PHANTOM MAPS AND HOMOLOGY THEORIES

J. DANIEL CHRISTENSEN AND NEIL P. STRICKLAND

Abstract. We study phantom maps and homology theories in a stable homotopy category 𝒮 via a certain Abelian category 𝒜. We express the group 𝒫(X, Y) of phantom maps X ⟶ Y as an Ext group in 𝒜, and give conditions on X or Y which guarantee that it vanishes. We also determine 𝒫(X, HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis’s axiomatisation conjecture to an extension problem. We show that a certain functor 𝒮 ⟶ 𝒜 is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.

Contents

1. Introduction 1 2. Axiomatic stable homotopy theory 4 3. Homology theories 6 4. Phantom maps 9 5. Margolis’s axiomatisation conjecture 15 6. Phantom cohomology 16 7. Universal homology theories 21 References 25

1. Introduction

In this paper we collect together a number of results about the homotopy category of spectra. A central theme is the problem of reconstructing this category from the category of finite spectra or (what is almost equivalent) from the category of generalised homology theories. A central result (to be explained in more detail below) is that the category of spectra is a non-split linear extension of the category of homology theories by a certain square-zero ideal, the ideal of phantom maps.

Many of our results hold not only for the category of spectra but also for other categories with similar formal properties. In Section 2, we give a list of axioms which are sufficient for most of the theory. Let 𝒮 be a category satisfying these axioms, and ℱ the full subcategory of finite objects. In Section 3 we study the category 𝒜 of additive functors from ℱ to the category 𝒜b of Abelian groups, with emphasis on the homology theories. We also study the functor h: 𝒮 ⟶ 𝒜 that sends a spectrum X to the homology theory h_{X} it represents.

1991 Mathematics Subject Classification. Primary 55P42; Secondary 55N20, 55U35, 55U99, 18E30.

Key words and phrases. phantom map, stable homotopy theory, spectrum, triangulated category.

This paper has appeared in Topology 37 (1998) 339–364. This version is dated March 19, 1999 and may have corrections.

The first author was partially supported by an NSF grant and an NSERC scholarship.

The second author was partially supported by an NSF grant.

In Section 4, we consider phantom maps: a map f: X ⟶ Y is called phantom if h_{f}: h_{X} ⟶ h_{Y} is zero, and the group of phantom maps from X to Y is written 𝒫(X, Y). In Section 5, we show how our results about phantoms give new evidence for a conjectured axiomatic characterisation of the classical stable homotopy category, due to Margolis. In Section 6, we analyse the groups 𝒫(X, HA), where X is an arbitrary spectrum and HA is an Eilenberg–Mac Lane spectrum. Finally, in Section 7, we show that the functor h: 𝒮 ⟶ 𝒜 is the universal example of a homology theory on 𝒮 with values in an Abelian category satisfying Grothendieck’s axiom AB 5. We also compare this with Freyd’s construction of a universal example without the AB 5 condition, and make some related remarks about pro-spectra and ind-spectra.

We next give a more detailed summary of our main results. First we show that there are several characterisations of phantom maps.

Proposition 1.1. Let f: X ⟶ Y be a map of spectra. Then the following conditions are equivalent:

(i) f is phantom, i.e., h_{X}(W) ⟶ h_{Y} (W) is zero for each finite W. (ii) H(f): H(X) ⟶ H(Y) is zero for each homology theory H. (iii) The composite W ⟶ X ⟶ Y is zero for each finite spectrum W and each

map W ⟶ X. (iv) The composite X ⟶ Y ⟶ IW is zero for each finite spectrum W and each

map Y ⟶ IW. (Here IW denotes the Brown–Comenetz dual of W; see Section 3.)

Another important result is the following.

Theorem 1.2. The composite of two phantom maps is zero (and thus the phantom maps form a square-zero ideal).

This is a result that is folklore, but as far as we are aware the only proof that works in this generality is the one presented here, which was independently discovered by Neeman [24]. Neeman also proved some parts of Propositions 1.4, 1.5 and 1.6. Ohkawa [25] has a proof of Theorem 1.2 which works in the stable homotopy category and uses CW-structures; it is not clear whether it goes through under our axiomatic assumptions. A simpler proof that works for a stricter notion of phantom map appears in Gray’s thesis [7] and is published in [9]. The two notions coincide when the source has finite skeleta.

It turns out that a number of interesting concepts can be described in terms of the homological algebra of the Abelian category 𝒜. As usual, an object F of 𝒜 is said to be projective if maps from F lift over epimorphisms, and injective if maps to F extend over monomorphisms. A spectrum X is 𝒜-projective if h_{X} is projective in 𝒜 and 𝒜-injective if h_{X} is injective in 𝒜.

Here are two of our main results.

Theorem 1.3. There is a natural isomorphism 𝒫(Σ^{−1}X, Y) ≅ Ext_{𝒜} (h_{X}, h_{Y}).

Proposition 1.4. Let F ∈ 𝒜. The following are equivalent:

(i) F has finite projective dimension. (ii) F has projective dimension at most one.

(iii) F is a homology theory. (iv) F has injective dimension at most one.

(v) F has finite injective dimension.

In view of the above, if an object F of 𝒜 is projective or injective, then it has the form h_{X} for some spectrum X (which is unique up to isomorphism). The following result describes those X for which h_{X} is projective or injective.

Proposition 1.5. Let X be a spectrum. Then the following are equivalent:

(i) X is 𝒜-projective. (ii) X is a retract of a wedge of finite spectra. (iii) 𝒫(X, Y) = 0 for each spectrum Y. Similarly, the following are equivalent:

(i) X is 𝒜-injective. (ii) X is a retract of a product of Brown–Comenetz duals of finite spectra. (iii) 𝒫(Y, X) = 0 for each spectrum Y.

We also prove the following facts:

Proposition 1.6. 1. The category 𝒜 has enough injectives and projectives.

2. Any spectrum X sits in a cofibre sequence P ⟶ Q ⟶ X ⟶ ΣP, where P and Q are 𝒜-projective and X ⟶ ΣP is phantom. The sequence h_{P} ⟶ h_{Q} ⟶ h_{X} is a short exact sequence in 𝒜. The map X ⟶ ΣP is weakly initial among phantom maps out of X.

3. Dually, any X sits in a cofibre sequence Σ^{−1}K ⟶ X ⟶ J ⟶ K, where J and K are 𝒜-injective and Σ^{−1}K ⟶ X is phantom. The sequence h_{X} ⟶ h_{J} ⟶ h_{K} is a short exact sequence in 𝒜. The map Σ^{−1}K ⟶ X is weakly terminal among phantom maps into X.

4. IX is 𝒜-injective for each X.

5. If π_{i}Y is finite for each i, then Y is 𝒜-injective.

6. If π_{i}Y is finitely generated for each i, then 𝒫(X, Y) is divisible for each X.

7. The group 𝒫(Hℤ/p, Y) is always a vector space over ℤ/p, and is nonzero (and thus not divisible) for some Y.

8. If X is 𝒜-projective and [X, W] = 0 for each finite W, then X = 0.

The above material appears in Sections 3 and 4. We warn the reader that while the results are for the most part self-dual, the proofs are not.

In Section 5 we show how the stable homotopy category can be viewed as a linear extension of the category of homology theories by the bimodule of phantom maps. Our point in making this rigorous is that both the category of homology theories and the bimodule of phantom maps are determined by the category of finite spectra, and so we see that the category of spectra is determined up to extension by the category of finite spectra. Moreover, the goal of Section 6 is to prove that the extension is not split. We begin with the following result on phantom cohomology classes. Here PExt denotes the subgroup of Ext consisting of the pure or phantom extensions, HB denotes the Eilenberg–Mac Lane spectrum with π_{0}HB = B, and H_{∗} denotes integral homology.

Theorem 1.7. For any spectrum X and Abelian group B we have 𝒫(X, HB) = PExt(H_{−1}X, B).

After submitting this paper, we discovered that this theorem is a special case of some earlier results. One such result is due to Huber and Meier [16]. They show that if E_{∗}(−) is a homology theory of finite type, B is an Abelian group, and F^{∗}(−) is a cohomology theory fitting into a natural short exact sequence

0 ⟶ Ext(E_{n−1}(X), B) ⟶ F^{n}(X) ⟶ Hom(E_{n}(X), B) ⟶ 0,

then the subgroup of phantom cohomology classes in F^{n}(X) is isomorphic to PExt(E_{n−1}(X), B). Taking E = H and F = HB gives our result. Pezennec [26] proves essentially the same result, while Yosimura [28] removes the finite type hypothesis on the homology theory E and concludes that the subgroup of phantom cohomology classes is isomorphic to \underleftarrow{lim}^{1}F^{n−1}(X_{α}), where the X_{α} range over the finite subspectra of X. Ohkawa [25] also comes to this conclusion, but without assuming the existence of E, B, or the short exact sequence. Another reference for this last result is [4].

Using the theorem we are able to calculate all phantom maps between Eilenberg– Mac Lane spectra.

Corollary 1.8. We have

𝒫(Σ^{k}HA, HB) = \begin{cases}PExt(A, B) & if k = −1 \\ 0 & otherwise. \end{cases}

When we take A = ℤ/p^{∞} and B = ⨁_{k} ℤ/p^{k} we can use the above result and an explicit calculation to show that the phantom sequence

0 ⟶ 𝒫(Σ^{−1}HA, HB) ⟶ 𝒮(Σ^{−1}HA, HB) ⟶ 𝒜(Σ^{−1}HA, HB) ⟶ 0

is not split. This implies that the linear extension is also not split.

In Section 7 our main result is that h: 𝒮 ⟶ 𝒜 is the universal example of a homology theory with values in an AB 5 category.

Proposition 1.9. Let 𝒞 be an AB 5 category, and K: 𝒮 ⟶ 𝒞 a homology theory. Then there is an essentially unique strongly additive exact functor K′: 𝒜 ⟶ 𝒞 such that K′ ∘ h ≃ K.

We also prove that the Ind completion of the category of finite spectra is the category of homology theories.

We are indebted to Haynes Miller, Mike Hopkins, Mark Hovey, and the rest of the MIT topology community not only for the many conversations about the work presented here, but also for providing such a stimulating environment.

2. Axiomatic stable homotopy theory

Many of the properties of the stable homotopy category follow from a collection of axioms which we state below. These axioms are a slight generalisation of those found in [22], and a specialisation of those studied in [15] (as one sees using [15, Theorem 1.2.1]). We shall say that an object X in an additive category 𝒮 is small if the functor 𝒮(X, −) preserves all coproducts that exist in 𝒮.

Definition 2.1. A monogenic Brown category is a category 𝒮 (whose objects are called spectra and whose morphism sets are denoted [−, −] or 𝒮(−, −)) satisfying the following axioms:

1. 𝒮 is triangulated (and satisfies the octahedral axiom). Triangles are sometimes called cofibre sequences.

2. 𝒮 has set-indexed coproducts. The coproduct is usually written ⋁.

3. 𝒮 is closed symmetric monoidal [20]. The multiplication is called the smash product and is denoted ∧, the unit is denoted S^{0}, and the function spectra are denoted F (X, Y). The smash product and function spectrum functors are required to be compatible with coproducts and the triangulated structure, and all diagrams that one would expect to commute are required to. See [15, Appendix A] for more details.

4. S^{0} is small.

5. S^{0} is a graded weak generator for 𝒮: if π_{n}X = 0 for each n ∈ ℤ then X = 0, where π_{n}X is defined to be [S^{n}, X] and S^{n} is Σ^{n}S^{0}.

6. Homology theories and maps between them are representable — see Section 3 for an explanation of this axiom.

Note 2.2. If we replace axioms 4 and 5 with the weaker assumption that there exists a set of small graded weak generators, we get the notion of a Brown category. Most, if not all, of what we discuss here goes through in this more general setting; we restrict ourselves to the monogenic setting only for simplicity. In fact, one can get a long way without a symmetric monoidal structure.

The classical stable homotopy category, the derived category of a countable commutative ring, the homotopy category of G-equivariant spectra (for G a compact Lie group) and suitable categories of comodules over countable cocommutative Hopf algebras all form Brown categories, the first two being monogenic.

An important subcategory of a monogenic Brown category 𝒮 is the category ℱ of finite spectra which we define below. Its importance stems from the fact that a homology functor on 𝒮 is determined by how it behaves on finite spectra. Later, we will see that even more of the structure of 𝒮 is captured by ℱ.

We first make some auxiliary definitions.

Definition 2.3. A thick subcategory 𝒞 of a triangulated category 𝒮 is a full subcategory which is closed under cofibres and retracts. That is, if X ⟶ Y ⟶ Z is a cofibre sequence with two of X, Y, and Z in 𝒞, then so is the third; and if X is in 𝒞 and Y is a retract of X, then Y is in 𝒞. If D is a class of spectra in 𝒮, then the thick subcategory generated by D is the intersection of all thick subcategories containing D.

The following definition was made and studied in [18], following work of Dold and Puppe.

Definition 2.4. Write DX = F (X, S^{0}). A spectrum X is strongly dualizable if the natural map DX ∧ Y ⟶ F (X, Y) is an isomorphism for each Y.

It is not hard to see that the following conditions on a spectrum X are equivalent. For a proof, see [15, Theorem 2.1.3].

1. X lies in the thick subcategory generated by S^{0}.

2. X is small.

3. X is strongly dualizable.

Definition 2.5. We say that a spectrum X is finite if it satisfies the above conditions, and we write ℱ for the category of finite spectra.

One can show that ℱ has a small skeleton ℱ′. One can also show that ℱ is closed under the functor D, and that there is a natural map X ⟶ D^{2}X that is an isomorphism when X is finite, so that D gives an equivalence ℱ^{op} ≃ ℱ. We call this equivalence Spanier–Whitehead duality.

In the case of the classical stable homotopy category, a spectrum is finite if and only if it is a possibly desuspended suspension spectrum of a finite CW-complex.

3. Homology theories

An additive functor from a triangulated category to an Abelian category is exact if it sends cofibre sequences to exact sequences. A homology theory on a triangulated category 𝒮 is an exact functor to an Abelian category which preserves the coproducts that exist in 𝒮. Unless we state otherwise, the target category will always be taken to be the category 𝒜b of Abelian groups. It is shown in [15, Section 4] that a homology theory defined on ℱ has an essentially unique extension to a homology theory defined on all of 𝒮, so the categories of homology theories on ℱ and 𝒮 are equivalent. More precisely, we have the following result.

Proposition 3.1. For each spectrum X there is a naturally defined small diagram Λ(X) = {X_{α} | α ∈ A(X)} of small spectra with compatible maps X_{α} ⟶ X such that for any homology theory H on 𝒮, the induced map \underrightarrow{lim}_{α}H(X_{α}) ⟶ H(X) is an isomorphism. Moreover, if K is a homology theory defined on ℱ and we define K̂(X) = \underrightarrow{lim}_{α}K(X_{α}) then K̂ is the unique homology theory on 𝒮 extending K (up to canonical isomorphism).

Corollary 3.2. If W is finite then [W, −] is a homology theory so

[W, X] = \underrightarrow{lim}_{α}[W, X_{α}].

In particular, we see that any map W ⟶ X factors through some X_{α}.

In fact, if we take ℱ′ to be a small skeleton of ℱ, we can define A(X) to be the category of pairs (U, u) where U ∈ ℱ′ and u: U ⟶ X. The diagram Λ(X) is then just the functor A(X) ⟶ ℱ sending (U, u) to U.

Definition 3.3. The homology theory h_{X}: ℱ ⟶ 𝒜b represented by a spectrum X is the functor h_{X}(W) = π_{0}(X ∧ W). We shall write h(X) instead of h_{X} where this is typographically convenient. We use the same symbol h_{X} for the unique extension of this to a homology theory on all of 𝒮, which is again given by h_{X}(W) = π_{0}(X∧W) = h_{W} (X).

We also write 𝒜 for the Abelian category of additive functors from ℱ to 𝒜b. This category has small Hom sets since ℱ has a small skeleton. Note that h gives a functor 𝒮 ⟶ 𝒜. Note also that if W is a finite spectrum then h_{W} (Z) = [DW, Z]; it follows easily that [V, W] = 𝒜(h_{V} , h_{W}) when V and W are finite.

We now give a more complete statement of Axiom 6 of Definition 2.1. This follows from the other axioms if π_{∗}S^{0} is countable, but not otherwise. (See [24] and [15, Section 4] for details.)

Axiom 3.4. If H is a homology theory on ℱ (taking values in 𝒜b), then there is a spectrum Y in 𝒮 and a natural isomorphism h_{Y} ⟶ H. Moreover, a natural transformation from h_{Y} to h_{Z} is always induced by a map from Y to Z. (This map need not be unique. It turns out that a spectrum Y representing a given homology theory is unique up to a non-unique isomorphism.)

Note 3.5. A cohomology theory with values in an Abelian category ℬ is a homology theory with values in ℬ^{op}. It does follow from the first five axioms that every cohomology theory on 𝒮 with values in 𝒜b is of the form [−, Y] for some Y. By the Yoneda lemma, natural transformations are uniquely representable.

We record some basic facts about the functor h.

Proposition 3.6. The functor h: 𝒮 ⟶ 𝒜 preserves both products and coproducts, and it sends cofibre sequences to exact sequences.

Proof. It is easy to see that limits and colimits in a functor category such as 𝒜 are computed pointwise. Thus, the first claim is that

h(∏_{i} X_{i})(W) = ∏_{i} h(X_{i})(W)

for each small W. This follows easily using h(Y)(W) = [DW, Y]. The second claim is that

h(⋁_{i} X_{i})(W) = ⨁_{i} h(X_{i})(W),

which follows similarly using the smallness of DW. The third claim is that for any cofibre sequence X ⟶ Y ⟶ Z, the resulting sequence

π_{0}(X ∧ W) ⟶ π_{0}(Y ∧ W) ⟶ π_{0}(Z ∧ W)

is exact, and this is clear.

We can now start our study of homological algebra in the category 𝒜.

Lemma 3.7. A finite spectrum W is 𝒜-projective. Hence, a retract of a wedge of finite spectra is 𝒜-projective.

Proof. Let W be a finite spectrum and suppose that α : h_{W} = [DW, −] ⟶ G is a natural transformation. By the Yoneda Lemma it corresponds to an element of GDW. If β : F ⟶ G is an epimorphism then F DW ⟶ GDW is as well, so α factors through β. Thus h_{W} is projective for W finite. But projectives are closed under coproducts and retracts, so if X is a retract of a wedge of finite spectra, then h_{X} is projective.

We can use this to show that 𝒜 has enough projectives.

Lemma 3.8. The category 𝒜 has enough projectives.

Proof. Let F: ℱ ⟶ 𝒜b be an additive functor, and choose a small skeleton ℱ′ of ℱ. Then the natural map

⨁_{W ∈ℱ′} ⨁_{α∈F W}[W, −] ⟶ F

is clearly an epimorphism. Using the fact that [W, −] = h_{DW}, we see that the source is projective.

Note that the source of the above epimorphism is just h_{X}, where

X = ⋁_{W ∈ℱ′} ⋁_{α∈F W} W.

Moreover, h_{X} is not just projective, but free in the following sense. Let 𝒞 be the category of ob(ℱ′)-indexed families of sets, and consider the evident forgetful functor 𝒜 ⟶ 𝒞. This has a left adjoint, whose image consists of the functors h_{X}, where X is a wedge of finite spectra; it is natural to regard these as the free objects of 𝒜. As usual, an object is projective if and only if it is a retract of a free object; it follows that projective objects are homology theories.

Lemma 3.9. A map f: X ⟶ Y is an isomorphism if and only if h_{f}: h_{X} ⟶ h_{Y} is an isomorphism. The same holds with “isomorphism” replaced by “split monomorphism” or “split epimorphism”.

Proof. Suppose that h_{f}: h_{X} ⟶ h_{Y} is an isomorphism. As π_{k}(X) = h_{X}(S^{−k}), we see that π_{∗}(f) is an isomorphism, so f is an isomorphism.

If h_{f}: h_{X} ⟶ h_{Y} is a split monomorphism, choose a splitting, which by Brown Representability is of the form h_{g}. The composite h_{g} ∘ h_{f} is the identity, so gf is an isomorphism. By composing g with the inverse of this isomorphism we get a splitting of f.

The case when h_{f} is a split epimorphism is dual.

Proposition 3.10. A spectrum X is 𝒜-projective if and only if it is a retract of a wedge of finite spectra.

Proof. ⇐: This is Lemma 3.7.

⇒: If h_{X} is projective, it is a retract of h_{Y} with Y a wedge of finite spectra. By Brown Representability and the previous lemma, X is a retract of Y.

The dual picture. We first recall the basic facts about duality for Abelian groups.

Definition 3.11. For any Abelian group A, we write 𝕀(A) = Hom(A, ℚ/ℤ). It is well-known that this is a contravariant exact functor which converts sums to products, and that the natural map A ⟶ 𝕀^{2}(A) is a monomorphism. Moreover, if A is finitely generated then 𝕀^{2}(A) is the profinite completion of A; in particular, if A is finite then 𝕀^{2}(A) = A.

Given a spectrum X consider the contravariant functor from 𝒮 to 𝒜b sending Y to 𝕀(π_{0}(X ∧ Y)); this is clearly a cohomology theory. There is thus a representing object IX such that 𝕀(π_{0}(X ∧ Y)) ≃ [Y, IX]; we call this the Brown–Comenetz dual of X [3].

Proposition 3.12. For each spectrum X, IX is 𝒜-injective.

Proof. Fix a spectrum X. As in Corollary 3.2, we have a diagram {X_{α}} of finite spectra such that [W, X] = \underrightarrow{lim}_{α}[W, X_{α}] for all finite W. We temporarily write 𝒜′ for the category of contravariant additive functors from ℱ to 𝒜b. If F is in 𝒜 we have

𝒜(F, h_{IX}) = 𝒜(F, 𝕀[−, X])
= 𝒜′([−, X], 𝕀F)
= 𝒜′(\underrightarrow{lim}[−, X_{α}], 𝕀F)
= \underleftarrow{lim} 𝒜′([−, X_{α}], 𝕀F)
= \underleftarrow{lim} 𝕀F X_{α}
= 𝕀(\underrightarrow{lim} F X_{α}).

Suppose now that F ⟶ G is a monomorphism in 𝒜. We must show that the map 𝒜(F, h_{IX}) ⟵ 𝒜(G, h_{IX}) is a surjection. Each map F X_{α} ⟶ GX_{α} is monic, and a filtered colimit of monomorphisms is monic, so the map 𝕀(\underrightarrow{lim} F X_{α}) ⟵ 𝕀(\underrightarrow{lim} GX_{α}) is surjective, since ℚ/ℤ is injective. Thus 𝒜(F, h_{IX}) ⟵ 𝒜(G, h_{IX}) is surjective.

Corollary 3.13. If Y has finite homotopy groups, then Y ≅ I^{2}Y and so Y is 𝒜-injective. Moreover, for any family {X_{i}} of spectra, the product ∏_{i} I(X_{i}) = I(⋁_{i} X_{i}) is 𝒜-injective, as is any retract of such a product.

Proposition 3.14. 𝒜 has enough injectives.

Proof. For finite W a natural transformation from G ∈ 𝒜 to h_{IW} corresponds to an element of 𝕀G(W). Let ℱ′ be a small skeleton of ℱ, so there is a natural map

G ⟶ ∏_{W ∈ℱ′} ∏_{α∈𝕀G(W)} h_{IW} .

Since ℚ/ℤ is an injective cogenerator in the category of Abelian groups, one can show that this map is a monomorphism.

In fact, the target of the monomorphism is the homology theory represented by a product of Brown–Comenetz duals of finite spectra. In 𝒜, being injective is equivalent to being a retract of such a functor. In particular, injectives are homology theories.

Proposition 3.15. A spectrum X is 𝒜-injective if and only if it is a retract of a product of Brown–Comenetz duals of finite spectra.

Proof. ⇐: This follows from Proposition 3.12.

⇒: If h_{X} is injective, it is a retract of h(∏ IW_{α}) with each W_{α} finite. As in the proof of Proposition 3.10, this implies that X is a retract of ∏ IW_{α}.

4. Phantom maps

There is a class of maps that we cannot see, at least not easily.

Proposition 4.1. The following conditions on a map f: X ⟶ Y are equivalent:

(i) The natural transformation h_{f}: h_{X} ⟶ h_{Y} is zero. (ii) For each homology theory H, we have H(f) = 0. (iii) The composite W ⟶ X ⟶ Y is zero for each finite spectrum W and each

map W ⟶ X. (A fourth equivalent condition appears in Proposition 4.12.)

Proof. (iii)⇒(ii): Let Λ(X) = {X_{α}} be as in Proposition 3.1, so that H(X) = \underrightarrow{lim}_{α}H(X_{α}). The composite X_{α} ⟶ X ⟶ Y is zero by (iii), so H(X_{α}) ⟶ H(Y) is zero. It follows that H(f): H(X) ⟶ H(Y) is zero.

(ii)⇒(i): Suppose that H(f) = 0 for each homology theory H. Then for each finite spectrum W, the map h_{W} (f): π_{0}(X ∧ W) ⟶ π_{0}(Y ∧ W) is zero. In other words, the natural map h_{f} is zero at W.

(i)⇒(iii): Suppose that (i) holds and that W is finite. Then DW is also finite, and f induces the zero map [W, X] = π_{0}(DW ∧ X) ⟶ π_{0}(DW ∧ Y) = [W, Y].

Definition 4.2. A map X ⟶ Y satisfying the equivalent conditions of the proposition is called phantom or 𝒜-null. The collection of phantom maps from X to Y is denoted 𝒫(X, Y) and is a subgroup of [X, Y]. Similarly, we say that a map X ⟶ Y is 𝒜-monic or 𝒜-epic if the natural transformation h_{X} ⟶ h_{Y} is monic or epic, respectively.

If {X_{α}} is an indexed collection of spectra, then the map ⋁ X_{α} ⟶ ∏ X_{α} is 𝒜monic, and hence its fibre is phantom. As an example of this in the classical stable homotopy category, let C be the cokernel of the map from the sum of countably many copies of ℤ to the product. The fibre of the map H(⨁ ℤ) ⟶ H(∏ ℤ) between Eilenberg–Mac Lane spectra is a phantom map Σ^{−1}HC ⟶ H(⨁ ℤ). It is non-zero because the short exact sequence 0 ⟶ ⨁ ℤ ⟶ ∏ ℤ ⟶ C ⟶ 0 is not split. To see that this sequence is not split, notice that the coset of the quotient containing (1, 2, 4, 8, 16, …) is non-zero and is divisible by 2^{k} for each k, since initial terms may be dropped without changing the coset. But ∏ ℤ contains no such elements, so C could not be a summand. We learned this argument from Dan Dugger, who credits it to [10].

As further evidence of the ubiquity of phantom maps, it can be shown that in the classical stable homotopy category there are uncountably many phantom maps from ℂP^{∞} to S^{3}. Gray [8] has a proof for spaces which simplifies when read stably.

Note 4.3. It is not hard to see that phantom maps form an ideal in 𝒮: if f, g and h are composable and g is phantom, then fg and gh are phantom; and if f and g are parallel phantom maps, then f + g is phantom. This means that there is a welldefined additive category 𝒮/𝒫 having the same objects as 𝒮 and with 𝒮/𝒫(X, Y) := 𝒮(X, Y)/𝒫(X, Y). We have a natural isomorphism 𝒜(h_{X}, h_{Y}) ≅ 𝒮/𝒫(X, Y), so h gives an equivalence between 𝒮/𝒫 and the category ℋ of homology theories.

Lemma 4.4. For any spectrum X there is a weakly initial phantom map

δ: X ⟶ X̃

from X. By ‘weakly initial’ we mean that any other phantom map from X factors through δ, but we don’t insist upon uniqueness.

Proof. Let Λ(X) = {X_{α}} be as in Proposition 3.1. For each α we have a given map X_{α} ⟶ X, so we get a map ⋁_{α} X_{α} ⟶ X. Let δ: X ⟶ X̃ be the cofibre of this map. Corollary 3.2 tells us that every map from a finite spectrum W to X factors through ⋁_{α} X_{α}, so the composite W ⟶ X ⟶^{δ} X̃ is zero. It follows that δ is phantom. Moreover, any phantom map from X is zero when restricted to ⋁_{α} X_{α} and so factors through δ.

Corollary 4.5. Let X be a spectrum. Then the following are equivalent:

(i) X is 𝒜-projective. (ii) X is a retract of a wedge of finite spectra. (iii) 𝒫(X, Y) = 0 for each spectrum Y.

Proof. If there are no phantom maps from X, then the weakly initial phantom map X ⟶ X̃ is zero, and so X is a retract of the wedge of finite spectra ⋁ X_{α}. Conversely, if X is a retract of a wedge of finite spectra, then it is clear that there are no phantoms from X.

Proposition 3.10 tells us that being a retract of a wedge of finite spectra is equivalent to being 𝒜-projective.

Proposition 4.6. Any spectrum X sits in a cofibre sequence P ⟶ Q ⟶ X ⟶ ΣP, where P and Q are 𝒜-projective and X ⟶ ΣP is phantom. The sequence h_{P} ⟶ h_{Q} ⟶ h_{X} is a short exact sequence in 𝒜. The map X ⟶ ΣP is weakly initial among phantom maps out of X.

Proof. Consider the diagram

[figure: ⋁_{α⟶β} X_{α}; 1−s; ⋁_{α} X_{α}; Y; P; ⋁_{α} X_{α}; X .]

The map called 1 includes the α ⟶ β summand into the α summand via the identity map, while the map s (for ‘shift’) sends the α ⟶ β summand to the β summand via the map X_{α} ⟶ X_{β}. The map ⋁_{α} X_{α} ⟶ X is the map considered in Lemma 4.4. The spectra Y and P are defined to make the rows cofibre sequences, so X̃ (from the lemma) is ΣP. The composite ⋁_{α⟶β} X_{α} ⟶ ⋁_{α} X_{α} ⟶ X is null, so there is a map of cofibre sequences in the downward direction. Now consider the following natural transformation from h_{X} to h_{Y}. Let W be a finite spectrum. An element of h_{X}(W) is a map DW ⟶ X. DW is finite, so this map is X_{γ} ⟶ X for some γ. We have a map ⋁_{α} X_{α} ⟶ Y, so in particular we have a map X_{γ} ⟶ Y. That is, we have a map DW ⟶ Y, or an element of h_{Y} (W). This defines a natural transformation h_{X} ⟶ h_{Y}, and by Brown Representability this natural transformation is induced by a map X ⟶ Y. By definition, the square commutes up to phantoms, but since ⋁_{α} X_{α} is 𝒜-projective, the square commutes. One thus obtains a fill-in map P ⟶ ⋁_{α⟶β} X_{α}. Also, one can check that the composite X ⟶ Y ⟶ X is an isomorphism, and it follows that the composite P ⟶ ⋁_{α⟶β} X_{α} ⟶ P is an isomorphism as well. Thus, P is a retract of a wedge of finite spectra, and we have demonstrated that X is the cofibre of a map between 𝒜-projective spectra. We saw in Lemma 4.4 that the map X ⟶ ΣP is weakly initial.

We now get an easy proof of a result that is folklore. The method of proof presented in this section was independently discovered by Neeman [24]. A proof for the special case of the classical stable homotopy category was given by Ohkawa [25]. A proof assuming that the source has finite skeleta appears in [7] and [9]. (See the introduction for more detailed comments.)

Corollary 4.7. The composite of two phantom maps is zero.

Proof. Suppose that X ⟶^{f} Y and Y ⟶^{g} Z are phantom. Factor f through δ:

\xymatrix{X \ar[r]^{δ} \ar[d]^{f} & ΣP \ar@{-->}[dl]^{f′} \\ Y \ar[d]^{g} \\ Z .}

The 𝒜-projectivity of ΣP implies that gf′ = 0 and so gf = 0.

We can now characterise homology theories in terms of the homological algebra of the category 𝒜.

Proposition 4.8. A functor in 𝒜 is a homology theory if and only if it has finite projective dimension if and only if it has projective dimension at most one.

Proof. First, consider a short exact sequence F ⟶ G ⟶ H in 𝒜, in which two of F, G and H are homology theories. We claim that the third is also. Indeed, consider a cofibre sequence X ⟶ Y ⟶ Z. By applying F, we get a chain complex

⋯ ⟶ F (Σ^{−1}Z) ⟶ F X ⟶ F Y ⟶ F Z ⟶ F (ΣX) ⟶ · · · .

By doing the same with G and H, we obtain a short exact sequence of chain complexes. By assumption, two of the three chain complexes are exact; it follows easily that the third is also, as required.

We have seen that projective functors are homology theories. It follows easily from the above that functors of finite projective dimension are homology theories (by induction on dimension).

Consider a homology theory H. There exists a spectrum X such that H = h_{X}, and a cofibre sequence P ⟶ Q ⟶ X ⟶ ΣP as in Proposition 4.6. This gives a projective resolution 0 ⟶ h_{P} ⟶ h_{Q} ⟶ h_{X} = H ⟶ 0, so H has projective dimension at most one.

We can now describe the phantom maps in terms of 𝒜.

Theorem 4.9. The group 𝒫(Σ^{−1}X, Y) of phantom maps is naturally isomorphic to Ext_{𝒜} (h_{X}, h_{Y}).

Proof. Consider the usual projective resolution 0 ⟶ h_{P} ⟶ h_{Q} ⟶ h_{X} ⟶ 0 of h_{X} in 𝒜. The first cohomology group of the left column of

[figure: 0; 0; 𝒜(h_{P} , h_{Y}); [P, Y]; 𝒜(h_{Q}, h_{Y}); [Q, Y]; 0; 0]

is the Ext group in question, and the left column can be identified with the right column since P and Q are 𝒜-projective. But the first cohomology of the right column is 𝒫(Σ^{−1}X, Y) because every phantom Σ^{−1}X ⟶ Y extends to P, and the difference between two such extensions factors through Q.

It is easy to see that the isomorphism is natural in X and Y.

Note 4.10. The above proposition can also be proved using the definition of Ext in terms of equivalence classes of short exact sequences. The isomorphism sends a phantom map f: Σ^{−1}X ⟶ Y to the short exact sequence

0 ⟶ h(Y) ⟶ h(cofibref) ⟶ h(X) ⟶ 0.

The dual picture. Now we prove the dual results, making use of what came above.

Proposition 4.11. For any spectra X and Y, we have 𝒫(X, IY) = 0.

Proof. By Theorem 4.9, 𝒫(Σ^{−1}X, IY) = Ext_{𝒜} (h_{X}, h_{IY}). But h_{IY} is injective, so this is zero.

One can prove this directly from the definition of IY as well.

With this we can now prove our fourth characterisation of phantom maps.

Proposition 4.12. A map X ⟶ Y is phantom if and only if the composite X ⟶ Y ⟶ IW is null for each finite W and each map Y ⟶ IW.

Proof. By the previous proposition, every phantom map is null when composed with a map Y ⟶ IW.

Conversely, suppose that X ⟶ Y is such that (X ⟶ Y ⟶ IW) = 0 for all Y ⟶ IW. Consider the spectrum

Z = ∏_{W ∈ℱ′} ∏_{Y ⟶IW} IW.

The evident map Y ⟶ Z is 𝒜-monic, as in Proposition 3.14. Since

h_{X} ⟶ h_{Y} ⟶ h_{Z}

is null by assumption, the map h_{X} ⟶ h_{Y} must also be null, so X ⟶ Y is phantom.

Lemma 4.13. There is a natural map X ⟶ I^{2}X, which is 𝒜-monic for all X.

Proof. Consider [X, I^{2}X]. By using the definition of I twice, we find [X, I^{2}X] = 𝕀(π_{0}(X ∧ IX)) = [IX, IX], and so there is a natural map X ⟶ I^{2}X corresponding to the identity map in [IX, IX].

We need to show that for W finite, the map [W, X] ⟶ [W, I^{2}X] is monic. We can calculate the latter group and we find that it is 𝕀^{2}[W, X]. The map [W, X] ⟶ [W, I^{2}X] is the natural inclusion of [W, X] into its double dual; since ℚ/ℤ is an injective cogenerator, this is monic.

Proposition 4.14. Any spectrum X sits in a cofibre sequence Σ^{−1}K ⟶ X ⟶ J ⟶ K, where J and K are 𝒜-injective and Σ^{−1}K ⟶ X is phantom. The sequence h_{X} ⟶ h_{J} ⟶ h_{K} is a short exact sequence in 𝒜. The map Σ^{−1}K ⟶ X is weakly terminal among phantom maps into X.

Proof. Let J = I^{2}X and let K be the cofibre of the natural map X ⟶ I^{2}X. Similarly, let L = I^{2}K and form the cofibre sequence K ⟶ L ⟶ M. By Lemma 4.13 the maps Σ^{−1}K ⟶ X and Σ^{−1}M ⟶ K are phantom and so the cofibre sequences X ⟶ J ⟶ K and K ⟶ L ⟶ M become short exact in 𝒜. Thus Ext_{𝒜}^{1} (h_{M}, h_{K}) = Ext_{𝒜}^{2} (h_{M}, h_{X}), which vanishes as h_{M} has projective dimension at most one (Proposition 4.8). Therefore the extension h_{K} ⟶ h_{L} ⟶ h_{M} splits in 𝒜 and hence h_{K} is injective.

We showed in Proposition 4.11 that 𝒫(−, J) = 0, and it follows easily that Σ^{−1}K ⟶ X is weakly terminal.

The following corollary is an easy consequence of the above constructions.

Corollary 4.15. The following are equivalent:

(i) X is 𝒜-injective. (ii) X is a retract of a product of Brown–Comenetz duals of finite spectra. (iii) 𝒫(Y, X) = 0 for each spectrum Y.

For example, this means that the completed Johnson-Wilson spectrum { \atop E(n)} is 𝒜-injective. Indeed, if W is finite then { \atop E(n)}^{∗}W is compact Hausdorff in the I_{n}-adic topology. The inverse limit functor is exact for inverse systems of compact Hausdorff topological groups, and one can deduce from this that there are no phantom maps

to { \atop E(n)}.

Summarising our homological results gives:

Theorem 4.16. Let F ∈ 𝒜. Then the following are equivalent:

(i) F has finite projective dimension. (ii) F has projective dimension at most one. (iii) F is a homology theory. (iv) F is in the image of h.

(v) F has finite injective dimension. (vi) F has injective dimension at most one.

Divisibility. To start with, we recall a result that is well-known to the experts.

Proposition 4.17. If π_{i}Y is a finitely generated Abelian group for each i, then 𝒫(X, Y) is divisible for each X.

Proof. Let Z be the cofibre of the natural map Y ⟶ I^{2}Y. We have seen that the resulting map Σ^{−1}Z ⟶ Y is a weakly terminal phantom map, so that 𝒫(X, Y) is a quotient of [X, Σ^{−1}Z]. It will thus be enough to show that [X, Σ^{−1}Z] is a rational vector space.

The induced map π_{k}(Y) ⟶ π_{k}(I^{2}Y) is just the inclusion of π_{k}(Y) into its double dual with respect to ℚ/ℤ, which is the same as its profinite completion (as π_{k}(Y) is finitely generated). It follows that π_{k}(Z) is a finite direct sum of copies of ℤ̂/ℤ, which is well-known to be a rational vector space. It follows that any nonzero integer n induces an isomorphism π_{∗}(Z) ⟶ π_{∗}(Z), and thus an isomorphism Z ⟶ Z. It follows that [X, Σ^{−1}Z] is a rational vector space, as required.

It is not the case that 𝒫(X, Y) is always divisible, however. Indeed, we have the following result.

Proposition 4.18. Let 𝒮 be the classical stable homotopy category, and Hℤ/p the mod p Eilenberg–Mac Lane spectrum in 𝒮. Then 𝒫(Hℤ/p, Y) is a vector space over ℤ/p, and there exist spectra Y for which it is nonzero (and thus not divisible).

Proof. As p times the identity map of Hℤ/p is zero, we see that [Hℤ/p, Y] is a vector space over ℤ/p, so the same is true of 𝒫(Hℤ/p, Y). Next, recall that [Hℤ/p, W] = 0 for each finite spectra W. Ravenel proves this in [27] by showing that Hℤ/p is E-acyclic (dissonant) and that finite spectra are E-local (harmonic), where E = ⋁_{p,n} K(n). It was also proved earlier by Margolis in [21] and by Lin in [19] using the Adams spectral sequence, and can be found in Margolis’s book [22, Cor. 16.27]. If 𝒫(Hℤ/p, Y) were zero for all Y, then the following proposition would imply that Hℤ/p = 0, a contradiction.

Proposition 4.19. If X is 𝒜-projective and [X, W] = 0 for each W ∈ ℱ then X = 0.

Proof. We can write X as a retract of a wedge of finite spectra:

\xymatrix{⋁_{α} X_{α} \ar[d] \\ X . \ar[u]_{i}}

Consider X ⟶^{i} ⋁ X_{α} ⟶^{j} ∏ X_{α}, where j is the natural map. As X has no maps to finite spectra, the composite ji is zero. But π_{∗}(j): ⨁_{α} π_{∗}X_{α} ⟶ ∏_{α} π_{∗}X_{α} is monic, so we see that π_{∗}(i) = 0. As i is a split monomorphism, we know that π_{∗}(X) is the image of π_{∗}(i). It follows that X = 0.

5. Margolis’s axiomatisation conjecture

The Spanier–Whitehead category ℱ of finite spectra (in the classical, topological sense) can be constructed quite simply. However, all known constructions of the homotopy category 𝒮 of all spectra are rather intricate. Moreover, there are a number of apparently different constructions of this category, all giving the same result up to equivalence. (In this section, equivalences of categories are tacitly required to preserve triangulations and symmetric monoidal structures.) It is thus natural to look for a system of axioms that characterises 𝒮 uniquely in terms of ℱ. Margolis [22] conjectured such a characterisation, which translates into our language as follows: if 𝒮′ is a monogenic Brown category whose subcategory ℱ′ of finite objects is equivalent to ℱ, then 𝒮′ is equivalent to 𝒮. As a first approximation to this conjecture, Margolis showed that 𝒮′/𝒫′ is equivalent to the category ℋ of homology theories on ℱ, or equivalently to 𝒮/𝒫. Of course, Note 4.3 is just a generalisation of this.

We can now come somewhat closer to a proof of Margolis’s conjecture. To explain this, we recall some of the theory of linear extensions of categories. Our treatment is inspired by [1], but is different in detail as we only consider additive categories. Let ℬ be an additive category. A bimodule over ℬ consists of Abelian groups D(A, B) (for every pair of objects A, B in ℬ) together with a trilinear composition operation

ℬ(A, B) ⊗ D(B, C) ⊗ ℬ(C, D) ⟶ D(A, D)

written

f ⊗ u ⊗ g ↦ f^{∗}g_{∗}u = g_{∗}f^{∗}u.

This operation is supposed to have the obvious functoriality properties. As an example, because the composite of two phantom maps is trivial, there is a welldefined composition

𝒮/𝒫(A, B) ⊗ 𝒫(B, C) ⊗ 𝒮/𝒫(C, D) ⟶ 𝒫(A, D).

This makes 𝒫 into a bimodule over 𝒮/𝒫.

If we have an additive functor F: 𝒜 ⟶ ℬ and a bimodule D over ℬ, then we can define a bimodule F^{∗}D over 𝒜 by F^{∗}D(A, B) = D(F A, F B). If F is naturally isomorphic to G then one can check that F^{∗}D and G^{∗}D are isomorphic as bimodules.

A linear extension of ℬ by a bimodule D is a category 𝒞 with the same objects as ℬ, together with short exact sequences

D(A, B) ⟶^{j} 𝒞(A, B) ⟶^{p} ℬ(A, B)

such that p is a functor and j(p(f)^{∗}p(g)_{∗}u) = g ∘ j(u) ∘ f. Two such extensions are considered equivalent if there is a functor ϵ: 𝒞 ⟶ 𝒞′ with p′ϵ = p and ϵj = j′ (strict equalities of functors, not just natural isomorphisms). We write M(ℬ, D) for the collection of equivalence classes of linear extensions of ℬ by D. The main example of interest to us is of course the extension 𝒫 ⟶ 𝒮 ⟶ 𝒮/𝒫.

Suppose again that we have an additive functor F: 𝒜 ⟶ ℬ and a linear extension D ⟶ 𝒞 ⟶ ℬ. Given objects A, B in 𝒜 we define F^{∗}𝒞(A, B) by the pullback diagram

\xymatrix{F^{∗}𝒞(A, B) \ar[d] \ar[r] & 𝒞(F A, F B) \ar[d]^{p} \\ 𝒜(A, B) \ar[r]_{F} & ℬ(F A, F B) .}

One can check that F^{∗}𝒞 becomes a linear extension of 𝒜 by F^{∗}D. Moreover, if G is naturally isomorphic to F then G^{∗}𝒞 is equivalent to F^{∗}𝒞 as a linear extension. Thus, a natural equivalence class of functors 𝒜 ⟶ ℬ induces a map M(ℬ, D) ⟶ M(𝒜, F^{∗}D). It is clear that this is essentially functorial, and thus M(ℬ, D) ≃ M(𝒜, F^{∗}D) if F is an equivalence of categories.

A procedure analogous to the Baer sum of extensions makes M(ℬ, D) into an Abelian group. For any pair of objects A, B in ℬ, the evident map M(ℬ, D) ⟶ Ext(ℬ(A, B), D(A, B)) is a homomorphism. Unfortunately, this is almost all the information that we have about the group M(ℬ, D) in the cases of interest. We do not even know whether M(ℬ, D) is a set or a proper class.

We now return to the context of the Margolis conjecture. We have an equivalence F: 𝒮/𝒫 ≃ 𝒮′/𝒫′. It follows from Theorem 4.9 that there is a canonical equivalence 𝒫 ≃ F^{∗}𝒫′ of bimodules over 𝒮/𝒫. Thus, Margolis’s conjecture is true up to an extension problem. Together with F, the above equivalence induces a canonical isomorphism M(𝒮′/𝒫′, 𝒫′) ≃ M(𝒮/𝒫, 𝒫). We need to know whether the class u(𝒮′) in M(𝒮′/𝒫′, 𝒫′) that classifies the extension 𝒫′ ⟶ 𝒮′ ⟶ 𝒮′/𝒫′ maps to the analogous class u(𝒮) ∈ M(𝒮/𝒫, 𝒫). This would follow from Margolis’s conjecture. Conversely, it would almost imply the conjecture, apart from possible questions about preservation of the triangulation and the monoidal structure.

We shall show in the next section that for each p we can choose spectra A and B such that the image of u(𝒮) in Ext(𝒮/𝒫(A, B), 𝒫(A, B)) is not divisible by p, and is not annihilated by any integer n > 0. It follows that the same is true of u(𝒮) itself. In particular, we will see that u(𝒮) is non-zero. This implies that there is no functorial way to choose a representing spectrum for a homology theory.

6. Phantom cohomology

In this section we restrict attention to the classical stable homotopy category; a more axiomatic approach would yield only a small amount of extra generality. Recall that for each Abelian group A there is an essentially unique spectrum HA with π_{0}HA = A and π_{k}HA = 0 for all k ≠ 0, and that [X, HA] = H^{0}(X; A). These objects are called Eilenberg–Mac Lane spectra. We shall study phantom cohomology classes, in other words, phantom maps from arbitrary spectra to Eilenberg– Mac Lane spectra.

We start with some algebraic preliminaries.

Definition 6.1. A monomorphism B ⟶ C of Abelian groups is said to be pure if for each n > 0 the induced map B/n ⟶ C/n is monic. If we regard B as a subgroup of C, this says that nC = (nB) ∩ C. A short exact sequence B ⟶ C ⟶ A is said to be pure if the map B ⟶ C is.

Let B ⟶ C ⟶ A be a short exact sequence. The six term exact sequence involving Hom(ℤ/n, −) and Ext(ℤ/n, −) reads

0 ⟶ _{n}B ⟶ _{n}C ⟶ _{n}A ⟶ B/n ⟶ C/n ⟶ A/n ⟶ 0,

where we use the notation

nA := {a ∈ A | na = 0}

and the identifications _{n}A = Hom(ℤ/n, A) and A/n = Ext(ℤ/n, A). Thus it is clear that pureness of the short exact sequence is equivalent to the requirement that _{n}A ⟶ B/n be zero, or that _{n}C ⟶ _{n}A be epic.

Now we present an algebraic proposition which summarises results that can be found, for example, in [6].

Proposition 6.2. Consider an element u ∈ Ext(A, B), corresponding to an extension B ⟶ C ⟶ A. The following are equivalent:

(a) The extension is pure.

(b) For each map A′ ⟶ A with A′ finitely generated, the image of u in Ext(A′, B) is zero.

(c) For each n > 0, u ∈ n Ext(A, B). That is, u is in ⋂_{n} n Ext(A, B), the first Ulm subgroup of Ext(A, B).

(d) For each map B ⟶ B′ with B′ finite, the image of u in Ext(A, B′) is zero.

We define the phantom Ext group PExt(A, B) to be the subgroup of Ext(A, B) consisting of all elements u satisfying the above conditions. These are the phantom maps from A to ΣB in D(ℤ), the derived category of the integers. It is easy to see that PExt is a subfunctor of Ext.

Proof. (a)⇒(b): If suffices to prove (b) when A′ = ℤ/n, as any finitely generated group is a sum of cyclic groups, and ℤ is projective. Given a map f : ℤ/n ⟶ A, the class f^{∗}u in Ext(ℤ/n, B) is zero if and only if f factors through C ⟶ A. Now f corresponds to an element of _{n}A, and since we are assuming u is pure, we know that _{n}C ⟶ _{n}A is epic and can therefore factor f through C. Thus f^{∗}u = 0.

(b)⇒(c): We will show that u is in the image of the endomorphism of Ext(A, B) induced by n : A ⟶ A. Consider the inclusion i : _{n}A ⟶ A. By [6, Lemma 17.2], a bounded torsion group is a sum of cyclic groups. Thus i^{∗}u = 0. Now in the diagram

\xymatrix{Ext(A, B) \ar[dr]^{n} \ar@{->>}[d] \\ Ext(nA, B) \ar[r] & Ext(A, B) \ar[r]^{i^{∗}} & Ext(_{n}A, B) ,}

the row is exact and the vertical map is an epimorphism (because Ext^{2} = 0), so u is in the image of multiplication by n.

(c)⇒(d): Let f : B ⟶ B′ be a map with B′ finite. To see that the image of u in Ext(A, B′) is zero, it suffices to check this when B′ is ℤ/n, since a finite group is a product of finite cyclic groups. But n kills Ext(A, ℤ/n), so if u is a multiple of n, then f_{∗}u = 0.

(d)⇒(a): Finally, assume that for any map B ⟶ B′ with B′ finite, the image of u in Ext(A, B′) is zero. Choose an element b ∈ B with b ∉ nB. We will show b ∉ nC. (For notational simplicity we regard B as a subgroup of C.) Let K be a maximal subgroup of B containing nB but not b. The quotient B/K can be shown to be “cocyclic” and so by [6, Section 3] B/K is isomorphic to ℤ/p^{k} for some prime p and some k with 1 ≤ k ≤ ∞. Therefore, by assumption (for finite k) or since ℤ/p^{∞} is divisible, the quotient map B ⟶ B/K extends over B ⟶ C. By the choice of K, the image of b in B/K is non-zero, but the image of nC is zero, so b ∉ nC.

Note 6.3. Clearly, if A is finitely generated, or if there is an integer n such that nA = 0, then PExt(A, B) = 0 for all B.

More generally, if A is a torsion group then A = \underrightarrow{lim}_{n} _{n!}A so there is a short exact sequence

⨁_{n}_{n!}A ⟶ ⨁_{n}_{n!}A ⟶ A

and a resulting short exact sequence

\underleftarrow{lim}^{1} Hom(_{n!}A, B) ⟶ Ext(A, B) ⟶ \underleftarrow{lim} Ext(_{n!}A, B).

Using part (b) of the definition of PExt(A, B), we see that

PExt(A, B) = \underleftarrow{lim} _{n}^{1} Hom(_{n!}A, B).

Our reason for introducing the phantom Ext groups is the following theorem, in which H denotes the integral Eilenberg–Mac Lane spectrum. See the introduction for references to more general results.

Theorem 6.4. For any spectrum X and Abelian group B we have 𝒫(X, HB) = PExt(H_{−1}X, B).

Proof. We begin by describing a map 𝒫(X, HB) ⟶ PExt(H_{−1}X, B). Let u: X ⟶ HB be a phantom map. If Y is the cofibre of u, then we have a short exact sequence 0 ⟶ B ⟶ H_{0}Y ⟶ H_{−1}X ⟶ 0, since H_{0}(HB) = B by the Hurewicz theorem and since H_{∗}(u) = 0 by Proposition 4.1. We claim that this is a phantom extension, and we prove this by showing that for each n the map_{n}(H_{0}Y) ⟶ _{n}(H_{−1}X) is surjective. Let a be an element of H_{−1}X with na = 0. This corresponds to a map S^{−1} ⟶ H∧X which can be extended to give a map a′: S^{−1}/n ⟶ H ∧ X. Since phantoms form an ideal under the smash product, the composite S^{−1}/n ⟶ H ∧ X ⟶ H ∧ HB is null and a′ factors through H ∧ Y. Thus S^{−1} ⟶ S^{−1}/n ⟶ H ∧ Y represents a class in _{n}(H_{0}Y) mapping to a.

Conversely, consider the composite

PExt(H_{−1}X, B) ⟶ Ext(H_{−1}X, B) ⟶ [X, HB] = H^{0}(X; B),

where the first map is the inclusion and the second map comes from the universal coefficient sequence. We claim that a map u in the image of this composite is a phantom map. Indeed, if W is a finite spectrum and W ⟶ X is a map, then by naturality the restriction of u to W lies in the image of PExt(H_{−1}W, B), which is trivial because H_{−1}W is finitely generated. It follows that u is a phantom map.

We leave it to the reader to check that the two maps we have constructed are inverses.

This allows us to calculate all phantom maps between Eilenberg–Mac Lane spectra.

Corollary 6.5. We have

𝒫(Σ^{k}HA, HB) = \begin{cases}PExt(A, B) & if k = −1 \\ 0 & otherwise. \end{cases}

Proof. For j < 0 we have H_{j}HA = 0, and H_{0}HA = A by the Hurewicz theorem. Given this, the claim follows for k ≥ −1 by a simple application of Theorem 6.4. For j > 0 we may have H_{j}HA ≠ 0, but we claim that PExt(H_{j}HA, B) = 0 nonetheless; this will cover the case k < −1. To see this, fix j > 0 and let {A_{α}} be the directed set of finitely generated subgroups of A. The natural map \underrightarrow{lim}_{α}(HA_{α})_{∗}X ⟶ (HA)_{∗}X is an isomorphism for each X, since it is when X is a sphere, and both sides are homology theories. Taking X = H we find that H_{j}HA = \underrightarrow{lim}_{α}H_{j}HA_{α}. By working rationally, we see that H_{j}HA is a torsion group, so it is the direct sum of its localisations at different primes. We claim that H_{j}(HA_{α})_{(p)} is a vector space over ℤ/p. Using the fact that H_{j}HA_{α} = (HA_{α})_{j}H and the fact that the universal coefficient sequence splits, we are reduced to proving that H_{j}H is killed by p. This is a classical calculation; an account appears in [17]. This implies that H_{j}HA is a direct sum of (prime) cyclic groups; it follows easily that PExt(H_{j}HA, B) = 0 as required.

We next study a special case in which the short exact sequence

𝒫(Σ^{−1}HA, HB) ⟶ 𝒮(Σ^{−1}HA, HB) ⟶ 𝒜(Σ^{−1}HA, HB)

can be understood explicitly. We choose a prime p and take

A = ℤ/p^{∞} = ℚ/ℤ_{(p)} = \underrightarrow{lim}_{k}ℤ/p^{k}.

For the moment we consider an arbitrary Abelian group B. As in Note 6.3, we have a short exact sequence

PExt(A, B) ⟶ Ext(A, B) ⟶ \underleftarrow{lim}_{k} Ext(ℤ/p^{k}, B).

Note that Ext(ℤ/p^{k}, B) = B/p^{k}, so the third term is just the p-completion B̂ of B. The middle term is the Ext-p-completion of B, as studied in [2]; we shall denote it by B̃. And it is clear that the first term is

p^{∞}B̃ = ⋂_{k} p^{k}B̃,

since everything is p-local. Using the fact that 𝒮(Σ^{−1}HA, HB) = Ext(A, B), we find that our phantom sequence is just

p^{∞}B̃ ⟶ B̃ ⟶ B̂.

It is tempting to believe that p^{∞}B̃ is a divisible group, but this is never true unless p^{∞}B̃ = 0. Any element of p^{∞}B̃ is divisible by p in B̃ but not necessarily in p^{∞}B̃.

Let B be a free Abelian group, say B = ⨁_{k=0}^{∞} ℤ. Then Hom(ℤ/p^{j}, B) = 0 so p^{∞}B̃ = \underleftarrow{lim} _{j}^{1} Hom(ℤ/p^{j}, B) = 0 so B̃ = B̂. Let v(a) denote the p-adic valuation of a p-adic integer a ∈ ℤ_{p}. It is not hard to see that

Ext(ℤ/p^{∞}, B) = B̃ = B̂ = \{\underline{a} ∈ ∏_{k} ℤ_{p} | v(a_{k}) ⟶ ∞\}.

One can also see directly that Hom(ℤ/p^{∞}, B) = 0.

Now consider the case B = ⨁_{k} ℤ/p^{k}. We then have a short exact sequence

⨁_{k} ℤ ⟶^{f} ⨁_{k} ℤ ⟶ B

where f is multiplication by p^{k} on the k’th factor. One can again see directly that Hom(ℤ/p^{∞}, B) = 0. The six-term exact sequence obtained by applying the functors Hom(ℤ/p^{∞}, −) and Ext(ℤ/p^{∞}, −) to the above presentation of B therefore collapses to a short exact sequence

Ext(ℤ/p^{∞}, ⨁_{k} ℤ) ⟶^{f} Ext(ℤ/p^{∞}, ⨁_{k} ℤ) ⟶ B̃.

It follows using the previous paragraph that

B̃ = \{\underline{a} | v(a_{k}) ⟶ ∞\}/\{\underline{a} | 0 ≤ v(a_{k}) − k ⟶ ∞\}.

One can also see directly that

B̂ = \{\underline{a} | v(a_{k}) ⟶ ∞\}/\{\underline{a} | 0 ≤ v(a_{k}) − k\}.

It follows that p^{∞}B̃ (which is the kernel of the map B̃ ⟶ B̂) is given by

p^{∞}B̃ = \{\underline{a} | 0 ≤ v(a_{k}) − k\}/\{\underline{a} | 0 ≤ v(a_{k}) − k ⟶ ∞\},

and this can also be expressed as

p^{∞}B̃ = ∏_{k} ℤ_{p}/\{\underline{b} ∈ ∏_{k} ℤ_{p} | v(b_{k}) ⟶ ∞\}

(where a_{k} = p^{k}b_{k}). It is easy to see from this that p^{∞}B̃ is nonzero and torsion-free.

We now return to the case of a general Abelian group B. Let w ∈ Ext(B̂, p^{∞}B̃) be the element classifying the canonical sequence p^{∞}B̃ ⟶ B̃ ⟶ B̂, and let

δ: Hom(ℤ/p, B̂) ⟶ Ext(ℤ/p, p^{∞}B̃)

be the obvious connecting homomorphism.

Proposition 6.6. Let B be an Abelian group. The following are equivalent:

(i) p^{∞}B̃ = 0. (ii) The natural map B̃ ⟶ B̂ is an isomorphism. (iii) w = 0. (iv) w is divisible by p.

(v) δ = 0. (vi) δ is divisible by p.

Proof. (i)⇒(ii)⇒(iii) ⇒(iv)⇒(vi): easy.

(vi)⇒(v): This is also clear, as the source and target of δ are killed by p.

(v)⇒(i): The next map in the sequence is Ext(ℤ/p, p^{∞}B̃) ⟶ Ext(ℤ/p, B̃), which can be identified with the natural map (p^{∞}B̃)/p ⟶ B̃/p. But this latter map is clearly zero, so the connecting homomorphism δ is epic. Its image is (p^{∞}B̃)/p, so this group is zero, so p^{∞}B̃ is p-divisible. This means that p^{∞}B̃ = Hom(ℤ, p^{∞}B̃) is a quotient of Hom(ℤ[\frac{1}{p}], p^{∞}B̃), which is a subgroup of Hom(ℤ[\frac{1}{p}], B̃). However, [2, VI.3.4] tells us that Hom(ℤ[\frac{1}{p}], B̃) = 0; it follows that p^{∞}B̃ = 0.

If B = ⨁_{k} ℤ/p^{k}, then we have p^{∞}B̃ ≠ 0 and thus w is not divisible by p. Our next result will show that w has infinite order.

Proposition 6.7. Let B be an Abelian group such that p^{k}w = 0. Then p^{k}p^{∞}B̃ = 0.

Proof. Let i: p^{∞}B̃ ⟶ B̃ and q: B̃ ⟶ B̂ be the usual maps. Let C be the pullback of B̃ along the map p^{k}: B̂ ⟶ B̂, so C = {(a, b) ∈ B̃ ×B̂ | q(a) = p^{k}b}. The hypothesis p^{k}w = 0 means that the evident sequence p^{∞}B̃ ⟶ C ⟶ B̂ is split; the splitting map B̂ ⟶ C necessarily has the form c ↦ (f(c), c), where qf = p^{k}: B̂ ⟶ B̂. The functor A ↦ p^{∞}A preserves split exact sequences, and p^{∞}B̂ = 0 (directly from the definitions) so p^{∞}C = p^{∞}p^{∞}B̃. On the other hand, suppose that b ∈ p^{∞}B̃, say b = p^{i}b_{i} for each i, with b_{i} ∈ B̃. Then (p^{k}b_{i}, q(b_{i})) ∈ C and p^{i}(p^{k}b_{i}, q(b_{i})) = (p^{k}b, 0). It follows that p^{k}p^{∞}B̃ ≤ p^{∞}C = p^{∞}p^{∞}B̃. This means that p^{k}p^{∞}B̃ is a divisible subgroup of B̃; as in the proof of Proposition 6.6, we conclude that p^{k}p^{∞}B̃ = 0.

Now take B = ⨁_{k} ℤ/p^{k} again. We saw previously that p^{∞}B̃ is non-trivial and torsion-free. It follows easily that p^{k}w ≠ 0 for all k.

We can now prove a result stated in Section 5. Recall that we defined there a group M(𝒮/𝒫, 𝒫) and an element u ∈ M(𝒮/𝒫, 𝒫) that classifies the linear extension of categories 𝒫 ⟶ 𝒮 ⟶ 𝒮/𝒫. The image of u under a certain homomorphism M(𝒮/𝒫, 𝒫) ⟶ Ext(B̂, p^{∞}B̃) is w. It follows that u is not divisible by p for any prime, so u is not divisible by any integer n > 1. If u were annihilated by any m > 0 then the image of u in any p-local group (such as Ext(B̂, p^{∞}B̃)) would be annihilated by some power of p. Thus, we conclude that u does not have finite order.

7. Universal homology theories

Although one is mostly interested in homology theories with values in the category of Abelian groups, one can also consider more general Abelian categories. In this section, we recall a construction of Freyd [5] which gives a universal example of an Abelian category ℬ equipped with a homology theory 𝒮 ⟶ ℬ. We also show that the functor h: 𝒮 ⟶ 𝒜 is the universal example of a homology theory with values in an Abelian category satisfying Grothendieck’s axiom AB 5.

At one point we need a fact that holds in all monogenic Brown categories that we care about, but which we have not been able to deduce from the axioms (although we suspect that it does follow). For simplicity, we therefore restrict attention to the classical stable homotopy category.

Let ℬ be the following category. The objects of ℬ are just the morphisms of 𝒮. Given a map u: W ⟶ X in 𝒮, we shall write I(u) for u thought of as an object of ℬ. The group ℬ(I(u), I(v)) is the quotient of the group of commutative squares

\xymatrix{W \ar[d]_{u} \ar[r]^{f} & Y \ar[d]^{v} \\ X \ar[r]_{g} & Z}

by the subgroup of squares for which the map vf = gu vanishes. This gives a category ℬ in an obvious way. There is a full and faithful embedding J: 𝒮 ⟶ ℬ sending X to I(1_{X}). Freyd shows that ℬ is an Abelian category and that J is a homology theory. Given a morphism u: W ⟶ X in 𝒮, the image of the morphism Ju: J(W) ⟶ J(X) is just I(u). Moreover, the image of J is the subcategory of injective objects in ℬ, which is the same as the subcategory of projective objects. Freyd also shows that for any Abelian category 𝒞 and any homology theory K: 𝒮 ⟶ 𝒞, there is an essentially unique strongly additive exact functor K′: ℬ ⟶ 𝒞 such that K′J ≃ K. (We say that a functor is strongly additive if it preserves all coproducts. Freyd actually proves the corresponding result without strong additivity but the necessary modifications are trivial.) In fact, K′I(u) is just the image of the morphism Ku in 𝒞. In particular, this construction gives a functor ℬ ⟶ 𝒜.

The following result is analogous to Theorem 4.16.

Proposition 7.1. In ℬ, any object of finite projective or injective dimension is both projective and injective, and thus lies in the image of J.

Proof. Suppose that X has projective dimension at most n > 0. There is then a short exact sequence Y ⟶ P ⟶ X, where Y has projective dimension at most n−1; by induction, we may assume that Y is projective. As projectives are injective, the sequence splits, so X is a retract of P and thus is projective.

While this seems a pleasant construction, the finiteness properties of the category ℬ are poor. We believe that every nonzero object has a proper class of subobjects, for example.

Next, recall that an Abelian category is said to satisfy AB 5 if set-indexed colimits exist and filtered colimits are exact [11]. The category of Abelian groups satisfies AB 5, as does the functor category 𝒜.

Proposition 7.2. If 𝒞 is an Abelian category satisfying AB 5 and K: 𝒮 ⟶ 𝒞 is a homology theory, then KX = \underrightarrow{lim}_{Λ(X)}KX_{α}. Thus, Kf is zero for any phantom map f.

Proof. Define K̂X = \underrightarrow{lim}_{Λ(X)}KX_{α}. Here we will need to use the fact (mentioned after Corollary 3.2) that Λ(X) is the diagram of all pairs (U, u), where U lies in some small skeleton of ℱ and u: U ⟶ X. Using this we see that K̂ is an additive functor 𝒮 ⟶ 𝒞, and that there is an evident natural map K̂ ⟶ K (compare [15, Proposition 2.3.9]). If X is finite then Λ(X) has a terminal object, so that K̂X = KX. If we can show that K̂ preserves coproducts and sends cofibre sequences to exact sequences, then the usual argument will show that K̂X = KX for all X. Consider a cofibre sequence X ⟶ Y ⟶ Z. We may assume that X is a CW subspectrum of Y, and that Z is the quotient. Let {Y_{α} | α ∈ I} be the directed set of finite subspectra of Y. Write X_{α} = Y_{α} ∩ X and Z_{α} = Y_{α}/X_{α}, so we have a cofibre sequence X_{α} ⟶ Y_{α} ⟶ Z_{α} for each α. It is easy to see that the evident functors from I to Λ(X), Λ(Y) and Λ(Z) are cofinal, so that K̂X = \underrightarrow{lim}_{I}KX_{α} and so on. As direct limits are exact, we conclude that the sequence K̂X ⟶ K̂Y ⟶ K̂Z is exact as required.

We next verify that K̂ preserves coproducts. Consider a family of spectra {X_{i} | i ∈ I}. Let Λ be the full subcategory of ∏_{I} Λ(X_{i}) consisting of those objects (Z_{i})_{i∈I} such that Z_{i} = 0 for almost all i. It is not hard to see that this is a filtered category, and that the projections Λ ⟶ Λ(X_{i}) are cofinal functors. The functor from Λ to Λ(⋁_{i} X_{i}) is also cofinal. By writing K̂(X_{i}) and K̂(⋁_{i} X_{i}) as colimits indexed by Λ, we see that K̂(⋁_{i} X_{i}) = ⨁_{i} K̂(X_{i}) as required.

In a more general monogenic Brown category, it is more difficult to prove that K̂ is an exact functor. An obvious approach is to replace the sequences X_{α} ⟶ Y_{α} ⟶ Z_{α} considered above by the category of all cofibre sequences of finite objects equipped with a map to the sequence X ⟶ Y ⟶ Z. However, it is not clear that this is a filtered category. The difficulty is related to the existence of maps of cofibre sequences that are not good in the sense of Neeman [23].

We can deduce from the above that h: 𝒮 ⟶ 𝒜 is the universal example of a homology theory with values in an AB 5 category.

Proposition 7.3. Let 𝒞 be an AB 5 category, and K: 𝒮 ⟶ 𝒞 a homology theory. Then there is an essentially unique strongly additive exact functor K′: 𝒜 ⟶ 𝒞 such that K′ ∘ h ≃ K.

Proof. First, let ℋ be the category of homology theories, so ℋ ⊂ 𝒜 and ℋ ≃ 𝒮/𝒫. By Proposition 7.2, we know that K kills phantom maps, so it factors in an essentially unique way through h: 𝒮 ⟶ ℋ. We write K again for the resulting functor ℋ ⟶ 𝒞. As the cofibre of an 𝒜-epimorphism is phantom, we see that K sends epimorphisms of homology theories to epimorphisms, and similarly for monomorphisms.

Consider an object F ∈ 𝒜. We know that 𝒜 has enough projectives and injectives, so we can choose maps P ⟶^{f} F ⟶^{g} I where f is epic, g is monic, P is projective and I is injective. In particular, P and I are homology theories, so K(P) and K(I) are defined. We would like to define K′(F) to be the image of the map K(gf): K(P) ⟶ K(I); we need only check that this is well-defined. Indeed, if we chose a different epimorphism f′: P′ ⟶ F then we could use the projectivity of P and P′ to show that f and f′ factor through each other; it follows easily that K(gf) and K(gf′) have the same image, regarded as a subobject of K(I). A similar argument shows that our definition is essentially independent of g.

Next, consider a morphism v: F ⟶ G in 𝒜. Choose sequences P ⟶ F ⟶ I and Q ⟶ G ⟶ J as above. Using the projectivity of P and the injectivity of J, we can choose maps u: P ⟶ Q and w: I ⟶ J compatible with v. These induce a map K(F) ⟶ K(G), which we would like to call K′(v). We must check that this does not depend on the choice of u and w. An easy argument reduces us to the case v = 0; this implies that the diagonal map in the square

\xymatrix{P \ar[d] \ar[r]^{u} & Q \ar[d] \\ I \ar[r]_{w} & J}

is zero, and thus the induced map image(K(P ⟶ I)) ⟶ image(K(Q ⟶ J)) is zero as required. Our definition of K′(v) is thus unambiguous, and it is easy to see that it gives a functor.

Suppose that F is a homology theory. Then P ⟶ F is an epimorphism of homology theories, so K(P) ⟶ K(F) is epic. Similarly, K(F) ⟶ K(I) is monic. It follows directly that K′(F) = K(F). Thus, K′ is an extension of K.

If v: F ⟶ G is a monomorphism then we may choose I = J and w = 1; this makes it clear that K′(v) is a monomorphism. Similarly, K′ preserves epimorphisms.

We next show that K′ preserves kernels. Consider a map v: F ⟶ G. Choose an epimorphism f: P ⟶ F and a monomorphism g: G ⟶ J. As P and J are homology theories, we can choose a map of spectra inducing the map gvf: P ⟶ J, and let j: H ⟶ P be its fibre. As H ⟶ P ⟶ J is zero, we see that H ⟶ P ⟶ F factors through ker(gv) = ker(v). As K′ preserves monomorphisms and epimorphisms, we obtain a diagram as follows:

[figure: K′(H); K′j; K′(P); K′f; K′(ker(v)); K′(F); K′v; K′(G); K′g; K′(J) .]

As H ⟶ P ⟶ J comes from a cofibre sequence of spectra, we know that K′(H) ⟶ K′(P) ⟶ K′(J) is exact. A diagram chase (using elements in the sense of [20], for example) now shows that K′(ker(v)) ⟶ K′(F) ⟶ K′(G) is exact as required.

Similarly, we see that K′ preserves cokernels; it is thus an exact functor.

Finally, we need to show that K′ preserves coproducts. Consider a family {F_{i}} of objects of 𝒜, and choose maps P_{i} ⟶ F_{i} ⟶ I_{i} in the usual way. Write P = ⨁_{i} P_{i} and F = ⨁_{i} F_{i} and I = ⨁_{i} I_{i}, so we have an epimorphism P ⟶ F and a monomorphism F ⟶ I (but I need not be injective). As K′ is exact, we see that K′(F) is the image of K′(P) ⟶ K′(I). As K preserves coproducts of spectra, we see that K′ preserves coproducts of homology theories, so K′(P) = ⨁_{i} K′(P_{i}). Similarly, K′(I) = ⨁_{i} K′(I_{i}). It follows that K′(F) = ⨁_{i} K′(F_{i}) as required.

It is also clear that any extension of K that preserves images (in particular, any exact extension of K) must be equivalent to K′.

We conclude this section with an interesting, if somewhat disconnected result. Consider an essentially small additive category ℱ. Recall that there is an essentially unique category Ind(ℱ) (the Ind completion of ℱ) equipped with a full and faithful embedding ℱ ⟶ Ind(ℱ) (thought of as an inclusion) such that

(i) Ind(ℱ) has colimits for all small filtered diagrams. (ii) Every object of Ind(ℱ) is the colimit of a small filtered diagram of objects of

ℱ. (iii) If X is an object of ℱ then the functor Ind(ℱ)(X, −) preserves filtered colimits. The Ind completion of a category was introduced in [12] and was described in detail in [13].

This category can be constructed in (at least) two ways. The first way is to consider pairs (I, X) where I is a small filtered category and X is a functor I ⟶ ℱ. We define Ind(ℱ) to be the category of such pairs, with morphisms

Ind(ℱ)((I, X), (J, Y)) = \underleftarrow{lim}_{I} \underrightarrow{lim}_{J}ℱ(X_{i}, Y_{j}).

Alternatively, we can embed ℱ in the category ℬ of additive functors ℱ^{op} ⟶ 𝒜b by X ↦ [−, X]. We then define Ind(ℱ) to be the subcategory of all functors F ∈ ℬ that can be written as a filtered colimit of a small diagram of objects of ℱ. It is equivalent to require that the category of pairs (X, a) (where X ∈ ℱ and a ∈ F X) is filtered.

Theorem 7.4. Let ℱ be the category of finite spectra. Then there is an equivalence of categories Ind(ℱ) = ℋ (where ℋ is the category of homology theories on ℱ).

Proof. We use the second description of Ind(ℱ), as a subcategory of ℬ = [ℱ^{op}, 𝒜b]. Composition with the Spanier-Whitehead duality functor gives an equivalence of ℬ with 𝒜 = [ℱ, 𝒜b], which sends [−, X] to h_{X}. Thus, Ind(ℱ) is equivalent to the category of those functors ℱ ⟶ 𝒜b that can be written as small filtered colimits of functors of the form h_{X} where X is small. As filtered colimits are exact, every such functor is a homology theory. Conversely, every homology theory is of the form h_{Y} for some Y. Since h_{Y} = \underrightarrow{lim}_{Λ(Y)}h_{Y_{α}}, it follows that every homology theory lies in Ind(ℱ).

We conclude that the Ind completion of a triangulated category need not be a triangulated category. For example, consider a monogenic Brown category with a non-zero phantom map f: X ⟶ Y. If Z is the cofibre of f, then the map h_{Y} ⟶ h_{Z} is monic but not split. However in a triangulated category all monics split, so ℋ is not triangulated. (Hartshorne mentions in [14] that the Ind completion may not be triangulated, but he does not indicate a proof.) We also gain some insight into the Pro completion of the category of spectra, which has been used by various people for various purposes, mainly concentrating on towers of spectra rather than more general inverse systems. The Pro completion of a category 𝒞 is just Ind(𝒞^{op})^{op}. As ℱ ≃ ℱ^{op} (by Spanier-Whitehead duality) we see that the Pro category of finite spectra is equivalent to the opposite of the category of homology theories. The subcategory consisting of towers of finite spectra is equivalent to the opposite of the category of homology theories with countable coefficients.

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Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218

   E-mail address: jdc@math.jhu.edu

Trinity College, Cambridge CB2 1TQ, England

   E-mail address: neil@pmms.cam.ac.uk