The integral cohomology of S^0 is Q=Z, concentrated in degree 1. In particular, all maps from S^0to HZ are phantom, but there are non-phantom maps from S^0^ M (p) = M (p) to HZ. Proof. We use the universal coefficient theorem to calculate H* (S^0) from the fact that H*(S^0) = ^Z, the pro-finite completion of Z, concentrated in degree 0. We have isomor- phisms H0(S^0) ~= Hom (^Z, Z) and H1(S^0) ~= Ext(^Z, Z), while the other groups vanish. Suppose we have a nonzero homomorphism f : ^Z! Z. Then the image of f must be the subgroup mZ for some nonzero m. If we define g = ( 1_m)f , then g is a well-defined ho- momorphism whose image is all of Z. By choosing a section of g, we find that ^Z~= Z A for some abelian group A. We have already seen that this is impossible in the paragraph following Proposition 2.1. Hence Hom (^Z, Z) = 0. In order to calculate H1(S^0), we use some topology. First, (HZ^)*(S^0) ~=(HZ^)*(S0) ~= ^Zconcentrated in degree 0. In particular, (HZ^)1(S^0) is zero. It follows from Propo- sition 2.1 (iii) that the group H1(S^0) is all phantom, and so is divisible. More- over, any non-torsion element of H1(S^0) would survive to give a nonzero element of (HQ)1(S^0) = 0. Thus H1(S^0) is a direct sum of copies of Q=Z(p) for various primes p. On the other hand, (HFp)*(S^0) = Fp concentrated in degree 0. Hence there must be exactly one summand of the form Q=Z(p) for each prime p, and so H1(S^0) = Q=Z, as claimed. |___| There are certain situations where the phenomenon of Proposition 2.5 can not occur. We say that a spectrum X is bounded above if its homotopy groups are bounded above; i.e., if there is an N such that ssiX = 0 for i > N . Theorem 2.6. Let Y be a spectrum. Then [HFp, Y ]* = 0 for all primes p if and only if [X, Y ] is rational for each bounded above X. Moreover, if Y has finite type, then a third equivalent condition is that all maps from every bounded above X to Y are phantom. Proof. It is clear that [HFp, Y ]* = 0 for all primes p if and only if [HZ, Y ] is rational. If [HZ, Y ] is rational, then so is [X, Y ] for any X in the localizing subcategory generated by HZ, since rational abelian groups are closed under kernels, cokernels, extensions and products. If X is bounded above then X is the homotopy colimit of its Postnikov sections, each of which has only finitely many nonzero homotopy groups. A spectrum with only finitely many nonzero homotopy groups is in the thick subcategory generated by HZ, so a bounded above spectrum is in the localizing subcategory generated by HZ. This proves the first statement. Now assume that Y has finite type. If [X, Y ] is rational for bounded above X, then by Theorem 2.2 all maps from every bounded above X to Y are phantom. And if all maps from every bounded above X to Y are phantom, then using Theorem 2.2 again we see that [HFp, Y ] is divisible and hence zero. This proves the second statement. |___| Note that our proof of Theorem 2.6 applies more generally to spectra X in the localizing subcategory generated by HZ. 6 J. DANIEL CHRISTENSEN AND MARK HOVEY Recall that a spectrum W is dissonant if its Morava K-theory is trivial, so K(n)*W = 0 for all n (and all primes p). A spectrum Y is harmonic if [W, X]* = 0 for all dissonant spectra W . Since HFp is dissonant for all p [16 , Theorem 4.7], [HFp, Y ]* = 0 for all harmonic spectra Y , and this gives rise to many examples. Recall that Hopkins and Ravenel proved that suspension spectra are harmonic [4]. Therefore, a finite type suspension spectrum and its desuspensions satisfy the hypotheses on Y in the theorem. In particular, a finite spectrum satisfies the hypotheses on Y and so a special case of Theorem 2.6 is that all maps from an Eilenberg-Mac Lane spectrum to a finite spectrum are phantom. We now deduce a variant. Note that an Eilenberg-Mac Lane spectrum has bounded below cohomotopy groups [9, Theorem 4.2]. Corollary 2.7. Let X have bounded below cohomotopy groups and let Y be a finite spectrum. Then all maps from X to Y are phantom, and [X, Y ] = Ph (X, Y ) is rational. In particular, if X has bounded below cohomotopy groups, then these groups consist of phantom maps and are rational. Proof. Let V be a finite spectrum and let V ! X be a map. We must show that the restriction map [X, Y ] ! [V, Y ] is zero, so it suffices to prove that the map F (X, Y ) ! F(V, Y ) is zero in homotopy groups. But, by induction over the cells of Y , one can prove that F (X, Y ) has bounded above homotopy groups. And since F (V, Y ) is finite, the result follows from Theorem 2.6. |___| 2.2. Rational homotopy. In this section we give stable analogs of results of McGibbon and Roitberg. First we have the analog of part (i) of [14 , Theorem 2]. Proposition 2.8. If X ! X0 induces a monomorphism on rational homotopy and Y is of finite type, then the natural map Ph (X0, Y ) ! Ph (X, Y ) is surjective. Proof. Let f : X ! Y be a phantom map. By the proof of Theorem 2.2, f factors through the rationalization XQ of X. Since f induces a monomorphism on rational homotopy groups, the map fQ : XQ ! X0Q is split monic. Thus f factors through the composite X ! XQ ! X0Q. And so f extends over X0 by composing with the universal map X0 ! X0Q. The resulting map X0 ! X0Q! Y is divisible and hence phantom. |___| As a consequence we get a stable version of part of [14 , Theorem 1]. Corollary 2.9. If there exists a rational monomorphism from X to a wedge of finite spectra, then Ph (X, Y ) = 0 for all finite type Y . |___| Note that if X ! X0 induces an epimorphism on rational homotopy, we can not conclude that Ph (X0, Y ) -! Ph (X, Y ) is a monomorphism. Indeed, consider the rational isomorphism S0 -! HQ. Then Ph (S0, Y ) = 0 for any Y , but Ph (HQ, Y ) is often nonzero, for example when Y = S1. Next we have the stable analog of part (ii) of [14 , Theorem 2]. PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 7 Proposition 2.10. If Y and Y 0are of finite type and f : Y ! Y 0induces an epimor- phism on rational homotopy, then the natural map Ph (X, Y ) ! Ph (X, Y 0) is surjective for all X. Proof. Let F denote the fiber of Y ! I2Y and let F 0denote the fiber of Y 0! I2Y 0. Then there is a unique induced map F ! F 0, since F is acyclic and I2Y 0is local with respect to the wedge of all the M (p) as p runs through the primes. We first claim that the map I2f : I2Y ! I2Y 0is also an epimorphism on rational homotopy. Indeed, by hypothesis, the cokernel of ssn f is a (necessarily finite) torsion group. Since applying I2 tensors the homotopy of a finite type spectrum with ^Z, the cokernel of ssn I2f is also a bounded torsion group, and so I2f induces an epimorphism on rational homotopy. The five lemma then implies that F ! F 0induces an epimorphism on rational homotopy, since we have a short exact sequence 0 ! ssn Y ! ssn I2Y ! ssn-1 F ! 0 and a similar one for Y 0. Since F and F 0are rational spectra, the map F ! F 0is a split epimorphism. Given a phantom map X ! Y 0, it must factor through a map X ! F 0. This map then factors through F , and so the original phantom map factors through Y , as required. |___| Once again, if Y ! Y 0induces a monomorphism on rational homotopy, we can not conclude that Ph (X, Y ) ! Ph (X, Y 0) is a monomorphism. Take Y = Y 0= HZ, with the map Y ! Y 0being multiplication by n, which is a rational isomorphism. Take X = S^0. Then we have seen in Proposition 2.5 that Ph (X, Y ) = Q=Z, on which multiplication by n is surjective but not injective. 2.3. Preserving nothing. The final part of this section is concerned with the following question, which is due to Neil Strickland. Suppose we know that no maps of any degree from X to Y are phantom. Is it necessarily the case that no maps from V ^ X to Y are phantom for any finite V ? We know of no case where the answer is no. The answer is yes when Y is an Eilenberg-Mac Lane spectrum and also when Y is a finite type spectrum. We begin with the Eilenberg-Mac Lane spectrum case. Theorem 2.11. Let X be a spectrum and G an abelian group such that there are no phantom maps of any degree from X to HG. Then there are no phantom maps from V ^ X to HG for any finite V . In order to prove this, we need to understand phantom maps from a spectrum X to an Eilenberg-Mac Lane spectrum. Recall that the universal coefficient theorem gives us a (split) monomorphism Ext (H-1 X, G) ,! [X, HG] whose image consists of those maps sent to zero by the integral homology functor H0(-). Thus the group of phantom maps from X to HG can be thought of as a subgroup of Ext (H-1 X, G). Theorem 6.4 of [3] identifies this subgroup as PExt (H-1 X, G), where we write PExt (A, B) for the 8 J. DANIEL CHRISTENSEN AND MARK HOVEY subgroup of Ext (A, B) consisting of the divisible elements. (These correspond to the "pure" extensions; see [3, Section 6].) Since the monomorphism above is split, divisibility in the Ext group is equivalent to divisibility in [X, HG], so we can make the slightly sharper statement that a map X ! HG is phantom if and only if it is divisible and is sent to zero by H0(-). To summarize: Proposition 2.12. Let X be a spectrum and let G be an abelian group. A map X ! HG is phantom if and only if it is divisible and is sent to zero by H0(-). The group of phantoms is isomorphic to PExt (H-1 X, G). |___| Proof of Theorem 2.11. By the K"unneth theorem, H*(V ^ X) is a (non-canonical) sum of tensor and torsion products of H*V and H*X. So the theorem follows from the following lemma. |___| Lemma 2.13. Let A, B and C be abelian groups with C finitely generated. If PExt (A, B) = 0, then PExt (A C, B) = 0 and PExt (A * C, B) = 0, where * denotes the torsion product. Proof. We can assume without loss of generality that C is cyclic. If it is infinite cyclic, then A C = A and A * C = 0, so we are done. If it is Z=n, then both A C and A * C are killed by n, and so the Ext groups have no divisible elements. |___| We now consider the finite type case. Theorem 2.14. Let X and Y be spectra such that there are no phantom maps of any degree from X to Y . Let V be a finite spectrum. Then Ph (V ^ X, Y )* is a bounded torsion graded group. In particular, if Y has finite type, then there are no phantom maps from V ^ X to Y . Proof. For any finite spectrum V , there is a cofiber sequence ` Sffi! V ! T, i where the wedge of spheres is finite and T is a finite torsion spectrum. In particular, nT = 0 for some n. Smashing this cofiber sequence with X, we get a cofiber sequence ` ffiX ! X ^ V ! X ^ T. i Our hypothesis implies that Ph (_ ffiX, Y ) = 0. Hence any phantom map f : X ^ V ! Y factors through a map g :X ^ T ! Y . Since ng = 0, it follows that nf = 0, so Ph (X ^ V, Y ) is a bounded torsion group, as required. The last sentence of the theorem then follows immediately from Theorem 2.2. |___| Note that the second half of the theorem only relies on the divisibility of Ph (V ^X, Y ). PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 9 3. Chromatic phantom maps In this section we consider a generalization of phantom maps. It is much simpler to work p-locally for this generalization, where p is some fixed prime, so we will do so. The symbol X^ will therefore refer to LM(p) X, the p-adic completion of X, rather than the pro-finite completion of X. Similarly, if A is an abelian group, A^ will refer to its p-completion. Recall that a finite spectrum V is said to be of type n if K(n - 1)*V = 0 but K(n)*(V ) is nonzero. Let Cn denote the thick subcategory of spectra of type at least n. Then the Cn form the complete list of nonzero thick subcategories of finite spectra [5]. Definition 3.1. A map X ! Y is said to be n-phantom if for every finite spectrum V 2 Cn and every map V ! X the composite V ! X ! Y is zero. We write n-Ph (X, Y ) for the group of n-phantom maps from X to Y . Note that a 0-phantom map is the same thing as a phantom map. In general, every n-phantom map is also an n + 1-phantom map. The n-phantom maps form a two-sided ideal of morphisms, so that if f is n-phantom, then gf and f h are also n-phantom whenever we can form the compositions. Unlike ordinary phantom maps, this ideal is not nilpotent when n 1. Indeed, if there are no maps at all from a type n spectrum to X, then the identity map of X is n-phantom. For example, the identity map of K(n-1) is n-phantom but not n - 1-phantom. This also shows that nonzero homotopy classes can be n-phantom if n > 0, though there are no nonzero n-phantoms whose domain is a type n finite spectrum, by definition. Just as with phantom maps, there is a universal n-phantom map out of X. To see this, let C0ndenote a countable skeleton of Cn . Let P denote the coproduct over all maps V -! X of V , where V runs through spectra in C0n. Let f : X ! Y denote the cofiber of the obvious map P ! X. Then f is obviously n-phantom, and if g :X ! Z is another n-phantom map, then g factors (non-uniquely) through f . In particular, there are no n-phantom maps out of X if and only if X is a retract of a wedge of finite spectra of type at least n. For example, there is a nonzero n-phantom map out of any finite spectrum of type n - 1 or less. There is also a universal n-phantom into Y , which will be discussed in the next part of this section. 3.1. Brown-Comenetz duality and universal n-phantom maps. To characterize n-phantom maps, we must recall the finite localization functor Lfn-1 on the stable homotopy category. See [15 ] and [7, Section 3.3] for details. Recall that Lfn-1 is a smashing localization whose acyclics consist of the localizing subcategory generated by Cn and whose local objects are those spectra Y such that [V, Y ] = 0 for every V in Cn . The fiber Cfn-1X of the map X ! Lfn-1X is the closest one can get to X using only finite spectra of type at least n. More precisely, let C0ndenote a countable skeleton of Cn , and let n X denote the category whose objects are maps F ! X where F 2 C0n, and 10 J. DANIEL CHRISTENSEN AND MARK HOVEY whose morphisms are commutative triangles. Then n X is a filtered category, and the minimal weak colimit of the obvious functor from n X to finite spectra is Cfn-1X [7, Remark 2.3.18]. As a special case, Lf-1X = 0 and Cf-1X = X. Since there are no maps from a finite V in Cn to an Lfn-1-local spectrum Y , the iden- tity map of such a Y is n-phantom. More generally, we have the following proposition. Proposition 3.2. A map g :X ! Y is n-phantom if and only if Cfn-1g :Cfn-1X ! Cfn-1Y is phantom. Proof. Suppose first that Cfn-1g is phantom. Suppose V 2 Cn , and h :V ! X is a map. The composite V ! X ! Lfn-1X is trivial, since V is Lfn-1-acyclic. Hence h factors through a map V ! Cfn-1X. Since Cfn-1g is phantom, the composite V ! Cfn-1X ! Cfn-1Y is trivial. Hence the composite V ! X ! Y is trivial, so g is n-phantom. Conversely, suppose that g is n-phantom, V is finite, and V ! Cfn-1X is a map. Since Cfn-1X is the minimal weak colimit of the finite spectra in Cn mapping to X, the map V ! Cfn-1X must factor through a map Z ! Cfn-1X where Z is in Cn . Thus, we may as well assume that V is in Cn . Then, since g is n-phantom, the composite V ! Cfn-1X ! Cfn-1Y factors through a map V ! -1 Lfn-1Y . But V is Lfn-1-acyclic, so this means the composite V ! Cfn-1X ! Cfn-1Y is zero. |___| Corollary 3.3. If g is an n-phantom map, then so is g ^ Z for all spectra Z. Proof. Since Lfn-1 is a smashing localization, we have Cfn-1(g ^ Z) = Cfn-1(g) ^ Z. The result now follows from the fact that ordinary phantoms form an ideal under the smash product, which is easily proved using Proposition 2.1. |___| The fact that ordinary phantom maps compose to zero [3, Corollary 4.7] then yields the following corollary. Corollary 3.4. Suppose f : X ! Y and g :Y ! Z are n-phantom maps. Then the composite gf factors through a map Lfn-1X ! Z. |___| There is another localization functor which is relevant to n-phantom maps. Let F (n) denote a finite spectrum of type n. We can consider the localization functor LF (n)with respect to F (n) and the associated acyclization functor CF (n), both of which depend only on n. These are orthogonal to Lfn-1 and Cfn-1 in the sense that the F (n)-acyclic spectra are the same as the Lfn-1-local spectra. (Such orthogonality is analyzed in [7, Theorem 3.3.5].) In particular, Lfn-1X is F (n)-acyclic for all X, and CF (n)X is Lfn-1- local for all X. The spectrum LF (n)X should be thought of as the (p, v1, . .,.vn-1 )- completion of X. If X is connective, then LF (n)X = LM(p) X [1, Theorem 3.1]. In particular, the preceding corollary implies that if f : X ! Y and g :Y ! Z are n-phantom maps, and either Z is F (n)-local or X is Lfn-1-acyclic, then the composite gf is null. Proposition 3.5. There are no nonzero n-phantom maps to Y if and only if there are no nonzero phantom maps to Y and Y is F (n)-local. PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 11 Proof. Suppose first that there are no nonzero n-phantom maps to Y . Then obviously there are no nonzero phantom maps to Y . Also, the identity map of CF (n)Y is n- phantom, since CF (n)Y is Lfn-1-local. Hence the map CF (n)Y ! Y must be 0, so Y is a retract of LF (n)Y . This implies that Y is F (n)-local. Conversely, suppose there are no nonzero phantom maps to Y and Y is F (n)-local. Let X ! Y be an n-phantom map. Then the composite Cfn-1X ! X ! Y is phantom by Proposition 3.2. Hence it is 0, so our original map factors through a map Lfn-1X ! Y . But Lfn-1X is F (n)-acyclic, so this map must be 0. |___| Proposition 3.5 suggests the following definition, which is due to Mahowald and Rezk [11 ]. We write IY for the Brown-Comenetz dual of Y , as described at the beginning of Section 2. Definition 3.6. Given n 0, define a functor Wn-1 on (p-local) spectra by Wn-1 Y = ICfn-1Y = F(Y, Wn-1 S0). Note that W-1 Y = IY , and that Wn-1 Y = 0 if and only if Y is F (n)-acyclic, or equivalently, Lfn-1-local. Note as well that the natural map Cfn-1Y ! Y gives rise to a natural map IY ! Wn-1 Y . Lemma 3.7. The natural map IY ! Wn-1 Y is the F (n)-localization of IY . Proof. The fiber of the natural map IY ! Wn-1 Y is ILfn-1Y . This is an Lfn-1S0- module spectrum, so is F (n)-acyclic. It therefore suffices to show that Wn-1 Y is F (n)- local. Suppose Z is F (n)-acyclic. Then [Z, Wn-1 Y ] = [Z ^ Cfn-1Y, I] = 0. Indeed, Cfn-1Y is built out of finite spectra of type at least n, so since Z ^ F (n) = 0, we must have Z ^ Cfn-1Y = 0. |___| Corollary 3.8. There are no nonzero n-phantom maps to Wn-1 Y . Proof. We have Wn-1 Y = ICfn-1Y = LF (n)IY , so there are no nonzero phantom maps to Wn-1 Y and Wn-1 Y is F (n)-local. Proposition 3.5 completes the proof. |___| It follows that there are no nonzero n-phantom maps from X to IY if Lfn-1(X ^Y ) = 0. For if X ! IY is an n-phantom map, then so is its adjoint X ^ Y ! I = IS0, as it factors through X ^ Y ! IY ^ Y . By Corollary 3.8, the map X ^ Y ! I factors through CF (n)I, which is Lfn-1-local. But X ^ Y is Lfn-1-acyclic, so the map is zero. As a special case of the Corollary, there are no nonzero n-phantom maps to IV = Wn-1 V when V is a finite spectrum of type at least n, since Cfn-1V = V . Proposition 3.9. There is a universal n-phantom map to Y . Proof. Define Q to be the product over all maps Y ! IV of IV , where V runs through spectra in C0n. Let g : F ! Y denote the fiber of the obvious map Y ! Q. Then any n-phantom map to Y will have to factor through g, since there are no non-zero n-phantom maps to Q. We claim that g is itself n-phantom. Indeed, suppose V ! Y is 12 J. DANIEL CHRISTENSEN AND MARK HOVEY a nonzero map, where V is a finite spectrum of type at least n. This map corresponds to a homotopy class in Y ^ DV , and so there is some map Y ^ DV ! I which does not send this homotopy class to 0. The adjoint of this map is a map Y ! IDV , and the composite V ! Y ! IDV is nonzero, since its adjoint V ^ DV ! I sends the unit to a nontrivial class in Q=Z. Thus, if V ! Y is nonzero, so is the composite V ! Y ! Q, and so g is n-phantom. |___| This construction shows that there are no n-phantom maps to Y if and only if Y is a retract of a product of Brown-Comenetz duals of finite spectra in Cn . Just as for Brown-Comenetz duality, there is a natural map Y ! Wn2-1Y adjoint to the evaluation map Y ^ Wn-1 Y ~= Y ^ F(Y, Wn-1 S0) ! Wn-1 S0. Note that this map is the composite Y ! I2Y ! LF (n)I2Y . Indeed, we have Wn2-1Y = LF (n)I2Cfn-1Y . But I2Lfn-1Y is F (n)-acyclic, so LF (n)I2Cfn-1Y ~= LF (n)I2Y . Proposition 3.10. The fiber of the natural map Y ! Wn2-1Y = LF (n)I2Y is a uni- versal n-phantom map into Y . Proof. Corollary 3.8 implies that there are no nonzero n-phantom maps into Wn2-1Y . To complete the proof, it therefore suffices to show that the map [V, Y ] ! [V, Wn2-1Y ] is injective for all finite spectra V of type at least n. It is easy to check that [V, Y ] ! [V, I2Y ] is injective for all finite spectra. But [V, I2Y ] ! [V, LF (n)I2Y ] is an isomorphism for all finite spectra of type at least n. |___| An interesting consequence of Proposition 3.10 is that, if there are no nonzero phan- tom maps to Y , then there are no nonzero n-phantom maps to LF (n)Y . Indeed, if there are no nonzero phantom maps to Y , then Y ! I2Y is a split monomorphism. It follows that LF (n)Y ! LF (n)I2Y is a split monomorphism, and so, by Proposition 3.10, there are no nonzero n-phantom maps to LF (n)Y . These results suggest that Wn-1 is the right notion of Brown-Comenetz duality in the F (n)-local stable homotopy category. Recall from [8, Appendix B] that this category is an algebraic stable homotopy category in its own right, though the sphere LF (n)S0 = S^0 is not small. Any finite spectrum of type n is a small weak generator. The natural notion of phantom map in this setting is therefore the notion of an n-phantom map. The comments preceding Proposition 3.5 imply that the composition of two n-phantoms in this local category is null, which also follows from [7, Theorem 4.2.5]. Ordinary Brown- Comenetz duality will not stay in this local subcategory, but Wn-1 obviously does do so. Note that the Brown-Comenetz duality ^Iin the K(n)-local category studied in [8, Section 10] is just the restriction of Wn-1 . Indeed, by definition, ^IY = IMn Y for a K(n)- local spectrum Y . But, for an Ln -local spectrum Y , Mn Y = Cfn-1Y , so ^IY = Wn-1 Y . We now introduce one possible analog of the finite type condition, appropriate for chromatic phantom maps. PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 13 Definition 3.11. Define a spectrum Y to be of n-finite type if Y ^ V has finite ho- motopy groups for all V of type at least n. Note that the collection of all V such that Y ^ V has finite homotopy groups is a thick subcategory, so that Y is of n-finite type if and only if there is a type n finite spectrum V such that Y ^ V has finite homotopy groups. If Y is of n-finite type, then Y is of (n+1)-finite type. The Johnson-Wilson spectrum E(n), whose homotopy groups are given by E(n)* ~=Z(p)[v1, . .,.vn-1 ][vn , v-1n], is of n-finite type but not of (n - 1)-finite type. We also have the following lemma, whose validity is the main reason for introducing spectra of n-finite type. Lemma 3.12. The spectrum Ln S0 is of n-finite type for n 1. Proof. Suppose V is a type n finite spectrum. Then Ln S0 ^ V = Ln V = LK(n) V . Corollary 8.12 of [8] shows that sskLK(n) V is finite for all k. |___| It follows that Ln Y is of n-finite type for any finite spectrum Y and any n 1. Any dualizable spectrum in the K(n)-local category is of n-finite type by [8, Theorem 8.9]. A connective p-complete spectrum is of n-finite type if and only if it is an fp-spectrum of fp-type at most n - 1, in the sense of [11 ]. In contrast to Lemma 3.12, the calculations of [10 ] suggest that Lf2S0 is not of 2-finite type. It is widely believed that in fact Ln S0 is of 1-finite type for all n. This is true for L2S0 if p 5, by the results of Shimomura [17 ] as explained in [8, Theorem 15.1]. This conjecture is implied by [11 , Conjecture 3.8], and by the chromatic splitting conjecture of [6]. For us, the main advantage of spectra of n-finite type is the following theorem. Theorem 3.13. Suppose Y is of n-finite type. Then the map Y ! Wn2-1Y is F (n)- localization. In particular, there are no n-phantom maps to LF (n)Y , and the inclusion CF (n)Y ! Y is a universal n-phantom map to Y . Proof. Consider the map Y ! Wn2-1Y = LF (n)I2Y . The target of this map is obviously F (n)-local. On the other hand, if we smash with F (n), we have LF (n)I2Y ^ F (n) = LF (n)I2(Y ^ F (n)) = Y ^ F (n), since Y is of n-finite type. Thus Y ! Wn2-1Y is F (n)-localization. The remaining statements follow from Proposition 3.10. |___| A corollary of this theorem is that Wn-1 defines a contravariant self-equivalence of the category of F (n)-local spectra of n-finite type. This generalizes [11 , Corollary 8.3], since for n 1 a connective spectrum is F (n)-local if and only if it is p-complete. If we apply Theorem 3.13 to Ln S0 we get the following corollary. 14 J. DANIEL CHRISTENSEN AND MARK HOVEY Corollary 3.14. The kernel of the map ss*Ln S0 ! ss*LK(n) S0 is the subgroup of n- phantom homotopy classes. Furthermore, there are no n-phantom maps to LK(n) S0. Similarly, there are no n-phantom maps to LK(n) E(n) = LF (n)E(n), and the kernel of the map E(n)*X ! (LK(n) E(n))*(X) is the subgroup of n-phantom cohomology classes. If Ln S0 has 1-finite type, then the k-phantom homotopy classes in ss*Ln S0 form the kernel of the map ss*Ln S0 ! ss*LF (k)Ln S0 = ss*LK(k)_..._K(n)S0. The chromatic splitting conjecture [6], if true, implies that this kernel is built from ss*LjS0 for j < k in a predictable way. In particular, the 1-phantom subgroup of ss*Ln S0 should be the Q=Z(p) summands (Q for n = 0), and the chromatic splitting conjecture would tell us that there are 2n - 1 of these located in specific dimensions. 3.2. Divisibility. We now describe the notion of divisibility which is relevant to n- phantom maps. Definition 3.15. A map X ! Y is vn-1 -divisible if for every finite spectrum V of type at least n - 1, every vn-1 -self map v :V ! -d V , and every map V ! X, the composite V ! Y is v-divisible. By v-divisible we mean that the map factors through vk : V ! -dk V for each k. The centrality of vn-1 -self maps [5] implies that testing divisibility against any single vn-1 -self map of V ensures divisibility with respect to any other vn-1 -self map of V . Proposition 3.16. A map X ! Y is vn-1 -divisible if and only if it is n-phantom. Proof. If X ! Y is vn-1 -divisible and V ! X is a map from a finite spectrum of type at least n, then the zero map is a vn-1 -self map of V . Therefore the composite V ! Y is zero. Conversely, suppose that X ! Y is n-phantom and let V ! X be a map from a finite spectrum of type at least n - 1. Let v :V ! -d V be a vn-1 -self map of V and write V =vk for the cofiber of vk. This cofiber has type at least n, and so the composite -1 V =vk ! V ! X ! Y is trivial. Thus V ! X ! Y is divisible by vk. |___| A divisible map is clearly v0-divisible, so we obtain the following corollary, which can easily be proved directly. Corollary 3.17. A divisible map is 1-phantom. Since there are non-divisible 1-phantoms, there are non-divisible maps which are v0-divisible. Sometimes, the converse of the Corollary holds. PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 15 Proposition 3.18. If X ^ M (p) is finite or Y has 1-finite type, then a map X ! Y is divisible if and only if it is v0-divisible if and only if it is 1-phantom. In case Y has 1-finite type, 1-Ph (X, Y ) is the divisible subgroup of [X, Y ]. Proof. We have already observed that divisible implies v0-divisible and that v0-divisible is equivalent to being 1-phantom. Suppose that X ! Y is a 1-phantom map. If X ^M (p) is finite, then so is X ^M (pn ) for all n. Hence the composite X ^ -1 M (pn ) ! X ! Y must be 0 for all n, so the map X ! Y must be divisible by pn for all n. On the other hand, if Y has 1-finite type, then there are no 1-phantom maps to LM(p) Y . Hence any 1-phantom map factors through the rational spectrum CM(p) Y , so is divisible as a 1-phantom map. |___| For example, take Y = L1S0 = Lf1S0 [16 , Theorem 10.12]. Then all the homotopy groups of L1S0 are finitely generated, except ss-2 , which has a Q=Z(p) summand [16 , Theorem 8.10]. This summand is the 1-phantom subgroup. Here is a variant of Proposition 3.16. Proposition 3.19. Let V be a finite spectrum of type at least n - 1, where n 1, and let v :V ! -d V be a vn-1 -self map. Then a map f : V ! Y is n-phantom if and only if it is v-divisible. Furthermore, if Y has n-finite type, then the group of n-phantom maps from V to Y is itself v-divisible. Proof. By the centrality of vn-1 -self maps [5], f is v-divisible if and only if it is vn-1 - divisible, and so the first part follows from Proposition 3.16. Now suppose that Y has n-finite type and f : V ! Y is n-phantom. A map to Y is n-phantom if and only if it factors through CF (n)Y , which is Lfn-1-local. Hence f factors through Lfn-1V = v-1 V . Since v is a self-equivalence of v-1 V , f is v-divisible as a map into v-1 V , and hence as a map into CF (n)V , and hence as an n-phantom map. |___| Now we introduce another finite type condition. Definition 3.20. A spectrum Y has vn -finite type if for each finite spectrum V of type n and each vn -self map v :V ! -d V , the graded group [V, Y ]* contains no nonzero v-divisible elements. Testing this condition against any single vn -self map of V ensures that it holds with respect to all other vn -self maps of V . A spectrum of finite type is always of v0-finite type. However, the converse is false, as one can see by taking a large product of suitable finite type spectra. In the same way, one sees that vn -finite type does not imply n-finite type. For n 1 it is also the case that n-finite type does not imply vn -finite type. Indeed, K(n) has n-finite type but not vn -finite type. Similar examples show that when m 6= n, vm -finite type and vn -finite type are unrelated. 16 J. DANIEL CHRISTENSEN AND MARK HOVEY It is clear that products and coproducts of spectra of vn -finite type also have vn -finite type. We will see below that the sphere has vn -finite type for each n. Since there exist spectra which do not have vn -finite type, it follows that for each n the collection of spectra of vn -finite type is not thick. For an explicit example which shows that the collection of v0-finite type spectra is not thick, chose a projective resolution of Q and look at the corresponding cofiber sequence of Eilenberg-Mac Lane spectra. Our reason for introducing the vn -finite type condition is the next result. Proposition 3.21. The following conditions on a spectrum Y are equivalent: (i)Y has vn -finite type. (ii)For every X and every map f : X ! Y , f is n-phantom if and only if it is n + 1- phantom. Proof. (i) =) (ii): First recall that every n-phantom is n + 1-phantom. So assume that X ! Y is n + 1-phantom and let V ! X be a map from a type n finite spectrum with a vn -self map v. By Proposition 3.16, the composite V ! Y is v-divisible. Since Y has vn -finite type, this composite is trivial. Thus the original map is n-phantom. (ii) =) (i): Assume that a map X ! Y is n-phantom if and only if it is n + 1- phantom. We will prove that Y has vn -finite type. Let V be a type n finite spectrum with a vn -self map v, and let f : V ! Y be a v-divisible map. We must show that f is trivial. By the centrality of vn -self maps, f is vn -divisible. By Proposition 3.16, this implies that f is n + 1-phantom. By assumption, this implies that f is n-phantom. But V is type n, so f must be trivial. |___| There are examples of spectra of vn -finite type. In order to describe them, we let D denote the class of abelian groups which are the sum of a finitely generated free group and a bounded torsion group. One can show that D is closed under kernels, cokernels and extensions. Proposition 3.22. (i) If Y is bounded below, then Y has vn -finite type for each n 1 and a map X ! Y is n-phantom if and only if it is 1-phantom. (ii)If Y is such that, for each k, sskY is in D, then Y has v0-finite type and a map X ! Y is 1-phantom if and only if it is phantom. Combining the above, we see that if Y is a bounded below spectrum such that each sskY is in D, then Y has vn -finite type for each n and a map X ! Y is n-phantom if and only if it is phantom. Proof. (i) If Y is bounded below and V is finite, then for large enough m there are no nonzero maps from -m V to Y . Thus Y has vn -finite type for n 1. That n-phantoms are 1-phantoms follows from Proposition 3.21. (ii) If each sskY is in D, then for any finite V the group [V, Y ] is in D as well. But the groups in D have no divisible elements, so Y has v0-finite type. That 1-phantoms are phantoms follows from Proposition 3.21. |___| PHANTOM MAPS AND CHROMATIC PHANTOM MAPS 17 Propositions 3.18 and 3.22 give another proof of Theorem 2.2. Note that there exist spectra Y of v0-finite type for which it is not the case that sskY Q is in D. For example, take Y = lHZ=pl. Consider the case Y = E(n). Let I denote the ideal (p, v1, . .,.vn-1 ) 2 E(n)*. Define T a class in E(n)*(X) to be I-divisible if it is in I1 E(n)*(X) = kIk E(n)*(X). Then the I-divisible cohomology classes should be the n-phantom cohomology classes. We can not completely prove this, but we do have the following proposition. Proposition 3.23. If a map X ! E(n) is I-divisible, then it is n-phantom. Proof. Suppose that f : X ! E(n) is I-divisible, and g : V ! X is a map where V 2 Cn . Then gf is an I-divisible element of E(n)*(V ). However, for a particular type n spectrum of the form M = M (pi0, vi11, . .,.vin-1n-1), one can easily see that E(n)*(M ) is all killed by a power of I. It follows by a thick subcategory argument that E(n)*(V ) is all killed by a power of I for any V 2 Cn . Thus any I-divisible element in E(n)*(V ) is zero, and so f is n-phantom. |___| We do not know if the converse to this proposition holds. It is certainly true that if X ! E(n) is n-phantom, then the composite X ! E(n) ! E(n)^M (pi0, vi11, . .,.vin-1n-1) is trivial, since the second map factors through LF (n)E(n) and there are no n-phantom maps to LF (n)E(n). But it is not clear that this implies that our map is I-divisible. Finally, if Y is an Eilenberg-Mac Lane spectrum, we have the following characteriza- tion of n-phantom maps, analogous to the results of [3, Section 6]. Proposition 3.24. Suppose B is an Abelian group. If n 1, the n-phantom subgroup of [X, HB] is precisely the subgroup of divisible elements. Furthermore, an n-phantom map X ! HB is phantom if and only if the map H0X ! B is zero. Proof. Since HB is bounded below, the n-phantom subgroup coincides with the 1- phantom subgroup, by Proposition 3.22. Every divisible map is 1-phantom by Propo- sition 3.18. So suppose X ! HB is 1-phantom. Then the composite X ! HB ! HB ^M (pn ) is 1-phantom, and hence phantom by Proposition 3.22. But HB ^M (pn ) = H(B=pn ) _ HC, where C is the group of pn -torsion elements in B. In particular, there are no phantom maps to HB ^M (pn ). Indeed, a phantom map from X to an Eilenberg- Mac Lane spectrum HA is divisible, by Proposition 2.12. Since both H(B=pn ) and HC are killed by pn , they are not the target of a nonzero phantom map. Hence the composite X ! HB ! HB ^ M (pn ) is null for all n, and so X ! HB is a divisible map. The last statement is just Proposition 2.12. |___| For example, if X = HA, then Ph (HA, HB) is PExt (A, B) concentrated in degree 1, and 1-Ph (HA, HB) is PExt (A, B) in degree 1 and the subgroup of divisible elements in Hom (A, B) in degree 0. 18 J. 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Boletin de la Sociedad Matematica Mexicana, 37:383-390, 1992. This is a special volume in memory of Jos'e Adem, and is really a book. The editor is Enr* *ique Ram'irez de Arellano. [16]Douglas C. Ravenel. Localization with respect to certain periodic homology theories. Amer. J. Math., 106:351-414, 1984. [17]Katsumi Shimomura. On the Adams-Novikov spectral sequence and products of fi-elements. Hi- roshima Math. J., 16:209-224, 1986. Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 E-mail address: jdc@math.jhu.edu Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org