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MORAVA K-THEORIES AND LOCALISATION

MARK HOVEY AND NEIL P. STRICKLAND

Abstract. We study the structure of the categories of K(n)-local and E(n)- local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)-nilpotent spectra. We give a number of useful extensions to the theory of v_{n} self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)-local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the Brown-Comenetz dual of M_{n}S lies in the Picard group. We give a detailed analysis of some examples when n = 1 or 2, and a list of open problems.

Introduction

The stable homotopy category 𝒮 is extraordinarily complicated. However, there is a set of approximations to it that are much simpler and closer to algebra. The stable homotopy category is somewhat analogous to the derived category of a ring R, except that R is replaced by the stable sphere S^{0}. In practice, we always consider the p-local stable homotopy category and the p-local sphere for some prime p, without changing the notation. It is very common to study R-modules using the fields over R, and in recent years there has been much work on studying the stable homotopy category via its fields. These fields are referred to as Morava K-theories, denoted by K(n), and were introduced by Morava in the early 1970’s. See [Mor85] for a description of Morava’s earlier work.

Associated to the Morava K-theories are various (homotopy) categories of local spectra that are the approximations to the stable homotopy category mentioned above. There is the category ℒ of spectra local with respect to K(0) ∨ ⋯ ∨ K(n) and the category 𝒦 of spectra local with respect to K(n). (We will always have a fixed n with 0 < n < ∞ in mind in this paper). These categories are themselves stable homotopy categories, in the sense of [HPS95]. The purpose of this paper is to study the structure of these categories. We will show that the category 𝒦 is in a certain sense irreducible; it has no nontrivial further localisations. On the other

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page

Received by the editors April 20, 1997.

1991 Mathematics Subject Classification. 55P42, 55P60, 55N22,55T15.

Key words and phrases. Morava K-theory, stable homotopy category, Bousfield localisation, Picard group, phantom maps, Landweber exact homology theories, Adams spectral sequence, spectrum, Brown-Comenetz duality, thick subcategory.

The first author was partially supported by an NSF Postdoctoral Fellowship.

The second author was partially supported by an NSF Grant.

hand, there are a number of results (such as the Chromatic Convergence Theorem of Hopkins and Ravenel [Rav92a], or the proof by the same authors that suspension spectra are harmonic [HR92]) which indicate that an understanding of 𝒦 for all n and p will give complete information about 𝒮. Hopkins’ Chromatic Splitting Conjecture [Hov95a], if true, would be a still stronger result in this direction.

The first half of the paper is mostly concerned with issues we need to resolve before beginning our study of 𝒦. In Section 1, we define the basic objects of study: the ring spectra E(n), E = { \atop E(n)} and K = K(n), the categories ℒ and 𝒦 and so on. In Section 2, we study E-cohomology. We prove that E^{∗}X is complete, using a slightly modified notion of completeness which turns out to be more appropriate than the traditional one. We show that a well-known quotient of the ring of operations in E-cohomology is a non-commutative Noetherian local ring. We also show that there are no even degree phantom maps between evenly graded Landweber exact spectra such as E, a result that has been speculated about for a long time. In Section 3, we prove some basic results about the (very simple) category of K-injective spectra. We then turn in Section 4 to the study of generalised Moore spectra, and prove a number of convenient and enlightening extensions to the theory developed by Hopkins and Smith [HS, Rav92a] and Devinatz [Dev92]. In Section 5, we assemble some (mostly well-known) results about the Bousfield classes of a number of spectra that we will need to study. In Section 6, we study the E(n)-local category ℒ. We prove some results about nilpotence, and we classify the thick subcategories of small objects, the localising subcategories and the colocalising subcategories.

MORAVA K-THEORIES AND LOCALISATION 3

In the second half of the paper, we concentrate on 𝒦. In Section 7, we prove our most basic results about the K(n)-local category 𝒦. In particular, we prove that it is irreducible, in the sense that the only localising or colocalising subcategories are {0} and 𝒦 itself. We also study the localisation functor L̂ = L_{K} and some related functors, describing them in terms of towers of generalised Moore spectra. In Section 8 we study two different notions of finiteness in 𝒦, called smallness and dualisability. The (local) sphere is dualisable but not small; some rather unexpected spectra are dualisable, such as the localisation of BG for a finite group G. We prove a number of different characterisations of smallness and dualisability; in particular, we give convenient tests in terms of computable cohomology theories. We also show that dualisable spectra lie in the thick subcategory generated by E, and that 𝒦small spectra lie in the thick subcategory generated by K. In Section 9 we study homology and cohomology theories on 𝒦, and prove that both are representable in a suitable sense. In Section 10 we study a version of Brown-Comenetz duality appropriate to the K(n)-local setting, and we prove that the Brown-Comenetz dual of the monochromatic sphere is an element of the Picard group. This result was stated in [HG94]. In Section 11, we introduce a natural topology on the groups [X, Y] for X and Y in 𝒦, and prove a number of properties. In Section 12, we return to the study of the category 𝒟 of dualisable spectra. We prove a nilpotence theorem and an analogue of the Krull-Schmidt theorem, saying that every dualisable spectrum can be written as a wedge of indecomposables in an essentially unique way. We make some remarks about ideals in 𝒟, but we have not been able to prove the obvious conjectures about them. In Section 13 we study K-nilpotent spectra, proving a number of interesting characterisations of them. In Section 14 we study the problem of grading homotopy groups over the Picard group of invertible spectra, rather than just over the integers. We have a satisfactory theory for homotopy groups of 𝒦-small spectra, but the general case seems less pleasant. We show that the Picard group is profinite in a precise sense, but we do not know if it is finitely generated over the p-adics. Section 15 is devoted to the study of the simplest examples. Even when n = 1 and p is odd, there are simple counterexamples to plausible conjectures. We also consider the case n = 2 and p > 3, showing that the Picard graded homotopy groups of the Moore spectrum are mostly infinite. We conclude the main body of the paper with Section 16, which contains a list of interesting questions that we have been unable to answer, some of them old and others new.

We also have two appendices: the first addresses the sense in which the Ecohomology of a K-local spectrum is complete, as mentioned above, and the second shows that some other interesting localisations of the category of spectra have a rather different behaviour, in that they contain no nonzero small objects at all.

We have chosen to rely on a minimum of algebraic prerequisites; in particular, we say almost nothing about formal groups or the Morava stabiliser groups. We have preferred to use thick subcategory arguments rather than spectral sequences where possible. We have chosen our methods very carefully to avoid having to say anything special when p = 2. For this reason we have generally used E rather than K, as E is commutative at all primes (for example).

Our debt to Mike Hopkins will be very obvious to anyone familiar with the subject. We have been heavily influenced by his point of view and a large number of our results were previously known to him. We also thank Matthew Ando, Dan Christensen, Chun-Nip Lee, John Palmieri and Hal Sadofsky for helpful conversations about the subject matter of this paper.

Contents

Introduction 1 1. Basic definitions 4 2. E theory 7 3. K-injective spectra 19 4. Generalised Moore spectra 21 5. Bousfield classes 32 6. The E(n)-local category 33 7. General properties of the K(n)-local category 41 8. Smallness and duality 45 9. Homology and cohomology functors 50 10. Brown-Comenetz duality 52 11. The natural topology 55 12. Dualisable spectra 57 13. K-nilpotent spectra 64 14. Grading over the Picard group 67 15. Examples 74 16. Questions and conjectures 81 Appendix A. Completion 82 Appendix B. Small objects in other categories 90 References 95 Index 97

1. Basic definitions

Fix a prime p and an integer n > 0. We shall localise all spectra at p; in particular, we shall write MU for what would normally be called MU_{(p)}. We write 𝒮 for the category of p-local spectra.

1.1. The spectra E(n), Ê(n) and K(n). We next want to define the spectra E = { \atop E(n)} and K = K(n). It is traditional to do this using the Landweber exact functor theorem and Baas-Sullivan theory. Here we will use the more recent techniques of [EKMM96] instead of the Baas-Sullivan construction.

It is well-known that the integral version of MU has a natural structure as an E_{∞} ring spectrum or (essentially equivalently) a commutative S-algebra in the sense of [EKMM96]. It follows from [EKMM96, Theorem VIII.2.2] that the same applies to our p-local version. We can thus use the framework of [EKMM96, Chapters II-III] to define a topological closed model category ℳ_{MU} of MU-modules. The associated homotopy category (obtained by inverting weak equivalences) is called the derived category of MU-modules and written 𝒟_{MU}. It is a stable homotopy category in the sense of [HPS95]. There is a “forgetful” functor 𝒟_{MU} ⟶ 𝒮 and a left adjoint MU ∧ (−): 𝒮 ⟶ 𝒟_{MU}.

Let w_{k} ∈ π_{2(p^{k}−1)}MU be the coefficient of x^{p^{k}} in the p-series of the universal formal group law over MU_{∗} (so w_{0} = p) and write I_{n} = (w_{0}, …, w_{n−1}). We can construct an object w_{n}^{−1}MU/I_{n} of 𝒟_{MU} by the methods of [EKMM96, Chapter V] (see also [Str96]). Using [EKMM96, Theorem VIII.2.2] again, we can Bousfield-localise MU with respect to w_{n}^{−1}MU/I_{n} to get a strictly commutative MU-algebra which we call M̂U. As explained in [GM95a], the homotopy ring of M̂U is (w_{n}^{−1}MU_{∗})_{I_{n}}^{∧}.

Next, consider the graded ring

E(n)_{∗} = Z_{(p)}[v_{1}, …, v_{n}][v_{n}^{−1}] |v_{k}| = 2(p^{k} − 1).

There is a unique p-typical formal group law over this ring with the property that

[p]_{F} (x) = exp_{F} (px) +_{F} ∑_{0<k≤n}^{F} v_{k}x^{p^{k}}.

(Thus we take v_{k} to be a Hazewinkel generator rather than an Araki generator.) This gives a map MU_{∗} ⟶ E(n)_{∗}, and one can check that this makes Z_{(p)}[v_{1}, …, v_{n}] into the quotient of MU_{∗} by a regular sequence (see [Str96, Proposition 8.15] for details). Moreover, the image of w_{k} is v_{k} modulo I_{k} = (v_{0}, …, v_{k−1}). It follows from the results of [EKMM96, Chapter V] that there is an MU-module E(n) ∈ 𝒟_{MU} with a given map MU ⟶ E(n) inducing an isomorphism π_{∗}E(n) = E(n)_{∗} of MU_{∗}- modules. It is shown in [Str96] that this is unique up to non-canonical isomorphism under MU, and that it admits a non-canonical associative product. The resulting E(n)_{∗}-module structure on E(n)_{∗}X (for any spectrum X) is nonetheless canonical, because it is derived from the MU-module structure of the spectrum E(n) ∧ X ∈ 𝒟_{MU}. When p > 2 there is a unique commutative product on E(n) ∈ 𝒟_{MU}, but we avoid using this here so that we can handle all primes uniformly.

We now define

E = { \atop E(n)} = E(n) ∧_{MU} M̂U.

This is clearly a module over M̂U with a given map M̂U ⟶ E, and one can check that this gives an isomorphism

E_{∗} = (E(n)_{∗})_{I_{n}}^{∧} = Z_{p}[v_{1}, v_{2}, …, v_{n−1}, v_{n}^{±1}]_{I_{n}}^{∧}.

It is again well-defined up to non-canonical isomorphism under M̂U, and it admits a non-canonical associative ring structure. If p > 2 then there is a unique commutative product on E as an object of 𝒟M̂U.

Because E(n) is an MU-module under MU, there is a canonical map

E(n)_{∗} ⊗_{MU_{∗}} MU_{∗}X ⟶ E(n)_{∗}X.

This is of course an isomorphism, by the Landweber Exact Functor Theorem [Lan76]. Similarly, we have an isomorphism

E_{∗} ⊗_{MU_{∗}} MU_{∗}X ⟶ E_{∗}X.

It follows that the homology theory represented by E(n) is independent of the choice of object E(n) ∈ 𝒟_{MU} up to canonical isomorphism, and thus the underlying spectrum of E(n) is well-defined up to an isomorphism that is canonical mod phantoms (see [HPS95, Section 4] for a discussion of phantoms and representability). We shall show later that the relevant group of phantoms is zero, so as a spectrum E(n) is well-defined up to canonical isomorphism. We shall also show that there is a canonical commutative ring structure on this underlying spectrum. Similar remarks apply to E.

We can also define MU-modules MU/I_{k} ∈ 𝒟_{MU} for 0 ≤ k ≤ n in the evident way, and then define

E(n)/I_{k} = MU/I_{k} ∧_{MU} E(n)
E/I_{k} = MU/I_{k} ∧_{MU} E
K = K(n) = E(n)/I_{n} = E/I_{n}.

It is clear that π_{∗}(E(n)/I_{k}) = E(n)_{∗}/I_{k} and so on. In particular we have K(n)_{∗} = E(n)_{∗}/I_{n} = E_{∗}/I_{n} = F_{p}[v_{n}^{±1}]. These MU-modules admit (non-unique) associative products, so (E(n)/I_{k})_{∗}X is canonically a module over E(n)_{∗}/I_{k}. Similar remarks apply to E/I_{k}.

There are evident cofibrations

Σ^{2(p^{k}−1)}E/I_{k} ⟶^{v_{k}} E/I_{k} ⟶ E/I_{k+1},

and similarly for E(n)/I_{k}.

We also know from [Bak91] that there is an essentially unique A_{∞} structure on the spectrum E. It is widely believed that this can be improved to an E_{∞} structure, and that the maps MU ⟶ E characterised by Ando [And95] (which do not include the map MU ⟶ E considered above) can be improved to E_{∞} maps. Unfortunately, proofs of these things have not yet appeared. Nonetheless, just using the A_{∞} structure we can still use the methods of [EKMM96] to define a derived category 𝒟_{E} of left E-modules. This is a complete and cocomplete triangulated category, with a smash product functor ∧: 𝒮 × 𝒟_{E} ⟶ 𝒟_{E}.

1.2. Categories of localised spectra. We use the following notation.

Notation 1.1.

1. 𝒮 is the (homotopy) category of p-local spectra. We write S for S^{0}.

2. ℒ = ℒ_{n} is the category of E(n)-local spectra, and L = L_{n} : 𝒮 ⟶ ℒ is the localisation functor. The corresponding acyclisation functor is written C, so there is a natural cofibre sequence CX ⟶ X ⟶ LX.

3. ℳ = ℳ_{n} is the monochromatic category. This is defined to be the image of the functor M_{n} = C_{n−1}L_{n}: 𝒮 ⟶ 𝒮. Note that there is a natural fibration M_{n}X ⟶ L_{n}X ⟶ L_{n−1}X = L_{n−1}L_{n}X.

4. 𝒦 = 𝒦_{n} is the category of K-local spectra, and L̂ = L̂_{n} : 𝒮 ⟶ 𝒦 is the localisation functor. The corresponding acyclisation functor is written Ĉ.

5. F (m) denotes a finite spectrum of type m, and T (m) = v_{m}^{−1}F (m) is its telescope. Recall from [Rav92a, Chapter V] or [HS] that any two F (m)’s generate the same thick subcategory and have the same Bousfield class, so it usually does not matter which one we use. Note also that the Spanier-Whitehead dual of an F (m) is again an F (m).

There are topological closed model categories whose homotopy categories are 𝒮, ℒ and 𝒦 [EKMM96, Chapter VIII].

1.3. Stable homotopy categories. In this section we collect some basic definitions from the theory of stable homotopy categories developed in [HPS95]. The reader will be familiar with most of these: we assemble them here as a convenient reference. We will not recall the definition of a stable homotopy category, except to say that a stable homotopy category is a triangulated category with a closed symmetric monoidal structure which is compatible with the triangulation and which has a set of generators in an appropriate sense. The symmetric monoidal structure is written X ∧ Y and the closed structure is written F (X, Y). The unit for the smash product is written S.

Definition 1.2. A full subcategory 𝒟 of any triangulated category is thick if it is closed under retracts, cofibres, and suspensions. That is, 𝒟 is thick if both of the following conditions hold.

(a) If X ∨ Y ∈ 𝒟, then both X and Y are in 𝒟; and

(b) If

X ⟶ Y ⟶ Z ⟶ ΣX

is a cofibre sequence and two of X, Y, and Z are in 𝒟, then so is the third.

Certain kinds of thick subcategories come up frequently.

Definition 1.3. Let 𝒞 be a thick subcategory of a stable homotopy category 𝒟.

(a) 𝒞 is a localising subcategory if it is closed under arbitrary coproducts.

(b) 𝒞 is a colocalising subcategory if it is closed under arbitrary products.

(c) 𝒞 is an ideal if, whenever X ∈ 𝒟 and Y ∈ 𝒞 we have X ∧ Y ∈ 𝒞.

(d) 𝒞 is a coideal if, whenever X ∈ 𝒟 and Y ∈ 𝒞 we have F (X, Y) ∈ 𝒞.

Whenever the localising subcategory generated by S is all of 𝒟, every (co)localising subcategory is a (co)ideal [HPS95, Lemma 1.4.6]. This is true in 𝒮, ℒ, and 𝒦 (and any other localisation of 𝒮).

The ideal generated by a ring object is particularly important.

Definition 1.4. Let 𝒞 be a stable homotopy category, and R a ring object in 𝒞. We say that an object X ∈ 𝒞 is R-nilpotent if it lies in the ideal generated by R.

We now recall some different notions of finiteness in a stable homotopy category. We have replaced “strongly dualisable” by “dualisable” for brevity. Also recall that DZ = F (Z, S) is the usual duality functor.

Definition 1.5. Let 𝒞 be a stable homotopy category, and Z an object of 𝒞. We say that Z is

(a) small if for any collection of objects {X_{i}}, the natural map ⨁[Z, X_{i}] ⟶ [Z, ∐ X_{i}] is an isomorphism.

(b) F -small if for any collection of objects {X_{i}}, the natural map ∐ F (Z, X_{i}) ⟶ F (Z, ∐ X_{i}) is an isomorphism.

(c) 𝒜-finite (for any family 𝒜 of objects of 𝒞) if Z lies in the thick subcategory generated by 𝒜.

(d) dualisable if for any X, the natural map DZ ∧X ⟶ F (Z, X) is an equivalence.

A triangulated category with a compatible closed symmetric monoidal structure is an algebraic stable homotopy category if there is a set 𝒢 of small objects such that the localising subcategory generated by 𝒢 is the whole category. An algebraic stable homotopy category is called monogenic if we can take 𝒢 = {S}.

Finally, we recall that limits and colimits generally do not exist in triangulated categories, but sometimes suitable weak versions do exist. In particular, given a sequence X_{0} ⟶ X_{1} ⟶ …, we can form the sequential colimit as the cofibre of the usual self-map of ⋁ X_{i}. We denote this by holim_{-→} X_{i} , as it is the homotopy colimit of a suitable lift of the sequence to a model category. Similarly, we denote the sequential limit of a sequence … ⟶ X_{1} ⟶ X_{0} by holim_{←-} X_{i}.

In case we have a more complicated functor F : ℐ ⟶ 𝒞 to a stable homotopy category, we say that a weak colimit X of F is a minimal weak colimit if the induced map lim_{-→} H ∘ F ⟶ HX is an isomorphism for all homology functors H. We write X = mwlim_{-→} F. Minimal weak colimits are unique in algebraic stable homotopy categories when they exist, and are extremely useful. See [HPS95, Section 2] for details.

2. E theory

In this section we assemble some basic results about E theory. We begin with the fact that E^{∗}X is L-complete in the sense of Appendix A. In Section 2.1 we show that the ring structure on E is canonical by showing there are no phantom maps between evenly graded Landweber exact spectra. We recall the modified Adams spectral sequence briefly in Section 2.2. Finally, we briefly discuss operations in E-theory in Section 2.3.

Proposition 2.1. If X is a finite spectrum and R is one of E, E(n), E/I_{k}, E(n)/I_{k} or K(n) then R_{∗}X is finitely generated over R_{∗}.

Proof. We first recall that in each case R has an associative ring structure, so that R_{∗}X is a module over R_{∗}. The ring structure is not canonical but the module structure is induced by the MU_{∗}-module structure so it is canonical. In each case R_{∗} is Noetherian and the claim follows easily by induction on the number of cells.

Proposition 2.2. For any spectrum X, there is a natural topology on E^{0}X making it into a profinite (and thus compact Hausdorff ) Abelian group. Moreover, if Λ(X) is the category of pairs (Y, u) where Y is finite and u: Y ⟶ X, then E^{0}X is homeomorphic to lim_{←-}_{(Y,u)∈Λ(X)} E^{0}Y.

Proof. If X is a finite spectrum then E^{∗}X is a finitely generated module over E^{∗}, and it follows easily that the I_{n}-adic topology on E^{0}X is profinite. For an arbitrary spectrum X, define F^{0}X = lim_{←-}_{(Y,u)∈Λ(X)} E^{0}Y, with the inverse limit topology.

By [HPS95, Proposition 2.3.16], F is a cohomology theory with values in the category of profinite groups and continuous homomorphisms. There is an evident map E^{0}X ⟶ F^{0}X which is an isomorphism when X is finite (because (X, 1_{X}) is a terminal object of Λ(X) in that case). It follows easily that E^{0}X = F^{0}X for all X.

Corollary 2.3. For any spectrum X, the module E^{∗}X is L-complete in the sense of Definition A.5.

Proof. This is immediate when X is finite. It thus follows for general X because the category of L-complete modules is closed under inverse limits (by Theorem A.6).

Proposition 2.4. For any spectrum X, the module E^{∗}X is finitely generated over E^{∗} if and only if K^{∗}X is finitely generated over K^{∗}.

Proof. The cofibration Σ^{2(p^{k}−1)}E/I_{k} ⟶^{v_{k}} E/I_{k} ⟶ E/I_{k+1} gives a short exact sequence

(E/I_{k})^{∗}(X)/v_{k} ↣ (E/I_{k+1})^{∗}(X) ↠ ann(v_{k}, (E/I_{k})^{∗−2p^{k}+1}(X)).

It follows that if (E/I_{k})^{∗}X is finitely generated then the same is true of (E/I_{k+1})^{∗}X. Conversely, suppose that (E/I_{k+1})^{∗}X is finitely generated over E^{∗}, and write M = (E/I_{k})^{∗}X. The above sequence shows that M/v_{k}M is finitely generated. This means that there is a finitely generated free module F over E^{∗}/I_{k} and a map f: F ⟶ M such that the induced map F/v_{k}F ⟶ M/v_{k}M is surjective. If we let N be the cokernel of f, we conclude that N is an L-complete module over E^{∗} with N = v_{k}N. It follows from Proposition A.8 that N = 0, so f is surjective and M is finitely generated.

It follows that K^{∗}X = (E/I_{n})^{∗}X is finitely generated if and only if E^{∗}X = (E/I_{0})^{∗}X is finitely generated.

Proposition 2.5. Let X be a spectrum. Suppose E^{∗}X is pro-free (in the sense of Definition A.10). Then K^{∗}X = (E^{∗}X)/I_{n}. Conversely, if K^{∗}X is concentrated in even degrees, then E^{∗}X is pro-free and concentrated in even degrees.

Proof. The first statement follows from the fact that the sequence (v_{0}, …, v_{n−1}) is regular on E^{∗}X, by Theorem A.9. Conversely, suppose that (E/I_{k+1})^{∗}X is concentrated in even degrees. Consider the short exact sequence

(E/I_{k})^{∗}(X)/v_{k} ↣ (E/I_{k+1})^{∗}(X) ↠ ann(v_{k}, (E/I_{k})^{∗−2p^{k}+1}(X)).

It follows that (E/I_{k})^{odd}(X)/v_{k} = 0. As (E/I_{k})^{odd}(X) is L-complete, we conclude from Proposition A.8 that it must be zero. It also follows from the sequence that ann(v_{k}, (E/I_{k})^{even}(X)) = 0, so that v_{k} acts injectively on (E/I_{k})^{∗}(X). Finally, we also see from the sequence that (E/I_{k+1})^{∗}(X) = (E/I_{k})^{∗}(X)/v_{k}.

By an evident induction we conclude that E^{∗}X is concentrated in even degrees, that the sequence {v_{0}, …, v_{n−1}} is regular on E^{∗}X and that K^{∗}(X) = E^{∗}(X)/(v_{0}, …, v_{n−1}) = K^{∗} ⊗_{E^{∗}} E^{∗}X. It follows from Theorem A.9 that E^{∗}X is pro-free.

2.1. Landweber exactness. We next recall the theory of Landweber exact homology theories, and prove some convenient extensions. Many of the theorems we prove were proved by Franke [Fra92] in the case where π_{∗}M is countable; our methods are a generalisation of his.

Definition 2.6. An MU_{∗}-module M_{∗} is said to be Landweber exact if the sequence (w_{0}, w_{1}, …) is regular on M_{∗}. We write ℰ_{∗} for the category of Landweber exact modules that are concentrated in even degrees. We also write ℰ for the category of MU-module spectra M such that π_{∗}(M) ∈ ℰ_{∗}. Maps in ℰ are MU-module maps. Finally, we write ℰℱ for the category of finite spectra X such that H_{∗}X is free and concentrated in even degrees. We refer to such an X as an even finite spectrum. Note that any even finite spectrum has a finite filtration where the filtration quotients are finite wedges of even spheres.

The basic result is as follows.

Theorem 2.7 (Landweber). If M_{∗} ∈ ℰ_{∗} then the functor M_{∗} ⊗_{MU_{∗}} MU_{∗}X is a homology theory. Thus (by Brown representability), there is a spectrum M equipped with a natural isomorphism M_{∗}X ≃ M_{∗} ⊗_{MU_{∗}} MU_{∗}X. This M is unique up to isomorphism, and the isomorphism is canonical modulo phantoms.

Proof. See [Lan76, Theorem 2.6].

The following result summarises Proposition 2.20 and Proposition 2.19, which are proved below. It justifies the statements made in Section 1 about the uniqueness of E and E(n) and their ring structures.

Theorem 2.8. The functor π_{∗}: ℰ ⟶ ℰ_{∗} is an equivalence of categories. The inverse functor sends commutative MU_{∗}-algebras to commutative MU-algebra spectra.

It is convenient to introduce a new category ℰ′ at this point; it will follow from the theorem that ℰ′ = ℰ.

Definition 2.9. ℰ′ is the category of spectra M such that M_{∗} is concentrated in even degrees, with a given MU_{∗}-module structure on M_{∗} and a stable natural isomorphism M_{∗}⊗_{MU_{∗}} MU_{∗}X ⟶ M_{∗}X which is the identity when X = S. Morphisms of ℰ′ must preserve the module structure.

The converse of the Landweber exact functor theorem [Rud86] shows that if M ∈ ℰ′, then M_{∗} ∈ ℰ_{∗}. Theorem 2.7 says that π_{∗}: ℰ′ ⟶ ℰ_{∗} is essentially surjective on objects.

In order to show that π_{∗} is an equivalence of categories, we introduce the following definition.

Definition 2.10. A spectrum X is evenly generated if and only if, for every finite spectrum Z and every map Z ⟶^{f} X, there is an even finite spectrum W and a factorisation Z ⟶^{g} W ⟶^{h} X of f.

Every even finite spectrum is evenly generated. The collection of evenly generated spectra is closed under even suspensions, coproducts, and retracts, but does not form a thick subcategory. We will see in Proposition 2.18 that evenly generated spectra are closed under the smash product.

Lemma 2.11. MU is evenly generated.

Proof. Any map from a finite spectrum to MU factors though a skeleton of MU. Any skeleton of MU is an even finite.

The following result is essentially due to Hopkins.

Proposition 2.12. Suppose M ∈ ℰ′. Then M is evenly generated.

Proof. Suppose f : Z ⟶ M is a map from a finite spectrum to M. Then f is a class in M^{0}Z. Spanier-Whitehead duality implies that M^{∗}Z = M^{∗} ⊗_{MU^{∗}} MU^{∗}Z. We can thus write f = ∑_{i=1}^{m} b_{i} ⊗ c_{i} say. As M^{∗} is concentrated in even degrees we see that |c_{i}| = −|b_{i}| is even. Each map c_{i} thus has a factorisation

c_{i} = (Z ⟶^{g_{i}} W_{i} ⟶^{e_{i}} Σ^{|b_{i}|}MU),

where W_{i} is a skeleton of Σ^{|b_{i}|}MU, and so is an even finite. Write W = W_{1} ∨ … ∨ W_{m}, let g: Z ⟶ W be the map with components g_{i} and let h: W ⟶ M be the map with components b_{i} ⊗ e_{i} ∈ M^{∗} ⊗_{MU_{∗}} MU^{∗}W_{i} = [W_{i}, M]^{∗}. This gives the desired factorisation.

There are several different characterisations of evenly generated spectra.

Proposition 2.13. The spectrum X is evenly generated if and only if X can be written as the minimal weak colimit [HPS95, Section 2.2] of a filtered system {M_{α}} of even finite spectra.

Proof. First suppose that X can be written as such a minimal weak colimit. Then, by smallness, any map from a finite to X will factor through one of the terms in the minimal weak colimit, and so through an even finite. Thus X is evenly generated.

Conversely, suppose X is evenly generated. We replace ℰℱ by a small skeleton of ℰℱ without change of notation. Let Λ_{ℰℱ} (X) be the category of pairs (U, u), where U ∈ ℰℱ and u: U ⟶ X. Let Λ(X) be the category of pairs (W, w), where W is any finite spectrum and w: W ⟶ X. We know from [HPS95, Theorem 4.2.4] that X is the minimal weak colimit of Λ(X). It will therefore be enough to show that the obvious inclusion Λ_{ℰℱ} (X) ⟶ Λ(X) is cofinal.

We first show that Λ_{ℰℱ} (X), like Λ(X), is filtered. Consider two objects (U, u) and (V, v) of Λ_{ℰℱ} (X). We need to show that there is an object (W, w) and maps (U, u) ⟶ (W, w) ⟵ (V, v) in Λ(X). Clearly we can take W = U ∨ V, and let w: W ⟶ X be the map with components u and v. We also need to show that when f, g: (U, u) ⟶ (V, v) are two maps in Λ_{ℰℱ} (X), there is an object (W, w) and a map h: (V, v) ⟶ (W, w) with hf = hg. To see this, let W′ be the cofibre of f − g and h′: V ⟶ W′ the evident map. We have vf = u = vg so v(f − g) = 0 so v = w′h′ for some w′: W′ ⟶ X. Because X is evenly generated, the map w′ factors as W′ ⟶^{k} W ⟶^{w} X for some even finite W. We can evidently take h = kh′.

It is now easy to check that the inclusion Λ_{ℰℱ} (X) ⟶ Λ(X) is cofinal, as required.

Corollary 2.14. A spectrum X is evenly generated if and only if there is a cofibre sequence

P ⟶ Q ⟶ X ⟶^{δ} ΣP,

where P and Q are retracts of wedges of even finite spectra and δ is a phantom map.

Proof. If there is such a cofibre sequence, then there is a short exact sequence

[Z, P] ↣ [Z, Q] ↠ [Z, X]

for all finite Z, since δ is phantom. It follows easily that X is evenly generated. The converse follows from the Proposition and the construction in [CS96, Proposition 4.6].

Corollary 2.15. Suppose X is evenly generated and Y is a spectrum such that Y_{∗} is concentrated in even degrees. Let 𝒫^{∗}(X, Y) be the graded group of phantom maps from X to Y. Then

𝒫^{2k}(X, Y) = 0
𝒫^{2k−1}(X, Y) = [X, Y]^{2k−1}.

In particular, this holds if X, Y ∈ ℰ′.

Proof. Proceeding by induction, one proves easily that Y^{∗}W is concentrated in even degrees for all even finite W. In particular, in the cofibre sequence of Corollary 2.14, Y^{∗}Q and Y^{∗}P are concentrated in even degrees. Any phantom map in Y^{2k}X would be in the image of Y^{2k−1}P = 0, so there are no even phantoms. On the other hand, every class in Y^{2k+1}X is in the image of Y^{2k}P, so is phantom.

Although this characterisation of phantoms is the main fact that we need, we will prove something rather sharper.

Proposition 2.16. Suppose R is a ring spectrum, S is an R-module spectrum, and X is evenly generated. Suppose as well that R_{∗} and S_{∗} are concentrated in even degrees. Then R_{∗}X is flat, has projective dimension at most 1, and is concentrated in even degrees. Furthermore, we have

S_{∗}X = S_{∗} ⊗_{R_{∗}} R_{∗}X
S^{2k}X = Hom_{R_{∗}}^{2k}(R_{∗}X, S_{∗})
S^{2k−1}X = Ext_{R_{∗}}^{1,2k}(R_{∗}X, S_{∗}).

Proof. Choose a cofibre sequence P ⟶ Q ⟶ X ⟶^{δ} ΣP as in Proposition 2.14. If W is an even finite, it is easy to see by induction on the (even) cells that R_{∗}W is free over R_{∗} and concentrated in even degrees. Since R_{∗}X = lim_{-→}_{Λ_{ℰℱ} (X)} R_{∗}W, we

see that R_{∗}X is also concentrated in even degrees and is a filtered colimit of free modules, so is flat. One can also check by induction on the cells, using the fact that S_{∗} is evenly graded, that the natural map S_{∗} ⊗_{R_{∗}} R_{∗}W ⟶ S_{∗}W is an isomorphism for W an even finite. Taking colimits, we find that S_{∗} ⊗_{R_{∗}} R_{∗}X ⟶ S_{∗}X is also an isomorphism.

Similarly, one can check by induction on the cells that when W is an even finite, the natural map [W, S]^{∗} ⟶ Hom_{R_{∗}}(R_{∗}W, S_{∗}) is an isomorphism. As P and Q are retracts of wedges of such W, we see that R_{∗}P and R_{∗}Q are projective over R_{∗} and that [Q, S]^{∗} = Hom_{R_{∗}}(R_{∗}Q, S_{∗}) and [P, S]^{∗} = Hom_{R_{∗}}(R_{∗}P, S_{∗}). In particular, these groups are concentrated in even degrees. Because δ is phantom, we have a short exact sequence

R_{∗}P ↣ R_{∗}Q ↠ R_{∗}X,

which is a projective resolution of R_{∗}X. We now apply the functor [−, S]^{∗} to the cofibration P ⟶ Q ⟶ X to get an exact sequence

Hom_{R_{∗}}^{∗+1}(R_{∗}Q, S_{∗}) ⟶ Hom_{R_{∗}}^{∗+1}(R_{∗}P, S_{∗}) ⟶ [X, S]^{∗} ⟶

Hom_{R_{∗}}^{∗}(R_{∗}Q, S_{∗}) ⟶ Hom_{R_{∗}}^{∗}(R_{∗}P, S_{∗}),

and thus a short exact sequence

Ext_{R_{∗}}^{1,∗+1}(R_{∗}X, S_{∗}) ↣ [X, S]^{∗} ↠ Hom_{R_{∗}}^{∗}(R_{∗}X, S_{∗}).

The first term is concentrated in odd degrees and the last one in even degrees, so the sequence splits uniquely.

Note that this proposition implies that spectra such as P (n) and K(n) are not evenly generated for n > 0. Indeed, P (n)_{∗}P (n) and K(n)_{∗}K(n) both contain a Bockstein element in degree 1. Similarly, HZ and HF_{p} are not evenly generated. In fact, the only MU-module spectra that are evenly generated are the even Landweber exact MU-module spectra. One can prove this by using the fact that X ∧ F (n) is a retract of (MU ∧ F (n)) ∧ X and is therefore evenly graded. Applying this to the spectra S/I of Section 4 shows that v_{n} acts injectively on X_{∗}/I.

Corollary 2.17. Let M and N be MU-module spectra in ℰ′. Then

[M, N]^{2k} = Hom_{MU_{∗}MU}^{2k}(MU_{∗}M, MU_{∗}N)
= Hom_{MU_{∗}}^{2k}(MU_{∗}M, N_{∗})
[M, N]^{2k−1} = Ext_{MU_{∗}MU}^{1,2k}(MU_{∗}M, MU_{∗}N)
= Ext_{MU_{∗}}^{1,2k}(MU_{∗}M, N_{∗}).

If s > 1 then Ext_{MU_{∗}MU}^{s,∗}(MU_{∗}M, MU_{∗}N) = 0.

Proof. By Landweber exactness we have X_{∗}N = X_{∗}MU ⊗_{MU_{∗}} N_{∗} and in particular MU_{∗}N = MU_{∗}MU ⊗_{MU_{∗}} N_{∗}. This is an extended comodule, so for any comodule C_{∗} we have

Hom_{MU_{∗}MU}(C_{∗}, MU_{∗}N) = Hom_{MU_{∗}}(C_{∗}, N_{∗}).

More generally, if we resolve N_{∗} by injective MU_{∗}-modules and apply the functor MU_{∗}MU ⊗_{MU_{∗}} (−) we get a resolution of MU_{∗}N by injective comodules. Using this it is not hard to check that

Ext_{MU_{∗}MU}^{s,∗}(C_{∗}, MU_{∗}N) = Ext_{MU_{∗}}^{s,∗}(C_{∗}, N_{∗})

for all s. The rest of the corollary is proved in Proposition 2.16.

Proposition 2.18. Suppose X and Y are evenly generated. Then X ∧ Y is evenly generated.

Proof. It is clear that if W and Z are even finites, so is W ∧ Z. Now suppose X is evenly generated and W is an even finite. Then X ∧ W is the minimal weak colimit of the functor on Λ_{ℰℱ} (X) which takes (U, u) to U ∧ W. Thus X ∧ W is evenly generated. Now suppose Y is also evenly generated. Choose a cofibre sequence P ⟶ Q ⟶ Y ⟶^{δ} ΣP as in Corollary 2.14. Then X ∧ P and X ∧ Q are evenly generated, as they are retracts of wedges of terms of the form X ∧ W. Furthermore, 1_{X} ∧ δ is still a phantom map (using the characterisation of phantom maps as those maps not seen by any homology theory). It follows that X ∧ Y is evenly generated.

In fact, it is also true that ℰ′ is closed under the smash product, though we do not need this.

Proposition 2.19. If M ∈ ℰ′ then M admits a canonical structure as an MUmodule spectrum (in the traditional homotopical sense). If N is another spectrum in ℰ′ then the degree-zero MU-module maps M ⟶ N biject with the MU_{∗}-module maps M_{∗} ⟶ N_{∗}.

Proof. For any spectrum X we have a natural map

ϵ: MU_{∗}(MU ∧ X) = MU_{∗}MU ⊗_{MU_{∗}} MU_{∗}X ⟶ MU_{∗}X

of left MU_{∗}-modules, and thus a natural map

(M ∧ MU)_{∗}X = M_{∗}(MU ∧ X) = M_{∗} ⊗_{MU_{∗}} MU_{∗}(MU ∧ X) ⟶ M_{∗} ⊗_{MU_{∗}} MU_{∗}X = M_{∗}X.

By Brown representability, we get a map ν: M ∧ MU ⟶ M which is unique mod phantoms. It is not hard to check that this is associative and unital mod phantoms. However, Proposition 2.18 shows that M ∧ MU and M ∧ MU ∧ MU are evenly generated, so Corollary 2.15 tells us that there are no degree-zero phantom maps M ∧ MU ⟶ M or M ∧ MU ∧ MU ⟶ M. Thus ν is unique, associative and unital, and M ∈ ℰ.

Now let N be another spectrum in ℰ′, and consider the following diagram.

[M, N]^{MU}  Hom_{MU_{∗}}(M_{∗}, N_{∗}) [M, N]  Hom_{MU_{∗}MU}(MU_{∗}M, MU_{∗}N).^{π_{∗}} w v u  v u_{f}_{_{\begin{matrix}≃ \\ MU_{∗} \end{matrix}} w}

Here [M, N]^{MU} denotes the group of MU-module maps, and all the groups are groups of degree-zero maps. The bottom map is an isomorphism by Proposition 2.17. The map f sends a map u: M_{∗} ⟶ N_{∗} to

1 ⊗ u: MU_{∗}MU ⊗_{MU_{∗}} M_{∗} = MU_{∗}M ⟶ MU_{∗}N = MU_{∗}MU ⊗_{MU_{∗}} N_{∗}.

It is easy to check that this is injective and that the diagram commutes. It follows that π_{∗} is injective. Now suppose we have a map v: M_{∗} ⟶ N_{∗} of MU_{∗}-modules. We then have a unique map u: M ⟶ N such that MU_{∗}u = f(v). A diagram chase shows that u is a map of MU-module spectra (up to a phantom term which is zero as usual), and f(π_{∗}(u)) = f(v) so π_{∗}(u) = v. Thus π_{∗} is an isomorphism.

Proposition 2.20. If A ∈ ℰ and π_{∗}(A) is a commutative MU_{∗}-algebra then there is a unique product on A making it into a commutative MU-algebra spectrum.

Proof. For any X and Y we have a pairing MU_{∗}X ⊗_{MU_{∗}} MU_{∗}Y ⟶ MU_{∗}(X ∧ Y) and thus a pairing

A_{∗}(X) ⊗ A_{∗}(Y) = A_{∗} ⊗_{MU_{∗}} MU_{∗}X ⊗ A_{∗} ⊗_{MU_{∗}} MU_{∗}Y ⟶

A_{∗} ⊗ MU_{∗}(X ∧ Y) = A_{∗}(X ∧ Y).

This is easily seen to be commutative, associative and unital. By writing A as a minimal weak colimit of finite spectra A_{α} and taking X = DA_{α}, Y = DA_{β} we construct a map A ∧ A ⟶ A which is well-defined, commutative, associative and unital up to phantoms. However, all the relevant phantom groups vanish by Proposition 2.18 and Corollary 2.15.

Proposition 2.21. If A is a Landweber exact ring spectrum and M is an A-module spectrum then there are universal coefficient spectral sequences of A_{∗}-modules

Ext_{A_{∗}}^{s,t}(A_{∗}X, M_{∗}) ⟹ M^{t+s}X
Tor_{s,t}^{A_{∗}}(M_{∗}, A_{∗}X) ⟹ M_{t+s}X.

Proof. The first spectral sequence follows from Proposition 2.12, Proposition 2.13 and [Ada74, Theorem 13.6]. The second is constructed by the same methods.

2.2. The E-based Adams spectral sequence. We next briefly recall an approach to the E-based Adams spectral sequence that we learnt from Mike Hopkins, which is explained in more detail in [Dev96b]. This approach is also used by Franke [Fra96], who attributes it to Brinkmann. Let A be an even Landweber exact commutative ring spectrum; in our applications, A will be E or E(n). By Proposition 2.16, A_{∗}A is flat as a left module over A_{∗}. Similarly, it is flat as a right module. It follows in the usual way that it is a Hopf algebroid, so we can think about comodules over A_{∗}A. If I_{∗} is an injective module over A_{∗} then the extended comodule A_{∗}A ⊗_{A_{∗}} I_{∗} is injective. It follows that there are enough injective comodules, and that they can be used to define Ext groups. If J_{∗} is an injective comodule then Brown representability gives a spectrum W such that [X, W] = Hom_{A_{∗}A}(A_{∗}X, J_{∗}) for all X. The identity map of W corresponds to a map A_{∗}W ⟶ J_{∗}, which we claim is an isomorphism. Indeed, when X ∈ ℰℱ we know that A_{∗}X is free over A_{∗} and using duality we find that

X_{∗}W = [DX, W] = Hom_{A_{∗}A}(A_{∗}DX, J_{∗}) = Hom_{A_{∗}A}(A_{∗}, A_{∗}X ⊗_{A_{∗}} J_{∗}).

By Proposition 2.13, we can write A = mwlim_{-→} A_{α} for A_{α} ∈ ℰℱ. By taking X = A_{α} and passing to the limit we find that

A_{∗}W = Hom_{A_{∗}A}(A_{∗}, A_{∗}A ⊗_{A_{∗}} J_{∗}) = J_{∗}.

There is some inconsistency in the literature about what to call spectra such as W. We will use the following terminology.

Definition 2.22. Let A be a ring spectrum. We say that a spectrum X is Ainjective if it is a retract of A ∧ Y for some Y. Suppose in addition that A_{∗}A is flat as a module over A_{∗}. We say that X is strongly A-injective if A_{∗}X is an injective comodule and the natural map [Z, X] ⟶ Hom_{A_{∗}A}(A_{∗}Z, A_{∗}X) is an isomorphism for all Z. Note that if X is A-injective then A_{∗}X is injective relative to A_{∗}-split exact sequences, but not necessarily absolutely injective.

If X = X_{0} is any spectrum then we can embed A_{∗}X_{0} in an injective comodule J_{∗} and define W = W_{0} as above. We then have a map X_{0} ⟶ W_{0}, and we let X_{1} be the fibre. Continuing in the obvious way, we get a tower

X = X_{0} ⟵ X_{1} ⟵ X_{2} ⟵ ….

For any spectrum Y we can apply the functor [Y, −] to get a spectral sequence

E_{2}^{s,t} = Ext_{A_{∗}A}^{s,t}(A_{∗}Y, A_{∗}X) ⟹ [Y, L_{A}X]_{t−s}.

We call this the modified Adams spectral sequence (MASS) . It need not converge without additional assumptions. We will prove a convergence result in Proposition 6.5.

We will also have occasion to use the (unmodified) Adams spectral sequence. For this, we choose a complex X ⟶ I_{0} ⟶ I_{1} ⟶ … of E-injective spectra which becomes a split exact sequence after applying the functor E ∧ (−). Such a complex is unique up to chain homotopy equivalence under X. It can be converted into a tower X = X_{0} ⟵ X_{1} ⟵ … just as above, and by applying [Y, −] we get a spectral sequence, called the Adams spectral sequence. If X is E-nilpotent then this converges to [Y, X]. If E_{∗}Y is projective over E_{∗} then the modified and unmodified Adams spectral sequences coincide from the E_{2} page onwards [Dev96b], and in particular the E_{2} page can be identified as an Ext group.

2.3. Operations in E theory. We now turn to the study of operations in Etheory and K-theory. There is a well-known connection between this and the study of the Morava stabiliser group, but no really adequate account of this. For this and a variety of technical reasons we have chosen to use a more traditional approach.

Write Σ̂^{∗} = E^{∗}E for the (non-commutative) ring of operations in E-cohomology. Note that K^{∗}E = Hom(K_{∗}E, K_{∗}) (by Proposition 2.16 or by the general theory of modules over a field spectrum) and K_{∗}E = K_{∗}BP ⊗_{BP_{∗}}E_{∗} because E_{∗} is Landweber exact. Because I_{n} is an invariant ideal, it is easy to check that K_{∗}E is the same as the ring Σ_{∗} = Σ(n)_{∗} studied in [Rav86, Chapter VI]. We thus have

K_{∗}E = Σ_{∗} = F_{p}[t_{k} | k > 0]/(t_{k}^{p^{n}} − v_{n}^{p^{k}−1}t_{k}).

Here |t_{k}| = 2(p^{k} − 1), so K_{∗}E and Σ^{∗} = K^{∗}E are in even degrees. It follows from Proposition 2.5 that Σ̂^{∗} = E^{∗}E is pro-free, and that Σ^{∗} = Σ̂^{∗}/I_{n}Σ̂^{∗}. Moreover, because I_{n} is an invariant ideal in BP_{∗} one can check that I_{n}Σ̂^{∗} = Σ̂^{∗}I_{n}, and thus that Σ^{∗} is a quotient ring of Σ̂^{∗}.

Our main result is as follows.

Theorem 2.23. The ring Σ^{∗} is left Noetherian in the graded sense.

It is also true that Σ̂^{∗} and K^{∗}K are Noetherian, but we do not need this so we will not prove it.

As the theory of non-commutative Noetherian rings is less familiar than the commutative version, we start with some elementary remarks. Let R be a possibly non-commutative ring. Unless otherwise specified, we shall take “ideal” to mean “left ideal” and “module” to mean “left module”. As in the commutative case, one checks easily that the following are equivalent:

(a) Every ideal J ≤ R is finitely generated.

(b) Every ascending chain J_{0} ≤ J_{1} ≤ … of ideals is eventually constant.

(c) Any submodule of a finitely generated module over R is finitely generated.

(d) Every ascending chain of submodules of a finitely generated module over R is eventually constant.

If so, we say that R is (left) Noetherian. If R is a graded ring and all ideals and modules are required to be homogeneous, then the corresponding conditions are again equivalent; if they hold, we say that R is Noetherian in the graded sense.

Lemma 2.24. Let R ⟶ S be a map of rings such that S is finitely generated and free both as a left R-module and a right R-module. Then S is left Noetherian if and only if R is left Noetherian. Similarly for the graded case.

Proof. Suppose that R is Noetherian. Then any ascending chain of ideals in S is a chain of R-submodules of a finitely generated R-module, and thus eventually constant.

Conversely, suppose that S is Noetherian, and that S = ⨁_{i=0}^{d−1} a_{i}R. Then the map

J ↦ SJ = ⨁_{i} a_{i}J ≃ ⨁_{i} J

embeds the lattice of ideals in R into the Noetherian lattice of ideals in S, so R is Noetherian.

Lemma 2.25. Let R^{∗} be a graded algebra over K^{∗} (so that K^{∗} is central in R^{∗}). Write \overline{R} = R^{∗}/(v_{n} −1) and π: R^{∗} ⟶ \overline{R} for the projection map. If \overline{R} is Noetherian, then R^{∗} is Noetherian in the graded sense.

Proof. It is enough to show that the map J ↦ πJ embeds the lattice of homogeneous ideals of R^{∗} into the Noetherian lattice of ideals in \overline{R}, and thus enough to show that the set of homogeneous elements in π^{−1}πJ is just J. As R^{∗}/J is a graded module over the graded field K^{∗}, it is free, generated by elements e_{i} of degree d_{i} say. Suppose that a ∈ R^{∗} is homogeneous of degree d; then a = ∑_{i} a_{i}v_{n}^{(d−d_{i})/|v_{n}|}e_{i} (mod J) where a_{i} ∈ F_{p}, and a_{i} is zero if the indicated exponent of v_{n} is not an integer. If π(a) ∈ J then ∑_{i} a_{i}π(e_{i}) = 0, but it is clear that {π(e_{i})} is a basis for \overline{R}/πJ, so that a_{i} = 0 for all i and thus a ∈ J.

Lemma 2.26. Let R be ring, and {I_{s}} a decreasing filtration such that I_{0} = R and I_{s}I_{t} ≤ I_{s+t}. Suppose that R/I_{s} is a finite set for all s, that R = lim_{←-}_{s} R/I_{s}, and

that the associated graded ring R′ = E_{0}R = ∏_{s} I_{s}/I_{s+1} is Noetherian. Then R is Noetherian.

Proof. Let J be a left ideal in R. Then J′ = ∏_{s}(J ∩ I_{s})/(J ∩ I_{s+1}) is a left ideal in R′. It is thus finitely generated, so there are elements a_{i} ∈ J ∩ I_{d_{i}} (for i = 1, …, m say) whose images generate J′. This means that for any element a ∈ J ∩ I_{s} there are elements b_{i} such that a = ∑_{i} b_{i}a_{i} (mod J ∩ I_{s+1}). In other words, if K ≤ J is the ideal generated by {a_{1}, …, a_{m}}, then J ∩ I_{s} ≤ K + J ∩ I_{s+1}. It follows easily that J = J ∩ I_{0} is contained in ⋂_{s}(K + I_{s}), which is the closure of K in the evident topology given by the ideals I_{s}. On the other hand, as R/I_{s} is finite, we see that R is I-adically compact and Hausdorff. As K is the image of an evident continuous map R^{m} ⟶ R, we see that K is compact and thus closed. It follows that J = K = (a_{1}, …, a_{m}), which is finitely generated as required.

We next recall that for each k > 0, the ideal (t_{j} | 0 < j < k) ⊲ Σ_{∗} is a Hopf ideal, so that Σ(k)_{∗} = Σ_{∗}/(t_{j} | 0 < j < k) is a Hopf algebra, and Σ(k)^{∗} is a quotient Hopf algebra of Σ^{∗}. We also write S = Σ_{∗}/(v_{n} − 1) and S(k) = Σ(k)_{∗}/(v_{n} − 1). We write S^{∗} = Hom(S, F_{p}) and S(k)^{∗} = Hom(S(k), F_{p}). Note that Ravenel [Rav86] calls these objects Σ(n, k)_{∗}, S(n, k) and so on.

Proposition 2.27. If k > pn/(p−1) then the Hopf algebra S(k)^{∗} can be filtered so that the associated graded ring is a commutative formal power series algebra over F_{p} on n^{2} generators.

Proof. In this proof, all theorem numbers and so on refer to Ravenel’s book [Rav86]. Our proposition is essentially Theorem 6.3.3. That theorem appears to apply to S rather than S^{∗}, but this is a typo; this becomes clear if we read the preceding paragraph. Some modifications are necessary to replace S^{∗} by S(k)^{∗}, and anyway Ravenel does not give an explicit proof of his theorem, so we will fill in some details.

The Hopf algebra filtration of S given by Theorem 6.3.1 clearly induces a filtration on S(k). It is easy to see that

E^{0}S(k) = T [t_{ij} | i ≥ k, j ∈ Z/n]

as rings, where T [t] = F_{p}[t]/t^{p} and t_{ij} corresponds to t_{i}^{p^{j}}. There is therefore an automorphism F on E^{0}S(k) that takes t_{i,j} to t_{i,j+1} which has order n. Moreover, this is a connected graded Hopf algebra (using the grading coming from the filtration, so that the degree of t_{ij} is the integer d_{n,i} of Theorem 6.3.1). The coproduct is given by Theorem 4.3.34, which says that Δ(t_{ij}) is the sum of the elements in a certain unordered list \overline{Δ}_{ij}. These lists are used in such a way that we may ignore any terms which have filtration less than that of t_{ij}. We start with the list

\overline{M}_{ij} = \{t_{ij} ⊗ 1, 1 ⊗ t_{ij}\}.

(This comes from Lemma 4.3.32, using the fact that k > pn/(p−1).) We also recall the Witt polynomials w_{J} defined in Lemma 4.3.8. We are working modulo p so we have w_{J} = w_{|J|}^{p^{∥J∥−|J|}}. In E^{0}S(k) we must interpret this as w_{J} = F^{∥J∥−|J|}w_{|J|}. We have set v_{n} = 1 and killed all other v’s, so v_{J} = 0 unless J has the form J = J_{r} = (n, …, n) (with r terms), in which case v_{J} = 1. With these observations, Lemma 4.3.33 becomes

\overline{Δ}_{ij} = \overline{M}_{ij} ∪ \{F^{−r}w_{r}(\overline{Δ}_{i−nr,j}) | r > 0\}.

As \overline{M}_{ij} is invariant under the twist map, we see that the same holds for \overline{Δ}_{ij}, and thus that E^{0}S(k) is cocommutative. We also see that

Δ(t_{ij}) = t_{ij} ⊗ 1 + 1 ⊗ t_{ij} + w_{1}(t_{i−n,j−1} ⊗ 1, 1 ⊗ t_{i−n,j−1}) (mod t_{rs} | r < i − n).

Here w_{1} is given by

w_{1}(x_{1}, x_{2}, …) = (∑_{t} x_{t}^{p} − (∑_{t} x_{t})^{p})/p.

It is not hard to conclude that the Verschiebung is

V (t_{ij}) = t_{i−n,j−1} (mod t_{rs} | r < i − n).

We also observe that the degree of t_{ij} is less than that of t_{kl} whenever i < k (this follows easily from the definition in Theorem 6.3.1). It follows by induction on i that each t_{ij} lies in the image of V, so that V is surjective.

We now dualise, and conclude that E_{0}S(k)^{∗} is a bicommutative connected graded Hopf algebra for which the Frobenius map is injective. We can thus apply the Borel structure theory [Spa66, Section 5.8][Bor53] to conclude that E_{0}S(k)^{∗} is a formal power series algebra. One can check that there must be n^{2} generators, corresponding to the t_{k+i,j} for 0 ≤ i < n.

We can now prove as promised that Σ^{∗} is Noetherian.

Proof of Theorem 2.23. According to Lemma 2.25, it is enough to check that S^{∗} is Noetherian. By Lemma 2.26, it is enough to check that E_{0}S^{∗} is Noetherian. We have an extension of Hopf algebras

S′(k) = F_{p}[t_{1}, …, t_{k−1}]/(t_{j}^{p^{n}} − t_{j}) ↣ S ↠ S(k),

and thus an injective map of connected graded Hopf algebras E_{0}S(k)^{∗} ↣ E_{0}S^{∗}. It follows from the Milnor-Moore theorem (the dual of [Rav86, Corollary A1.1.20]) that E_{0}S^{∗} is a free (as a left or right module) over E_{0}S(k)^{∗}. The rank is just dim(S′(k)) < ∞. Thus, by Lemma 2.24, it is enough to check that E^{0}S(k)^{∗} is Noetherian, and this follows from Proposition 2.27.

We next show that Σ^{∗} is local in a suitable sense. Let I be the kernel of the augmentation map Σ^{∗} ⟶ K^{∗}.

Proposition 2.28. Σ^{∗} = lim_{←-}_{k} Σ^{∗}/I^{k}.

Proof. This follows easily from the fact [Rav86, Theorem 6.3.3] that S^{∗} has a filtration whose associated graded ring is a graded connected Hopf algebra of finite type.

This gives us a version of Nakayama’s lemma.

Corollary 2.29. Let M be a finitely generated graded module over Σ^{∗}. If IM = M then M = 0.

Proof. If M = Σ^{∗}m is cyclic and IM = M, then m = am for some a ∈ I. This means that (1 − a)m = 0, but the sum ∑_{k} a^{k} converges to an inverse for (1 − a), so M = 0 as required.

More generally, suppose M = Σ^{∗}{m_{1}, …, m_{k}} and IM = M. Write M′ = Σ^{∗}{m_{1}, …, m_{k−1}}, so that N = M/M′ is cyclic and IN = N. It follows that N = 0, so M = M′, and the claim follows by induction on k.

This means that the usual theory of minimal projective resolutions applies.

Corollary 2.30. Let M be a finitely generated graded module over Σ^{∗}. Then M has a resolution P_{∗} ⟶ M by finitely generated free modules over Σ^{∗}, such that the maps P_{k}/IP_{k} ⟶ P_{k−1}/IP_{k−1} are zero. This is a retract of any other projective resolution, and is unique up to non-canonical isomorphism.

Proof. Suppose that the first k stages

M ⟵ P_{0} ⟵ … ⟵ P_{k−1}

have been constructed. Let N be the kernel of the differential P_{k−1} ⟶ P_{k−2}. As P_{k−1} is finitely generated and Σ^{∗} is Noetherian, we see that N is finitely generated. We can thus choose finitely many elements n_{1}, …, n_{r} in N giving a basis for N/IN over K^{∗}. Let P_{k} be a direct sum of r copies of Σ^{∗} (suitably shifted in degree), and let P_{k} ⟶ N ⟶ P_{k−1} be the obvious map. We leave it to the reader to check that this does the job.

3. K-injective spectra

We say that a spectrum X is K-injective if it is equivalent to a wedge of suspensions of copies of K. We recall a number of facts about such spectra, and prove a few more.

Remark 3.1. It will follow from the results below that this is consistent with Definition 2.22.

We start by considering K-module spectra. If p > 2 then K is commutative, and if p = 2 it has a canonical antiautomorphism χ with χ^{2} = 1 (see [Nas96]), so we can convert freely between left and right modules.

Proposition 3.2. Suppose that R is a ring spectrum, M is an R-module spectrum, and X is arbitrary. Then there are natural R-module structures on M ∧X, F (X, M) and F (M, X). Moreover, if R = K, then M is K-injective.

Proof. For the first part, let μ: R ∧ M ⟶ M be the R-module structure map. We can make M ∧ X into a R-module with structure map μ ∧ 1_{X}. We can use the following composite as a structure map for F (X, M):

R ∧ F (X, M) = F (S, R) ∧ F (X, M) ⟶^{∧} F (X, R ∧ M) ⟶^{μ_{∗}} F (X, M).

Finally, the structure map R ∧ F (M, X) ⟶ F (M, X) is adjoint to the following composite:

M ∧ R ∧ F (M, X) ⟶^{τ∧1} R ∧ M ∧ F (M, X) ⟶^{μ∧1} M ∧ F (M, X) ⟶^{eval} X.

For the last part, note that K_{∗} is a graded field, so M_{∗} is necessarily a free module over K_{∗}, say M_{∗} = K_{∗}{e_{i} | i ∈ I}. Each map e_{i}: S^{|e_{i}|} ⟶ M gives a K-module map Σ^{|e_{i}|}K ⟶ M, so we have a map ⋁_{i} Σ^{|e_{i}|}K ⟶ M which is an equivalence by construction.

The following proposition (most of which appears in [HS]) summarises the main facts that we need.

Proposition 3.3.

(a) For any spectrum X, the smash product K ∧ X is K-injective.

(b) A spectrum X is K-injective if and only if it admits a K-module structure.

(c) Any retract of a K-injective spectrum is K-injective.

(d) If X is K-injective, then π_{∗}X has a natural structure as a K_{∗}-module. Any map f : X ⟶ Y of K-injective spectra induces a K_{∗}-linear map f_{∗} : π_{∗}X ⟶ π_{∗}Y.

(e) If f : X ⟶ Y is a map of K-injective spectra such that f_{∗} is a split monomorphism of K_{∗}-modules, then f is a split monomorphism and the cofibre of f is K-injective. If f is compatible with given module structures on X and Y, then the splitting may also be chosen compatibly. Similarly for epimorphisms.

(f) If {X_{i}} is a family of K-injective spectra, then the natural map ⋁_{i} X_{i} ⟶ ∏_{i} X_{i} is a split monomorphism.

Proof. (a) K ∧ X is clearly a K-module, and thus K-injective by Proposition 3.2.

(b) It is clear that a K-injective spectrum admits a module structure; the converse is just Proposition 3.2 again.

(c) This is proved after (d).

(d) Any choice of K-module structure on X gives a K_{∗}-module structure on π_{∗}X. Suppose that we have two different module structures, given by maps α, β: K∧ X ⟶ X. We then have a map

γ = (K ∧ K ∧ X ⟶^{1∧α} K ∧ X ⟶^{β} X).

The two different v_{n}-multiplications are given by composing γ with the two maps

α′ = η ∧ v_{n} ∧ 1: Σ^{|v_{n}|}X ⟶ K ∧ K ∧ X
β′ = v_{n} ∧ η ∧ 1: Σ^{|v_{n}|}X ⟶ K ∧ K ∧ X

and applying π_{∗}. If X = K then a well-known calculation shows that π_{∗}(α′) = π_{∗}(β′), so this remains true for any K-injective spectrum X. It follows that the two K_{∗}-module structures coincide.

Before proving the rest of (d), we need to prove a part of (f). Namely, given a set I and integers d_{i} for i ∈ I, we show that the natural map ⋁_{i} Σ^{d_{i}}K ⟶ ∏_{i} Σ^{d_{i}}K is a split monomorphism. Note that this map is a map of K-module spectra, so induces the usual monomorphism of K_{∗}-modules ⨁_{i} Σ^{d_{i}}K_{∗} ⟶ ∏_{i} Σ^{d_{i}}K_{∗}. Extend the usual basis {e_{i}}of ⨁_{i} Σ^{d_{i}}K_{∗} to a basis of ∏_{i} Σ^{d_{i}}K_{∗} by adding new basis elements f_{j} for j ∈ J. Just as in the proof of Proposition 3.2, we get an equivalence ⋁_{i} Σ^{d_{i}}K∨⋁_{j} Σ^{|f_{j}|}K ⟶ ∏_{i} Σ^{d_{i}}K. This gives an obvious left inverse to the inclusion of the wedge, as required.

Now consider a map f: X ⟶ Y of K-injective spectra. We claim that f_{∗}: π_{∗}X ⟶ π_{∗}Y is K_{∗}-linear. Note that X is a wedge of K’s, and we have just shown that Y is a retract of a product of K’s; this reduces us easily to the case X = Y = K (up to suspension). The usual theory of flat ring spectra shows that the graded map r_{∗}: E_{∗} ⟶ E_{∗} induced by a graded map r: E ⟶ E (that is, an element r ∈ E^{∗}E = Hom_{E_{∗}}(E_{∗}E, E_{∗})) is the composite E_{∗} ⟶^{η_{R}} E_{∗}E ⟶^{r} E_{∗}. Normally, η_{R} is a homomorphism of right E_{∗}-modules, but for E = K we have η_{R} = η_{L}. Thus, r_{∗} is K_{∗}-linear.

(c) Let X be K-injective, and e: X ⟶ X an idempotent. By (d) we know that e_{∗} is K-linear, and thus π_{∗}X splits as a direct sum π_{∗}eX ⊕ π_{∗}(1 − e)X of K_{∗}- modules. By choosing a basis for π_{∗}X adapted to this splitting and proceeding as in the proof of Proposition 3.2, we see that eX is K-injective.

(e) Let f: X ⟶ Y be a map of K-injective spectra such that f_{∗}: X_{∗} ⟶ Y_{∗} is a monomorphism. Note that the image is a K_{∗}-submodule by (d). Choose elements {e_{i} ∈ Y_{∗} | i ∈ I} giving a basis for Y_{∗}/f_{∗}X_{∗}. Choose a K-module structure on Y, and use it to convert the maps e_{i}: S^{|e_{i}|} ⟶ Y into module maps Σ^{|e_{i}|}K ⟶ Y, and thus a map Z = ⋁_{i} Σ^{|e_{i}|}K ⟶ Y. By construction, the evident map X ∨ Z ⟶ Y is an equivalence, so that X ⟶ Y is a split monomorphism with cofibre Z. Clearly, if f is compatible with given module structures on X and Y, then the splitting will also be compatible. The proof for epimorphisms is similar.

(f) This follows immediately from (e).

Recall that Σ_{∗} = K_{∗}E = (E_{∗}E)/I_{n}, using the invariance of I_{n}. There is thus a natural coaction of Σ_{∗} on (E_{∗}X)/I_{n} induced from the coaction of E_{∗}E. In particular, if X is such that E_{∗}X is annihilated by I_{n} then E_{∗}X is a comodule over Σ_{∗}.

Proposition 3.4. Let M be a K-module spectrum, and X a spectrum. Then there are natural isomorphisms

M_{∗}X ≃ M_{∗} ⊗_{K_{∗}} K_{∗}X
[X, M]_{∗} ≃ Hom_{K_{∗}}(K_{∗}X, M_{∗})
E_{∗}M ≃ Σ_{∗} ⊗_{K_{∗}} M_{∗}
M_{∗} ≃ Prim_{Σ_{∗}} E_{∗}M

Proof. The map

M ∧ K ∧ X ⟶^{mult∧1} M ∧ X

induces a natural map M_{∗} ⊗_{K_{∗}} K_{∗}X ⟶ M_{∗}X, which is visibly an isomorphism when X is a sphere. Both sides are homology theories because M_{∗} is free over K_{∗}, so the map is an isomorphism for all X. The second part is similar.

The third part follows from the first part, since we have

E_{∗}M = M_{∗}E = M_{∗} ⊗_{K_{∗}} K_{∗}E = M_{∗} ⊗_{K_{∗}} Σ_{∗}.

A diagram chase shows that this is actually an isomorphism of comodules (where M_{∗} has trivial coaction) from which the fourth part is immediate.

4. Generalised Moore spectra

We now discuss generalised Moore spectra. The most important ideas are due to Hopkins and Smith [HS], and are also explained in [Rav92a, Chapter 6]. Here we offer some convenient technical improvements.

In this section, if we write w: X ⟶ Y then we always allow the possibility that w has nonzero degree, but we always insist that the degree should be even.

We shall consider v_{n} self maps with two extra properties. The first is strong centrality in the following sense:

Definition 4.1. Let X be a finite spectrum. We say that a self map w: X ⟶ X is strongly central if 1_{X} ∧ w is central in End(X ∧ X)_{∗}.

It seems that this condition was first considered by Ethan Devinatz [Dev92]. It turns out to have surprisingly strong implications. To explain this, we need the following definition.

Definition 4.2. If X is a finite spectrum, we let 𝒥_{X} be the full subcategory of (possibly infinite) spectra Y such that Y may be written as a retract of some spectrum of the form X ∧ Z. (Note that we do not take such a retraction as part of the structure of Y.)

Remark 4.3. Let R be the finite ring spectrum End(X) = F (X, X) = DX ∧ X. If Y is an R-injective spectrum then Y is a retract of R ∧ Y = X ∧ (DX ∧ Y), so Y ∈ 𝒥_{X}. Conversely, if Y ∈ 𝒥_{X} then Y is a retract of some spectrum X ∧ Z, which is an R-module (because X is). It follows that 𝒥_{X} is precisely the category of R-injective spectra, and thus that it is closed under products, coproducts, and retracts. Moreover, if Y ∈ 𝒥_{X} and U is an arbitrary spectrum then Y ∧ U, F (U, Y) and F (Y, U) are all in 𝒥_{X} (by Proposition 3.2).

Later in this section we will prove the following result.

Proposition 4.4. Let v be a strongly central self-map of a finite spectrum X. Then there is a unique natural transformation v_{Y} : Y ⟶ Y for Y ∈ 𝒥_{X}, such that v_{X∧Z} = v ∧ 1_{Z} for all spectra Z. Furthermore, v_{Y ∧Z} = v_{Y} ∧ 1_{Z} for all spectra Z.

This means that when we work with spectra in 𝒥_{X} we can pretend that v is an element of π_{∗}S. We will often write v instead of v_{Y}.

Our second extra condition involves the group of MU-module self maps of MU ∧ X. This can be interpreted in two different ways. On the one hand, we can work entirely in the homotopy category of spectra. The spectrum MU is a ring object in this category, so we can consider the category of modules over it; we shall call this 𝒞_{MU}. On the other hand, we can work in the derived category 𝒟_{MU} of strict MU-modules, as explained in Section 1. In the present context it does not matter which interpretation we use, because

𝒞_{MU}(MU ∧ X, M)_{∗} = 𝒟_{MU}(MU ∧ X, M)_{∗} = [X, M]_{∗}.

In particular,

𝒞_{MU}(MU ∧ X, MU ∧ X)_{∗} = [X, X ∧ MU]_{∗} = MU_{∗}(DX ∧ X).

Definition 4.5. Let X be a p-local finite spectrum of type at least n > 0. We say that w: Σ^{p^{k}|v_{n}|}X ⟶ X is a good v_{n} self map if

1. 1 ∧ w = v_{n}^{p^{k}} ∧ 1 in 𝒞_{MU}(MU ∧ X, MU ∧ X).

2. w is strongly central.

Every good v_{n} self map is a v_{n} self map in the sense of Hopkins and Smith [HS]. Indeed, we shall eventually prove the following result:

Theorem 4.6. Any finite spectrum X of type at least n > 0 admits a good v_{n} self map, and if v and w are two such maps then v^{p^{i}} = w^{p^{j}} for some i and j. Moreover, for any MU-module spectrum M ∈ 𝒞_{MU} we have

1 ∧ w = v_{n}^{p^{k}} ∧ 1: M ∧ X ⟶ M ∧ X.

If M is an object of 𝒟_{MU}, then the above equation also holds when interpreted in 𝒟_{MU}. In any case, it follows that the induced map w_{∗}: M_{∗}X ⟶ M_{∗}X is just multiplication by v_{n}^{p^{k}}.

We now start work on the proofs.

Lemma 4.7. Let w be a strongly central self map of a finite spectrum X. Then w ∧ 1_{X} = 1_{X} ∧ w in End(X ∧ X)_{∗}. Moreover, for any spectra Y, Z and any map u: X ∧ Y ⟶ X ∧ Z , the following diagram commutes:

X ∧ Y  X ∧ Z X ∧ Y  X ∧ Z_{_{w∧1} u}  ^{u} w _{u_{w∧1}} w_{u}

(In particular, by taking Z = Y we see that w ∧ 1 is central in End(X ∧ Y)_{∗}; in particular, by taking Y = S we see that w is central in End(X)_{∗}.)

Proof. Let w be a strongly central self map of a finite spectrum X. Let τ ∈ End(X ∧ X) be the twist map. As 1_{X} ∧ w is central, we have

1_{X} ∧ w = τ ∘ (1_{X} ∧ w) ∘ τ = w ∧ 1_{X}.

Now consider a map u: X ∧ Y ⟶ X ∧ Z, and write

g = u ∘ (w ∧ 1_{Y}) − (w ∧ 1_{Z}) ∘ u,

so the claim is that g = 0. By rewriting 1_{X}∧w as w∧1_{X}, we see that both composites in the following diagram are w ∧ u, and thus that the diagram commutes.

X ∧ X ∧ Y  X ∧ X ∧ Z X ∧ X ∧ Y  X ∧ X ∧ Z_{_{1∧w∧1} u}  ^{1∧u} w  _{u_{1∧w∧1}} w_{1∧u}

This means that 1_{X} ∧ g = 0; a fortiori we have 1_{DX∧X} ∧ g = 0. On the other hand, X ∧ Z is a module-spectrum over DX ∧ X = End(X), so the map

η ∧ 1_{X∧Z}: X ∧ Z ⟶ DX ∧ X ∧ X ∧ Z

is a split monomorphism. By considering the following diagram, we conclude that g = 0 as claimed.

X ∧ Y  X ∧ Z DX ∧ X ∧ X ∧ Y  DX ∧ X ∧ X ∧ Z v u_{η∧1}  ^{g}  w v u_{η∧1} w_{1∧g=0}

Proof of Proposition 4.4. Let Y be a spectrum in 𝒥_{X}. We can choose a spectrum Z and maps Y ⟶^{j} X ∧ Z ⟶^{q} Y such that qj = 1_{Y}. By naturality, we must define v_{Y} = q(v ∧ 1)j: Y ⟶ Y. Suppose that we have a different spectrum Z′ and maps

Y ⟶^{j′} X ∧ Z′ ⟶^{q′} Y with q′j′ = 1_{Y}. We claim that q′(v ∧ 1)j′ = q(v ∧ 1)j, so that our definition is independent of the choice of Z, j and q. To see this, note that Lemma 4.7 gives j′q(v ∧ 1_{Z}) = (v ∧ 1_{Z′})j′q. We thus have

q(v ∧ 1)j = q′j′q(v ∧ 1)j = q′(v ∧ 1)j′qj = q′(v ∧ 1)j′

as required. It follows that v_{Y ∧Z} = v_{Y} ∧ 1_{Z} as well.

Now suppose that we have a morphism f: Y_{0} ⟶ Y_{1} in 𝒥_{X}. We need to show that v is natural with respect to f, or in other words that the central square in the following diagram commutes:

X ∧ Z_{0}  Y_{0}  Y_{1}  X ∧ Z_{1} X ∧ Z_{0}  Y_{0}  Y_{1}  X ∧ Z_{1}_{_{v∧1} u}  ^{q_{0}} ww_{_{v_{Y_{0}}} u}  ^{f} w _{u_{v_{Y_{1}}}}v  w^{j_{1}}  _{u_{v∧1}} ww_{q_{0}}  w_{f} v  w_{j_{1}}

Because Y_{0}, Y_{1} are in 𝒥_{X}, we can choose spectra Z_{0}, Z_{1} and a split monomorphism j_{1} and split epimorphism q_{0} as shown. By the definition of v given above, the outer squares commute. The total rectangle commutes by Lemma 4.7. It follows easily that the middle square commutes, as required.

Definition 4.8. A μ-spectrum is a spectrum X equipped with maps S ⟶^{η} X ⟵^{μ} X ∧ X such that

μ ∘ (η ∧ 1) = 1 : X ⟶ X.

(We reserve the term ring spectrum for examples in which μ is associative and η is a two-sided unit.) If X is a μ-spectrum, then π_{∗}X is a (possibly non-associative) graded ring, with possibly only a left unit. A module spectrum over X is a spectrum Y equipped with a map ν : X ∧ Y ⟶ Y such that ν ∘ (η ∧ 1) = 1 : Y ⟶ Y. Note that this is independent of μ. If X ⟶ Y is a map of μ-spectra, then Y has an obvious structure as an X-module; in particular, X is an X-module. We shall say that a μ-spectrum X is atomic if every map f: X ⟶ X such that fη = η is an isomorphism.

Definition 4.9. Given a finite μ-spectrum X and an element v ∈ π_{d}X, we define

λ(v) = (Σ^{d}X ⟶^{v∧1} X ∧ X ⟶^{μ} X) ∈ [X, X]_{d}.

If λ(v) is a strongly central self map of X, we say that v is a strongly central element of π_{∗}X. If in addition the Hurewicz image of v in MU_{∗}X is (the image under the unit map of) v_{n}^{p^{k}} for some k, we say that v is a good v_{n} element . Given a self map w: Σ^{d}X ⟶ X, we define η^{∗}(w) = w ∘ η ∈ π_{d}X.

Proposition 4.10. Let X be a finite, atomic μ-spectrum. Then every strongly central self map of X is a map of X-modules. Moreover, the set of such maps forms a commutative and associative ring under composition, which is isomorphic to the ring of strongly central elements of π_{∗}X via λ and η^{∗}. This also induces a bijection between good v_{n} self maps and good v_{n} elements.

Proof. Write A_{∗} for the set of strongly central self maps of X; this is clearly a commutative and associative graded ring under composition. Write B_{∗} for the set of strongly central elements of π_{∗}X, so we have a map λ: B_{∗} ⟶ A_{∗}.

Consider w ∈ A_{∗}. We first claim that w is a map of X-modules, in other words that the following diagram commutes.

X ∧ X  X ∧ X X  X^{1∧w} w_{_{μ} u}  _{u_{μ}} w_{w}

This is just Lemma 4.7 (with Y = X, Z = S, u = μ).

Next, we claim that w = λ(η^{∗}w). To see this, write v = η^{∗}w and consider the following diagram.

X  X X ∧ X  X ∧ X  X._{_{η∧1} u} _{[ [ [ [ [ [ [ [[]}_{v∧1=η∧w}^{w}  w_{u_{η∧1}} _{[ [ [ [ [ [ [ [[]}_{1} w_{w∧1=1∧w}  w_{μ}

We remarked earlier that w∧1 commutes with the twist map and thus w∧1 = 1∧w. By thinking of this map as 1 ∧ w we see that the square commutes and that the diagonal map is η ∧ w. By thinking of it as w ∧ 1 instead we see that the diagonal can also be described as v ∧ 1. The right hand triangle commutes by the axiom for a μ-spectrum. By looking at the long composite in the diagram we see that w = λ(η^{∗}w) as claimed.

This means that v ∈ B_{∗}, so we have maps

A_{∗} ⟶^{η^{∗}} B_{∗} ⟶^{λ} A_{∗}

with λη^{∗} = 1.

Next, consider two strongly central self maps w, w′, and write v = η^{∗}w and v′ = η^{∗}w′. We claim that η^{∗}(ww′) = vv′ (which means μ ∘ (v ∧ v′)). This follows by inspecting the following diagram:

S  X ∧ X X  X  X._{_{η} u} _{[ [ []}_{v′}^{v∧v′}  w _{u_{μ}} w_{w′} ^{ȷ ȷ ȷ fi}_{v∧1} w_{w}

Thus, η^{∗} is a ring map from A_{∗} to B_{∗}.

Next, consider the map θ = μ ∘ (1 ∧ η): X ⟶ X. As we do not assume that η is a two-sided unit for μ, this need not be the identity. However, it is easy to see that θη = η; as X is atomic, we conclude that θ is an isomorphism. For any v ∈ B_{∗}, we have a commutative diagram as follows:

S  X X  X ∧ X  X^{v} w_{_{η} u}  _{_{1∧η} u} _{[ [ [ []}_{θ} w_{v∧1}  w_{μ}

This shows that η^{∗}λ(v) = θ ∘ v, so that λ: B_{∗} ⟶ A_{∗} is injective. As λη^{∗} = 1, it is not hard to see that λ and η^{∗} are mutually inverse isomorphisms.

We leave it to the reader to check that good v_{n} self maps go to good v_{n} elements and vice versa.

In view of the above proposition, we allow ourselves to write v for λ(v), when v is a strongly central homotopy element for a finite, atomic μ-spectrum.

Proposition 4.11. Let X be a finite atomic μ-spectrum, and v ∈ π_{∗}X a strongly central element. Suppose that k > 1, and let X/v^{k} be the cofibre of v^{k}: X ⟶ X. Then X/v^{k} can be made into a μ-spectrum in such a way that the map q: X ⟶ X/v^{k} is a map of μ-spectra and the maps

X ⟶^{v^{k}} X ⟶^{q} X/v^{k} ⟶^{d} ΣX

are maps of X-modules.

Proof. This is essentially due to Ethan Devinatz [Dev92]. His Theorem 1 states that if w: X ⟶ X is a self map of a finite μ-spectrum and

(i) w is strongly central (ii) w^{2} is a map of X-modules then X/w^{2} can be made into a μ-spectrum, with properties essentially above. Our Proposition 4.10 shows that hypothesis (ii) is redundant. We assume that X is atomic so that we can translate cleanly between (powers of) strongly central self maps and homotopy elements. This proves the case k = 2 of our claim. The general case is essentially the same; in Devinatz’ Lemma 5, one simply has to apply Verdier’s

axiom to the maps X ⟶^{v^{k−1}} X ⟶^{v} X rather than X ⟶^{v} X ⟶^{v} X.

Definition 4.12. Consider an ideal I ≤ MU_{∗} of the form

I = (v_{0}^{a_{0}}, …, v_{n−1}^{a_{n−1}}),

where each a_{i} is a power of p. We call n the height of I. We give a recursive definition of “generalised Moore spectra of type S/I”; such a thing will be a certain kind of finite, atomic μ-spectrum. The only spectrum of type S/0 is S, equipped with the obvious structure. Consider an ideal I as above, and write J = (v_{0}^{a_{0}}, …, v_{n−2}^{a_{n−2}}). A μ-spectrum X has type S/I if there is a μ-spectrum Y of type S/J with a good v_{n−1} self map v of degree |v_{n−1}^{a_{n−1}}| and a cofibration X ⟶^{v} X ⟶^{q} Y ⟶^{d} ΣX such that q is a map of μ-spectra and q and d are maps of X-modules.

If a spectrum of type S/I exists, we shall refer to it as S/I; this is an abuse, as there may be many non-isomorphic spectra of type S/I. It is easy to see that the unit map S ⟶ S/I induces an isomorphism MU_{∗}/I ≃ MU_{∗}(S/I). It follows that any map f: S/I ⟶ S/I with f ∘ η = η satisfies MU_{∗}f = 1 and thus is an isomorphism; this means that S/I is atomic. We can also construct an object MU/I ∈ 𝒟_{MU} by the methods of [EKMM96, Chapter V] or [Str96], and it is easy to check that MU ∧ S/I is isomorphic in 𝒟_{MU} to MU/I.

Proposition 4.13. Suppose that there is a spectrum X of type S/I; then it has a good v_{n} element, and thus S/(I, v_{n}^{p^{j}}) exists for j ≫ 0.

Proof. This is based on results in [HS]; we refer to Ravenel’s account [Rav92a] as it is more readily available.

By [Rav92a, Theorem 1.5.4], there exists k ≥ 0 and a map w ∈ End(X)_{∗} such that K(m)_{∗}w = 0 for m ≠ n, and K_{∗}w = K(n)_{∗}w is an isomorphism. By [Rav92a, Lemma 6.1.1], we may assume that K_{∗}w = v_{n}^{r} for some r. By examining the proofs, we see that r may be assumed to have the form p^{k}. It follows that 1 ∧ w ∈ End(X ∧ X)_{∗} also has K(m)_{∗}(1 ∧ w) = 0 for m ≠ n and K_{∗}(1 ∧ w) = v_{n}^{p^{k}}. Thus, by [Rav92a, Lemma 6.1.2], the map 1 ∧ w^{p^{i}} is central in End(X ∧ X)_{∗} for large i. After replacing w by w^{p^{i}}, we may assume that w is a strongly central self map of X. Write v = η^{∗}w ∈ π_{∗}X, so that v is a strongly central element. Note that v maps to a primitive element v′ in the MU_{∗}MU-comodule MU_{∗}X = MU_{∗}/I. As Prim(MU_{∗}/I_{n}) = F_{p}[v_{n}], we conclude that v′ = αv_{n}^{p^{k}} (mod I_{n}) for some α ∈ F_{p}. Recall that K_{∗}v = v_{n}^{p^{k}}; by comparing the unit maps, we conclude that α = 1. Using [HS, Lemma 3.4], we see that (v′)^{p^{i}} = v_{n}^{p^{k+i}} for i ≫ 0; we may replace v by v^{p^{i}} and thus assume that v′ = v_{n}^{p^{k}}. This means that v is a good v_{n} element, as required. Using Proposition 4.11, we see that S/(I, v_{n}^{p^{j}}) = X/v^{p^{j−k}} exists when j > k.

Corollary 4.14. For any ideal J ≤ MU_{∗} with radical I_{n}, there exists an ideal I = (v_{0}^{a_{0}}, …, v_{n−1}^{a_{n−1}}) such that I ≤ J and S/I exists.

Corollary 4.15. Any finite spectrum X of type at least n admits a good v_{n} self map.

Proof. Let 𝒞 be the category of those X that admit a good v_{n} self map. Spectra of type S/I (where I has height n) lie in 𝒞, so it will suffice to check that 𝒞 is thick. It is clearly closed under suspensions. Suppose that X ∈ 𝒞 and that e: X ⟶ X is idempotent. Write Y = eX and Z = (1 − e)X, so that X = Y ∨ Z. Choose a good v_{n} self map w: X ⟶ X. As w is central, it commutes with e, so it must have the form u ∨ v where u: Y ⟶ Y and v: Z ⟶ Z. It is easy to check that u and v are good v_{n} self maps, so 𝒞 is closed under retracts.

Now consider a cofibration X ⟶^{f} Y ⟶^{g} Z with X, Z ∈ 𝒞. For k ≫ 0, we can choose good v_{n} self maps u: X ⟶ X and w: Z ⟶ Z with |u| = |w| = |v_{n}^{p^{k}}|. We may also choose a compatible fill-in map v: Y ⟶ Y. After replacing u, v and w by suitable iterates, we may assume that v is strongly central. Indeed, u and w are strongly central, so 1 ∧ v − v ∧ 1 is nilpotent. It follows from [HS, Lemma 3.4] that v^{p^{j}} is strongly central for large j. For typographical convenience, we write x = v_{n}^{p^{k}}: MU ⟶ MU. We thus have a commutative diagram

MU ∧ X  MU ∧ Y  MU ∧ Z MU ∧ X  MU ∧ Y  MU ∧ Z^{1∧f} w_{_{x∧1} u}  ^{1∧g} w_{_{1∧v} u}  _{u_{x∧1}} w_{1∧f}  w_{1∧g}

Write d = 1∧v−x∧1 ∈ End(MU ∧Y)_{∗}. Note that d∘(1∧f) = 0 and (1∧g)∘d = 0, so that d can be written in the form (1 ∧ g) ∘ r and also in the form s ∘ (1 ∧ f). Thus d^{2} factors through (1 ∧ f)(1 ∧ g) = 0, so d^{2} = 0. As x ∧ 1 commutes with 1 ∧ v, the same is true of d. Thus we have

1 ∧ v^{p^{i}} = x^{p^{i}} ∧ 1 + p^{i}(x^{p^{i}−1} ∧ 1)d.

Moreover, we may assume that n > 0 so that End(MU ∧ Y)_{∗} is a torsion group. It follows that

1 ∧ v^{p^{i}} = x^{p^{i}} ∧ 1 = v_{n}^{p^{k+i}} ∧ 1

for i ≫ 0, as required.

Proof of Theorem 4.6. Let X be a finite spectrum of type at least n. We know from Corollary 4.15 that X admits a good v_{n} self map v, with |v| = |v_{n}^{p^{k}}| say. Let w be another such map. By the asymptotic uniqueness of v_{n} self maps [Rav92a, Lemma 6.1.3], we know that v^{N} = w^{M} for some N and M. By inspecting the proof (bearing in mind that MU_{∗}v and MU_{∗}w are powers of v_{n} and not merely unit multiples of powers of v_{n}) we see that M and N may be taken to be powers of p.

Now let M be a module-spectrum over MU in the classical sense, so M ∈ 𝒞_{MU}; let ν: M ∧MU ⟶ M be the structure map. Using the bijection 𝒞_{MU}(MU ∧X, MU ∧ X) = [X, MU ∧ X], we find that

η ∧ v = v_{n}^{p^{k}} ∧ 1: X ⟶ MU ∧ X.

By applying M ∧ (−) and composing with ν ∧ 1_{X}, we find that

1 ∧ v = v_{n}^{p^{k}} ∧ 1: M ∧ X ⟶ M ∧ X

as claimed.

On the other hand, let N be an MU-module in the strict sense, so N ∈ 𝒟_{MU}. We know that

1 ∧ v = v_{n}^{p^{k}} ∧ 1 ∈ 𝒞_{MU}(MU ∧ X, MU ∧ X) = 𝒟_{MU}(MU ∧ X, MU ∧ X).

We can apply the functor N ∧_{MU} (−): 𝒟_{MU} ⟶ 𝒟_{MU} to this equation and conclude that

1 ∧ v = v_{n}^{p^{k}} ∧ 1 ∈ 𝒟_{MU}(N ∧ X, N ∧ X)

as claimed.

Proposition 4.16. If S/I is a generalised Moore spectrum and X is an S/Imodule then S/I ∧ X is a wedge of finitely many suspended copies of X.

Proof. We may assume that I has nonzero height, so S/I = S/(J, v) for some generalised Moore spectrum S/J and some good v_{n} element v. Note that there is a map S/J ⟶ S/I of μ-spectra so X is a module over S/J. Thus X ∈ 𝒥_{S/J} and v_{X} is defined. By induction we may assume that W = X ∧ S/J is a wedge of suspended copies of X, and thus that X ∧ S/I is the cofibre of v: W ⟶ W. However, it is clear that v_{S/I} = 0 and X is a retract of S/I ∧ X so v_{X} = 0 so v_{W} = 0. It follows that X ∧ S/I is a wedge of suspended copies of X, as claimed.

Proposition 4.17. Let X be a spectrum. Then the following are equivalent:

(a) X is a retract of some spectrum of the form Y ∧Z, where Z is a finite spectrum of type at least n.

(b) There is a generalised Moore spectrum S/I of type n such that X is a module over S/I.

Moreover, the category of such X is an ideal.

Proof. (a)⇒(b): It is enough to show that Z is a module over some S/I of type n. By induction, we may assume that it is a module over some S/J of type n − 1. Let v be a good v_{n−1} self map of S/J. As K(n − 1)_{∗}Z = 0 and K(m)_{∗}v = 0 for all m ≠ n − 1, the Nilpotence Theorem tells us that v ∧ 1: S/J ∧ Z ⟶ S/J ∧ Z is nilpotent, say v^{p^{N}} ∧ 1 = 0. It follows that Z is a module over S/(J, v^{p^{N}}).

(b)⇒(a): Clear.

Suppose that X ⟶ Y ⟶ Z is a cofibration and that X and Z satisfy (b). We claim that Y satisfies (b) as well. By induction, we may assume that X, Y and Z are all modules over some S/J of type n − 1. Let v be a good v_{n−1} self map of S/J. By the argument above we see that v_{X} and v_{Z} are nilpotent, and it follows easily that v_{Y} is nilpotent, say v_{Y}^{p^{N}} = 0. It follows that Y is a module over S/(J, v^{p^{N}}). This shows that the category in question is thick. Using (a) it is trivial to see that it is an ideal.

It is sometimes convenient to know that S/I is self-dual.

Proposition 4.18. Let X be a spectrum of type S/I, and let d be the dimension of its top cell. Then there is an isomorphism Σ^{d}DX = X.

Proof. Write I = (J, v_{n}^{p^{k}}), so there is a generalised Moore spectrum Y of type S/J and a good v_{n} self map v of Y such that X = Y/v. We can assume by induction that Y is self-dual. Given any map f: U ⟶ V, Df is the composite

DV ⟶^{1∧η} DV ∧ U ∧ DU ⟶^{1∧f∧1} DV ∧ V ∧ DU ⟶^{ϵ∧1} DU

In particular, 1_{DY} = D(1_{Y}) = (ϵ ∧ 1) ∘ (1 ∧ η), so DY is in the category 𝒥_{Y} and we have a self map v_{DY} of DY. Using the naturality of v with respect to 1 ∧ η and ϵ ∧ 1 we see that v_{DY} = Dv. Now let t : Y ⟶ DY be a self-duality isomorphism, where we have omitted the suspension. By naturality, we have t ∘ v = Dv ∘ t. Hence t induces an isomorphism from Y/v to (a suspension of) DY/Dv = D(Y/v), as required.

4.1. Towers of generalised Moore spectra. We next try to build a tower of spectra S/I(j), where the I(j) all have height n, their intersection is trivial, and the maps are compatible with the unit maps S ⟶ S/I(j). We start with some generalities about towers (or more general pro-systems) of finite spectra. These ideas also appear in [CS96].

Let ℱ be the category of finite spectra. Its pro-completion is the dual of the category of those functors F: ℱ ⟶ Sets that can be written as a small filtered colimit of representable functors. Consider two such functors, say F Z = lim_{-→}_{i∈I}[X_{i}, Z] and

GZ = lim_{-→}_{j∈J}[Y_{j}, Z]. Using the Yoneda lemma, we see that

Pro(ℱ)(F, G) = [ℱ, Sets](G, F) = [ℱ, Sets](lim_{-→}_{j}[Y_{j}, −], lim_{-→}_{i}[X_{i}, −]) = lim_{←-}_{j} lim_{-→}_{i}[X_{i}, Y_{j}].

An alternate definition of Pro(ℱ) is as the category of pairs (I, X), where I is a small filtered category, {X_{i}} is an I^{op}-indexed system of finite spectra, and the morphisms from (I, X) to (J, Y) are given by the above formula. This clearly gives a category equivalent to the one described previously.

Note that the Yoneda functor X ⟶ [X, −] gives a full and faithful covariant embedding of ℱ in Pro(ℱ), which we think of as an inclusion. Every object in Pro(ℱ) is an inverse limit of objects of ℱ (because inverse limits in Pro(ℱ) are the same as direct limits in the functor category Pro(ℱ)^{op}).

There is a popular full subcategory Tow(ℱ) of Pro(ℱ) consisting of limits of towers (or equivalently, countable filtered systems with countably many morphisms) of representable functors. In this case we can be more explicit about the morphisms. Let Seq(ℱ) be the category of inverse sequences of finite spectra, like

X_{0} ⟵ X_{1} ⟵ X_{2} ⟵ ….

The maps from X_{•} to Y_{•} are just systems of maps f_{k}: X_{k} ⟶ Y_{k} making the evident diagrams commute. Let U be the directed set of strictly increasing maps N ⟶ N. If u ∈ U and X_{•} ∈ Seq(ℱ) then we define u^{∗}X_{•} ∈ Seq(ℱ) by (u^{∗}X_{•})_{k} = X_{u(k)}. The maps X_{u(k)} ⟵ X_{u(k+1)} are the evident composites of the maps X_{j} ⟵ X_{j+1}. It is not hard to check that

Tow(ℱ)(lim_{←-}_{i} X_{i}, lim_{←-}_{j} Y_{j}) = lim_{-→}_{U} Seq(ℱ)(u^{∗}X_{•}, Y_{•}).

In particular, a map f: X_{•} ⟶ Y_{•} becomes zero in the Pro category if and only if there exists u ∈ U such that the composite

X_{u(k+1)} ⟶ X_{u(k)} ⟶^{f_{u(k)}} Y_{u(k)}

is zero for all k. It is equivalent to say that for all i there exists j > i such that the composite X_{j} ⟶ X_{i} ⟶^{f_{i}} Y_{i} vanishes. In particular (taking f = 1) we see that lim_{←-}_{i} X_{i} = 0 in Pro(ℱ) if and only if for all i there exists j > i such that the map X_{j} ⟶ X_{i} is zero. Note that this certainly implies that holim_{←-} X_{i} = 0 in 𝒮, but is a much stronger statement.

It turns out that Pro(ℱ) is a rather familiar category in disguise.

Proposition 4.19. Pro(ℱ) is the dual of the category of homology theories (or equivalently, spectra mod phantoms). Moreover, Tow(ℱ) corresponds to the subcategory of homology theories with countable coefficients.

Proof. Any homology theory defined on finite spectra has an essentially unique extension to the category of all spectra, so it doesn’t matter which we meant.

As filtered colimits are exact, it is clear that any filtered colimit of representable functors is a homology theory. Conversely, let H be a homology theory. It is not hard to check that the category I of pairs (X, u) (where X ∈ ℱ and u ∈ HX) is filtered, and it is formal that H = lim_{-→}_{I}[X, −]. Moreover, I is essentially small because ℱ is.

If H has countable coefficients, then I is easily seen to be countable, so there is a cofinal functor N ⟶ I, so H ∈ Tow(ℱ). The converse is also clear.

Remark 4.20. As the smash product of a phantom map and any other map is phantom, there is an induced smash product on the category of spectra mod phantoms. We can transfer this to the equivalent category of homology theories, or the dual category Pro(ℱ). One can easily check that with this definition we have

(lim_{←-}_{i} X_{i}) ∧ (lim_{←-}_{j} Y_{j}) = lim_{←-}_{i,j} X_{i} ∧ Y_{j} = lim_{←-}_{i} lim_{←-}_{j} X_{i} ∧ Y_{j}.

Remark 4.21. One can see either directly from the definitions or from Proposition 4.19 that an object X ∈ Pro(ℱ) is zero if and only if [X, W] = 0 for all W ∈ ℱ, and that a map f: X ⟶ Y in Pro(ℱ) is an isomorphism if and only if the induced map [X, W] ⟵ [Y, W] is an isomorphism for all W ∈ ℱ.

It follows from the above proposition that Pro(ℱ) is not a triangulated category. Indeed, in a triangulated category every monomorphism is split, but any map of spectra with phantom fibre gives a non-split monomorphism of represented homology theories. Nonetheless, some features of triangulated categories remain. In particular, suppose that we have an inverse system of cofibre sequences X_{i} ⟶^{f_{i}} Y_{i} ⟶^{g_{i}} Z_{i} ⟶^{h_{i}} ΣX_{i}. Write X = lim_{←-}_{i} X_{i} ∈ Pro(ℱ) and so on, so we

have maps X ⟶^{f} Y ⟶^{g} Z ⟶^{h} ΣX. For any W ∈ ℱ we have an exact sequence [X_{i}, W] ⟵ [Y_{i}, W] ⟵ [Z_{i}, W] ⟵ [ΣX_{i}, W], and passage to direct limits gives an ex-

act sequence [X, W] ⟵^{f^{∗}} [Y, W] ⟵^{g^{∗}} [Z, W] ⟵^{h^{∗}} [ΣX, W]. In view of Remark 4.21, we see that f is an isomorphism if and only if Z = 0.

We next construct some special objects of Pro(ℱ). Fix an integer n and let ℱ_{n} be the category of finite spectra of type at least n. Let Λ_{n} be the category of pairs (X, u) where X ∈ ℱ_{n} and u: S ⟶ X. It is easy to see that Λ_{n}^{op} is a filtered category, so we have an object T_{n} = lim_{←-}_{(X,u)∈Λ_{n}} X of Pro(ℱ_{n}). It is formal to check that this

is the initial object in S ↓ Pro(ℱ_{n}).

Proposition 4.22. Fix an integer n. Then there exists a tower

… ⟶^{g_{3}} X_{2} ⟶^{g_{2}} X_{1} ⟶^{g_{1}} X_{0}

such that (a) X_{i} is a generalised Moore spectrum of type S/J(i) for some ideal J(i) of height

n.

(b) We have g_{i} ∘ η_{i} = η_{i−1} for all i, where η_{i}: S ⟶ X_{i} is the unit map of the μ-spectrum X_{i}.

(c) ⋂_{i} J(i) = 0.

(d) For any Z ∈ ℱ_{n} we have Z = lim_{←-}_{i} Z ∧ X_{i} in Pro(ℱ_{n}).

Note that the unit maps η_{i} give a map from S to lim_{←-}_{i} X_{i} in Pro(ℱ), which is

implicitly used in (d). For any such tower, the corresponding object of S ↓ Pro(ℱ_{n}) is isomorphic to T_{n}.

Remark 4.23. We shall often talk about towers of the form

… ⟶ S/J(3) ⟶ S/J(2) ⟶ S/J(1) ⟶ S/J(0),

by which we mean a tower with the properties mentioned in the above proposition. We will also use notation like holim_{←-}_{J} S/J to refer to the homotopy inverse limit of

any such tower. This depends only on the isomorphism class of the tower in Pro(ℱ) and thus is independent of the choices made.

Proof. We assume that n > 1 and that the result holds for n−1; small modifications are required for n = 1, but we leave them to the reader. By induction, there is a tower {W_{i}} of the required type for the n − 1 case. Choose a spectrum Y of type n. Write W′_{i} for the cofibre of the map S ⟶ W_{i}. These cofibres can be assembled into a tower. Because Y ⟶ lim_{←-}_{i} Y ∧ W_{i} is an isomorphism, we see that lim_{←-}_{i} Y ∧ W′_{i} = 0.

After replacing {W_{i}} and {W′_{i}} by cofinal subtowers, we may assume that the maps Y ∧ W′_{i} ⟶ Y ∧ W′_{i−1} are all zero.

We shall recursively build a tower {X_{i}} and cofibrations as shown below, such that w_{i} is a good v_{n−1} self map, and 1_{Y} ∧ w_{i} = 0: Y ∧ W_{i} ⟶ Y ∧ W_{i}.

W_{i}  W_{i}  X_{i}  ΣW_{i} W_{i−1}  W_{i−1}  X_{i−1}  ΣW_{i−1}^{w_{i}} w_{_{w_{i−1}^{p^{k}−1}∘f_{i}} u}  w_{_{f_{i}} u}  w_{u_{g_{i}}}  _{u_{w_{i−1}^{p^{k}−1}∘f_{i}}} w_{w_{i−1}}  w  w

Suppose we have constructed all this up to the (i − 1)’th level. Choose a good v_{n} self map w_{i} of W_{i}. After replacing w_{i} by w_{i}^{p^{j}} for some j > 0, we may assume that w_{i} is compatible with w_{i−1}, say f_{i} ∘ w_{i} = w_{i−1}^{p^{k}} ∘ f_{i} for some k. We may also assume that k > 0 and 1_{Y} ∧ w_{i} = 0. We let X_{i} be the cofibre of w_{i}; the usual comparison of cofibrations gives a map g_{i} as shown. Moreover, X_{i} can be made into a μ-spectrum of type S/J(i) for a suitable ideal J(i), as in Proposition 4.11. Thus (a) holds. The maps W_{i} ⟶ X_{i} are maps of μ-spectra and thus carry units to units. By the induction hypothesis, f_{i} carries the unit of W_{i} to that of W_{i−1}. It follows that g_{i} carries the unit of X_{i} to that of X_{i−1}, so (b) holds. The assumption k > 0 above ensures that |w_{i}| ⟶ ∞ as i ⟶ ∞, and thus that ⋂_{i} J(i) = 0, so (c) holds.

Clearly, when we smash the towers at either end of the above diagram with Y, all the maps become zero. It follows that lim_{←-}_{i} Y ∧ X_{i} = lim_{←-}_{i} Y ∧ W_{i} = Y in Pro(ℱ), as claimed.

It is not hard to check that {Z | Z = lim_{←-}_{i} Z ∧ X_{i}} is a thick subcategory of ℱ, so

it contains all of ℱ_{n}. This proves (d).

Similarly, for Z ∈ ℱ_{n} we have

Pro(ℱ)(lim_{←-}_{i} X_{i}, Z) = Pro(ℱ)(lim_{←-}_{i} DZ ∧ X_{i}, S) = Pro(ℱ)(DZ, S) = [S, Z].

It follows that for any object Y = lim_{←-}_{j} Z_{j} ∈ Pro(ℱ_{n}) we have

Pro(ℱ)(lim_{←-}_{i} X_{i}, Y) = lim_{←-}_{j}[S, Y_{j}] = Pro(ℱ)(S, Y).

This means that lim_{←-}_{i} X_{i} is initial in S ↓ Pro(ℱ_{n}), so lim_{←-}_{i} X_{i} = T_{n}.

Corollary 4.24. We have T_{n}∧T_{n} = T_{n} in Pro(ℱ) (because T_{n}∧T_{n} = lim_{←-}_{I} S/I ∧T_{n}

and S/I ∧ T_{n} = S/I).

Remark 4.25. In any tower as above, the spectrum X_{i} = S/J(i) can be made into a module over X_{j} for j > i. Indeed, as X_{i} is a μ-spectrum we know that the map

X_{i} ⟶^{η_{i}∧1} X_{i} ∧ X_{i} is a split monomorphism. This can be factored as X_{i} ⟶^{η_{j}∧1} X_{j} ∧ X_{i} ⟶^{g∧1} X_{i} ∧ X_{i}, so η_{j} ∧ 1 is a split monomorphism, as required.

5. Bousfield classes

In this section, we recall some known results about Bousfield classes that we need in the rest of this paper. Recall that in a stable homotopy category 𝒞, the Bousfield class ⟨X⟩ of an object X is the collection of X-local objects. We order Bousfield classes by inclusion. (It is more traditional to define ⟨X⟩ to be the category of Xacyclic objects, and to order Bousfield classes by reverse inclusion. Either approach gives the same lattice of Bousfield classes, but the former has some conceptual advantages.) We begin with the crucial result of Hopkins and Ravenel.

Theorem 5.1 ([Rav92a, Theorem 7.5.6]). The localisation functor L is smashing. That is, the natural map LS ∧ X ⟶ LX is an isomorphism.

Note that the Bousfield class in ℒ of an L-local spectrum X is the same as the Bousfield class in 𝒮, since any X-local spectrum is L-local. Indeed, since L is a smashing localisation functor, if W is L-acyclic, then X ∧ W = X ∧ LS ∧ W = X ∧ LW = 0.

If X and Y have the same Bousfield class, we write X ∼ Y.

Recall that C_{j} is the acyclisation functor corresponding to L_{j}, and M_{j} = C_{j−1}L_{j} is the fibre of the natural map L_{j} ⟶ L_{j−1} = L_{j−1}L_{j}.

Lemma 5.2. We have L̂E(n) = E.

Proof. Let F be the fibre of the evident map E(n) ⟶ E. Choose a generalised Moore spectrum S/I of type n. By Landweber exactness we have E(n)_{∗}(S/I) = E(n)_{∗}/I and E_{∗}(S/I) = E_{∗}/I. But E(n)_{∗}/I = E_{∗}/I so F ∧ S/I = 0. Thus K_{∗}(F) ⊗_{K_{∗}} K_{∗}(S/I) = 0 and K_{∗}(S/I) ≠ 0 so K_{∗}F = 0. Thus, the map E(n) ⟶ E is a K-equivalence. On the other hand, one can use the techniques of [EKMM96, Chapter V] to build a tower of spectra E/I ∈ 𝒟_{MU} with π_{∗}(E/I) = E_{∗}/I. There is a natural map from E to the sequential limit of this tower, which is easily seen to be an isomorphism. The spectra E/I lie in the thick subcategory generated by E/I_{n} = K, so the sequential limit is K-local. Thus E is a K-local spectrum that is K_{∗}-equivalent to E(n), so E = L̂E(n).

Proposition 5.3. We have the following equalities of Bousfield classes:

⟨LS⟩ = ⟨L̂S⟩ = ⟨E(n)⟩ = ⟨E⟩ = ⟨K(0)⟩ ∨ … ∨ ⟨K(n)⟩
⟨L_{j}S⟩ = ⟨K(0)⟩ ∨ … ∨ ⟨K(j)⟩
⟨C_{j}LS⟩ = ⟨LF (j + 1)⟩ = ⟨K(j + 1)⟩ ∨ … ∨ ⟨K(n)⟩
⟨M_{j}S⟩ = ⟨K(j)⟩ = ⟨LT (j)⟩

Proof. It is proved in [Rav84, Theorem 2.1] that ⟨E(n)⟩ = ⟨K(0)⟩ ∨ … ∨ ⟨K(n)⟩. By Theorem 5.1, LX = LS ∧ X. It is immediate from this that ⟨LS⟩ = ⟨E(n)⟩. By Landweber exactness we have E_{∗}X = E_{∗} ⊗_{E(n)_{∗}} E(n)_{∗}X, and E_{∗} is faithfully flat over E(n)_{∗} so ⟨E⟩ = ⟨E(n)⟩. We have ring maps LS ⟶ L̂S ⟶ L̂E(n) = E so ⟨E⟩ ≤ ⟨L̂S⟩ ≤ ⟨LS⟩ but we have seen that ⟨E⟩ = ⟨LS⟩ so ⟨L̂S⟩ = ⟨LS⟩ also.

Next, we claim that ⟨C_{j}LS⟩ = ⟨K(j + 1)⟩ ∨ … ∨ ⟨K(n)⟩. Indeed, because L_{j} is smashing, we know that

C_{j}LS = C_{j}S ∧ LS ∼ C_{j}K(0) ∨ … ∨ C_{j}K(n)

If i ≤ j then K(i) is E(j)-local. Indeed, K(i) is self-local, and ⟨K(i)⟩ ≤ ⟨E(j)⟩, so K(i) is also E(j)-local. Hence C_{j}K(i) = 0. If j < i ≤ n then

L_{j}K(i) = L_{j}S ∧ K(i) ∼ (K(0) ∧ K(i)) ∨ … ∨ (K(j) ∧ K(i)).

We know from [Rav84, Theorem 2.1] that K(k) ∧ K(i) = 0 unless i = k. Thus, if j < i ≤ n, then L_{j}K(i) = 0 and C_{j}K(i) = K(i). It follows that C_{j}LS ∼ K(j + 1) ∨ … ∨ K(n) as claimed. Clearly M_{j}S = C_{j−1}L_{j}S, so ⟨M_{j}S⟩ = ⟨K(j)⟩.

By the definition of a type j spectrum, we have K(i) ∧ F (j) = 0 if and only if i < j. Moreover, if i ≥ j then K(i) ∧ F (j) is a nontrivial wedge of suspensions of K(i), hence Bousfield equivalent to K(i). It follows that

LF (j) ∼ (K(0) ∧ F (j)) ∨ … ∨ (K(n) ∧ F (j)) ∼ K(j) ∨ … ∨ K(n)

Now let v : Σ^{d}F (j) ⟶ F (j) be a v_{j}-self map, so that v^{−1}F (j) = T (j). By definition, v induces a nilpotent endomorphism of K(i)_{∗}F (j) when i ≠ j, so that K(i)_{∗}T (j) = 0. It follows easily that ⟨LT (j)⟩ ≤ ⟨K(j)⟩. On the other hand, the Telescope Lemma [Rav84, Lemma 1.34] says that LF (j) ∼ LT (j) ∨ LF (j + 1). By smashing with K(j) we deduce that ⟨K(j)⟩ ≤ ⟨LT (j)⟩ and thus that K(j) ∼ LT (j).

6. The E(n)-local category

In this section, we discuss the general properties of ℒ. Some of this material also appears in [Dev96a] or [HS95].

Proposition 6.1. ℒ is a monogenic stable homotopy category in the sense of [HPS95]. In particular, it is a triangulated category with coproducts and compatible smash products and function spectra, giving a closed symmetric monoidal structure. Coproducts, minimal weak colimits, smash products and function spectra are the same as in 𝒮. The unit for the smash product is LS, which is a small graded weak generator. This last statement means that for any family of spectra X_{i} ∈ ℒ we have

[LS, ⋁ X_{i}] = ⨁[LS, X_{i}],

and that if X ∈ ℒ satisfies [LS, X]_{∗} = 0 then X = 0.

Proof. Apply [HPS95, Theorems 3.5.1 and 3.5.2].

Theorem 6.2.

(a) Homology and cohomology functors ℒ ⟶ Ab are representable.

(b) Any spectrum X ∈ ℒ is the minimal weak colimit of LΛ(X), where Λ(X) is the category of finite ordinary spectra over X.

(c) Suppose that X ∈ ℒ. The following are equivalent: 1. X is small.

2. X is F -small.

3. X is LS-finite.

4. X is dualisable.

5. X is a retract of LY for some finite spectrum Y ∈ 𝒮.

See Definition 1.5 for definitions.

(d) Every spectrum in ℒ is E(n)-nilpotent (Definition 1.4).

Proof. Part (a) follows from [HPS95, Theorem 3.5.2] and [HPS95, Theorem 4.1.5]. For part (b), recall that X is the minimal weak colimit in 𝒮 of Λ(X), by [HPS95, Theorem 4.2.4] or [Mar83, Chapter 5]. It follows from part (e) of [HPS95, Proposition 2.2.4] that X = LS∧X is the minimal weak colimit in 𝒮 of LS∧Λ(X) = LΛ(X). Moreover, minimal weak colimits are the same in 𝒮 or ℒ. For part (c), Theorem 2.1.3 of [HPS95] implies that the first four conditions are equivalent. Moreover, if Y ∈ 𝒮 is finite, then it is easy to see that LY (and hence, any retract of LY) is LS-finite. Conversely, suppose that X ∈ ℒ is small, so that [X, −] is a homology theory on ℒ. It follows from part (b) that [X, X] = lim_{-→}_{Y ∈Λ(X)}[X, LY], so the identity map

1_{X} ∈ [X, X] factors through LY, in other words, X is a retract of LY. Part (d) is proved in [HS95].

Proposition 6.3. Suppose that X, Y ∈ ℒ are small. Then [X, Y]_{∗} is countable.

Proof. By an evident thick subcategory argument, we need only show that π_{∗}LS is countable. This is proved in [Hov95a, Lemma 5.5] (by showing that the E_{1} term of the Adams-Novikov spectral sequence converging to π_{∗}LS is countable in each bidegree, and that the E_{∞} term has only finitely many lines).

For calculations when n = 1, see [Bou79] or [Rav84]. For the case n = 2, see the many papers by Katsumi Shimomura and his coworkers, for example [SY95]. See also Section 15.

We can now prove a convergence theorem for the modified Adams spectral sequence described in Section 2.2.

Lemma 6.4. Let 𝒞 be a stable homotopy category, and R a ring object in 𝒞. Then the ideal of R-nilpotent objects is the same as the thick subcategory generated by the R-injective objects.

Proof. Let 𝒩 be the category of R-nilpotent spectra, which is by definition the ideal generated by R. Let ℐ be the category of R-injectives, and 𝒩′ the thick subcategory generated by ℐ. Write

𝒩′′ = {X | X ∧ Y ∈ 𝒩′ for all Y }.

Clearly ℐ ⊆ 𝒩 and 𝒩 is thick so 𝒩′ ⊆ 𝒩. Clearly R ∈ 𝒩′′ and 𝒩′′ is an ideal so 𝒩 ⊆ 𝒩′′. By putting Y = S in the definition we see that 𝒩′′ ⊆ 𝒩′. Thus 𝒩′ = 𝒩 = 𝒩′′ as required.

Proposition 6.5. There are constants r_{0}, s_{0} such that for all spectra X and Y, in the modified Adams spectral sequence

Ext_{E(n)_{∗}E(n)}^{s,t}(E(n)_{∗}X, E(n)_{∗}Y) ⟹ [X, LY]_{t−s}

(see Section 2.2) we have E_{r}^{s,t} = 0 whenever r ≥ r_{0} and s ≥ s_{0}. Moreover, the spectral sequence converges.

Proof. Let i: \overline{E(n)} ⟶ LS be the fibre of the unit map LS ⟶ E(n). It is easy to see that i ∧ 1_{E(n)} : \overline{E(n)} ∧ E(n) ⟶ E(n) is null, and thus that i ∧ 1_{I} = 0 for any E(n)-injective spectrum I. It follows using Lemma 6.4 that for any E(n)-nilpotent

spectrum X we have i^{(N)} ∧ 1_{X} = 0: \overline{E(n)}^{(N)} ∧ X ⟶ X for some N. However, we

know that LS is E(n)-nilpotent, so that i^{(N)} : \overline{E(n)}^{(N)} ⟶ LS is null for some N. Now suppose that X is such that E(n)_{∗}X is a projective module over E(n)_{∗}. We then have an ordinary Adams spectral sequence for [X, LY]_{∗} obtained by mapping X to the tower

LY ⟵ \overline{E(n)} ∧ LY ⟵ \overline{E(n)}^{(2)} ∧ LY ⟵ ….

As every composite in the tower of length at least N vanishes, one can show that the spectral sequence converges, and that it stops at the E_{N} page with a flat vanishing line. One also knows from [Dev96b, Proposition 1.9 and Remark 1.10] that this spectral sequence is isomorphic to the modified Adams spectral sequence from the E_{2} page onwards, and that the resulting filtrations of [X, LY]_{∗} are also the same. In particular, in the filtration arising from the MASS we have F^{s} = 0 for s ≥ N.

Now choose a tower LY = Y^{0} ⟵ Y^{1} ⟵ Y^{2} ⟵ … of the type used in the MASS. By the above, we see that when E(n)_{∗}X is projective, the map [X, Y^{s}]_{∗} ⟶ [X, Y^{0}]_{∗} is null when s ≥ N. Moreover, if we truncate the tower below Y^{t} we get a tower of the type that computes the MASS for [X, Y^{t}]_{∗} so we can apply the same logic and deduce that the map [X, Y^{t+s}]_{∗} ⟶ [X, Y^{t}]_{∗} is null for s ≥ N.

Now consider a general spectrum X. By the methods of [Ada74, Theorem 13.6] (taking account of Proposition 2.13) we can choose a tower

X = X_{0} ⟶ X_{1} ⟶ … ⟶ X_{n}

and triangles Σ^{−1}X_{k+1} ⟶ P_{k} ⟶ X_{k} such that E(n)_{∗}P_{k} is projective and the map E(n)_{∗}P_{k} ⟶ E(n)_{∗}X_{k} is surjective. As E(n)_{∗} = Z_{(p)}[v_{1}, …, v_{n}^{±1}] has global dimension n (in the graded sense) we see that E(n)_{∗}X_{n} is projective. It is not hard to show by induction that [X_{n−j}, Y^{t+(j+1)N}]_{∗} ⟶ [X_{n−j}, Y^{t}]_{∗} is null and thus that [X, Y^{t+(n+1)N}]_{∗} ⟶ [X, Y^{t}]_{∗} is null. It follows that the MASS for [X, LY]_{∗} converges, and that it stops with a flat vanishing line at the E_{(n+1)N} page.

Jens Franke has shown [Fra96, Theorem 10] that when 2p − 2 > n^{2} + n, the category ℒ is equivalent to a purely algebraically defined derived category of periodic complexes of comodules. He shows that (in contrast to Quillen’s algebraicisation of rational unstable homotopy) the equivalence does not capture all the higherorder homotopical phenomena of the model category underlying ℒ, but that the approximation improves as p increases. His result implies for example that there is an ℒ-small spectrum X with E(n)_{∗}X = K(n)_{∗}. There is some reason to suspect that this is not of the form LY for any finite spectrum Y (although it must be a retract of some such LY) but the situation is rather unclear at the moment. Ethan Devinatz has also constructed the spectrum X by a more direct argument.

6.1. Nilpotence and thick subcategories. In this situation, we have the simple nilpotence theorem proved in [HPS95, Section 5]. Recall that a map f : X ⟶ Y is called smash nilpotent if f^{∧m} : X^{∧m} ⟶ Y^{∧m} is null for large enough m.

Corollary 6.6 (Nilpotence theorem).

(a) Let F be an ℒ-small spectrum, and X an arbitrary spectrum in ℒ. A map f : F ⟶ X is smash nilpotent if and only if K(i)_{∗}f = 0 for all i ≤ n.

(b) Let X be an ℒ-small spectrum. A map f : X ⟶ X is nilpotent if and only if K(i)_{∗}f is nilpotent for all i ≤ n.

(c) Let R be a ring spectrum in ℒ. An element α ∈ π_{∗}R is nilpotent if and only if K(i)_{∗}α is nilpotent in the ring K(i)_{∗}R for all i ≤ n.

Proof. Most of this is proved in [HPS95, Section 5]. The only thing left to prove is that if f : F ⟶ X is a smash nilpotent map from an ℒ-small spectrum F to X, then K(i)_{∗}f = 0. This follows easily from the Künneth theorem since K(i)_{∗}(f^{∧N}) = (K(i)_{∗}f)^{⊗N}.

Note that this nilpotence theorem follows directly from the Bousfield decomposition of LS, but that Bousfield decomposition depends on the smashing theorem of Hopkins-Ravenel, whose proof requires the much more difficult nilpotence theorem of Devinatz-Hopkins-Smith [DHS88].

The nilpotence theorem can be used to classify thick subcategories (Definition 1.2) of ℒ-small spectra. For this, we need a definition of the support of a spectrum or full subcategory of ℒ.

Definition 6.7. If 𝒳 is any class of spectra (in particular if 𝒳 ⊆ ℒ), we write

K(m) ∧ 𝒳 = {K(m) ∧ X | X ∈ 𝒳}
supp(𝒳) = {m | K(m) ∧ 𝒳 ≠ {0}}.

We refer to supp(𝒳) as the support of 𝒳. Similarly, the cosupport of 𝒳 is

cosupp(𝒳) = {m | F (K(m), 𝒳) ≠ {0}}.

We have to classify the possible supports of ℒ-small spectra. Since not every such spectrum is the localisation of an ordinary finite spectrum, this does not immediately follow from knowledge of possible supports in 𝒮. However, the same result does hold.

Proposition 6.8. If X is small in ℒ and j < n, then dim K(j)_{∗}X ≤ dim K(j + 1)_{∗}X. Thus supp(X) = {m, m + 1, …, n} for some m with 0 ≤ m ≤ n + 1 (where the case m = n + 1 means supp(X) = 0 and thus X = 0).

Proof. The proof is the same as that of [Rav84, Theorem 2.11].

Now, given an integer j such that 0 ≤ j ≤ n + 1, let 𝒞_{j} denote the thick subcategory of ℒ consisting of all small spectra X such that K(i)_{∗}X = 0 for all i < j. Then we have

𝒞_{0} ⊇ 𝒞_{1} ⊇ ⋯ ⊇ 𝒞_{n+1} = {0}.

Note that 𝒞_{0} consists of all ℒ-small spectra. All of the inclusions are strict, since LF (j) is in 𝒞_{j} but not 𝒞_{j+1}.

We now have the following theorem, which is a corollary of the general thick subcategory theorem proved in [HPS95, Section 5] and the classification of supports proved above.

Theorem 6.9 (Thick Subcategory Theorem). If 𝒞 is a thick subcategory of ℒ, then 𝒞 = 𝒞_{j} for some j such that 0 ≤ j ≤ n + 1.

The telescope conjecture also holds in ℒ. More explicitly, consider the following three functors:

L_{i} : ℒ ⟶ ℒ  =  localisation with respect to K(0) ∨ … ∨ K(i)
L_{i}^{lf} : ℒ ⟶ ℒ  =  finite localisation away from LF (i + 1)
L_{i}^{f} : 𝒮 ⟶ 𝒮  =  finite localisation away from F (i + 1)

(See [HPS95, Section 3.3] for discussion of finite localisation). The following corollary is the telescope conjecture for ℒ.

Corollary 6.10. If L′ is a smashing localisation functor on ℒ, then L′ = L_{i} for some i. Moreover, L_{i} = L_{i}^{lf} = LL_{i}^{f}.

Proof. Since L′ is smashing, L′ is Bousfield localisation with respect to L′S. Note that if L′S ∧ K(j) is nonzero then it is a wedge of suspensions of K(j) and thus has the same Bousfield class as K(j). We thus have

⟨L′S⟩ = ⟨L′S⟩ ∧ ⋁_{j=0}^{n} ⟨K(j)⟩ = ⋁_{j∈supp(L′S)} ⟨K(j)⟩.

Now, if j ∈ supp(L′S), then L_{K(j)}S is a module over L′S so we have

⟨L′S⟩ ≥ ⟨L_{K(j)}S⟩ = ⟨K(0)⟩ ∨ … ∨ ⟨K(j)⟩

(using Proposition 5.3). Thus L′ = L_{i} for the largest i in the support of L′S.

Next, recall that there is a cofibre sequence C_{i}^{f}S ⟶ S ⟶ L_{i}^{f}S, in which C_{i}^{f}S has a cellular tower built from F (i + 1) (see [HPS95, Theorem 3.3.3]). Moreover, [F (i + 1), L_{i}^{f}S]_{∗} = 0, so DF (i + 1) ∧ L_{i}^{f}S = 0. It follows that LC_{i}^{f}S has a cellular tower built from LF (i + 1), so it is L_{i}^{lf}-acyclic. Moreover,

[LF (i + 1), LL_{i}^{f}S]_{∗} = [LS, L(DF (i + 1) ∧ L_{i}^{f}S)]_{∗} = 0.

It follows that LL_{i}^{f}S is L_{i}^{lf}-local, so LL_{i}^{f} = L_{i}^{lf}.

We know that L_{i}^{lf} is again a smashing localisation. Thus, to see that L_{i}^{lf} = L_{i}, it is enough to check that supp(L_{i}^{lf}S) = {0, …, i}. We know from Proposition 5.3 that L_{i}^{f}S ∼ T (0) ∨ … ∨ T (i), so

L_{i}^{lf}S ∼ LT (0) ∨ … ∨ LT (i) ∼ K(0) ∨ … ∨ K(i)

The claim follows.

Corollary 6.11. If X ∈ 𝒞_{j} then X is a retract of LY for some finite spectrum Y of type at least j.

Proof. We have seen that X lies in the thick subcategory generated by LF (i), so X = C_{i}^{lf}X, but we have also seen that this is the same as LC_{i}^{f}X, which is a minimal weak colimit of terms of the form LY. As X is small, we conclude that it is a retract of one of these terms.

We conclude this section by showing that the theory of v_{j} self-maps (defined by the evident generalisation of Definition 4.5) works as expected in ℒ.

Theorem 6.12. Every spectrum X ∈ 𝒞_{j} admits a good v_{j} self map .

Proof. Choose a finite spectrum Y of type at least j such that X is a retract of LY = LS ∧ Y, so that X lies in the category 𝒥_{Y} of Definition 4.2. Choose a good v_{j} self map v: Σ^{d}Y ⟶ Y. Proposition 4.4 gives an induced map v_{X}: Σ^{d}X ⟶ X, and it is easy to check that this is a good v_{j} self map.

6.2. Localising and colocalising subcategories. In this section, we classify the localising and colocalising subcategories (Definition 1.3) of ℒ . Every such category is determined by the Morava K-theories which it contains.

We write loc⟨𝒳⟩ and coloc⟨𝒳⟩ for the localising and colocalising subcategories generated by a class 𝒳 of spectra. We also write

𝒳^{⊥} = {Y | [𝒳, Y]_{∗} = 0}
⊥𝒳 = {Y | [Y, 𝒳]_{∗} = 0}

(where [𝒳, Y]_{∗} = {[X, Y]_{∗} | X ∈ 𝒳} and so on). Note that 𝒳^{⊥} is a colocalising subcategory and ^{⊥}𝒳 is a localising subcategory. Also note that every (co)localising subcategory is a (co)ideal, since ℒ is monogenic.

Definition 6.13.

(a) We write 𝒫 for the lattice of subsets of {0, …, n}. For any S ∈ 𝒫 we write S^{c} = {0, …, n} \ S.

(b) K(S) = ⋁_{m∈S} K(m)

(c) 𝒞_{S} = {X ∈ ℒ | K(S^{c})_{∗}X = 0} = {K(S^{c})–acyclic spectra in ℒ}

(d) 𝒟_{S} = {K(S)–local spectra} ⊆ ℒ

Theorem 6.14.

(a) The lattice of localising subcategories is isomorphic to 𝒫, via the maps S ↦ 𝒞_{S} and 𝒞 ↦ supp(𝒞).

(b) The lattice of colocalising subcategories is isomorphic to 𝒫, via the maps S ↦ 𝒟_{S} and 𝒟 ↦ cosupp(𝒟).

(c) 𝒞_{S} = loc⟨K(S)⟩ = {X ∈ ℒ | ⟨X⟩ ≤ ⟨K(S)⟩}.

(d) 𝒟_{S} = coloc⟨K(S)⟩ = {X ∈ ℒ | ⟨X^{∗}⟩ ≤ ⟨K(S)^{∗}⟩ = ⟨K(S)⟩}.

(e) 𝒞_{S^{c}} = ^{⊥}𝒟_{S} and 𝒟_{S^{c}} = 𝒞_{S}^{⊥}.

The notation ⟨X^{∗}⟩ denotes the cohomological Bousfield class of X, the collection of X^{∗}-local spectra. A spectrum Y is said to be X^{∗}-local if [Z, Y]_{∗} = 0 for all spectra Z such that [Z, X]_{∗} = 0.

This theorem is similar to the analogous theorem proven as [HPS95, Corollary 6.3.4] for a Noetherian stable homotopy category (with some hypotheses). However, we have been unable to find a worthwhile common generalisation of the two theorems. We shall prove the theorem after some intermediate results.

Proposition 6.15. If X is a finite spectrum of type at least n, then LX is Knilpotent.

Proof. By a thick subcategory argument, it is sufficient to verify this for X = S/I. We know from [HS95] that LS is E(n)-nilpotent, and {Y | Y ∧S/I ∈ ideal⟨E(n)/I⟩} is an ideal containing E(n), so it contains LS. Thus, LS/I lies in the ideal generated by E(n)/I, which is easily seen to be the same as the ideal generated by K.

Remark 6.16. We shall show in Corollary 8.12 that the localisation of a finite spectrum of type n actually lies in the thick subcategory generated by K.

Proposition 6.17. For any X ∈ 𝒮, the spectrum M_{m}X = M_{m}S ∧ X lies in the localising subcategory generated by K(m), and F (M_{m}S, X) lies in the colocalising subcategory generated by K(m).

Proof. Observe that M_{m}X = C_{m−1}L_{m}X, which is L_{m−1}-acyclic. Recall that the functor L_{m−1} : ℒ_{m} ⟶ ℒ_{m−1} is simply finite localisation away from L_{m}F (m) (Corollary 6.10), so that the L_{m−1}-acyclics in ℒ_{m} are precisely loc⟨L_{m}F (m)⟩. Because L_{m}F (m) is K(m)-nilpotent (Proposition 6.15), this is contained in loc⟨K(m)⟩, so M_{m}X ∈ loc⟨K(m)⟩.

The spectrum F (K(m), X) is a K(m)-module, hence a wedge of suspensions of K(m). By part (f) of Proposition 3.3, the wedge splits off as a retract of the product, so it lies in coloc⟨K(m)⟩. The subcategory {Y | F (Y, X) ∈ coloc⟨K(m)⟩} is localising and contains K(m), so it contains M_{m}S. Thus F (M_{m}S, X) ∈ coloc⟨K(m)⟩.

Proposition 6.18. For any spectrum X ∈ ℒ we have

X ∈ loc⟨X⟩ = loc⟨K(supp(X))⟩

and

X ∈ coloc⟨X⟩ = coloc⟨K(cosupp(X))⟩.

Proof. We have a tower of spectra as follows, in which the vertical maps are the fibres of the horizontal maps.

M_{n}S  M_{n−1}S  M_{1}S  M_{0}S
↓  ↓  ↓  ↓
L_{n}S ⟶ L_{n−1}S  …  L_{1}S ⟶ L_{0}S ⟶ 0

After smashing this with X, we see that X ∈ loc⟨M_{0}X, …, M_{n}X⟩. Because ⟨M_{m}S⟩ = ⟨K(m)⟩ (by Proposition 5.3), we see that M_{m}X = 0 unless m ∈ supp(X). On the other hand, when m ∈ supp(X) we have M_{m}X ∈ loc⟨K(m)⟩ by Proposition 6.17. It follows easily that X ∈ loc⟨K(m) | m ∈ supp(X)⟩ = loc⟨K(supp(S))⟩, so loc⟨X⟩ ⊆ loc⟨K(supp(S))⟩. Moreover, if m ∈ supp(S) then loc⟨X⟩ contains K(m) ∧ X, which is a nonzero wedge of suspensions of K(m), so loc⟨X⟩ contains K(m). It follows that loc⟨X⟩ = loc⟨K(supp(S))⟩. For the colocalising case, we apply the functor F (−, X) to the above tower and argue in the same way.

Proof of Theorem 6.14. It is clear that S ↦ 𝒞_{S} and S ↦ 𝒟_{S} are order preserving maps from 𝒫 to the lattices of localising and colocalising subcategories. Next, recall from Proposition 5.3 that K(m)_{∗}K(n) ≠ 0 if and only if m = n. Using this, we see that K(S) ∈ 𝒞_{S}, and that supp(𝒞_{S}) = S. Moreover, because K(m)^{∗}X = Hom_{K(m)_{∗}}(K(m)_{∗}X, K(m)_{∗}), we see that K(m)^{∗}K(n) ≠ 0 if and only if m = n. Using this and the fact that K(S) ∈ 𝒟_{S}, we see that cosupp(𝒟_{S}) = S.

It is immediate from Proposition 6.18 that any localising subcategory 𝒞 ⊆ ℒ satisfies 𝒞 = loc⟨K(supp(𝒞))⟩, and similarly for colocalising subcategories. In particular, 𝒞_{S} = loc⟨K(S)⟩. Thus, for any 𝒞, we have 𝒞 = 𝒞_{supp(𝒞)} . This proves part (a) of the theorem, and part (b) is similar.

We have already proved the first equality in part (c). For the second, write 𝒞 = {X ∈ ℒ | ⟨X⟩ ≤ ⟨K(S)⟩}. It is easy to see that this is a localising subcategory. Suppose that ⟨X⟩ ≤ ⟨K(S)⟩. As K(S)∧K(S^{c}) = 0, we conclude that X∧K(S^{c}) = 0, so that X ∈ 𝒞_{S}. Thus 𝒞 ⊆ 𝒞_{S}. On the other hand, it is clear that K(S) ∈ 𝒞, so 𝒞_{S} = loc⟨K(S)⟩ ⊆ 𝒞. Thus 𝒞 = 𝒞_{S} as claimed. The proof of part (d) is similar.

For the first part of (e), observe that K(S)^{∗}X = 0 if and only if K(S)_{∗}X = 0, so 𝒞_{S^{c}} = {X | [X, K(S)]_{∗} = K(S)^{∗}X = 0}. In other words,

𝒞_{S^{c}} = ^{⊥}{K(S)} = ^{⊥} coloc⟨K(S)⟩ = ^{⊥}𝒟_{S}.

For the second part, recall that by definition

𝒟_{S^{c}} = {K(S)–local spectra} = {K(S)–acyclics}^{⊥} = 𝒞_{S}^{⊥}.

Note that Theorem 6.14 implies that we can localise with respect to any localising subcategory of ℒ (because we can localise with respect to the homology theory K(S)_{∗}).

6.3. The monochromatic category. As mentioned in the introduction, there are two different ways to investigate the difference between the E(n)-local category and the E(n−1)-local category. The first is to consider the fibre MX = M_{n}X of the map L_{n}X ⟶ L_{n−1}X, and the second is to observe that E(n) ∼ E(n−1)∨K(n) (where ∼ means Bousfield equivalence, as in Section 5) and thus consider L_{K(n)}X = L̂X. The purpose of this section is to demonstrate that these two approaches are essentially equivalent.

First, we observe that M_{n} (considered as a functor ℒ_{n} ⟶ ℒ_{n−1}) is just E(n−1)- acyclisation, so it is an idempotent exact functor.

Theorem 6.19. For any X ∈ 𝒮, there are natural equivalences ML̂X = MX and

L̂MX = L̂X. It follows that the functors ℳ \underrightarrow{L̂} 𝒦 ⟶^{M} ℳ are mutually inverse equivalences between the monochromatic category ℳ and the K-local category 𝒦.

Proof. We start by observing that for any X ∈ 𝒮, we have MX = MLX and L̂X = L̂LX.

Next, we know that L_{n−1}S ∼ K(0) ∨ … ∨ K(n − 1), so that K ∧ L_{n−1}S = 0, so L̂L_{n−1}S = 0. By applying L̂ to the cofibration MX ⟶ LX ⟶ L_{n−1}X, we conclude that L̂MX = L̂X. Next, using [Hov95a, Lemma 4.1] (or part (e) of Proposition 7.10) and the cofibration MS ⟶ LS ⟶ L_{n−1}S, we obtain a cofibration

F (L_{n−1}S, X) ⟶ LX = F (LS, LX) ⟶ L̂X = F (MS, LX).

We claim that F (L_{n−1}S, X) is E(n − 1)-local. Indeed, if Z is E(n − 1)-acyclic then

[Z, F (L_{n−1}S, X)]_{∗} = [L_{n−1}S ∧ Z, X]_{∗} = [L_{n−1}Z, X]_{∗} = 0

because L_{n−1}Z = 0. It follows easily that MF (L_{n−1}S, X) = 0, and thus that MX = ML̂X.

If X ∈ 𝒦 then L̂X = X so L̂MX = X. If X ∈ ℳ then MX = X so ML̂X = X. It follows that M and L̂ are mutually inverse equivalences between ℳ and 𝒦.

7. General properties of the K(n)-local category

This section begins the second part of the paper, in which we will study the Klocal category 𝒦. Unlike many other categories of localised spectra (see Appendix B) this category has a good supply of small objects. These are the K-localisations of finite type n spectra. We will show that these 𝒦-small spectra can be used as replacements for the sphere in many cases. That is, many of the constructions commonly used in stable homotopy categories, such as the cellular tower, have analogues in the K-local category. It is just that the cells are no longer spheres, but suspensions of LF (n).

Theorem 7.1. 𝒦 is an algebraic stable homotopy category in the sense of [HPS95]. The spectrum LF (n) is a graded weak generator, and loc⟨LF (n)⟩ = 𝒦. The coproduct, smash product and function objects in 𝒦 are

⋁_{𝒦}X_{i} = L̂(⋁_{𝒮}X_{i})

X ∧_{𝒦} Y = L̂(X ∧_{𝒮} Y)

F_{𝒦} (X, Y) = F_{𝒮}(X, Y).

Proof. This all follows from [HPS95, Theorem 3.5.1], except for the fact that LF (n) is a graded weak generator, which will be proved as Theorem 7.3. (It follows from this that loc⟨LF (n)⟩ = 𝒦, as in [HPS95, Theorem 1.2.1]).

We shall usually not write the subscript 𝒦 on the symbols for the coproduct and smash product. Instead, we shall say “in 𝒦, we have X = Y ∧ Z” (rather than “X = Y ∧_{𝒦} Z”).

Lemma 7.2. If X is a finite spectrum of type at least n and Y ∈ 𝒮 then L(X∧Y) = L̂(X ∧ Y) = X ∧ L̂Y.

Proof. For any finite spectrum X, we have L̂(X ∧ Y) = X ∧ L̂Y, and similarly for the functor L, as we see easily by induction on the number of cells. To complete the proof, it suffices to show that X ∧ LY is already K-local. Suppose Z is K-acyclic. Then Z ∧ DX is E-acyclic, since X, and thus DX, has type at least n. Hence [Z, X ∧ LY] = [Z ∧ DX, LY] = 0, as required.

Theorem 7.3. If X is a finite type n spectrum then LX = L̂X is a small graded weak generator in 𝒦. It follows that any spectrum Y ∈ 𝒦 is the sequential colimit of a sequence 0 = Y_{0} ⟶ Y_{1} ⟶ Y_{2} ⟶ … Y, in which the cofibre of Y_{k} ⟶ Y_{k+1} is a coproduct of suspensions of X (and therefore Y ∈ loc⟨X⟩).

Proof. Suppose that {Y_{i}} is a family of K-local spectra. We have

[L̂X, L̂(⋁ Y_{i})] = [X, L̂(⋁ Y_{i})]
= [S, L̂(⋁ Y_{i}) ∧ DX]
= [S, L(⋁ Y_{i} ∧ DX)]
= [S, ⋁ L(Y_{i} ∧ DX)]
= ⨁[S, L(Y_{i} ∧ DX)]
= ⨁[S, Y_{i} ∧ DX)]
= ⨁[L̂X, Y_{i}].

Here DX is the Spanier-Whitehead dual of X. The third and sixth equalities use Lemma 7.2, and the fact that Y_{i} = L̂Y_{i}. The fourth equality uses the fact that L is smashing. We conclude that LX is small.

Now suppose that Y ∈ 𝒦 has [LX, Y]_{∗} = 0. Observe that

[LX, Y]_{∗} = [X, Y]_{∗} = π_{∗}(DX ∧ Y).

It follows that DX ∧ Y = 0, so

0 = K_{∗}(DX ∧ Y) = Hom_{K_{∗}}(K_{∗}X, K_{∗}Y).

This means that K_{∗}Y = 0, and thus (because Y is K-local) that Y = 0. In other words, LX is a graded weak generator.

The remaining claims follow from [HPS95, Theorem 1.2.1].

7.1. K-nilpotent spectra. This short section concerns a small technicality. The class of K-nilpotent spectra is by definition the ideal generated by K. Because there are different smash products in ℒ and 𝒦, it might a priori make a difference whether we interpret this definition in 𝒦 or ℒ.

Lemma 7.4. The category 𝒩 of K-nilpotent spectra is the thick subcategory of 𝒦 generated by the K-injective spectra. It does not make a difference whether we interpret the definition in 𝒮, ℒ or 𝒦. If X ∈ 𝒩 and Y ∈ 𝒦 then X ∧_{𝒦} Y = X ∧_{𝒮} Y. Moreover, 𝒩 ⊆ loc_{𝒦} ⟨K⟩.

Proof. It is easy to see that K ∧_{𝒮} X ∈ 𝒦 for all X, and thus that the K-injectives are the same in 𝒮, ℒ or 𝒦. It follows immediately from Lemma 6.4 (applied in 𝒮, ℒ and 𝒦) that the K-nilpotents are the same in all three categories and that they are always the same as the thick subcategory generated by the K-injectives.

Now write 𝒞 = loc_{𝒦} ⟨K⟩, and

𝒟 = {X ∈ 𝒮 | ∀Y ∈ 𝒮 X ∧_{𝒮} Y ∈ 𝒞}.

It is easy to see that 𝒟 is an ideal in 𝒮 containing K, so 𝒩_{𝒮} ⊆ 𝒟. On the other hand 𝒟 ⊆ 𝒞 (take Y = S in the definition of 𝒟).

7.2. Localising and colocalising subcategories.

Theorem 7.5. The only localising or colocalising subcategories of 𝒦 are {0} and 𝒦.

Proof. It is easy to see that loc_{𝒦} ⟨L̂S⟩ = 𝒦 (because the analogous thing holds in ℒ), and thus that any (co)localising subcategory is a (co)ideal [HPS95, Lemma 1.4.6].

Suppose that 𝒞 ⊆ 𝒦 is a nontrivial localising subcategory, say 0 ≠ X ∈ 𝒞. Then K ∧_{𝒦} X = K ∧_{𝒮} X lies in 𝒞 and is a nontrivial wedge of suspensions of K. Thus K ∈ 𝒞, and therefore every K-nilpotent spectrum lies in 𝒞. In particular, LF (n) ∈ 𝒞. By Theorem 7.1, we conclude that 𝒞 = 𝒦.

Now suppose that 𝒟 ⊆ 𝒦 is a nontrivial colocalising subcategory, say 0 ≠ X ∈ 𝒦. Because 𝒟 is a coideal, the K-injective spectrum F (K, X) lies in 𝒟. Observe that {Y ∈ 𝒦 | [Y, X]_{∗} = 0} is a localising subcategory of 𝒦 which does not contain X, so it is zero. It follows that [K, X]_{∗} ≠ 0, so F (K, X) ≠ 0. Thus K is a retract of F (K, X), so K ∈ 𝒟. Any K-injective spectrum Z is a wedge of suspensions of K, and the wedge is a retract of the product by part (f) of Proposition 3.3, so Z ∈ 𝒟. It follows by Lemma 7.4 that any K-nilpotent spectrum lies in 𝒟.

Suppose that Y ∈ 𝒦. Consider 𝒞 = {U ∈ 𝒦 | F (U, Y) ∈ 𝒟}. This is clearly a localising subcategory. Because F (LF (n), Y) = LDF (n) ∧ Y is K-nilpotent, we see that F (n) ∈ 𝒞. We know that loc_{𝒦} ⟨F (n)⟩ = 𝒦, so 𝒞 = 𝒦. In particular, L̂S ∈ 𝒞, so Y ∈ 𝒟. Thus 𝒟 = 𝒦.

Corollary 7.6. If X, Y ∈ 𝒦 are both nontrivial, then F (X, Y) ≠ 0 (equivalently, [X, Y]_{∗} ≠ 0) and X ∧ Y ≠ 0.

Proof. 𝒞 = {Z | [Z, Y]_{∗} = 0} is a localising subcategory of 𝒦 not containing Y, so 𝒞 = {0}, so X ∉ 𝒞. Similarly, {Z | Z ∧ Y = 0} is a localising subcategory which does not contain L̂S, so it is trivial. More directly, K_{∗}X ≠ 0 ≠ K_{∗}Y, so K_{∗}(X ∧ Y) = K_{∗}X ⊗_{K_{∗}} K_{∗}Y ≠ 0.

As an aside, we point out that from this theorem it is very easy to prove that the Bousfield class of K is minimal among all cohomological Bousfield classes, as was conjectured in [Hov95b].

Theorem 7.7. Suppose τ : H ⟶ K is a natural transformation of homology or cohomology functors on 𝒦. If there is a nontrivial X such that τ_{X} is an isomorphism, then τ is a natural equivalence.

Proof. {X | τ_{X} is an isomorphism } is a nontrivial localising subcategory, hence is all of 𝒦.

Corollary 7.8. If H is a homology or cohomology functor on 𝒦 such that H_{∗}(X) = 0 or H^{∗}(X) = 0 for some nontrivial X, then H is trivial.

For cohomology functors, this corollary follows immediately from Corollary 7.6 and the fact that cohomology functors are representable (discussed in Section 9).

We do not yet have a complete classification of ideals in 𝒦, but we do have the following simple result.

Proposition 7.9. Suppose that 𝒟 is a nonzero ideal of 𝒦. Then 𝒟 contains the ideal of K-nilpotent spectra.

Proof. Suppose X is a nontrivial element of 𝒟. Then X ∧ K ∈ 𝒟, but X ∧ K is a wedge of suspensions of K. Therefore K ∈ 𝒟, so 𝒟 contains the ideal of K-nilpotent spectra.

7.3. The localisation functor L̂. In this section we recall some facts about the functor L̂ : 𝒮 ⟶ 𝒦. Let L^{f} : 𝒮 ⟶ 𝒮 be the finite localisation away from F (n), and C^{f} the corresponding acyclisation functor [Mil92] [HPS95, Section 3.3]. Thus, we have a cofibration

C^{f}X ⟶ X ⟶ L^{f}X

in which C^{f}X has an F (n)-cellular tower and [F (n), L^{f}X]_{∗} = 0.

Proposition 7.10. In the claims below, all homotopy colimits are calculated in 𝒮. (The homotopy limits are the same in 𝒮, ℒ or 𝒦). They are taken over a tower of generalised Moore spectra, as in Proposition 4.22.

(a) C^{f}X = holim_{-→}_{I} F (S/I, X) = holim_{-→}_{I} D(S/I) ∧ X.

(b) LL^{f}X = L_{n−1}X = L^{f}LX.

(c) LC^{f}X = MX = C^{f}LX = holim_{-→}_{I} D(S/I) ∧ LX.

(d) L_{F (n)}X = holim_{←-}_{I} X ∧ S/I = F (C^{f}S, X).

(e) L̂X = L_{F (n)}LX = holim_{←-}_{I} LX ∧ S/I = F (MS, LX).

Proof. By the Thick Subcategory Theorem [HS, Theorem 7] [Rav92a, Theorem 3.4.3], we know that D(S/I) lies in the thick subcategory generated by F (n). It follows that holim_{-→}_{I} D(S/I) ∧ X ∈ loc⟨F (n)⟩. On the other hand,

[F (n), holim_{-→}_{I} D(S/I) ∧ X] = lim_{-→}_{I}[F (n) ∧ S/I, X] = [F (n), X]

(the second equality by Proposition 4.22). It follows that holim_{-→}_{I} D(S/I) ∧ X ⟶ X

has universal property of C^{f}X ⟶ X, which gives (a). For part (b), we know that L and L^{f} are both smashing, so that LL^{f} = L^{f}L, and Corollary 6.10 tells us that L^{f}L = L_{n−1}. Part (c) follows immediately, as MX is by definition the fibre of LX ⟶ L_{n−1}X.

We now prove (d) (which is also proved as [Hov95a, Theorem 2.1]). The second equality is clear from (a). Suppose that Z is F (n)-acyclic. As C^{f}S ∈ loc⟨F (n)⟩, we see that [Z, F (C^{f}S, X)]_{∗} = [Z ∧ C^{f}S, X]_{∗} = 0. Thus F (C^{f}S, X) is F (n)-local. We have a cofibration

F (L^{f}S, X) ⟶ X = F (S, X) ⟶ F (C^{f}S, X).

We know that [U, L^{f}S]_{∗} = 0 whenever U is a finite type n spectrum. In particular π_{∗}(F (n) ∧ L^{f}S) = [DF (n), L^{f}S]_{∗} = 0, so F (n) ∧ L^{f}S = 0. It follows that [F (n), F (L^{f}S, X)]_{∗} = [F (n) ∧ L^{f}S, X]_{∗} = 0, so that X ⟶ F (C^{f}S, X) is an F (n)- equivalence. Claim (d) follows.

Finally, we prove (e). By part (c), we know that MS = holim_{-→}_{I} LD(S/I), which

implies that F (MS, LX) = holim_{←-}_{I} LX ∧ S/I, and this is the same as L_{F (n)}LX

by (d). Because X ⟶ LX is an LS-equivalence, and LX ⟶ L_{F (n)}LX is an F (n)- equivalence, we see that X ⟶ L_{F (n)}LX is an LS ∧ F (n)-equivalence. Because LS ∧ F (n) = LF (n) is Bousfield equivalent to K, it is a K-equivalence. Thus, we need only show that L_{F (n)}LX is K-local. This follows immediately from the equivalence L_{F (n)}LX = F (MS, LX) and the fact that MS is Bousfield equivalent to K.

Corollary 7.11. If X ∈ 𝒦 and we compute sequential (co)limits in 𝒦 then

X = holim_{-→}_{I} F (S/I, X) = holim_{←-}_{I} X ∧ S/I

Proof. Note that X = LX = L̂X. By part (c) of Proposition 7.10, we see that

MX = holim_{-→}_{I}^{𝒮}D(S/I) ∧ X.

By Theorem 6.19, we have X = L̂MX = holim_{-→}_{I}^{𝒦} D(S/I) ∧ X as claimed. It is

immediate from part (e) of Proposition 7.10 that X = holim_{←-}_{I} X ∧ S/I.

8. Smallness and duality

We next consider various notions of smallness in 𝒦. To show that things are not as simple as in 𝒮 or ℒ, we have the following lemma.

Lemma 8.1. The functor L̂ is not smashing, and L̂S is not small in 𝒦.

Proof. We know from Proposition 5.3 that ⟨L̂S⟩ = ⟨E(n)⟩ > ⟨K⟩. On the other hand, if L̂ were smashing we would have

L̂S ∧ X = 0 ⇔ L̂X = 0 ⇔ K_{∗}X = 0 ⇔ K ∧ X = 0

It would follow that ⟨L̂S⟩ = ⟨K⟩, a contradiction. If L̂S were small then L̂ would preserve smallness, so it would be smashing by [HPS95, Theorem 3.5.2], another contradiction.

In spite of this, we have a good understanding of two different notions of smallness, as indicated by the theorems below. We start with some definitions.

Definition 8.2. We say that a graded Abelian group A_{∗} is finite if A_{k} is a finite set for each k. (This is the most natural definition given that most of our graded Abelian groups are periodic.)

Definition 8.3. For any spectrum X we define E_{∗}^{∨}X = π_{∗}L̂(E ∧ X).

We will see below that E_{∗}^{∨}X is a better covariant analogue of E^{∗}X than E_{∗}X for X ∈ 𝒦, though E_{∗}^{∨}X is not a homology theory. For now we note the following analogue of some of the results in Section 2.

Proposition 8.4.

(a) The E_{∗}-module E_{∗}^{∨}X is L-complete.

(b) If X is finite in 𝒮 or in ℒ, then E_{∗}^{∨}X = E_{∗}^{∨}L̂X = E_{∗}X.

(c) If E_{∗}X is a free E_{∗}-module, then E_{∗}^{∨}X = (E_{∗}X)_{I_{n}}^{∧} (which is a pro-free E_{∗}- module).

(d) E_{∗}^{∨}X is finitely generated if and only if K_{∗}X is finite-dimensional.

(e) If E_{∗}^{∨}X is pro-free, then K_{∗}X = (E_{∗}^{∨}X)/I_{n}.

(f) If K_{∗}X is concentrated in even dimensions, then E_{∗}^{∨}X is pro-free and concentrated in even dimensions.

Proof. (a) Proposition 7.10 gives a Milnor exact sequence

lim_{←-}_{I}^{1}(E/I)_{∗+1}(X) ↣ E_{∗}^{∨}X ↠ lim_{←-}_{I}(E/I)_{∗}X.

Theorem A.6 then implies that E_{∗}^{∨}X is L-complete.

(b) If X is finite in 𝒮 or ℒ then E∧X is already K-local, so E_{∗}^{∨}X = E_{∗}^{∨}L̂X = E_{∗}X.

(c) This follows from the Milnor sequence and the fact that (E/I)_{∗}X = (E_{∗}X)/I, so there is no lim_{←-}^{1} term.

(d) This is very similar to the proof of Proposition 2.4, except that we replace (E/I_{k})^{∗}X by (E/I_{k})_{∗}^{∨}X = π_{∗}L̂(E/I_{k} ∧ X). Note that (E/I_{n})_{∗}^{∨}X = K_{∗}X and that (E/I_{k})_{∗}^{∨}X is L-complete (by induction on k).

(e) If E_{∗}^{∨}X is pro-free, then the sequence {v_{0}, …, v_{n−1}} is regular on E_{∗}^{∨}X, so K_{∗}X = (E/I_{n})_{∗}^{∨}X = (E_{∗}^{∨}X)/I_{n}.

(f) Proceed just as in Proposition 2.5 using the theories (E/I_{k})_{∗}^{∨}X.

Recall that we have different notions of finiteness in 𝒦, or any stable homotopy category (Definition 1.5). Throughout the rest of this paper, we will denote the category of small spectra in 𝒦 by ℱ and the category of dualisable spectra in 𝒦 by 𝒟. Note that if X is dualisable, we have E^{∗}DX = π_{∗}(E ∧_{𝒦} X) = E_{∗}^{∨}X.

Theorem 8.5. Suppose that X ∈ 𝒦. Consider the following statements:

(a) X is small.

(b) X is LF (n)-finite.

(c) X is a retract of LX′ = L̂X′ for some finite spectrum X′ of type at least n.

(d) E_{∗}^{∨}X is finite.

(e) E^{∗}X is finite.

(f) X is dualisable and K-nilpotent. Then (a),(b) and (c) are equivalent, and they imply (d),(e) and (f ). (We shall show in Corollary 12.16 that (d),(e) and (f ) are also equivalent to (a),(b) and (c).)

The category ℱ ⊆ 𝒟 is thick. Moreover, if X ∈ ℱ and Y ∈ 𝒟 then X ∧ Y, F (X, Y) and F (Y, X) lie in ℱ. In particular, DX = F (X, L̂S) ∈ 𝒟.

Proof. It follows from part (c) of [HPS95, Theorem 2.1.3] that (a)⇔(b), and that small implies dualisable. Because every finite spectrum of type at least n lies in the thick subcategory generated by F (n), we see that (c)⇒(b).

We now prove that (a)⇒(c). Suppose that X is small. By Corollary 7.11, we see that X = holim_{-→}_{I} D(S/I) ∧ X, so [X, X] = lim_{-→}_{I}[X, D(S/I) ∧ X], so X is a retract of

Y = D(S/I) ∧ X for some I. We claim that Y is small in ℒ. To see this, consider a family {Z_{i}} of spectra in 𝒦. By Proposition 5.3 we have L(S/I) ∼ K, which implies that

L(S/I) ∧ ⋁_{i}^{ℒ} Z_{i} = L(S/I) ∧ ⋁_{i}^{𝒦} Z_{i} = ⋁_{i}^{𝒦} S/I ∧ Z_{i}

It follows that

[Y, ⋁_{i}^{ℒ} Z_{i}]_{∗} = [X, ⋁_{i}^{𝒦} S/I ∧ Z_{i}]_{∗} = ⨁_{i}[X, S/I ∧ Z_{i}]_{∗} = ⨁_{i}[Y, Z_{i}]_{∗}

as claimed. Since K(i)_{∗}Y = 0 for i < n, Corollary 6.11 implies that Y, and hence X, is a retract of LZ = L̂Z for some finite spectrum Z of type at least n.

We now show that (c) implies (d),(e), and (f). By an evident thick subcategory argument, we need only show that E_{∗}^{∨}S/I and E^{∗}S/I are finite, and that S/I is dualisable and K-nilpotent. As S/I is finite, we see that E_{∗}^{∨}S/I = E_{∗}S/I = (E/I)_{∗}, which is clearly finite. Similarly, E^{∗}(S/I) = (E/I)^{∗} which is finite. Moreover, Proposition 6.15 shows that S/I is K-nilpotent, and it is easy to see that S/I is dualisable.

From (b) it is clear that ℱ is thick, and from (c) it is clear that ℱ is closed under the Spanier-Whitehead duality functor D. Part (a) of [HPS95, Theorem 2.1.3] says that ℱ is an ideal in 𝒟. This also means that if X ∈ ℱ and Y ∈ 𝒟 then F (X, Y) = DX ∧ Y and F (Y, X) = X ∧ DY lie in ℱ.

Theorem 8.6. Suppose that X ∈ 𝒦. The following are equivalent:

(a) X is F -small.

(b) X is dualisable.

(c) K_{∗}X is finite.

(d) K^{∗}X is finite.

(e) E_{∗}^{∨}X is finitely generated.

(f) E^{∗}X is finitely generated. The category 𝒟 of dualisable spectra is thick. Moreover, L̂S ∈ 𝒟 and if X, Y ∈ 𝒟 then X ∧ Y ∈ 𝒟 and F (X, Y) ∈ 𝒟. In particular, DX = F (X, L̂S) ∈ 𝒟.

Proof. It follows from part (c) of [HPS95, Theorem 2.1.3] that (a)⇔(b), and also that the subsidiary claims after (f) hold. Because

K^{∗}X = Hom_{K_{∗}}(K_{∗}X, K_{∗}),

we see that (c)⇔(d). Proposition 2.4 shows that (d)⇔(f), and Proposition 8.4 shows that (c)⇔(e).

We now prove that (a)⇒(c). Suppose that X is F -small. We know from Theorem 8.5 that LF (n) is small, and thus that F (n)∧X is small, and thus that F (n)∧X is a retract of LY for some finite type spectrum Y of type at least n. For any finite spectrum Y, it is easy to see that K_{∗}Y = K_{∗}LY is finite (by induction on the number of cells). It follows that K_{∗}(F (n) ∧ X) = K_{∗}F (n) ⊗_{K_{∗}} K_{∗}X is finite. As K_{∗}F (n) ≠ 0, we conclude that K_{∗}X is finite.

Finally, we prove that (c)⇒(b). Suppose that K_{∗}X is finite. There is a natural map

X ∧ F (X, U) ∧ Y ⟶^{ev∧1} U ∧ Y

with adjoint

ρ_{U,Y} : F (X, U) ∧ Y ⟶ F (X, U ∧ Y).

Write

𝒞 = {U ∈ 𝒦 | ∀Y ∈ 𝒦 ρ_{U,Y} is an isomorphism }.

we claim that every K-module M lies in 𝒞. Indeed, using Proposition 3.4 repeatedly, we see that

π_{∗}(F (X, M) ∧ Y) = Hom_{K_{∗}}(K_{∗}X, M_{∗}) ⊗_{K_{∗}} K_{∗}Y
π_{∗}(F (X, M ∧ Y)) = Hom_{K_{∗}}(K_{∗}X, M_{∗} ⊗_{K_{∗}} K_{∗}Y).

One can check that the map induced by ρ_{M,Y} is the obvious one, which sends f ⊗ y to x ↦ f(x) ⊗ y. Given that K_{∗}X is finite, we see that this map is an isomorphism.

As 𝒞 is clearly thick, we see from Lemma 7.4 that all K-nilpotent spectra lie in 𝒞, in particular LF (n) ∈ 𝒞. On the other hand, because LF (n) is dualisable, we can identify F (X, LF (n)) ∧ Y with LF (n) ∧ (F (X, L̂S) ∧ Y) and F (X, LF (n) ∧ Y) with LF (n) ∧ F (X, Y). Under these identifications, ρ_{LF (n),Y} becomes 1_{LF (n)} ∧ ρ_{L̂S,Y}. It follows that the smash product of LF (n) with the fibre of ρ_{L̂S,Y} is zero, so (by Corollary 7.6) ρ_{L̂S,Y} is an isomorphism. This means that X is dualisable.

Corollary 8.7. Let G be a finite group, and X a finite G-complex. Then L̂(EG∧_{G} X) is dualisable. Moreover, we have DL̂BG_{+} = L̂BG_{+}.

Proof. It is shown in [GS96b, Corollary 5.4] that K^{∗}(EG ∧_{G} X) is finite, so by the preceding theorem, L̂(EG ∧_{G} X) is dualisable. We next use the Greenlees-May Tate construction [GM95b] (see also [GS96a]). We can think of any spectrum X as a G-equivariant spectrum with trivial action indexed on a G-fixed universe, apply a change of universe functor to get to a complete universe and perform the Tate construction to get a G-spectrum t_{G}X. We write P_{G}X for the Lewis-May fixed-point spectrum (t_{G}X)^{G}. It fits into a natural cofibre sequence

BG_{+} ∧ X ⟶ F (BG_{+}, X) ⟶ P_{G}(X).

If we let X be L̂S and localise, we get a cofibre sequence

L̂BG_{+} ⟶ DL̂BG_{+} ⟶ L̂P_{G}L̂S.

Now in [HS96] it is shown that the Bousfield class of P_{G}L̂X is the same as that of L_{n−1}X if X is finite. In particular, its K-localisation is trivial. Thus L̂BG_{+} = DL̂BG_{+}.

Corollary 8.8. If X is a connected p-local loop space such that π_{k}(X) is finite for all k and zero for almost all k. Then L̂Σ^{∞}X is dualisable.

Proof. It is shown in [RW80] that K_{∗}K(A, q) is finite whenever A is a finite Abelian group, so it follows from the results of [HRW96] that K_{∗}X is finite.

Another important example of a dualisable spectrum is the Brown-Comenetz dual of MS; see Section 10.

Theorem 8.9. Every dualisable spectrum in 𝒦 is E-finite.

The proof depends on the following lemmas.

Lemma 8.10. Let X ∈ 𝒦 be a spectrum such that

(a) E^{∗}X is pro-free as a module over E.

(b) K^{∗}X is a finitely generated module over Σ^{∗}, of projective dimension 0 ≤ d < ∞.

Then there is a cofibration X ⟶ J ⟶ Y such that (i) J is a finite wedge of suspensions of E. (ii) K^{∗}J is the Σ^{∗}-projective cover of K^{∗}X. (iii) Y satisfies (a) and (b), with projective dimension d − 1 (and Y = 0 if d = 0).

Proof. Since E^{∗}X is pro-free, we have K^{∗}X = (E^{∗}X)/I_{n} by Proposition 2.5. In particular, Σ^{∗} = (E^{∗}E)/I_{n} acts on K^{∗}X, so (b) makes sense. Let I be the augmentation ideal in Σ^{∗}. Choose a (finite) basis \{\overline{m}_{i}\} for (K^{∗}X)/I over K^{∗}. Since K^{∗}X = (E^{∗}X)/I_{n}, we can lift the \overline{m}_{i} to get elements m_{i} ∈ E^{∗}X. Define J = ⋁_{i} Σ^{|m_{i}|}E, so the m’s give an evident map m: X ⟶ J. Define Y to be the cofibre. It is immediate, using the arguments of Corollary 2.30, that K^{∗}J ⟶ K^{∗}X is the Σ^{∗}-projective cover, and thus that the kernel K^{∗}Y has projective dimension d − 1. In the exceptional case d = 0, we see that K^{∗}X = K^{∗}J and thus K^{∗}Y = 0; as Y is K-local, we conclude that Y = 0. In any case, because Σ^{∗} is Noetherian, we see that K^{∗}Y is finitely generated over Σ^{∗}.

By construction, the image of E^{∗}J in (E^{∗}X)/I_{n} = K^{∗}X is K^{∗}X. As E^{∗}J and E^{∗}X are L-complete, we conclude from Theorem A.6 that E^{∗}J ⟶ E^{∗}X is epi. As E^{∗}X is pro-free, we conclude that this epimorphism splits. As E^{∗}J is also pro-free, we see that E^{∗}Y is again pro-free.

In the next lemma, we use the derived category 𝒟_{E} of E-modules defined as in [EKMM96]; see Section 1 for more discussion.

Lemma 8.11. Let M be an object of 𝒟_{E} such that π_{∗}M is a finitely generated module over E_{∗}. Then M is E-finite in 𝒟_{E} (and thus also in the category of spectra).

Proof. This is very similar to the last result. We can recursively construct finitely generated free modules P_{k} over the ring spectrum E and cofibrations M_{k+1} ⟶ P_{k} ⟶ M_{k} (with M_{0} = M), such that the modules π_{∗}M_{k} form a minimal projective resolution for π_{∗}M over E_{∗}. As E_{∗} has finite global dimension n, we see that M_{n+1} = 0, and thus M is E-finite as claimed.

We can now prove the theorem.

Proof of Theorem 8.9. Jeff Smith has constructed a finite p-local spectrum X with only even cells such that

Ext_{Σ^{∗}}^{k}(K^{∗}X, K^{∗}) = Ext_{Σ_{∗}}^{k}(K_{∗}, K_{∗}X) = 0 for k > n^{2}.

See [Rav92a, Section 8.3] for a proof of this. Ravenel actually shows that

Ext_{BP_{∗}BP}^{k} (BP_{∗}, v_{n}^{−1}BP_{∗}X/I_{n}) = 0

for large k, but the Miller-Ravenel change of rings theorem [Rav86, Theorem 6.6.1] and the fact that X has only even cells implies that this is equal to Ext_{Σ_{∗}}(K_{∗}, K_{∗}X). That we can choose the horizontal vanishing line to be n^{2} does not appear to be in the published literature, but is not important for this argument. As X has only even cells, the Atiyah-Hirzebruch spectral sequence collapses to show that E^{∗}X is free of finite rank over E^{∗}. It follows a fortiori that K^{∗}X = E^{∗}X/I_{n} is finitely generated over Σ^{∗}. Now choose a minimal projective resolution P_{∗} of K^{∗}X as in Corollary 2.30. We find that the complex Hom_{Σ^{∗}}(P_{∗}, K^{∗}) has zero differential, so the projective dimension of K^{∗}X is at most n^{2}. Thus Lemma 8.10 applies to X_{0} = L̂X to give a sequence of cofibrations X_{k} ⟶ J_{k} ⟶ X_{k+1}, where K^{∗}X_{k} has projective dimension at most n^{2} −k over R, and X_{k} = 0 for k > n^{2}. As each J_{k} is a finite wedge of E’s, we conclude that X_{0} = L̂X is E-finite. By a thick subcategory argument, we even see that L̂S is E-finite.

Now let Y ∈ 𝒦 be dualisable. By the above, Y = L̂S ∧_{𝒦} Y is in the thick subcategory generated by M = E ∧_{𝒦} Y. Moreover, because Y is dualisable, we know that π_{∗}M is a finitely generated module over π_{∗}E. It follows by Lemma 8.11 that M is E-finite in 𝒟_{E} (and thus in the category of spectra). It follows that everything in the thick subcategory generated by M is E-finite, and thus that Y is E-finite.

Corollary 8.12. If X ∈ 𝒦 is small, then X is K-finite, and in particular π_{k}X is finite for all k.

Proof. Because X is small it is easy to see using Theorem 8.5 and Proposition 4.17 that the unit map X ⟶ S/I ∧ X is a split monomorphism for some generalised Moore space S/I of height n. As L̂S is E-finite, we see that S/I ∧ X lies in the thick subcategory generated by E ∧ S/I ∧ X = E/I ∧ X. It is easy to see that this lies in the thick subcategory generated by K ∧ X, which is a finite wedge of suspensions of K. This implies that X is K-finite.

9. Homology and cohomology functors

In this section, we discuss the representability of (co)homology functors from 𝒦 to the category Ab of Abelian groups. We write 𝒦_{•} and 𝒦^{•} for the categories of homology and cohomology functors.

Theorem 9.1. Every cohomology functor H : 𝒦 ⟶ Ab is uniquely representable. More precisely, the Yoneda functor X ↦ [−, X] is an equivalence 𝒦 ≃ 𝒦^{•}.

Proof. Observe that H ∘ L̂ is a cohomology theory on 𝒮, and apply the Brown representability theorem for 𝒮. It is easy to check that the representing spectrum automatically lies in 𝒦.

Before we discuss the more complicated representability of homology functors, we discuss the closely related subject of realizing any object of 𝒦 as a minimal weak colimit of small spectra. Recall that colimits do not usually exist in triangulated categories, but that weak colimits always exist but are not unique. Minimal weak colimits, discussed in [HPS95, Section 2.2], often exist and are always unique (in algebraic stable homotopy categories).

We define

Λ′(X) = {X′ ⟶^{u} X | X′ ∈ 𝒦 is small }
Λ′′(X) = {X′′ ⟶^{u} X | X′′ ∈ 𝒮 is a finite spectrum of type at least n}.

As usual, we make Λ′(X) and Λ′′(X) into categories, in which the morphisms are commutative triangles. It is not hard to see that they are filtered (compare [HPS95, Proposition 2.3.9]). Recall that every 𝒦-small spectrum is a retract of LY for some finite spectrum Y of type at least n, and that there are only countably many isomorphism classes of finite spectra. It follows that Λ′(X) and Λ′′(X) both have only a set of isomorphism classes, in other words that they are essentially small.

Proposition 9.2. For any X ∈ 𝒦 we have

X = mwlim_{-→}_{Λ′(X)} X′ = mwlim_{-→}_{Λ′′(X)} L̂X′′

(where the minimal weak colimits are computed in 𝒦). Moreover, the evident functor L̂: Λ′′(X) ⟶ Λ′(X) is cofinal.

Proof. Let L^{f}X be the finite localisation of X away from F (n) and C^{f}X the corresponding acyclisation. We then have a cofibration C^{f}X ⟶ X ⟶ L^{f}X, and [F (n), L^{f}X] = 0, so F (n) ∧ L^{f}X = 0. As K_{∗}F (n) ≠ 0 and we have a Künneth isomorphism, we see that K_{∗}L^{f}X = 0, so L̂L^{f}X = 0 and L̂C^{f}X = L̂X = X.

By [HPS95, Proposition 2.3.17], we know that C^{f}X = mwlim_{-→}_{Λ′′(X)}^{𝒮} X′′. As local-

isation functors preserve minimal weak colimits [HPS95, Theorem 3.5.1], we find

that X = L̂C^{f}X = mwlim_{-→}_{Λ′′(X)}^{𝒦} L̂X′′.

We now prove that L̂ : Λ′′(X) ⟶ Λ′(X) is cofinal. Suppose that (X′ ⟶^{u} X) ∈ Λ′(X). Because X′ is small, [X′, −] is a homology functor on 𝒦. It follows that

[X′, X] = lim_{-→}_{Λ′′(X)}[X′, L̂X′′].

We can apply this fact to the element u ∈ [X′, X]. The conclusion is that there is a map X′ ⟶ L̂X′′ over X, for some X′′ ∈ Λ′′(X). Moreover, given two such maps X′ ⟶ L̂X′′_{i} (for i = 0, 1), there are maps X′′_{0} ⟶ X′′_{2} ⟵ X′′_{1} in Λ′′(X) such that the two composites X′ ⟶ L̂X′′_{2} are the same. This means precisely that L̂ : Λ′′(X) ⟶ Λ′(X) is cofinal. It follows easily that X = mwlim_{-→}_{Λ′(X)} X′.

Recall from [HPS95, Section 4.1] that a homology functor H : 𝒦 ⟶ Ab is representable if there is a spectrum X ∈ 𝒦 and a natural isomorphism H(Y) ≃ [S, X ∧ Y] for 𝒦-small spectra Y. (It does not matter whether the smash product is interpreted in 𝒮 or 𝒦, because Y is small). Recall also that for any X and Y in 𝒦 we have

M(X ∧_{𝒦} Y) = M(X ∧_{𝒮} Y) = MS ∧_{𝒮} X ∧_{𝒮} Y.

Lemma 9.3. A natural isomorphism H(Y) ≃ [S, X ∧ Y] for small spectra Y gives rise to a natural isomorphism H(Y) ≃ [S, M(X ∧ Y)] = MX_{0}Y for all Y ∈ 𝒦.

Proof. Write {Y_{i}} for the objects of Λ′′(Y) (as in Proposition 9.2). Thus each Y_{i} is a finite spectrum of type at least n, so LY_{i} = MY_{i} = L̂Y_{i} is K-nilpotent by Proposition 6.15. Moreover, Y = mwlim_{-→}_{i}^{𝒦} LY_{i} and thus HY = lim_{-→}_{i}[S, X ∧ Y_{i}].

Next, because X ∈ 𝒦 we see that X = LX, so there is a cofibration MX ⟶ X ⟶ L_{n−1}X. Because L_{n−1}X ∧_{𝒮} Y_{i} = X ∧_{𝒮} L_{n−1}Y_{i} = 0, we see that

X ∧ Y_{i} = MX ∧ Y_{i}.

We know that mwlim_{-→}_{i}^{𝒮} Y_{i} = C^{f}Y, using the notation of the proof of Proposition 9.2.

Moreover [F (n), C^{f}Y]_{∗} = [F (n), Y]_{∗}, so DF (n) ∧ C^{f}Y = DF (n) ∧ Y, so MC^{f}Y = MY (since MZ = MS ∧ Z and MS ∼ LS ∧ DF (n)). It follows that

mwlim_{-→}_{i}^{𝒮} MX ∧ Y_{i} = MX ∧ C^{f}Y = MX ∧ Y.

This gives an isomorphism

[S, MX ∧ Y] ≃ lim_{-→}_{i}[S, MX ∧ Y_{i}] ≃ lim_{-→}_{i}[S, X ∧ Y_{i}] ≃ H(Y).

Before stating the representability theorem for homology functors, it is convenient to record the definition of phantom maps.

Definition 9.4. A map f : X ⟶ Y in 𝒦 is phantom if (Z ⟶^{u} X ⟶^{f} Y) = 0 for all small Z and all u : Z ⟶ X. We also say that f is cophantom if (X ⟶^{f} Y ⟶^{u} Z) = 0 for all small Z and all u : Y ⟶ Z.

Theorem 9.5. Any homology functor H: 𝒦 ⟶ Ab is representable. More precisely, there is a spectrum X ∈ 𝒦 such that H(Y) = X_{0}Y = π_{0}(X ∧ Y) for all 𝒦-small Y, and H(Y) = MX_{0}Y = π_{0}(MX ∧ Y) for all Y ∈ 𝒦. The group of natural maps MX_{0} ⟶ MY_{0} is naturally isomorphic to [X, Y]/phantoms. Any composite of a phantom with another map is phantom, and any composite of two phantoms is zero.

Proof. This follows easily by combining the results of [HPS95, Section 4.1] with Lemma 9.3.

10. Brown-Comenetz duality

In this section, we define and study a good notion of Brown-Comenetz duality on the category 𝒦. We have been strongly influenced by the outline in [HG94]. Recall the usual definition of Brown-Comenetz duality. In keeping with our convention that everything is p-local, we write Q/Z for Q/Z_{(p)}. Because π_{0}X is a homology functor on 𝒮 and Q/Z is an injective Abelian group, we see that Hom(π_{0}Y, Q/Z) is a cohomology functor. By the Brown representability theorem, there is a spectrum I unique up to canonical isomorphism, with a natural isomorphism

Hom(π_{0}Y, Q/Z) ≃ [Y, I].

We also write IX = F (X, I), so there is a natural isomorphism Hom(X_{0}Y, Q/Z) ≃ [Y, IX]. Recall from Section 9 that any homology functor on 𝒦 has the form Y ↦ π_{0}(MX ∧ Y) for some X ∈ 𝒦. This motivates the following definition:

Definition 10.1. If X ∈ 𝒦, the Brown-Comenetz dual of X is

ÎX = IMX = F (MX, I).

We also write Î = ÎS, so that ÎX = F (X, Î).

Theorem 10.2. (a) There is a natural isomorphism

[Y, ÎX] ≃ Hom(MX_{0}Y, Q/Z).

(b) The correspondence X ↦ ÎX is a contravariant exact functor 𝒦^{op} ⟶ 𝒦.

(c) ÎK ≃ K.

(d) The natural map X ⟶ Î^{2}X is an isomorphism when π_{∗}(F (n) ∧ X) is finite in each degree (in particular, when X is dualisable, for example if X = S).

(e) The spectrum Î is invertible under the smash product.

(f) The functor X ↦ ÎX preserves the categories ℱ, 𝒟, ℐ and 𝒩 of small, dualisable, K-injective and K-nilpotent spectra.

The isomorphism classes of invertible spectra form the Picard group Pic, studied in Section 14 below and in [HMS94, Str92]. There is a natural homomorphism from Pic to an algebraically defined Picard group, as explained in [HMS94]. It is natural to ask where Î goes under this map. The answer is stated in [HG94], but we will not discuss this issue here.

We will prove Theorem 10.2 after two lemmas.

Lemma 10.3. Let {A_{k}} be an inverse system of finite-dimensional vector spaces of dimension at most d. Then lim_{←-}_{k} A_{k} has dimension at most d (and in particular

is finite-dimensional), and lim_{←-}_{k}^{1} A_{k} = 0.

Proof. This follows easily from the standard Mittag-Leffler construction. Explicitly, we define

A′_{k} = ⋂_{m>k} image(A_{m} ⟶ A_{k}).

As A_{k} has finite dimension we must have A′_{k} = image(A_{m} ⟶ A_{k}) for m ≫ 0. The groups A′_{k} form a subtower of A, and the quotient A/A′ is pro-trivial so lim_{←-}_{i}^{s} A_{i} =

lim_{←-}_{i}^{s} A′_{i} for s = 0, 1. The maps A′_{k} ⟶ A′_{k−1} are surjective so lim_{←-}_{i}^{1} A′_{i} = 0. The

dimension of A′_{i} is a nondecreasing function of i that is bounded by d, so for large i it is constant and thus the map A′_{i} ⟶ A′_{i−1} is an isomorphism. This means that lim_{←-}_{i} A′_{i} = A′_{m} for large m, so lim_{←-}_{i} A′_{i} has dimension at most d.

The next lemma is crucial.

Lemma 10.4. Let S/I be a localised generalised Moore spectrum of type n. Then [E, S/I]_{∗} is finite.

Proof. We can form the standard E-based Adams resolution of S/I and apply the functor [E, −] to get a spectral sequence whose E_{1} term is the group

C^{st}(I) = E_{1}^{s,t} = [E, E ∧ Σ^{−s}\overline{E}^{(s)} ∧ S/I]_{t−s} = [E, Σ^{−s}\overline{E}^{(s)} ∧ E/I]_{t−s}.

Here \overline{E} is the cofibre of the unit map S ⟶ E. As E_{∗}E is not a projective module over E_{∗}, the standard theorems are not enough to identify the E_{2} page as an Ext group but this turns out not to be important. What is important is that S/I is K-nilpotent and thus E-nilpotent, so the resulting spectral sequence converges to [E, S/I]_{∗} (by the argument of Proposition 6.5). It will thus be enough to show that the E_{2} page is finite in each total degree.

To do this, we define C^{st} = [E, E ∧ Σ^{−s}\overline{E}^{(s)}]_{t−s}. We make C^{∗∗} into a module over E^{∗} using the second copy of E. Using Proposition 8.4 (f) we see that π_{∗}L̂(E ∧ \overline{E}^{(s)}) is pro-free and in even degrees, so that L̂(E ∧ \overline{E}^{(s)}) is isomorphic as an Emodule to a spectrum of the form L̂(⋁_{i} Σ^{2d_{i}}E). This is a retract of ∏_{i} Σ^{2d_{i}}E so C^{s∗} is a retract of ∏_{i}[E, Σ^{2d_{i}}E]^{∗} and thus is pro-free. It is not hard to deduce that C^{∗∗}(I) = C^{∗∗}/IC^{∗∗} and that this admits a finite filtration with quotients C^{∗∗}/I_{n}C^{∗∗} = C^{∗∗}(I_{n}). It will thus be enough to show that H^{∗}C^{∗∗}(I_{n}) is finite in each total degree. But we have E/I_{n} = K, so by Proposition 3.4 we have

E_{1}^{s,t} = Hom_{K_{∗}}^{t−s}(K_{∗}E, Σ^{−s}K_{∗}(\overline{E}^{(s)})) = Hom_{K_{∗}}^{t}(Σ_{∗}, \overline{Σ}_{∗}^{⊗s}) = Hom_{Σ_{∗}}^{t}(Σ_{∗}, Σ_{∗} ⊗ \overline{Σ}_{∗}^{⊗s})

where \overline{Σ}_{∗} is the coaugmentation coideal of Σ_{∗}. It follows that

H^{∗}C^{∗∗}(I_{n}) = Ext_{Σ_{∗}}^{∗∗}(Σ_{∗}, K_{∗}).

Consider the subring Σ′(k)_{∗} < Σ_{∗} generated by t_{1}, …, t_{k−1}. It is easy to check that this is a sub Hopf algebra, and thus a subcomodule of Σ_{∗}. Moreover, we have Σ_{∗} = ⋃_{k} Σ′(k)_{∗}, which gives a short exact sequence of comodules

⨁_{k} Σ′(k)_{∗} ↣ ⨁_{k} Σ′(k)_{∗} ↠ Σ_{∗}.

This in turn gives a Milnor sequence relating Ext_{Σ_{∗}}(Σ_{∗}, K_{∗}) to lim_{←-}^{i} Ext_{Σ_{∗}}(Σ′(k)_{∗}, K_{∗}) for i = 0, 1. As Σ′(k) is a finite-dimensional cocommutative Hopf algebra, it is self-dual (up to a dimension shift) as a comodule over itself [LS69] and thus as a comodule over Σ_{∗}. We thus have

Ext_{Σ_{∗}}(Σ′(k)_{∗}, K_{∗}) = Ext_{Σ_{∗}}(K_{∗}, Σ′(k)_{∗}) = Ext_{Σ_{∗}//Σ′(k)_{∗}}(K_{∗}, K_{∗}).

We know from [Rav86, Theorem 6.3.7] that for k ≫ 0 this is an exterior algebra over K^{∗} on n^{2} generators, and thus has dimension 2^{n^{2}} independently of k. It follows from Lemma 10.3 and the Milnor sequence that Ext_{Σ_{∗}}^{∗∗}(Σ_{∗}, K_{∗}) has finite dimension over K_{∗}, as required.

Corollary 10.5. [K, S]_{∗} is finite.

Proof. It follows from the lemma that [E ∧ DS/I, S]_{∗} is finite, and it is easy to see that K lies in the thick subcategory generated by E ∧ DS/I.

Proof of Theorem 10.2. (a): This is clear.

(b): We first need to show that ÎX is K-local. Suppose that Z is K-acyclic. As MS ∼ K and MX = MS ∧ X, we see that MX ∧ Z = 0, so

[Z, ÎX]_{∗} = Hom(π_{∗}(MX ∧ Z), Q/Z) = 0

as required. Because ÎX = F (X, Î), the correspondence X ↦ ÎX is clearly a contravariant exact functor.

(c): Note that MK = K. Think of π_{0}K = F_{p} as a subgroup of Q/Z in the obvious way. We know that [X, K]_{∗} ≃ Hom_{K_{∗}}(K_{∗}X, K_{∗}). Looking in degree zero, we see that

[X, K] ≃ Hom(MK_{0}X, F_{p}) = Hom(MK_{0}X, Q/Z)

the first equality because v_{n} is a unit, and the second equality because MK_{0}X is a vector space over F_{p}. It follows that ÎK = K.

(d): The evaluation map X ∧ F (X, Î) ⟶ Î gives by adjunction a natural map κ : X ⟶ F (F (X, Î), Î) = Î^{2}X. Write F = F (n) and Y = F ∧X, so that π_{∗}Y is finite in each degree. Because F is dualisable, we have ÎY = DF ∧ÎX and Î^{2}Y = F ∧Î^{2}X. Under this identification, we have κ_{Y} = κ_{F ∧X} = 1_{F} ∧κ_{X}, so it is enough to show that κ_{Y} is an isomorphism. Because F and DF are K-nilpotent, we see that MY = Y and MÎY = ÎY. It follows that π_{∗}Î^{2}Y = Hom(Hom(π_{∗}Y, Q/Z), Q/Z), which is the same as π_{∗}Y because π_{∗}Y is finite in each degree. Thus κ_{Y} is an isomorphism, as required.

If X is dualisable then F ∧ X is small by Theorem 8.5, so π_{∗}(F ∧ X) is finite in each degree by Corollary 8.12, so the above applies.

(e): We know from Lemma 10.5 that [K, S]_{∗} is finite in each degree. We also know from (c) and (d) that κ_{K} and κ_{S} are isomorphisms, from which it follows that Î: [K, S]_{∗} ⟶ [Î, ÎK]_{∗} = K^{∗}Î is an isomorphism. Thus K^{∗}Î is finite, which implies by Theorem 8.6 that Î is dualisable. This implies that F (Î, Î) = DÎ ∧ Î, but we also know that F (Î, Î) = Î^{2}S = S. Thus DÎ∧ Î ≃ S, which means that Î ∈ Pic.

(f): By part (c), ÎK = K. Any K-injective spectrum X is a wedge of suspensions of K, so ÎX is a product of suspensions of K, which is still K-injective by Proposition 3.3. Thus X ↦ ÎX preserves ℐ, and so also 𝒩, by Lemma 7.4. Part (e) implies that Î ∈ 𝒟, so it follows from Theorems 8.6 and 8.5 that the functor X ↦ ÎX = DX ∧ Î preserves 𝒟 and ℱ.

Corollary 10.6. If X is an arbitrary spectrum in 𝒦, then ÎX ≃ DX ∧ Î.

Proof. Because Î is dualisable, we have F (X, Î) ≃ F (X, S) ∧ Î.

Recall the definition of phantom and cophantom maps from Definition 9.4.

Theorem 10.7. Suppose that X, Y ∈ 𝒦.

(a) There are no nonzero phantom maps X ⟶ IY.

(b) If π_{∗}(F (n) ∧ Y) is finite in each degree (in particular, if Y is dualisable, for example if Y = S) then there are no nontrivial phantom maps X ⟶ Y.

(c) A map u : X ⟶ Y is phantom if and only if it is cophantom.

Proof. (a): Suppose that u : X ⟶ IY is phantom. This means that the corresponding map u^{#} : π_{0}(X ∧ MY) = π_{0}(MX ∧ Y) ⟶ Q/Z vanishes on π_{0}(Z ∧ MY), for every small spectrum Z equipped with a map Z ⟶ X. On the other hand, X is the minimal weak colimit in 𝒦 of these Z’s (Proposition 9.2), and π_{0}(− ∧ MY) is a homology functor, so π_{0}(X ∧ MY) = lim_{-→}_{Z} π_{0}(Z ∧ MY). It follows that u^{#} = 0, and thus u = 0.

(b): By part (d) of Theorem 10.2, we know that Y = Î^{2}Y. It therefore follows from (a) that there are no nonzero phantoms X ⟶ Î^{2}Y = Y.

(c): In the following statements, Z runs over 𝒦-small spectra. Thus MDZ = DZ and

Hom([Z, X], Q/Z) = [X, Z ∧ Î].
u is phantom ⇔ ∀Z [Z, X] ⟶ [Z, Y] is zero
⇔ ∀Z Hom([Z, X], Q/Z) ⟵ Hom([Z, Y], Q/Z) is zero
⇔ ∀Z [X, Z ∧ Î] ⟵ [Y, Z ∧ Î] is zero
⇔ ∀Z [X, Z] ⟶ [Y, Z] is zero
⇔ u is cophantom

(The fourth implication uses the fact that Î ∈ Pic, so that Z ↦ Z ∧ Î is an automorphism of the category of small spectra).

11. The natural topology

As in [HPS95, Section 4.4], we can give [X, Y] a natural topology. For any map f : F ⟶ X with F small, we write U_{f} = {u : X ⟶ Y | u ∘ f = 0}, and then give [X, Y] the linear topology for which the sets U_{f} form a basis of neighbourhoods of 0. We recall from [HPS95, Proposition 4.4.1] the basic properties of this topology:

Proposition 11.1.

(a) The composition map [X, Y] × [Y, Z] ⟶ [X, Z] is continuous.

(b) Any pair of maps X′ ⟶ X and Y ⟶ Y′ induces a continuous map [X, Y] ⟶ [X′, Y′].

(c) If X is small then [X, Y] is discrete.

(d) The closure of 0 in [X, Y] is the set of phantom maps, so [X, Y] is Hausdorff if and only if there are no phantoms from X to Y.

(e) [X, Y] is always complete.

(f) [⋁_{i} X_{i}, Y] is homeomorphic to ∏_{i}[X_{i}, Y] with the product topology, but the natural topology on [X, ∏_{i} Y_{i}] is finer than the product topology in general.

Corollary 11.2. If we give Hom(π_{k}X, π_{k}Y) the compact-open topology, then it follows from (a) that the evident map

[X, Y] ⟶ Hom(π_{k}X, π_{k}Y)

is continuous.

We next prove some additional properties.

Proposition 11.3. The smash product map [W, X] × [Y, Z] ⟶ [W ∧ Y, X ∧ Z] is continuous.

Proof. Suppose we have maps u: W ⟶ X and v: Y ⟶ Z, a 𝒦-small spectrum A and a map A ⟶^{f} W ∧ Y, so that

N = {w: W ∧ Y ⟶ X ∧ Z | wf = (u ∧ v)f}

is a basic neighbourhood of u ∧ v in [W ∧ Y, X ∧ Z]. We know from Proposition 9.2 that W can be written as a minimal weak colimit W = mwlim_{-→}_{α} W_{α} with W_{α} small,

and similarly Y = mwlim_{-→}_{β} Y_{β}. Note that [A, W ∧ (−)] is a homology theory, so that

[A, W ∧ Y] = lim_{-→}_{β}[A, W ∧ Y_{β}], so f factors through a map g: A ⟶ W ∧ Y_{β} for some

β. By a similar argument, g factors through a map h: A ⟶ W_{α} ∧ Y_{β} for some α. Write i and j for the given maps W_{α} ⟶ W and Y_{β} ⟶ Y. Write

N′ = {(u′, v′) ∈ [W, X] × [Y, Z] | u′i = ui and v′j = vj}.

Then N′ is a neighbourhood of (u, v) and if (u′, v′) ∈ N′ then u′ ∧ v′ ∈ N. This means that the smash product is continuous at (u, v), as required.

Corollary 11.4. The adjunction map [X, F (Y, Z)] ⟶ [X ∧ Y, Z] is a continuous bijection, and a homeomorphism if Y is dualisable.

Proof. Let ϵ: F (Y, Z) ∧ Y ⟶ Z be the evaluation map. Then the adjunction map is the composite

[X, F (Y, Z)] ⟶^{(−)∧1_{Y}} [X ∧ Y, F (Y, Z) ∧ Y] ⟶^{ϵ_{∗}} [X ∧ Y, Z],

which is continuous by Proposition 11.3 and Proposition 11.1(a). It is also a bijection, by the defining property of F (Y, Z). If Y is dualisable and η: S ⟶ Y ∧ DY = F (Y, Y) is the unit map then we can identify the inverse of the adjunction map with the composite

[X ∧ Y, Z] ⟶^{(−)∧1} [X ∧ Y ∧ DY, Z ∧ DY] ⟶^{(1∧η)^{∗}} [X, Z ∧ DY],

which is again continuous.

Proposition 11.5. If [F (n), Y]_{∗} is finite, then for any X ∈ 𝒦 we have

[X, Y] = lim_{←-}{[X′, Y] | (X′ ⟶ X) ∈ Λ′(X)},

and this is a compact Hausdorff group.

Remark 11.6. This applies when Y is E-finite, and thus (by Theorem 8.9) when Y is dualisable.

Proof. Because X is the minimal weak colimit of Λ′(X), the map from [X, Y] to the inverse limit is surjective. The kernel is the set of phantom maps X ⟶ Y, which is zero by part (b) of Theorem 10.7.

Corollary 11.7. If X is dualisable and Y is small then [X, Y] is both finite (by Corollary 8.12) and compact Hausdorff, hence discrete.

Corollary 11.8. If X and Y are dualisable then the map D: [X, Y] ⟶ [DY, DX] is a homeomorphism.

Proof. As D^{2} = 1 it is enough to check that D is open. Consider a 𝒦-small spectrum Z and a map f: Z ⟶ X, so that U_{f} = {u: X ⟶ Y | uf = 0} is a basic open neighbourhood of 0 in [X, Y]. The image under D is {v: DY ⟶ DX | (Df)∘v = 0}. This is open in [DY, DX] because (Df)_{∗} is continuous and [DY, Z] is discrete.

Proposition 11.9. If X is dualisable, then the natural topology on [X, E] is the I_{n}-adic topology. This is also the same as the topology defined by the kernels of the maps [X, E] ⟶ [X, E/I] as I runs over a tower as in Proposition 4.22. Similar remarks apply to E_{∗}^{∨}X.

Proof. The natural topology is profinite by Proposition 11.5. The I_{n}-adic topology is profinite because E^{∗}X is finitely generated over E^{∗} by Theorem 8.6 and Proposition 2.4. Using Theorem 8.6 again and a thick subcategory argument we see that (E/I)^{∗}X is finite. We also know from Corollary 7.11 that E = holim_{←-}_{I} E/I and

there are no lim_{←-}^{1} terms because everything is finite so [X, E] = lim_{←-}_{I}[X, E/I] and the

topology defined by the kernels is again profinite. As the maps [X, E] ⟶ [X, E/I] are continuous, we see that the natural topology is at least as large as the topology defined by the E/I, and this is at least as large as the I_{n}-adic topology because the groups [X, E/I] are I_{n}-torsion. By a well-known lemma, comparable compact Hausdorff topologies are always equal. The result carries over to E_{∗}^{∨}X because of Corollary 11.4.

12. Dualisable spectra

In this section we prove some more results about the category 𝒟 of dualisable spectra in 𝒦, and its subcategory ℱ of 𝒦-small spectra.

First observe that ℱ is the same as the category 𝒞_{n} ⊆ ℒ considered in Section 6.1. The following three results thus follow from Theorem 6.9, Theorem 6.12 and Corollary 6.6.

Proposition 12.1. The only thick subcategories of ℱ are {0} and ℱ.

Proposition 12.2. Every small spectrum X has a good v_{n} self map v: Σ^{d}X ⟶ X (which is an isomorphism).

Proposition 12.3. Let u: Σ^{d}X ⟶ X be a self map of a 𝒦-small spectrum such that K_{∗}u is nilpotent. Then u is nilpotent.

For self maps of spectra that are dualisable but not small it is natural to consider self maps that are topologically nilpotent rather than nilpotent. We first need to define topological nilpotence.

Proposition 12.4. Let X be a spectrum in 𝒦 and u: Σ^{d}X ⟶ X a map. Let u^{−1}X denote the sequential colimit in 𝒦 of the sequence

X ⟶^{u} Σ^{−d}X ⟶^{u} Σ^{−2d}X ⟶ ….

Consider the following statements, in which Z and W run over 𝒦-small spectra.

(a) u^{−1}X = 0.

(b) For all Z and all z: Z ⟶ X we have u^{N} ∘ z = 0 for N ≫ 0.

(c) For all Z we have 1_{Z} ∧ u^{N} = 0 for N ≫ 0.

(d) For all W and all w: W ⟶ S, the following composite vanishes for N ≫ 0:

Σ^{Nd}W ∧ X ⟶^{w∧1} Σ^{Nd}X ⟶^{u^{N}} X.

Equivalently, the following adjoint map vanishes:

Σ^{Nd}W ⟶^{w} S^{Nd} ⟶^{(u^{#})^{N}} F (X, X).

(e) d = 0 and u^{N} ⟶ 0 in the natural topology on [X, X].

Then (a)⇔(b)⇐(c) and (b)⇐(d). If X is dualisable then (b)⇔(c)⇔(d). If d = 0 then (c)⇔(e).

Definition 12.5. If (a) and (b) hold in the above proposition, we say that u is topologically nilpotent.

Proof of Proposition 12.4. (a)⇔(b): We know that u^{−1}X = 0 if and only if we have [Z, u^{−1}X] = 0 for all Z, and as Z is small we have [Z, u^{−1}X] = u^{−1}[Z, X]. The claim follows easily.

(a)⇐(c): If (c) holds then we see that Z ∧ u^{−1}X = 0 and thus u^{−1}X = 0, so (a) holds. Conversely, suppose (a) holds and X is dualisable. Then Z ∧ X is small by Theorem 8.5, and (1 ∧ u)^{−1}(Z ∧ X) = 0. It follows that (c) holds.

(d)⇒(b): By Corollary 7.11 we have X = holim_{-→}_{I} D(S/I)∧X, so any map z: Z ⟶

X factors through D(S/I) ∧ X ⟶ X for some I. If (d) holds then D(S/I) ∧ X ⟶

X ⟶^{u^{N}} Σ^{−Nd}X vanishes for N ≫ 0 and thus so does Z ⟶ X ⟶^{u^{N}} Σ^{−Nd}X, so (b) holds.

We next assume that X is dualisable; then if (b) holds we can put Z = W ∧ X and deduce that (d) holds.

Finally, if d = 0 then (e) is a direct translation of (b).

Lemma 12.6. If u: Σ^{d}X ⟶ X is topologically nilpotent, then the same is true of u ∧ 1_{Y} for any Y.

Proof. The functor (−) ∧ Y preserves sequential colimits, so the claim is obvious from criterion (a).

Recall that a self map u : A ⟶ A of a topological Abelian group A is said to be topologically nilpotent if the sequence {u^{n}x} converges to 0 uniformly in x. We will apply this definition to E^{∗}X and E_{∗}^{∨}X for dualisable X, using the I_{n}-adic topology. In this case, both E^{∗}X and E_{∗}^{∨}X are finitely generated, so there is no difference between pointwise and uniform convergence.

Proposition 12.7. Let u: Σ^{d}X ⟶ X be a self map of a dualisable spectrum. Then the following are equivalent:

(a) K_{∗}u is nilpotent.

(b) K^{∗}u is nilpotent.

(c) E_{∗}^{∨}u is topologically nilpotent.

(d) E^{∗}u is topologically nilpotent.

(e) u is topologically nilpotent.

Proof. (a)⇔(b): Duality.

(a)⇔(e): Recall that u is topologically nilpotent if and only if 1_{Z} ∧ u is nilpotent for all small Z. This is equivalent to (a) by the Künneth theorem and Proposition 12.3.

(d)⇒(b): We check by induction on k that u induces a topologically nilpotent self map of (E/I_{k})^{∗}X. Indeed, consider the short exact sequence

(E/I_{k})^{∗}(X)/v_{k} ↣ (E/I_{k+1})^{∗}(X) ↠ ann(v_{k}, (E/I_{k})^{∗−2p^{k}+1}(X)).

The subspace topology on ann(v_{k}, (E/I_{k})^{∗}(X)) is the same as the I_{n}-adic topology by the Artin-Rees lemma. It is then easy to see that u induces a topologically nilpotent map on (E/I_{k+1})^{∗}X if it does on (E/I_{k})^{∗}X. In the case k = n we have (E/I_{n})^{∗}X = K^{∗}X and this is finite and thus discrete so the self map is nilpotent.

(b)⇒(d): It is easy to check that for any generalised Moore spectrum S/I of height n, the spectrum E/I is K-finite so (E/I)^{∗}u is nilpotent. It follows easily

that if a ∈ E^{∗}X then the composite Σ^{Nd}X ⟶^{u^{N}} X ⟶^{a} E ⟶ E/I vanishes for large N. We know from Proposition 11.9 that the natural topology is the same as that defined by the kernels of the maps E^{∗}X ⟶ (E/I)^{∗}X. It follows that a ∘ u^{N} ⟶ 0 in the I_{n}-adic topology. Thus the sequence (u^{N})^{∗} ∈ End_{E^{∗}}(E^{∗}X) converges to zero pointwise, and hence uniformly.

This shows that all our statements are equivalent except for (c). However, E_{∗}^{∨}u = E^{∗}Du and by the above applied to Du this is topologically nilpotent if and only if Du is, and Du is topologically nilpotent if and only if u is (by Corollary 11.8). Thus (c) is equivalent to (e) also.

Lemma 12.8. Let u: Σ^{d}X ⟶ X be a self map of a small spectrum. Then there is a unique idempotent e: X ⟶ X such that u is the wedge of an isomorphism Σ^{d}eX ⟶ eX with a nilpotent map Σ^{d}(1 − e)X ⟶ (1 − e)X.

Proof. Let v: Σ^{t}X ⟶ X be a good v_{n} self map. We can choose a > 0 and b ∈ Z such that w = u^{a}v^{b} has degree zero. By Corollary 8.12, we know that [X, X] = π_{0}(DX ∧ X) is a finite ring. It follows that there exist r < s such that w^{r} = w^{s}. We then find that w^{k(s−r)} = w^{(k+1)(s−r)} as long as k(s − r) > r. Thus, if t is a multiple of s − r which is greater than r, we find that e = w^{t} = u^{at}v^{bt} satisfies e^{2} = e. Thus e is an idempotent which commutes with u, which means that u respects the splitting X = eX ∨ (1 − e)X. As u^{at}v^{bt} = 1 on eX we see that u: Σ^{d}eX ⟶ eX is an isomorphism. On the other hand, we have u^{at}v^{bt} = 0 on (1 − e)X, so that u: (1 − e)X ⟶ (1 − e)X is nilpotent.

If e′ is a different idempotent with the required properties then e = u^{at}v^{bt} is the wedge of an isomorphism on e′X with a nilpotent self map on (1 − e′)X. As e is idempotent, it is easy to deduce that e′ = e.

Proposition 12.9. Let u: Σ^{d}X ⟶ X be a self map of a dualisable spectrum in 𝒦. Then there is a unique idempotent e: X ⟶ X such that u is the wedge of an isomorphism Σ^{d}eX ⟶ eX with a topologically nilpotent map Σ^{d}(1 − e)X ⟶ (1 − e)X.

Proof. Choose a tower of generalised Moore spectra S/I(j) as in Proposition 4.22. We will write X/I(j) for X ∧ S/I(j). For each j Lemma 12.8 gives a unique idempotent e_{j}: X/I(j) ⟶ X/I(j) such that u ∧ 1 is a wedge of an isomorphism and a nilpotent in terms of the splitting given by e_{j}. We claim that the following diagram commutes:

X/I(j)  X/I(j) X/I(j − 1)  X/I(j − 1)._{_{1∧g_{j}} u}  ^{e_{j}} w _{u_{1∧g_{j}}} w_{e_{j−1}}

To see this, let v be a good v_{n} self map of S/I(j). Using Remark 4.25 and Proposition 4.4, we see that v induces compatible v_{n} self maps of all spectra in the diagram (all of which we call v) and that v ∘ (1 ∧ g_{j}) = (1 ∧ g_{j}) ∘ v. Note also that (u ∧ 1) ∘ (1 ∧ g_{j}) = (1 ∧ g_{j}) ∘ (u ∧ 1). It follows from the proof of Lemma 12.8 that there exist integers a, b such that e_{j} = (u ∧ 1)^{a}v^{b} (interpreted as a self map of X/I(j)) and e_{j−1} = (u ∧ 1)^{a}v^{b} (interpreted as a self map of X/I(j − 1)). It follows easily that the diagram commutes.

We next recall from Proposition 7.10 that X = holim_{←-}_{k} X ∧ S/I(k). Moreover,

Corollary 8.12 shows that [X, X ∧ S/I(k)] = π_{0}(DX ∧ X ∧ S/I(k)) is finite, so there is no lim^{1} term and [X, X] = lim_{←-}_{k}[X, X ∧ S/I(k)]. It follows that there is a unique map e: X ⟶ X such that the following diagram commutes for each j:

X  X X/I(j)  X/I(j).^{e}  w_{_{1∧η_{j}} u}  _{u_{1∧η_{j}}} w_{e_{j}}

The map e^{2} also has this property, so e^{2} = e. We also know that

(1 ∧ η_{j}) ∘ (ue − eu) = ((u ∧ 1)e_{j} − e_{j}(u ∧ 1)) ∘ (1 ∧ η_{j}) = 0

for all j, and thus that ue = eu, so u is compatible with the splitting X = eX ∨ (1 − e)X. Similar considerations show that u is an isomorphism on eX and that u ∧ 1 is nilpotent on (1 − e)X/I(j) for all j. It follows by Proposition 12.7 that u is topologically nilpotent on (1 − e)X.

It follows from the previous proposition that the Krull-Schmidt theorem holds in 𝒟. (It was shown by Freyd [Fre66] that the Krull-Schmidt theorem holds for finite torsion spectra, and our argument is much the same.)

Definition 12.10. We say that a spectrum X ∈ 𝒟 is indecomposable if X ≠ 0 and there do not exist nontrivial spectra Y, Z ∈ 𝒟 with X ≃ Y ∨ Z.

We will see in Proposition 12.17 that there are only a set of isomorphism classes of dualisable spectra, and hence of indecomposable spectra.

Proposition 12.11. Let X ∈ 𝒟 be an indecomposable spectrum, and write R = [X, X]. Let I ⊆ R be the set of topologically nilpotent self maps of X. Then I is a two-sided ideal in R and R/I is a finite field.

Remark 12.12. Adams and Kuhn have shown [AK89] that in a slightly different context, all possible finite fields can occur. Presumably this is true for us too.

Proof. First, it follows immediately from Proposition 12.9 that every u ∈ R is either invertible or topologically nilpotent, so R = R^{×} ⨿ I. If a ∈ I and b ∈ R then a ∉ R^{×} so ab, ba ∉ R^{×} so ab, ba ∈ I. Now suppose that a, b ∈ I. We claim that c = a + b ∈ I. If not, then c is invertible. By our previous observation we have c^{−1}b ∈ I so c^{−1}b is topologically nilpotent. Hence ∑_{k}(c^{−1}b)^{k} converges to an inverse for 1 − c^{−1}b = c^{−1}a, so a is invertible. This contradicts the assumption that a ∈ I. Thus I is a two-sided ideal. It is clear that every nonzero element of R/I is invertible, so R/I is a division ring. Next, let J be the kernel of the evident map R ⟶ End(K_{∗}X). This is clearly a two-sided ideal, and Proposition 12.7 tells us that J ≤ I. Because K_{∗}X has finite dimension over K_{∗}, we see that R/J is finite and thus that R/I is finite. A well-known theorem of Wedderburn says that a finite division ring is a field.

Proposition 12.13. For any spectrum X ∈ 𝒟 there are unique integers n_{Y} = n_{Y} (X) ≥ 0, as Y runs through isomorphism classes of indecomposable spectra, such that n_{Y} = 0 for all but finitely many Y, and X is isomorphic to ⋁_{Y ∈ℐ} n_{Y} Y. (Here n_{Y} Y means the wedge of n_{Y} copies of the spectrum Y).

Proof. We may assume that this holds for all X′ such that dim(K_{∗}X′) < dim(K_{∗}X). It is clear that it also holds if X is indecomposable. If X is decomposable we can write X = V ∨ W with V ≠ 0 ≠ W and thus dim(K_{∗}V) > 0 and dim(K_{∗}W) > 0 and thus dim(K_{∗}V) < dim(K_{∗}X) and dim(K_{∗}W) < dim(K_{∗}X). By the induction hypothesis we have V ≃ ⋁_{Y} n_{Y} (V)Y and W ≃ ⋁_{Y} n_{Y} (W)Y. It follows that X ≃ ⋁_{Y} (n_{Y} (V) + n_{Y} (W))Y, so we may take n_{Y} (X) = n_{Y} (V) + n_{Y} (W).

We still need to prove uniqueness. For this, we change notation and assume that we have indecomposables U_{1}, …, U_{r} and V_{1}, …, V_{s} such that there exists an isomorphism f: U_{1} ∨ … ∨ U_{r} ≃ V_{1} ∨ … ∨ V_{s}. We need to show that r = s and that after reordering the V ’s we have U_{i} ≃ V_{i} for all i. Write g = f^{−1}, and f_{ij} for the component of f mapping U_{i} to V_{j}, and g_{ji} for the component of g mapping V_{j} to U_{i}. We then have ∑_{j} g_{ji}f_{kj} = δ_{ik}: U_{k} ⟶ U_{i}. In particular, we have ∑_{j} g_{j1}f_{1j} = 1: U_{1} ⟶ U_{1}. Recall that the maps U_{1} ⟶ U_{1} that are not isomorphisms form a two-sided ideal (and thus an additive subgroup) in [U_{1}, U_{1}]; it follows that g_{j1}f_{1j} is invertible for some j. After renumbering the V ’s, we may assume j = 1. This means that f_{11} : U_{1} ⟶ V_{1} is a split monomorphism but V_{1} is indecomposable so f_{11} is an isomorphism. Similarly g_{11} is an isomorphism. We now write U′ = U_{2} ∨ … ∨ U_{r} and V′ = V_{2} ∨ … ∨ V_{s} and redefine our notation slightly so that f_{12} is the component of f: U_{1} ∨ U′ ⟶ V_{1} ∨ V′ mapping U_{1} to V′ and so on. We thus have a matrix equation

( \begin{matrix}f_{11} & f_{21} \\ f_{12} & f_{22} \end{matrix} ) ( \begin{matrix}g_{11} & g_{21} \\ g_{12} & g_{22} \end{matrix} ) = ( \begin{matrix}1_{V_{1}} & 0 \\ 0 & 1_{V′} \end{matrix} ) .

One can deduce directly that f_{22} : U′ ⟶ V′ is an isomorphism, with inverse given by g_{22} −g_{12}g_{11}^{−1}g_{21}. By induction on the number of indecomposables, we can assume that r − 1 = s − 1 and that after reordering we have U_{i} ≃ V_{i} for i > 1. Thus r = s and U_{i} ≃ V_{i} for all i.

We next exhibit some ideals in 𝒟; we conjecture that they are all the ideals.

Definition 12.14. Given k ≤ n we write 𝒟_{k} for the category of spectra X ∈ 𝒟 such that X is a retract of Y ∧ Z for some Y ∈ 𝒟 and some finite spectrum Z of type at least k. It is easy to see that 𝒟_{n} = ℱ.

Proposition 12.15. 𝒟_{k} is an ideal in 𝒟 and is closed under D. Moreover, given X ∈ 𝒟, the following are equivalent.

(a) X ∈ 𝒟_{k}.

(b) X is a module over some generalised Moore spectrum S/I of height k.

(c) E_{∗}^{∨}X is I_{k}-torsion.

(d) E^{∗}X is I_{k}-torsion.

Proof. (a)⇔(b): This is Proposition 4.17.

(b)⇒(c): It is easy to see using the cofibrations which define S/I that E_{∗}^{∨}(X ∧ S/J) is I_{k}-torsion. It follows that E_{∗}^{∨}X is I_{k}-torsion.

(c)⇒(b): By induction we may assume that there is a generalised Moore spectrum S/J of type k − 1 such that X is a module over S/J. Let v be a good v_{k−1} self map of S/J. Let 𝒞 be the category of those spectra Y such that for each a ∈ π_{∗}(S/J ∧ X ∧ Y) we have v^{N}a = 0 for N ≫ 0. One sees easily that 𝒞 is thick and E ∈ 𝒞. Thus D(S/J ∧ X) ∈ 𝒞 by Theorem 8.9, and v acts nilpotently on 1 ∈ π_{0}(S/J ∧ X ∧ D(S/J ∧ X)) = End(S/J ∧ X). This means that v is nilpotent as a self map of S/J ∧ X, so that the evident map S/J ∧ X ⟶ S/(J, v^{p^{N}}) ∧ X is a split monomorphism for large N. As X ⟶ S/J ∧ X is also a split monomorphism, we see that X is a module over S/(J, v^{p^{N}}).

We also know from Proposition 4.17 that 𝒟_{k} is an ideal. If X is a retract of Y ∧ Z for dualisable Y and Z, then DX is a retract of D(Y ∧ Z) = DY ∧ DZ. Thus 𝒟_{k} is closed under D. It is then immediate from the isomorphism E^{∗}X = E_{∗}^{∨}DX that (c)⇔(d).

Corollary 12.16. If X ∈ 𝒦 then the following are equivalent:

(a) X is small.

(b) E^{∗}X is finite.

(c) E_{∗}^{∨}X is finite.

(d) X is dualisable and K-nilpotent.

Proof. Using Theorem 8.6 we see that any of (a) … (d) implies that X is dualisable, so we may assume this throughout. We saw in Theorem 8.5 that (a) implies (b),(c) and (d). By applying Proposition 12.15, we see that (b) implies (a) and (c) implies (a).

All that is left is to prove that (d) implies (a). Suppose that X is dualisable and K-nilpotent. Let ℐ be the category of those spectra Z ∈ 𝒦 such that Z is a module over S/I for some I of height n. We know from Proposition 4.17 that ℐ is an ideal, and thus from Proposition 7.9 that it contains all K-nilpotent spectra. In particular, we see that X is a retract of some S/I ∧ X, and S/I ∧ X is the smash product of a small spectrum with a dualisable one so it is small, so X itself is small.

12.1. The semiring π_{0}𝒟. Write π_{0}𝒟 for the set of isomorphism classes of spectra in 𝒟. We start by verifying that this is really a set.

Proposition 12.17. The category 𝒟 has 2^{ℵ_{0}} isomorphism classes.

Proof. There are only countable many finite spectra Y of type at least n, and for each one we know that [LY, LY] is finite, so LY has only finitely many retracts. It follows that π_{0}(ℱ) is countable. Moreover, if U, V ∈ ℱ then [U, V] is finite. It follows that there are at most ℵ_{0}^{ℵ_{0}} = 2^{ℵ_{0}} different towers of 𝒦-small spectra. If X ∈ 𝒟 then X ∧ S/I is small and X = holim_{←-}_{I} X ∧ S/I, so X is the inverse limit of one of these

towers. Thus |π_{0}𝒟| ≤ 2^{ℵ_{0}}.

Conversely, it is shown in [HMS94, Proposition 9.3] that the p-adic integers embed in the Picard group, which in turn embeds in π_{0}𝒟. Thus |π_{0}𝒟| ≥ 2^{ℵ_{0}}.

The set π_{0}𝒟 is a commutative semiring under ∨ and ∧. The group of units is the aforementioned Picard group of 𝒦. See [HMS94, HS95, Str92] for discussion and calculation of Picard groups. See also Section 14 and Section 15.1.

The subsets π_{0}𝒟_{k} are ideals. The maps n_{Y} : π_{0}𝒟 ⟶ N give an additive isomorphism

π_{0}𝒟 = ⨁_{Y ∈ℐ}N

where ℐ denotes the set of isomorphism classes of indecomposable spectra. There is also an interesting semiring homomorphism

d: π_{0}𝒟 ⟶ N[t]/(t^{|v_{n}|} − 1)

defined by

d(X) = ∑_{i=0}^{|v_{n}|−1} t^{i} dim_{F_{p}} K_{i}(X).

We can of course set t = 1 to get a semiring map π_{0}𝒟 ⟶ N, or set t = −1 to get a map π_{0}𝒟 ⟶ Z. This factors through the ring K_{0}𝒟, which is the quotient of π_{0}𝒟 in which we set Y = X + Z whenever there is a cofibre sequence X ⟶ Y ⟶ Z.

When X ∈ ℱ is small, we have two other measures of the “size” of X.

Definition 12.18. Given a finite Abelian p-group A we define the length to be len(A) = log_{p} |A|. Given a graded Abelian p-group A_{∗} which is periodic of period |v_{n}^{p^{K}} | for some K, we define

len(A_{∗}) = ∑_{0≤k<|v_{n}^{p^{K}} |} len(A_{k})/p^{K} ∈ Z[\frac{1}{p}].

We have normalised this so that len(K_{∗}) = 1, and thus len(A_{∗}) ∈ N when A_{∗} is a finite module over E_{∗}.

The length len(K_{∗}X) is defined whenever X is dualisable. It is not a very sensitive measure of the size of X, because len(K_{∗}(S/I)) = 2^{n} for all I. If X is small we can also define len(E_{∗}X) and len(π_{∗}X) (we have the required periodicity because of the existence of v_{n} self maps). The following result shows that len(E_{∗}X) is a good measure of size.

Theorem 12.19. For any integer k ≥ 0, the set of isomorphism classes of 𝒦-small spectra X with len(E_{∗}X) ≤ k is finite.

Proof. We will work with E(n) instead of E: since we are only considering finite modules and small spectra, this makes no difference. Note that there are only finitely many E(n)_{∗}-modules M with len M ≤ k. Now suppose we have a finite E(n)_{∗}- module M. If M ≅ E(n)_{∗}X for some K-local spectrum X, then Y is necessarily small. The filtration theorem proved in [HS95, Section 2] then tells us that M has a finite comodule filtration 0 = M_{0} ⊆ M_{1} ⊆ … ⊆ M_{t} where t ≤ k and M_{i}/M_{i−1} ≅ Σ^{j(i)}K_{∗} for some 0 ≤ j(i) < |v_{n}|. Given M_{i−1}, the number of possibilities for M_{i} is then determined by Ext_{E(n)_{∗}E(n)}^{1,j(i)}(K_{∗}, M_{i−1}). It follows that, as long as the groups Ext_{E(n)_{∗}E(n)}^{s,t}(K_{∗}, K_{∗}) are finite for all s and t, the number of realisable comodule structures on M is finite.

Now fix a realisable E(n)_{∗}E(n)-comodule structure on M. Consider the category 𝒞(M) of pairs (X, x), where X is a small spectrum and x: M ⟶ E(n)_{∗}X is an isomorphism of comodules. It is enough to show that 𝒞(M) has only finitely many isomorphism classes. Recall from Proposition 6.5 that there is an Adams spectral sequence

E_{2}^{st}(X, Y) = Ext_{E(n)_{∗}E(n)}^{s,t}(E(n)_{∗}X, E(n)_{∗}Y) ⟹ [X, Y]_{t−s},

with E_{N} = E_{∞} for some constant N independent of X and Y. Moreover, given maps E(n)_{∗}X ⟶^{f} E(n)_{∗}Y ⟶^{g} E(n)_{∗}Z which survive to E_{r}, we have d_{r}(gf) = g_{∗}d_{r}(f) + f^{∗}d_{r}(g) (see [Mos68, Theorem 2.1]. We say that objects (X, x) and (Y, y) in 𝒞(M) are E_{r}-equivalent if the element yx^{−1} ∈ Hom_{E(n)_{∗}E(n)}(E(n)_{∗}X, E(n)_{∗}Y) = E_{2}^{0,0}(X, Y) survives to E_{r}. Our formula for d_{r}(gf) shows that this is an equivalence relation. Clearly any two objects in 𝒞(M) are E_{2}-equivalent. If they are E_{r+1}- equivalent then they are E_{r}-equivalent, and if they are E_{N}-equivalent then they are isomorphic. It is thus enough to show that each E_{r}-equivalence class splits into finitely many E_{r+1}-equivalence classes.

If (X, x) and (Y, y) are E_{r}-equivalent, we define

δ(X, x, Y, y) = y_{∗}(x^{−1})^{∗}d_{r}(y^{−1}x) ∈ Ext_{E(n)_{∗}E(n)}^{r,r−1}(M, M).

As x and y are isomorphisms, we have δ(X, x, Y, y) = 0 if and only if (X, x) and (Y, y) are E_{r+1}-equivalent. Moreover, the formula for d_{r}(gf) implies that

δ(X, x, Z, z) = δ(X, x, Y, y) + δ(Y, y, Z, z)

whenever this makes sense. We conclude that the set of E_{r+1}-equivalence classes in a given E_{r}-equivalence class bijects with a subgroup of Ext_{E(n)_{∗}E(n)}^{r,r−1}(M, M), so it will be enough to show that this group is finite. Using our comodule filtration of M, we find once again that it suffices to show that Ext_{E(n)_{∗}E(n)}^{s,t}(K_{∗}, K_{∗}) is finite for all s and t.

It is easy to check using cobar resolutions that

Ext_{E(n)_{∗}E(n)}^{s,t}(E(n)_{∗}, K_{∗}) = Ext_{Σ_{∗}}^{s,t}(K_{∗}, K_{∗}).

This is finite in each bidegree by the proof of of [Rav86, Theorem 6.2.10(a)]. It follows inductively that Ext_{E(n)_{∗}E(n)}^{s,t}(E(n)_{∗}/I_{k}, K_{∗}) is finite and in particular that Ext_{E(n)_{∗}E(n)}^{s,t}(K_{∗}, K_{∗}) is finite, as required.

13. K-nilpotent spectra

In this section, we give a number of characterisations of K-nilpotent spectra.

Theorem 13.1. Suppose X is a spectrum in 𝒦. Then the following are equivalent.

(a) X is a retract of X ∧ Z for some finite spectrum Z of type at least n.

(b) X is K-nilpotent.

(c) X lies in the thick subcategory generated by the K-injectives.

(d) X ∧_{𝒮} Y is K-local for all Y ∈ 𝒮.

(e) The functor on 𝒦 which takes Y to π_{0}(X ∧_{𝒦} Y), is a homology functor.

Before proving this, we need to address a different problem. Let X be a finite spectrum of type m < n, with a good v_{m} self map v: Σ^{d}X ⟶ X. We need to understand what happens to X and v under K-localisation.

As usual, if Z is a spectrum with a self-map f, we denote the cofibre of f by Z/f.

Proposition 13.2. If Y ∈ 𝒦 and X is a finite spectrum of type m < n with v: Σ^{d}X ⟶ X a good v_{m} self map of X, then the natural map

X ∧ Y ⟶ holim_{←-}(X ∧ Y)/v^{k}

is an equivalence. Furthermore we have a short exact sequence

0 ⟶ lim_{←-}^{1} ker(v^{k}, π_{∗−dk−1}(X ∧ Y)) ⟶ π_{∗}(X ∧ Y) ⟶ π_{∗}(X ∧ Y)_{v}^{∧} ⟶ 0

Proof. First note that it does not matter whether we work in 𝒦 or in 𝒮. Indeed, X is finite, so X ∧ Y is already K-local, so we have X ∧ Y = L̂(L̂X ∧ Y). Also, the products in 𝒦 and in 𝒮 are the same. The homotopy inverse limit of a tower can be constructed as the fibre of a standard self map of the product of its terms, so it is also the same in 𝒦 and 𝒮.

We claim that the map X ∧Y ⟶^{g} holim_{←-}(X ∧Y)/v^{k} is an X/v-equivalence. Indeed,

as v is strongly central, the map X/v ∧ X ⟶^{1∧v^{k}} X/v ∧ X is trivial for k ≥ 2. Thus the map X/v ∧ X ∧ Y ⟶ X/v ∧ X/v^{k} ∧ Y is a split monomorphism for k ≥ 2, with cofibre a suspension of X/v ∧ X ∧ Y. Thus we have a short exact sequence

π_{i}(X/v ∧ X ∧ Y) ↣ π_{i}(X/v ∧ X/v^{k} ∧ Y) ↠ π_{i−dk−1}(X/v ∧ X ∧ Y)

and a corresponding six-term long exact sequence of lim_{←-} and lim_{←-}^{1} terms. However, the maps of the tower induce multiplication by 1 ∧ v on the cokernel terms, and using strong centrality again, we find that the tower of cokernels is pro-isomorphic to the zero tower. The tower of kernels is constant. Hence the map g is an X/vequivalence.

The map g is therefore an F (m + 1)-equivalence between K(n)-local, and hence F (n)-local, spectra. Since m + 1 ≤ n, we see that g is an equivalence.

We now consider the Milnor exact sequence

0 ⟶ lim_{←-}^{1}π_{∗+1}(X ∧ Y/v^{k}) ⟶ π_{∗}(X ∧ Y) ⟶ lim_{←-} π_{∗}(X ∧ Y/v^{k}) ⟶ 0.

To identify the terms in this sequence, note that we have short exact sequences

0 ⟶ (π_{∗}(X ∧ Y))/v^{k} ⟶ π_{∗}(X ∧ Y/v^{k}) ⟶ ker(v^{k}, π_{∗−dk−1}(X ∧ Y)) ⟶ 0.

Taking inverse limits gives us the usual 6-term exact sequence

0 ⟶ π_{∗}(X ∧ Y)_{v}^{∧} ⟶ lim_{←-} π_{∗}(X ∧ Y/v^{k}) ⟶ lim_{←-} ker(v^{k}, π_{∗−dk−1}(X ∧ Y))

⟶ lim_{←-}^{1}π_{∗}(X ∧ Y)/v^{k} ⟶ lim_{←-}^{1}π_{∗}(X ∧ Y/v^{k}) ⟶ lim_{←-}^{1} ker(v^{k}, π_{∗−dk−1}(X ∧ Y)) ⟶ 0.

Note that the sequence π_{∗}(X ∧ Y)/v^{k} is Mittag-Leffler, so its lim_{←-}^{1} term vanishes.

Thus lim_{←-}^{1} π_{∗}(X ∧ Y/v^{k}) = lim_{←-}^{1} ker(v^{k}, π_{∗−dk−1}(X ∧ Y)), so the first term in the Milnor sequence is as claimed.

It remains to prove that π_{∗}(X ∧ Y)_{v}^{∧} = lim_{←-} π_{∗}(X ∧ Y/v^{k}), or in other words that the first map in the six term sequence is surjective. This is true because the surjective map π_{∗}(X ∧ Y) ⟶ lim_{←-} π_{∗}(X ∧ Y/v^{k}) factors through it.

Proof of Theorem 13.1. (a)⇒(b): We know by Proposition 6.15 that LZ is Knilpotent. It follows that any retract of X ∧ Z = X ∧ LZ is also K-nilpotent.

(b)⇔(c): this is Lemma 7.4.

(b)⇒(d): If X is K-nilpotent, so is X ∧_{𝒮} Y for any Y, so in particular X ∧_{𝒮} Y is K-local.

(d)⇒(e): Suppose that X ∧_{𝒮} Y is K-local for all Y, so that X ∧_{𝒦} Y = X ∧_{𝒮} Y. It also follows that for any Z ∈ 𝒮, the spectrum X ∧_{𝒮} ĈZ is both K-local and K-acyclic, hence is zero. This implies that X ∧_{𝒦} L̂Z = X ∧_{𝒮} L̂Z = X ∧_{𝒮} Z. In particular, we find that

X ∧_{𝒦} ⋁_{i}^{𝒦} Y_{i} = X ∧_{𝒮} ⋁_{i}^{𝒮} Y_{i}.

Given this, it is trivial to verify that Y ↦ π_{0}(X ∧_{𝒦} Y) is a homology functor on 𝒦.

(e)⇒(a): This is the most difficult part of the argument. For the rest of the proof, all smash products, limits and so on are taken in 𝒦. The plan is to start with the obvious fact that X is a retract of X ∧ L̂S. We then show that if Z is a finite type m spectrum with a v_{m} self-map v, then X ∧ L̂Z is a retract of (X ∧ L̂Z)/v^{k} for some sufficiently large k. It will then follow by induction that X is a retract of X ∧ L̂Z for some finite type n spectrum Z.

So consider the functor H on 𝒦 defined by H(Y) = π_{0}(L̂Z ∧ X ∧ Y). By (e), H is a homology functor on 𝒦. Our first goal is to show that there is a k such that v^{k}H_{∗}(Y) is v-divisible for all Y. Indeed, suppose not. For each k, choose a spectrum Y_{k} and an element a_{k} ∈ H_{∗}(Y_{k}) such that b_{k} = v^{k}a_{k} is not v-divisible. We can assume that b_{k} is in degree 0. Let Y = ⋁ Y_{k}, so we have H_{∗}(Y) = ⨁ H_{∗}(Y_{k}). We know from Proposition 13.2 that H(Y) maps onto its v-completion. However, (b_{k}) defines an element of the v-completion of H(Y), since only finitely many are not in the image of any v^{l}. Furthermore, since no b_{k} is v-divisible, each maps nontrivially to the v-completion, so that (b_{k}) can’t be in the image of H(Y), which is a contradiction.

Hence there is a k such that v^{k}H(Y) is v-divisible for all Y. We will now show that v^{k}H(Y) is in fact 0 for all Y. Indeed, by Proposition 13.2, the kernel of the surjective map from H(Y) to its v-completion is lim_{←-}^{1} ker(v^{l}, H(Y)), where we have left out the dimension shift. But we claim this tower is in fact Mittag-Leffler, so that it has no lim_{←-}^{1} term. Indeed, suppose we have a class x in ker(v^{l}, H(Y)) which

is in the image of ker(v^{l+k}, H(Y)). Then in particular, x is in v^{k}H(Y) so is vdivisible. Then for all i ≥ k, there is a w such that v^{i}w = x. In particular, w is in ker(v^{i+l}, H(Y)), so x is in the image of ker(v^{l+i}, H(Y)) for all i ≥ k. Thus the tower is Mittag-Leffler, and we find that H(Y) is v-complete, for all Y. In particular, v^{k}H(Y) is a v-divisible subgroup of a v-complete group, so is trivial.

Now, take Y = DW for a finite type n spectrum W. Using Spanier-Whitehead duality in 𝒦, we find that for any map W ⟶ L̂Z ∧ X, the composite

W ⟶ L̂Z ∧ X ⟶^{v^{k}∧1} L̂Z ∧ X

is null. (Remember the smash product is taken in 𝒦). Thus,

v^{k} ∧ 1 : L̂Z ∧ X ⟶ L̂Z ∧ X

is phantom, so by the last part of Theorem 9.5, v^{2k} ∧ 1 is trivial. Thus, L̂Z ∧ X is a retract of L̂(Z/v^{2k}) ∧ X. Since Z/v^{2k} is a finite spectrum of type m + 1, we are done.

Proposition 13.3. If X is K-nilpotent and Y is arbitrary then X ∧ Y, F (X, Y) and F (Y, X) are K-nilpotent.

Proof. Let 𝒞 be the category of those X such that X ∧ Y, F (X, Y) and F (Y, X) are K-nilpotent; this is clearly thick. If X is a K-module, then X ∧ Y, F (X, Y) and F (Y, X) are K-modules and therefore K-nilpotent. This shows that 𝒞 contains all K-injective spectra, and hence (by part (c) of Theorem 13.1) all K-nilpotent spectra.

It follows from Proposition 13.3 that the K-nilpotent spectra form the coideal generated by K as well as the ideal generated by K. One might then guess that if X is in both the localising subcategory generated by K (equivalently, MX = X) and the colocalising subcategory generated by K (equivalently L̂X = X), then X is K-nilpotent. This is false, however: the spectrum Y considered in Section 15.1 is a counterexample.

14. Grading over the Picard group

Recall, from for example [HMS94, Str92], that the Picard group Pic = Pic(𝒦) consists of isomorphism classes of spectra in 𝒦 which are invertible under the smash product. In this section we will need to be very careful about the distinction between objects and isomorphism classes, so we make formal definitions as follows.

Definition 14.1. We say that a spectrum P ∈ 𝒦 is invertible if there is a spectrum Q ∈ 𝒦 such that P ∧ Q ≃ S. We write 𝒫 for the category of invertible spectra. Given P ∈ 𝒫 we write [P] for the isomorphism class of P. We write Pic for the collection of these isomorphism classes, which is a set (rather than a proper class) by [HMS94, Proposition 7.6] or by Proposition 12.17.

The following result is proved as [HMS94, Theorem 1.3] but we give a different argument here for the sake of completeness.

Proposition 14.2. Given P ∈ 𝒦, the following are equivalent:

(a) P ∈ Pic.

(b) K_{∗}P ≃ K_{∗} (up to suspension).

(c) E^{∗}P ≃ E^{∗} (up to suspension).

(d) E_{∗}^{∨}P ≃ E_{∗} (up to suspension).

Proof. (a)⇒(b): If P ∧ Q = S then K_{∗}(P) ⊗_{K_{∗}} K_{∗}(Q) = K_{∗}, so K_{∗}(P) has rank one and is isomorphic to K_{∗} up to suspension.

(b)⇔(c): We may assume that K_{∗}P ≃ K_{∗} so that K_{∗}P is in even degrees. Then E^{∗}P is pro-free by Proposition 2.5, and (E^{∗}P)/I_{n} = K^{∗}. It follows easily that E^{∗}P = E^{∗}. The converse also follows from Proposition 2.5.

(b)⇔(d): This is similar to (b)⇔(c), using Proposition 8.4.

(b)⇒(a): Since K_{∗}P is finite dimensional, we know that P is dualisable. Furthermore, the group K_{∗}DP = Hom_{K_{∗}}(K_{∗}P, K_{∗}) is also one-dimensional. Hence the unit S ⟶ DP ∧ P of the ring spectrum DP ∧ P is a K-equivalence, and thus an isomorphism.

There is an evident surjective homomorphism deg: Pic ⟶ Z/|v_{n}| which sends [P] to the degree in which K_{∗}P is concentrated.

The general properties of the Picard group are given in the following proposition.

Proposition 14.3.

(a) The homomorphism Z ⟶ Pic that takes m to L̂S^{m|v_{n}|} extends to an injective homomorphism Z_{p} ⟶ Pic.

(b) | Pic | = 2^{ℵ_{0}}.

(c) If X is small, the orbit of X under the action of Pic is finite.

(d) Pic is a profinite Abelian group, and the kernel of deg: Pic ⟶ Z/|v_{n}| is a pro-p-group of finite index in Pic.

Proof. Part (a) is proved in [HMS94, Section 9], and part (b) is an immediate corollary of part (a) and Proposition 12.17. Part (c) follows from Proposition 14.2 and Theorem 12.19. Part (d) can also be deduced from the results of [HMS94]. Here we will give an independent proof that Pic is profinite, but we rely on [HMS94] to tell us that ker(deg) is p-local.

To prove that Pic is profinite, choose a tower {S/J(i)} as in Proposition 4.22. Let G(i) be the stabiliser in Pic of S/J(i). Part (c) says that Pic /G(i) is finite. We claim first that G(i + 1) ⊆ G(i). Indeed, suppose P ∈ G(i + 1), so that P ∧ S/J(i + 1) ≃ S/J(i + 1). Then we have the composite

P ⟶ P ∧ S/J(i + 1) ⟶^{≃} S/J(i + 1) ⟶ S/J(i)

induced by the unit. Since S/J(i) is a μ-spectrum, there is an induced map P ∧ S/J(i) ⟶^{f} S/J(i). The map E_{∗}f is easily seen to be an isomorphism, so f is an equivalence.

To complete the proof, we need only show that if P ∈ G(i) for all i, then P ≃ S. Let M_{i} be the set of maps P ⟶ S/J(i) such that the induced map E_{∗}^{∨}P ⟶ E_{∗}/J(i) is surjective. Because P ∧ S/J(i) ≃ S/J(i), one can check that M_{i} is nonempty. It is also finite, and the sets M_{i} form an inverse system. It follows that the inverse limit is nonempty. The Milnor sequence gives us a map P ⟶ holim_{←-}_{i} S/J(i) = S.

It is easy to see that this induces an isomorphism E_{∗}^{∨}P ≃ E_{∗}^{∨}S, and thus that P ≃ S.

Note that Proposition 14.3 does not rule out the possibility that Pic contains an infinite product of copies of Z_{p}. The main unanswered question about Pic is whether it is finitely generated as a profinite group.

We next address two related problems. Firstly, we have seen that most of Pic is a module over Z_{p}. However, given P ∈ Pic and a ∈ Z_{p} we would like to be able to define P^{(a)} as an object rather than just an isomorphism class, or at least to define it up to canonical isomorphism. Secondly, given an element a ∈ Pic it is natural to choose P ∈ 𝒫 with [P] = a and define π_{a}(X) = [P, X]. One would like to choose the spectra P compatibly for all a and collect the groups π_{a}(X) into a graded object with commutative and associative pairings π_{a}(X) ⊗ π_{b}(Y) ⟶ π_{a+b}(X ∧ Y). This is not automatically possible: there is an obstruction in H^{3}(Pic; Aut(S)), and if it vanishes then the solutions form a principal homogeneous space for H^{2}(Pic; Aut(S)) (here Pic is acting trivially on Aut(S)). We believe that this fact is in the literature, perhaps in the theory of Tannaka categories, but we have not managed to find a reference. Rather than using this general theory, we will proceed more directly.

We have not been able to construct a grading over Pic itself, but Theorem 14.11 provides a tolerable substitute.

We start with a discussion of signs, most of which is taken from [Riv72, Chapter I]. We write R = [S, S], which is a commutative ring under composition (which is the same as the smash product). For any X ∈ 𝒦, the smash product gives a natural map R ⟶ [X, X]. If X = P ∈ 𝒫 then this map is an isomorphism (because [P, P] = [S, P^{−1} ∧ P] = [S, S]). We write t = t_{P} : [P, P] ⟶ R for the inverse. One can check that t_{P ∧Q}(u ∧ v) = t_{P} (u)t_{Q}(v) and t_{P} (v ∘ w) = t_{P} (v)t_{P} (w). Moreover, if we have maps P ⟶^{u} Q ⟶^{v} P then t_{P} (vu) = t_{Q}(uv).

Given a spectrum P ∈ 𝒫, we have a twist map τ_{P} : P ∧ P ⟶ P ∧ P and thus an element ϵ_{P} = t(τ_{P}) ∈ R. Of course for n ∈ Z we have ϵ_{S^{n}} = (−1)^{n}.

Lemma 14.4. If p > 2 then ϵ_{P} = (−1)^{deg[P]}. Even if p = 2, we have ϵ_{P}^{2} = 1 and the map [P] ↦ ϵ_{P} is a homomorphism Pic /2 ⟶ R^{×}.

Proof. As τ^{2} = 1 we have ϵ_{P}^{2} = 1. It is not hard to check that ϵ_{P ∧Q} = ϵ_{P} ϵ_{Q} and that ϵ_{P} = ϵ_{P′} if P ≃ P′. We thus get a homomorphism ϵ: Pic /2 ⟶ R^{×}. Now suppose that p > 2. Then K is commutative so external products in K homology are commutative up to the usual sign; it follows that K_{∗}ϵ_{P} = (−1)^{deg[P]}. For notational convenience we will assume that deg[P] is even so that K_{∗}ϵ_{P} = 1. It is not hard to see that e = (1 − ϵ_{P})/2 is an idempotent in R and K_{∗}e = 0. Thus K_{∗}eS = 0, so e = 0, so ϵ_{P} = 1 as required.

We write 𝒫^{0} for the category of those P ∈ 𝒫 such that deg[P] = 0 and ϵ_{P} = 1 (the second condition being redundant when p > 2). Because the symmetric group on k letters is generated by adjacent transpositions, we see that it acts trivially on P^{(k)} if P ∈ 𝒫^{0}. We write 𝒫′ for the category of pairs (P, u) such that P ∈ 𝒫^{0} and u generates K_{0}P. The morphisms (P, u) ⟶ (Q, v) are required to send u to v. We write Pic^{0} and Pic′ for the groups of isomorphism classes in 𝒫^{0} and 𝒫′.

If (P, u) ∈ 𝒫′ and X ∈ ℱ, we define

π_{k}(P, u)(X) = \begin{cases}[P^{(k)}, X] & if k ≥ 0 \\ [DP^{(|k|)}, X] & if k < 0 \end{cases}

These are easily seen to be finite groups. It is not hard to construct associative and commutative (without signs) pairings

π_{k}(P, u)(X) ⊗ π_{l}(P, u)(Y) ⟶ π_{k+l}(P, u)(X ∧ Y).

We would like to interpolate these groups for p-adic values of k. The main problem is to find a natural formulation of this not depending on any choices. Our solution involves the following definition.

Definition 14.5. Given a compact Hausdorff space X, let ℬ(X) be the category of locally-trivial bundles over X whose fibres are finite Abelian p-groups with the discrete topology. The morphisms are continuous bundle maps that are homomorphisms on each fibre.

Given a morphism f: A ⟶ B in ℬ(X), it is easy to see that for each x ∈ X there is a neighbourhood U of x and groups A′, B′ and a homomorphism f′: A′ ⟶ B′ such that the restriction of f over U is isomorphic to f′ × 1: A′ × U ⟶ B′ × U. It follows from this that the kernel, cokernel and image of f all lie in ℬ(X) and thus that ℬ(X) is an Abelian category.

A map ϕ: X ⟶ Y gives a pullback functor ϕ^{∗}: ℬ(Y) ⟶ ℬ(X) in an evident way. In particular, if G is an compact Hausdorff Abelian topological group and σ: G × G ⟶ G is the addition map, we have a functor σ^{∗}: ℬ(G) ⟶ ℬ(G × G). We also have an external tensor product functor ⊗: ℬ(G) × ℬ(G) ⟶ ℬ(G × G). In this context, a pairing from A and B to C (where A, B, C ∈ ℬ(G)) means a map A ⊗ B ⟶ σ^{∗}C.

We write F_{x}: ℬ(X) ⟶ Ab for the functor which sends A to the fibre A_{x} of A over x.

Proposition 14.6. Given (P, u) ∈ 𝒫′ there is a homology theory π(P, u): ℱ ⟶ ℬ(Z_{p}), together with natural pairings

π(P, u)(W) ⊗ π(P, u)(Z) ⟶ σ^{∗}π(P, u)(W ∧ Z)

that are commutative and associative in a suitable sense. Moreover, for m ∈ Z there are natural isomorphisms F_{m}π(P, u)(Z) = π_{m}(P, u)(Z) which are compatible with the above pairings.

This will be proved after Proposition 14.8. We will denote F_{m}π(P, u)(Z) by π_{m}(P, u)(Z) even for non-integer values of m. Note that, if Z is a specific small spectrum and m ∈ Z_{p}, then the local triviality of π(P, u)(Z) implies that π_{m}(P, u)(Z) = π_{i}(P, u)(Z) for some integer i sufficiently close to m in the p-adic topology. However, the i in question depends on Z.

Note also that π(P, u) is a homology theory defined only on small spectra. The category ℬ(Z_{p}) does not have colimits, so in order to interpret π(P, u) as a homology theory on 𝒦, we would need to consider sheaves over Z_{p} rather than locally trivial bundles.

Definition 14.7. Let X ∈ ℱ be a small spectrum, and (P, u) an object of 𝒫′. We say that a map v: P^{(p^{k})} ∧ X ⟶ X is a good (P, u) self map of X if

(a) The map v_{∗}: K_{∗}(P)^{⊗p^{k}} ⊗ K_{∗}(X) ⟶ K_{∗}(X) is given by v_{∗}(u^{⊗p^{k}} ⊗ x) = x for all x ∈ K_{∗}X.

(b) v ∧ 1 is central in the graded ring π_{∗}(P, u)F (X ∧ X, X ∧ X).

Note that there are no suspensions involved in the definition of a good (P, u) self map. Note also that a good (P, u) self map is an equivalence.

Proposition 14.8. Every small spectrum X admits a good (P, u) self map. If v and v′ are two such maps then v^{p^{i}} = (v′)^{p^{j}} for some i and j. If v is a good (P, u) self map of X and w is a good (P, u) self map of Y then there exist integers i and j such that w^{p^{j}}f = fv^{p^{i}} for all maps f: X ⟶ Y.

Proof. This is proved in a way which is closely parallel to the proof for integergraded v_{n} self maps [Rav92a, Chapter 6]. If v is a map satisfying part (a) of the definition, we form the ring spectrum R = F (X, X). The map v defines a map v : P^{(p^{k})} ⟶ R, so we can form the map ad(v): P^{(p^{k})} ∧ R ⟶ R which measures the difference between left multiplication and right multiplication by v. By our assumption, the map K_{∗}v : K_{∗}P^{(p^{k})} ⟶ K_{∗}R = Hom_{K_{∗}}(K_{∗}X, K_{∗}X) takes the generator u^{⊗p^{k}} to the identity map, which is obviously in the centre. Hence K_{∗} ad(v) = 0, so the telescope of the sequence

R ⟶^{ad(v)} DP^{(p^{k})} ∧ R ⟶^{ad(v)} DP^{(2p^{k})} ∧ R ⟶^{ad(v)} …

is zero. It follows using the smallness of R that ad(v) is nilpotent, and in the usual way we deduce that some power v^{p^{j}} is central in π_{∗}(P, u)F (X, X). We can apply the same argument to v∧1 and thus show that v^{p^{j}} ∧1 is central in π_{∗}(P, u)F (X^{(2)}, X^{(2)}) for large j, and thus that v^{p^{j}} is a good (P, u) self map.

Next, suppose that we have two good (P, u) self maps, say v and v′. We then have K_{∗}(v − v′) = 0 and thus v − v′ is nilpotent, by an argument with telescopes as above. Just as in [Rav92a], we deduce that v^{p^{i}} = (v′)^{p^{j}} for some i and j. Asymptotic naturality follows from asymptotic uniqueness, again just as in [Rav92a]. Given the asymptotic naturality, it is not hard to see that the category of small spectra that admit a good (P, u) self map is thick. It is thus enough so show that a generalised Moore spectrum S/I of type n admits a good (P, u) self map, or even just a map satisfying condition (a).

It follows from Proposition 14.3 that P^{(p^{k})} ∧ S/I ≃ S/I for some k. Choose an isomorphism v. Choose an element ũ ∈ E_{∗}^{∨}P^{(p^{k})} lifting u^{⊗p^{k}}. After replacing v by a p-adic unit multiple of v, we may assume that v_{∗}ũ = 1 (mod I_{n}). It follows using the cofibrations Σ^{|v_{k}|}E/I_{k} ⟶^{v_{k}} E/I_{k} ⟶ E/I_{k+1} that v^{p^{n}} satisfies condition (a), as required.

Proof of Proposition 14.6. Let (P, u) be an object of 𝒫′, and X ∈ ℱ a small spectrum. Let C be a coset of p^{k}Z_{p} in Z_{p} for some k. A good (P, u) self map v : P^{(p^{k})} ∧ X ⟶ X induces an isomorphism π_{j+p^{k}}(P, u)(X) ⟶ π_{j}(P, u)(X). Let π_{C}(P, u)(X) be the set of elements a ∈ ∏_{j∈C∩Z} π_{j}(P, u)(X) such that there exists a good (P, u) self map v: P^{(p^{k})} ∧ X ⟶ X such that a_{j} = va_{j+p^{k}} for all j ∈ C ∩ Z. If C′ is another coset and C′ ⊆ C then there is an evident restriction map π_{C}(P, u)(X) ⟶ π_{C′}(P, u)(X). For i ∈ Z_{p} we define

π_{i}(P, u)(X) = lim_{-→}_{C∋i} π_{C}(P, u)(X).

We shall see shortly that this is consistent with our earlier definition when i ∈ Z. If a ∈ π_{C}(P, u)(X) and a′ ∈ π_{C′}(P, u)(X) and i ∈ C ∩ C′ then we see using the asymptotic uniqueness of (P, u) self maps that there is a coset C′′ with i ∈ C′′ ⊆ C ∩ C′ such that the restriction of a + a′ lies in π_{C′′}(P, u)(X). Using this, we make π_{i}(P, u)(X) into an Abelian group. We define

π(P, u)(X) = ∐_{i∈Z_{p}} π_{i}(P, u)(X),

so there is an evident map q: π(P, u)(X) ⟶ Z_{p} with fibres π_{i}(P, u)(X). Given an element a ∈ π_{C}(P, u)(X), we get an element a_{i} ∈ π_{i}(P, u)(X) for each i ∈ C. We define U(C, a) = {(i, a_{i}) | i ∈ C} ⊆ π(P, u)(X), and we give π(P, u)(X) the smallest possible topology for which all sets of this form are open.

Suppose we have integers j, k with k ≥ 0 and a good (P, u) self map v: P^{(p^{k})} ∧ X ⟶ X. There is an evident map

θ = θ_{j,k,v}: π_{j}(P, u)(X) ⟶ π_{j+p^{k}Z_{p}}(P, u)(X),

sending a ∈ π_{j}(P, u)(X) to the system of elements (v^{m}a | m ∈ Z). Using asymptotic uniqueness again, it is not hard to check that this is an isomorphism. One can also deduce that π(P, u)(X) is a locally trivial bundle over Z_{p}, or in other words an object of the category ℬ(Z_{p}).

Given a triangle X ⟶ Y ⟶ Z of small spectra, we can choose good (P, u) self maps of X, Y and Z that are compatible with the maps of the triangle. Having done so, it is easy to check that π(P, u)(X) is an exact functor of X.

Similar arguments show that when i ∈ Z there is a canonical isomorphism between our old definition of π_{i}(P, u)(X) and our new one.

Suppose that i, j ∈ Z_{p}. We would like to construct a natural map

μ_{i,j}: π_{i}(P, u)(X) ⊗ π_{j}(P, u)(Y) ⟶ π_{i+j}(P, u)(X ∧ Y).

Suppose that v: P^{(p^{k})} ∧ X ⟶ X and w: P^{(p^{k})} ∧ Y ⟶ Y are good (P, u) self maps with v ∧ 1 = 1 ∧ w on X ∧ Y, and that i′, j′ are integers congruent to i, j modulo p^{k}. The smash product gives a pairing

ν_{i′,j′}: π_{i′}(P, u)(X) ⊗ π_{j′}(P, u)(Y) ⟶ π_{i′+j′}(P, u)(X ∧ Y).

The maps θ_{i′,k,v} and so on give isomorphisms

π_{i′}(P, u)(X) ≃ π_{i}(P, u)(X)
π_{j′}(P, u)(Y) ≃ π_{j}(P, u)(Y)
π_{i′+j′}(P, u)(X ∧ Y) ≃ π_{i+j}(P, u)(X ∧ Y).

It is natural to require that μ_{i,j} should be compatible with ν_{i′,j′} under these isomorphisms. It is not hard to check using the strong centrality of good (P, u) self maps that there is a unique map μ_{i,j} which has this compatibility for all choices of k, i′, j′, u and v. Given this, one can deduce the expected naturality, commutativity and associativity properties.

Corollary 14.9. Given an object (P, u) ∈ 𝒫′, there are objects (P^{(k)}, u^{⊗k}) in 𝒫′ for all k ∈ Z_{p}, defined up to canonical isomorphism, such that there is a natural isomorphism

π_{k}(P, u)(X) = [P^{(k)}, X]

for all small spectra X. Moreover, there are canonical and coherent isomorphisms P^{(k)} ∧ P^{(l)} = P^{(k+l)}. The spectrum P^{(k)} is independent of u up to unnatural isomorphism.

Proof. The functor X ↦ π_{−k}(P, u)(X) is a homology theory on small spectra, so by [HPS95, Corollary 2.3.11] it extends canonically to a homology theory on all of 𝒦. By Theorem 9.5 there is a spectrum P^{(k)} representing this functor, so in particular π_{−k}(P, u)(X) = π_{0}(P^{(k)} ∧ X) whenever X is small. This spectrum is unique up to isomorphism, and the isomorphism is unique up to phantoms. However, for any small spectrum Z we have [Z, P^{(k)}] = π_{−k}(P, u)DZ which is finite, so Theorem 10.7 tells us that there are no phantom maps into P^{(k)}. This shows that P^{(k)} is unique up to canonical isomorphism.

Let X be a nontrivial small spectrum, and v: P^{(p^{j})} ∧ X ⟶ X a good (P, u) self map. Choose an integer i congruent to k mod p^{j}. Using v we construct a natural isomorphism

π_{−k}(P, u)(X ∧ Y) = π_{−i}(P, u)(X ∧ Y) = π_{0}(P^{(i)} ∧ X ∧ Y)

for all small Y, and thus (by unique representability of homology theories) an isomorphism P^{(k)} ∧ X ≃ P^{(i)} ∧ X. We know that K_{∗}P has dimension one over K_{∗}, and it follows that K_{∗}P^{(k)} has dimension one, and thus that P^{(k)} ∈ 𝒫.

We still need to produce a canonical generator of K_{0}P^{(k)}, however. To do this, suppose that we have small spectra X and Y, a map f: X ⟶ Y, and good self maps v, w of X and Y such that wf = fv. If we construct isomorphisms P^{(k)} ∧ X ≃ P^{(i)} ∧ X and P^{(k)} ∧ Y ≃ P^{(i)} ∧ Y by the procedure outlined above, then we find that they commute with the maps 1 ∧ f. In particular, we can take X = S/I and Y = S/I ∧ S/I, choose a good self map v of S/I and take w = v ∧ 1 = 1 ∧ v. We can then take f = η ∧ 1 or 1 ∧ η, where η: S ⟶ S/I is the unit map. This gives us a commutative diagram

[figure: P^{(k)} ∧ S/I; P^{(k)} ∧ S/I ∧ S/I; P^{(i)} ∧ S/I; P^{(i)} ∧ S/I ∧ S/I.; \begin{matrix}η∧1 \\ 1∧η \end{matrix}; u; u; \begin{matrix}η∧1 \\ 1∧η \end{matrix}]

If we apply K_{∗}, then it is easy to see (just by calculating everything) that the equaliser of the top line is K_{∗}P^{(k)} and the equaliser of the bottom line is K_{∗}P^{(i)}, so we get an isomorphism K_{∗}P^{(i)} ≃ K_{∗}P^{(k)}. One can check that this is independent of the choices involved. We define u^{⊗k} ∈ K_{∗}P^{(k)} to be the image of u^{⊗i} under this isomorphism.

A similar argument shows that ϵ_{P^{(k)}} = 1. Indeed, one can see as above that the image of ϵ_{P^{(k)}} in [S, S/I] is the same as the image of ϵ_{P^{(i)}} for some integer i depending on I. Since this image is the unit map η of S/I, and [S, S] = lim_{←-}[S, S/I], we find that ϵ_{P^{(k)}} = 1.

We next produce an isomorphism P^{(k)} ∧ P^{(l)} ≃ P^{(k+l)}. Suppose that we have finite spectra X and Y of type at least n, and maps S ⟶ X and S ⟶ Y. This gives a map P^{(k)} ⟶ P^{(k)} ∧ X, or equivalently an element of π_{−k}(P, u)F (P^{(k)}, X). Similarly, we have an element of π_{−l}(P, u)F (P^{(l)}, Y). Using the pairings on the groups π_{∗}(P, u)(−) and the dualisability of P^{(k)} and P^{(l)} we get an element of π_{−k−l}(P, u)F (P^{(k)} ∧ P^{(l)}, X ∧ Y), and thus a map S ⟶ P^{(k+l)} ∧ F (P^{(k)} ∧ P^{(l)}, X ∧ Y), or equivalently a map P^{(k)} ∧ P^{(l)} ⟶ P^{(k+l)} ∧ X ∧ Y. One can show that these maps are compatible as X and Y run over the category of finite spectra of type n equipped with a map in of S. We can therefore pass to the inverse limit, which is just P^{(k+l)} by Remark 4.20 and Proposition 7.10. This gives a map P^{(k)} ∧ P^{(l)} ⟶ P^{(k+l)}. There are no lim^{1} terms because all groups involved are finite, so the map is unique. One can show that it induces our earlier pairing π_{−k}(P, u)Z ⊗ π_{−l}(P, u)W ⟶ π_{−k−l}(P, u)(Z ∧ W). Using this, one can check that the maps P^{(k)} ∧ P^{(l)} ⟶ P^{(k+l)} have the expected coherence properties.

Finally, we show that our definition of P^{(k)} is independent of u up to unnatural isomorphism. If j is an integer then P^{(j)} is just an iterated smash power of P or DP and clearly does not depend on u. For any generalised Moore spectrum S/I we can choose a good self map P^{(p^{i})} ∧ S/I ⟶ S/I and an integer j congruent to k modulo p^{i}. We then find that P^{(k)} ∧ S/I is isomorphic to P^{(j)} ∧ S/I independently of u. We are thus reduced to showing that if P ∈ Pic and P ∧ S/I ≃ S/I for all I then P ≃ S. We have shown this in the proof of Proposition 14.3.

We also want to be able to grade things over subgroups of Pic′ of rank greater than one. The following lemma will help us to do this.

Lemma 14.10. Let (P, a) and (Q, b) be objects of 𝒫′. Let X be a small spectrum, with good self maps v: P^{(p^{k})} ∧ X ⟶ X and w: Q^{(p^{l})} ∧ X ⟶ X. Then the following diagram commutes.

Q_{(p^{l})} ∧ P_{(p^{k})} ∧ X  P_{(p^{k})} ∧ Q_{(p^{l})} ∧ X  P^{(p^{k})} ∧ X Q^{(p^{l})} ∧ X  X^{τ∧1} w_{_{1∧v} u}  ^{1∧w} w _{u_{v}} w_{w}

Proof. We may assume that k = l = 0, to simplify the notation. It is easy to generalise Proposition 4.4 to show that w induces a natural transformation w′_{Y} : Q∧ Y ⟶ Y on the category of spectra Y which can be written as a retract of a spectrum of the form X ∧ Z. By applying naturality to the twist map τ: P ∧ X ⟶ X ∧ P, we see that w′_{P ∧X} is the composite

Q ∧ P ∧ X ⟶^{1∧τ} Q ∧ X ∧ P ⟶^{w∧1} X ∧ P ⟶^{τ^{−1}} P ∧ X.

General nonsense about symmetric monoidal categories tells us that this is the same as the top line of the diagram in the lemma (think about the analogous statement with vector spaces and tensor products). By applying naturality of w′ to the map v: P ∧ X ⟶ X, we conclude that the diagram commutes.

We finish with a theorem about grading over groups of rank greater than one.

Theorem 14.11. Given objects (P_{1}, u_{1}), …, (P_{r}, u_{r}) of 𝒫′, there is a homology theory F: ℱ ⟶ ℬ(Z_{p}^{r}) whose fibre over a point \underline{a} = (a_{1}, …, a_{r}) is

F\underline{a}(X) = [P_{1}^{(a_{1})} ∧ … ∧ P_{r}^{(a_{r})}, X].

There are also natural pairings from F (X) and F (Y) to F (X ∧ Y).

Note that this gives a grading over Z_{p}^{r} and not over the image of Z_{p}^{r} in Pic′. Moreover, our construction uses a choice of basis for Z_{p}^{r}; it is not clear whether this can easily be avoided.

Proof. We simply define F (X) to be the disjoint union of the sets F\underline{a}(X) as \underline{a} runs over Z_{p}^{r}. If we choose good (P_{i}, u_{i}) self maps for each i, these give us isomorphisms F\underline{a}(X) ≃ F\underline{b}(X) for all \underline{b} sufficiently close to \underline{a}. These isomorphism appear to depend on the order in which we use the different self maps, but Lemma 14.10 tells us that this is not the case. Moreover, if we use a different system of self maps then by asymptotic uniqueness we find that the isomorphisms F\underline{b}(X) ≃ F\underline{a}(X) do not change provided that \underline{b} is sufficiently close to \underline{a}. Thus, we get a bundle structure on F (X). We leave the rest of the details to the reader.

15. Examples

15.1. The case n = 1. The height one case is quite straightforward and amenable to calculation. However, we warn the reader that it rather atypical and often misleading as a guide to the case n > 1. The material in this section is well-known. See [Rav84] or [Bou79] for the closely related theory of localisation with respect to K theory.

We shall assume that p > 2 for simplicity. The spectrum E is the 2(p−1)-periodic Adams summand of KU_{p}^{∧}, and K = E/p. For each a ∈ 1 + pZ_{p} < Z_{p}^{×} we have an Adams operation ψ^{a}: E ⟶ E, with ψ^{a}ψ^{b} = ψ^{ab}. These are ring maps, and their action on E_{∗} = Z_{p}[v_{1}^{±1}] is given by ψ^{a}(v_{1}) = a^{p−1}v_{1}. We choose a topological generator a of the group 1 + pZ_{p} ≃ Z_{p} and define T = ψ^{a} − 1 ∈ E^{0}E. It turns out that E^{0}E = Z_{p}[[T]]; in particular, this ring is commutative. More generally, the ring E^{∗}E is the non-commutative power series ring E^{∗}[[T]] in which ψ^{b}v_{1} = b^{p−1}v_{1}ψ^{b} and thus T v_{1} = a^{p−1}v_{1}T + (a^{p−1} − 1)v_{1}.

For any d ∈ Z_{p} we define X_{d} to be the fibre of the map ψ^{a}−a^{d} = T +1−a^{d}: E ⟶ E. One checks that E^{∗}X_{d} = E^{∗}[[T]]/(T +1−a^{d}) = E^{∗} and that ψ^{b} acts on E^{0}X_{d} with eigenvalue b^{d} for all b ∈ 1+pZ_{p}. We can also determine π_{∗}X_{d}. Note that ψ^{a}−a^{d} acts on π_{2(p−1)k}E with eigenvalue a^{(p−1)k} − a^{d}, which is a unit multiple of 1 − a^{d−(p−1)k} or of p(d − (p − 1)k). It follows that π_{∗}X_{d} has a summand Z_{p}/(p(d − (p − 1)k)) in dimension 2k(p−1)−1, and a summand Z_{p} in dimension 2d if d is an integer divisible by p − 1. One can check that in this case the generator of π_{2d}X_{d} is a K-equivalence S^{2d} ⟶ X_{d}, so that X_{d} = L̂S^{2d}. More generally, it is not hard to show that X_{d} is just the p-adic smash power (S^{2p−2})^{(d/(p−1))} considered in Section 14. Note that

the fibration L̂S^{2d} ⟶ E ⟶^{ψ^{a}−a^{d}} E shows that L̂S^{2d} is E-finite, as predicted by Theorem 8.9. We know from [HMS94] that the Picard group consists of the spectra Σ^{k}X_{d} for d ∈ Z_{p} and 0 ≤ k < 2p−2, and that Pic ≃ lim_{←-}_{N} Z/|v_{1}^{p^{N}} | = Z/(2p−2)×Z_{p}.

We next define Y to be the fibre of T^{2} − p: E ⟶ E. This spectrum is a counterexample to a number of plausible conjectures, as pointed out to us by Mike Hopkins. Note that E^{∗}Y = E^{∗}[[T]]/(T^{2} − p) = E^{∗} ⊕ E^{∗}, so Y is dualisable. Because T acts on each homotopy group π_{d}E (for d ∈ Pic) with eigenvalue in pZ_{p} we find that the eigenvalue of T^{2} − p is a unit multiple of p and thus that π_{d}Y = F_{p} when d = −1 (mod 2p − 2) and π_{d}Y = 0 otherwise. Thus p: Y ⟶ Y is a map of dualisable spectra which induces the zero map of all Pic-graded homotopy groups. However, p ≠ 0 because E^{∗}Y = E^{∗} ⊕ E^{∗} is torsion-free. Moreover, using the fibration Y/p^{2} ⟶ E/p^{2} ⟶ E/p^{2} one finds that pπ_{d}(Y/p^{2}) = 0 for all d and thus that p: Y/p^{2} ⟶ Y/p^{2} is a map of small spectra that is nontrivial but acts trivially on all Pic-graded homotopy groups. This shows that all reasonable analogues of Freyd’s generating hypothesis [Fre66] in 𝒦 are false. See [Dev96a] for more discussion of the generating hypothesis from the chromatic point of view. Appendix 2 of that paper contains a different kind of counterexample to a certain analogue of the hypothesis. It appears that Devinatz does not consider this to be evidence against the original conjecture, and he is the expert.

We next show that Y does not lie in the thick subcategory generated by the Picard group. Indeed, for any Z in that category we see easily that [Z, Y] is a torsion group, but the identity map has infinite order in [Y, Y]. Another instructive proof is to observe that when Z lies in the thick subcategory generated by the Picard group, the group Q ⊗ E^{0}Z is a finite-dimensional vector space over Q_{p} on which T acts with eigenvalues in Q_{p}. However, the eigenvalues of T acting on E^{0}Y are ±\sqrt{p} ∉ Q_{p}.

One might expect that a dualisable spectrum with finite homotopy groups would be small, but again Y is a counterexample.

15.2. The case n = 2. We next analyse some calculations of Shimomura [Shi86] in the light of the theory developed above. See also [Sad93]. We take n = 2 and p ≥ 5. Amongst other things, we obtain the following thought-provoking result. Recall from [HMS94] that the map from Z to Pic sending n to S^{n} extends over the completion Ẑ = lim_{←-}_{N} Z/|v_{2}^{p^{N}} | = Z/(2p^{2} −2)×Z_{p}. Recall also that there is a unique

translation-invariant measure on Ẑ with μ(Ẑ) = 1, called the .

Theorem 15.1. For all a ∈ Z and all finite torsion spectra X the group π_{a}L̂X is finite. However,

(i) π_{a}L̂S/p is finite for all a ∈ Ẑ such that a = 0 (mod |v_{1}|). (ii) If b = −1, −2, −3 or −4 then {a ∈ Ẑ | a = b (mod |v_{1}|) and π_{a}L̂S/p is finite}

has Haar measure zero. (iii) If a ∉ {0, −1, …, −4} (mod |v_{1}|) then π_{a}L̂S/p = 0.

We have not yet understood the more recent calculation of π_{∗}L_{2}S due to Shimomura and Yabe [SY95].

Note that E(2)_{∗} and E(2)_{∗}E(2) are concentrated in degrees divisible by 2p − 2. Thus if E(2)_{∗}X is generated in degrees divisible by 2p−2 then the whole cobar complex which calculates H^{∗∗}E(2)_{∗}X = Ext_{E(2)_{∗}E(2)}^{∗∗}(E(2)_{∗}, E(2)_{∗}X) is concentrated in those degrees, as is the Ext group itself. Moreover, we know from [HS95] that H^{s∗}E(2)_{∗} = 0 when s > n^{2} + n = 6. The argument used there shows a fortiori that H^{s∗}E(2)_{∗}X = 0 when s > 6 and X is any of the spectra S/(p, v_{1}^{N}) used below, where N may be infinite. As 2p − 2 > 6 we see that the E(2)-based Adams spectral sequence collapses and π_{k}LX = H^{s,k+s}E(2)_{∗}X for the unique s with 0 ≤ s < 2p−2 and s = −k (mod 2p − 2).

Shimomura calculates H^{∗∗}M_{1}^{1}, where

M_{1}^{1} = E_{∗}/(p, v_{1}^{∞}) = E(2)_{∗}S/(p, v_{1}^{∞}) = E_{∗}M_{2}S^{1},

and thus H^{∗∗}M_{1}^{1} = π_{∗}M_{2}S^{1}. We would prefer to work with L̂S, so we start with some remarks about the necessary translation. We know that π_{∗}LS/(p, v_{1}^{N}) = H^{∗∗}E_{∗}/(p, v_{1}^{N}) is finite in each total degree, as the homotopy of any small object is finite, so there are no lim_{←-}^{1} terms and π_{∗}L̂S/p = lim_{←-}_{N} H^{∗∗}E_{∗}/(p, v_{1}^{N}). We shall

write H^{∗∗}E_{∗}/(p, v_{1}^{N}) as a direct sum of groups A_{i,N}, where for fixed i the groups A_{i,N} form a tower as N varies. As H^{∗∗}E_{∗}/(p, v_{1}^{N}) is finite we see that A_{i,N} = 0 for almost all i so the direct sum is the same as the product and thus π_{∗}L̂S/p =

lim_{←-}_{N} ∏_{i} A_{i,N} = ∏_{i} lim_{←-}_{N} A_{i,N}.

Let Ω^{∗∗} be the cobar resolution for E(2)_{∗}/p. We have a finite number of summands in H^{∗∗}M_{1}^{1} that are isomorphic to F_{p}[v_{1}]/v_{1}^{∞}. Each such summand is generated by elements x_{r}/v_{1}^{r} where x_{r} ∈ Ω^{s,t} say and d(x_{r}) is divisible by v_{1}^{r} and x_{r+1}/v_{1}^{r} is homologous to x_{r}/v_{1}^{r}. We refer to these as summands of type (∞, s, t). We also have summands isomorphic to F_{p}[v_{1}]/v_{1}^{α}, where α > 0. Each of these is generated by an element y/v_{1}^{α} where y ∈ Ω^{s,t} say and d(y) is divisible by v_{1}^{α}. We refer to these as summands of type (α, s, t). We now use the short exact sequence

Σ^{−N|v_{1}|}E(2)_{∗}/(p, v_{1}^{N}) ⟶^{v_{1}^{−N}} E(2)_{∗}/(p, v_{1}^{∞}) = M_{1}^{1} ⟶^{v_{1}^{N}} Σ^{−N|v_{1}|}M_{1}^{1}.

This gives a short exact sequence

U_{N} = H^{∗∗}(M_{1}^{1})/v_{1}^{N} ⟶ H^{∗∗}E(2)_{∗}/(p, v_{1}^{N}) ⟶ ann(v_{1}^{N}, H^{∗∗}M_{1}^{1}) = V_{N},

and thus a short exact sequence

lim_{←-}_{N} U_{N} ↣ π_{∗}L̂S/p ↠ lim_{←-}_{N} V_{N}.

Each summand of type (∞, s, t) in H^{∗∗}M_{1}^{1} contributes a summand of V_{N} generated by x_{N} and isomorphic to F_{p}[v_{1}]/v_{1}^{N}. As N varies these form a tower isomorphic to the evident tower

F_{p}[v_{1}]/v_{1} ↞ F_{p}[v_{1}]/v_{1}^{2} ↞ F_{p}[v_{1}]/v_{1}^{3} ↞ F_{p}[v_{1}]/v_{1}^{4} ↞ …,

so the (graded) inverse limit is F_{p}[v_{1}]. It is generated by a class x_{∞} ∈ H^{s,t}.

On the other hand, consider a summand of type (α, s, t) generated by y/v_{1}^{α}. Write β = min(N, α). We then get a summand of U_{N} generated by d(y)/v_{1}^{β} and isomorphic to F_{p}[v_{1}]/v_{1}^{β}. We also get an isomorphic summand in V_{N} generated by v_{1}^{N−β}y. As N varies, the summands in U_{N} form a pro-constant tower and the summands in V_{N} form a pro-trivial tower. In the inverse limit we are left with a summand in lim_{←-}_{N} U_{N} isomorphic to F_{p}[v_{1}]/v_{1}^{α}, generated by d(y)/v_{1}^{α} ∈ H^{s+1,t−α|v_{1}|}.

Feeding Shimomura’s calculations into this analysis gives the following description of π_{∗}L̂S/p. Firstly, there is a class ζ ∈ π_{−1}L̂S/p and an isomorphism π_{∗}L̂S/p = E[ζ]⊗A_{∗} for some other graded group A_{∗}. This group contains two copies of F_{p}[v_{1}], generated by 1 ∈ π_{0} and h_{0} ∈ π_{2p−3}. We write B_{∗} for the direct sum of the other summands in A_{∗}, which is a v_{1}-torsion group.

We need some new language to give a manageable description of B_{∗}. Firstly, we will write numbers in terms of their base p expansions, for example [123] = p^{2} + 2p + 3. We write \overline{k} for the digit p − k so that [\overline{1} 2] = (p − 1)p + 2 for example. We indicate repetitions by exponents, so [1^{n}] = ∑_{i=0}^{n−1} p^{i} = (p^{n}−1)/(p−1). We write ∗ for an undetermined string of digits, so that [∗11] means any integer congruent to p + 1 modulo p^{2}. We handle negative numbers by allowing infinite expansions, for example −p^{n} = [\overline{1}^{∞}0^{n}]. We write x ↑ n for x^{n} where typographically convenient.

Proposition 15.2 (Shimomura). The generators of B_{∗} are as follows. The entries will be explained and justified after the table.

generator  parameters  width  period degree
(a)  v_{2}^{[∗ t]}h_{1}  t ≠ 0  [1]  p  1 − 2p
(b) v_{2}^{[∗ t \overline{1} 0^{n−1}]}h_{0} t ≠ \overline{1} , n > 0  [1 0 \overline{1}^{n−1}]  p^{n+1} 2p − 3
(c)  v_{2}^{[∗ \overline{1}]}g_{0}  [\overline{1}]  p  2p − 4
(d) v_{2}^{[∗ t \overline{2}^{n}]}g_{1}v_{2} t ≠ \overline{1}, \overline{2} , n ≥ 0  [1 2^{n}] + 1  p^{n+1}  −2p
(e) v_{2}^{[∗\overline{2}\overline{1}\overline{2}^{n}]}g_{1}v_{2}  n ≥ 0  [1 0 0 2^{n}] + 1 p^{n+2}  −2p
(f) v_{2}^{[∗ t \overline{2}^{n}]}ρv_{2}  t ≠ \overline{2} , n ≥ 0  [1 0 \overline{1}^{n−1}]  p^{n+1}  −3

In part (f ) the width is 1 when n = 0.

What all this means is as follows. The group B_{∗} is a direct sum of modules isomorphic to F_{p}[v_{1}]/v_{1}^{w} generated by various elements x. These generators x come in v_{2}-periodic families. In the first column we identify the generator x by giving its image in H^{∗∗}E_{∗}/I_{2}. The entries in the first column depend on various parameters which are listed in the second column (except for the parameter “∗” which always occurs). It is implicit that t is an integer with 0 ≤ t < p. The third column is the integer w such that the summand in question is isomorphic to F_{p}[v_{1}]/v_{1}^{w}. If the entry in the fourth column is p^{k}, then the generators in question form a v_{2}^{p^{k}}- periodic family as ∗ varies. The last column is the degree of the generators modulo |v_{2}| = 2(p^{2} − 1).

We now deduce our table from Shimomura’s calculation of H^{∗∗}M_{1}^{1} in [Shi86].

Proof. We start by remarking that our degrees are the usual ones, which are 2(p−1) times larger than Shimomura’s degrees (see his (3.5.8)).

(a): This comes from Shimomura’s (4.1.5). He has summands F_{p}[v_{1}]⟨x_{n}^{s}/v_{1}^{a_{n}}⟩ in H^{∗∗}M_{1}^{1} for s ≠ 0 (mod p). Our part (a) is the case n = 0. In (3.6.4) he defines a_{0} = 1. By our earlier discussion, we get a generator in π_{∗}L̂S/p of width a_{0} = 1 by taking d(x_{0}^{s})/v_{1}. In (4.1.5) he states that d(x_{0}^{s})/v_{1} is a unit multiple of v_{2}^{s}h_{1} mod I_{2}. In (3.5.6) he defines h_{1} = t_{1}^{p}/v_{2} which has cohomological degree s = 1 and internal degree t = 2(p − 1)p − 2(p^{2} − 1) and thus total degree t − s = 1 − 2p. Our part (a) follows easily.

(b): This comes from the case n > 0 in Shimomura’s (4.1.5). We then have a_{n} = p^{n} +p^{n−1} −1 = [10\overline{1}^{n−1}] by (3.6.4). By (4.1.5) we see that d(x_{n}^{s}/v_{1}^{a_{n}}) is a unit multiple of v_{2}^{sp^{n}−p^{n−1}}h_{0}, so these elements are generators of π_{∗}L̂S/p of width a_{n}. As s ≠ 0 (mod p) we have t ∈ {0, …, p − 2} such that s = t + 1 (mod p) and thus sp^{n} −p^{n−1} = [∗t0^{n}]+p^{n} −p^{n−1} = [∗t\overline{1}0^{n−1}]. In (3.6.5) h_{0} is defined as t_{1} ∈ Ω^{1,2p−2} so the degree is 2p − 3.

(c): This comes from the summands F_{p}[v_{1}]⟨v_{2}^{t}V/v_{1}^{p−1}⟩ in Shimomura’s (4.1.6). Here t ∈ pZ and d(v_{2}^{t}V)/v_{1}^{p−1} = −v_{2}^{t+p−1}g_{0} mod I_{2}. We thus have generators v_{2}^{t+p−1}g_{0} of width p−1, and the general form of t+p−1 is [∗\overline{1}]. We learn from (3.5.7) that g_{0} ∈ Ω^{2,2p−2} so the total degree is 2p − 4.

(d): Shimomura has summands F_{p}[v_{1}]⟨y_{m}/v_{1}^{A(m)}⟩ for numbers m of the form sp^{n} where either s ∉ {0, −1} (mod p) or s = −1 (mod p^{2}). Our part (d) is the case s ∉ {0, −1} (mod p), so in Shimomura’s notation we have ϵ(s) = 0 and thus A(m) = 2 + (p + 1)(p^{n} − 1)/(p − 1). He shows that d(y_{m})/v_{1}^{A(m)} is a unit multiple of v_{2}^{e(m)}g_{1} mod I_{2}, where in the case ϵ(s) = 0 we have e(m) = m − (p^{n} − 1)/(p − 1). The general form of m − p^{n} is [∗t0^{n}] with t = 0, …, p − 3, so the general form of e(m) is [∗t0^{n}] + [10^{n}] − [1^{n}] = [∗t\overline{2}^{n}] + 1, so the generators are v_{2} ↑ [∗t\overline{2}^{n}](g_{1}v_{2}). (We have separated out one factor of v_{2} to avoid special behaviour when n = 0.) We learn from (3.5.7) that g_{1} ∈ Ω^{2,2−2p} so the total degree is −2p.

(e): This comes from the summands F_{p}[v_{1}]⟨y_{m}/v_{1}^{A(m)}⟩ where m = sp^{n} with s = −1 (mod p^{2}) so ϵ(s) = 1 and so A(m) = 2 + p^{n}(p^{2} − 1) + (p + 1)(p^{n} − 1)/(p − 1). Once again d(y_{m})/v_{1}^{A(m)} is a unit multiple of v_{2}^{e(m)}g_{1} mod I_{2}, but now e(m) = m − p^{n}(p − 1) − (p^{n} − 1)/(p − 1). The general form of m is [∗\overline{1}\overline{1}0^{n}] so e(m) = [∗\overline{1}\overline{1}0^{n}] − [\overline{1}0^{n}] − [1^{n}] = [∗\overline{2}\overline{1}\overline{2}^{n}] + 1. The total degree is −2p again.

(f): This comes from the summands F_{p}[v_{1}]⟨x_{n}^{s}G_{n}/v_{1}^{a_{n}}⟩ in Shimomura’s Theorem 4.4. Here n ≥ 0 and s ≠ −1 (mod p) and a_{n} = p^{n} + p^{n−1} − 1 as before, except that a_{0} = 1. His Proposition 4.3 shows that d(x_{n}^{s}G_{n})/v_{1}^{a_{n}} is a unit multiple of v_{2}^{e(n,s)}ρ, where e(n, s) = sp^{n}−(p^{n}−1)/(p−1) (see his Lemma 4.2). The general form of (s − 1)p^{n} is [∗t0^{n}] with t ≠ \overline{2} and then e(n, s) = [∗t0^{n}] + [10^{n}] − [1^{n}] = [∗t\overline{2}^{n}] + 1, so the generators are v_{2} ↑ [∗t\overline{2}^{n}](ρv_{2}). We know from (3.5.7) that ρ ∈ Ω^{3,0} so the total degree is −3.

Remark 15.3. Our table only gives the Z-graded homotopy groups of L̂S/p. How-

ever, if a ∈ Z and b ∈ Z_{p} then we still have π_{a+b|v_{2}|}L̂S/p = lim_{←-}_{N} π_{a+b|v_{2}|}LS/(p, v_{1}^{N}),

and the terms in the inverse limit and the maps between them are locally constant as b varies. One can deduce that our table also describes π_{a+b|v_{2}|}L̂S/p if we allow the symbol ∗ to denote an arbitrary infinite sequence of p-adic digits.

15.3. Finiteness questions. We next study the finiteness or otherwise of the groups π_{a+b|v_{2}|}L̂S/p. (These are of course vector spaces over F_{p} so that “finite” means the same as “finitely generated”.) It is convenient to formulate the problem more generally. Let M be a group graded over Z_{p}, for example M_{b} = π_{a+b|v_{2}|}L̂S/p. We assume that it is a graded module over F_{p}[u], where u has degree one; in our applications u = v_{1}^{p+1}.

Definition 15.4. Suppose that λ ≥ 0 and 0 ≤ ϕ, w < p^{λ}. Let A(ϕ, λ, w) be the Z_{p}- graded group which is a direct sum of copies of F_{p}[u]/u^{w} with one copy generated in each degree congruent to ϕ mod p^{λ}. Thus A(ϕ, λ, w) is a p-adically interpolated version of Σ^{ϕ|v_{2}|}F_{p}[u, v_{2}^{±p^{λ}}]/u^{w}. Given a with 0 ≤ a < w and b ∈ p^{λ}Z_{p} we write u^{a}v_{2}^{b}x for the generator of A(ϕ, λ, w)_{a+b+ϕ}. The mnemonic is that ϕ, λ and w are the phase, wavelength and width of the group. We also write supp(A(ϕ, λ, w)) = {b ∈ Z_{p} | A(ϕ, λ, w)_{b} ≠ 0}.

Definition 15.5. Given integers 0 ≤ r ≤ s and ϕ ∈ Z_{p} we let ϕ_{t} be the t-th p-adic digit of ϕ (so ϕ = ∑_{t≥0} ϕ_{t}p^{t} and 0 ≤ ϕ_{t} < p) and define

B(r, s; ϕ) = {ψ ∈ Z_{p} | ψ_{t} = ϕ_{t} for r ≤ t < s}.

We say that a set of this form is an (r, s)-block.

Lemma 15.6. If 2p^{μ} ≤ w then B(μ, λ; ϕ + p^{μ}) ⊆ supp(A(ϕ, λ, w)).

Proof. Write θ = ϕ+p^{μ}. If ψ ∈ B(μ, λ, θ) then we define k = p^{μ} +∑_{j=0}^{μ−1}(ψ_{t} −ϕ_{t})p^{t}. It is not hard to check that 1 ≤ k ≤ 2p^{μ} − 1 < w. Moreover, we have

ψ − k − ϕ = ψ − ∑_{t=0}^{μ−1}(ψ_{t} − ϕ_{t}) − θ = ∑_{t<μ}(ϕ_{t} − θ_{t})p^{t} + ∑_{t≥μ}(ψ_{t} − θ_{t})p^{t}.

It is clear that θ_{t} = ϕ_{t} when t < μ, and by assumption we have ψ_{t} = θ_{t} when μ ≤ t < λ. It follows that ψ − k − ϕ is divisible by p^{λ} so that u^{k}v_{2}^{ψ−k−ϕ}x defines a nonzero element of A(ϕ, λ, w)_{ψ}.

Proposition 15.7. Suppose that M = ⨁_{i} A(ϕ_{i}, λ_{i}, w_{i}), where the numbers λ_{i} are unbounded and the numbers p^{λ_{i}}/w_{i} are bounded above. Then {a ∈ Z_{p} | M_{a} is finite } has Haar measure zero.

Proof. As the numbers p^{λ_{i}}/w_{i} are bounded above, we can find α ≥ 0 such that 2p^{λ_{i}−α} ≤ w_{i} for all i. As the numbers λ_{i} are unbounded, we may assume (after passing to a subgroup and reindexing if necessary) that our direct sum is indexed by the natural numbers and that λ_{i} > λ_{i−1} + α (where λ_{−1} means 0). We now write

B_{i} = B(λ_{i} − α, λ_{i}, ϕ_{i} + p^{λ_{i}−α}) ⊆ supp(A(ϕ_{i}, λ_{i}, w_{i})).

It will be enough to show that C = {a | a lies in infinitely many of the sets B_{i}} has measure one. Note that the conditions a ∈ B_{i} depend on disjoint subsets of the digits of a (because λ_{i} > λ_{i−1} + α). It follows that given any finite set I of integers we have μ(⋂_{j∈I} B_{j}^{c}) = (1−p^{−α})^{|I|}, and thus that the corresponding measure is zero if I is infinite. In particular if J is finite then μ(⋂_{j∉J} B_{j}^{c}) = 0. There are only countably many such sets J, so we have μ(⋃_{J} ⋂_{j∉J} B_{j}) = 0. This union is just the complement of C, so μ(C) = 1.

Proposition 15.8. Suppose that M = ⨁_{i} A(ϕ_{i}, λ_{i}, w_{i}) and define ϕ′_{i} = p^{λ_{i}} −(w_{i} − 1) − ϕ_{i}. If ϕ_{i} ⟶ ∞ and ϕ′_{i} ⟶ ∞ then M_{a} is finite for all a ∈ Z.

Proof. If a ∈ Z then for almost all i we have ϕ_{i} > |a| and ϕ′_{i} > |a|, and in particular ϕ′_{i} > 0. It is clear that ϕ_{i} = |x_{i}| is the lowest positive degree in which A(ϕ_{i}, λ_{i}, w_{i}) is nonzero, and −ϕ′_{i} = |v_{2}^{−1}u^{w_{i}−1}x_{i}| is the highest negative degree in which A(ϕ_{i}, λ_{i}, w_{i}) is nonzero, so A(ϕ_{i}, λ_{i}, w_{i})_{a} = 0 for all but finitely many i. It is easy to see that A(ϕ_{i}, λ_{i}, w_{i})_{a} is finite for all i, so M_{a} is finite.

We now want to transfer these results to modules that are graded in the more usual way. For each a ∈ Z and b ∈ Z_{p} we can define π_{a+b|v_{2}|}L̂S/p = π_{b|v_{2}|}L̂S^{−a}/p using the p-adic powers of S^{|v_{2}|} ∈ 𝒫_{0} that we considered earlier. We can patch these groups together to get a group graded over Ẑ = lim_{←-}_{N} Z/|v_{2}^{p^{N}} | = Z/2(p^{2} − 1) × Z_{p}.

(We do not need to worry about making coherent choices of isomorphisms here, as we are simply counting group orders.) We now define A′(ϕ, λ, w) to be the obvious interpolation of the Z-graded group Σ^{ϕ}F_{p}[v_{1}, v_{2}^{±p^{λ}}]/v_{1}^{w}. Note that when w is large this is almost |v_{1}|-periodic. In this context we write ϕ′ = p^{λ}|v_{2}| − (w − 1)|v_{1}| − ϕ = −|v_{2}^{−p^{λ}}v_{1}^{w−1}x|, where x is the generator of A′(ϕ, λ, w).

It is not hard to deduce the following from Propositions 15.7 and 15.8.

Proposition 15.9. Suppose that M = ⨁_{i} A′(ϕ_{i}, λ_{i}, w_{i}), where

(a) The integers ϕ_{i} are all congruent to a fixed number ϕ modulo |v_{1}| = 2p − 2.

(b) The sequences λ_{i}, w_{i}, ϕ_{i} and ϕ′_{i} all tend to infinity.

(c) The numbers p^{λ_{i}}/w_{i} are bounded above. Then M_{a} is finite for all a ∈ Z, but {a ∈ Ẑ | a = ϕ (mod |v_{1}|) and M_{a} is finite } has measure zero.

We can now prove our result about π_{∗}L̂S/p.

Proof of Theorem 15.1. By a thick subcategory argument it suffices to verify the finiteness statement for X = S/p, so we work with S/p throughout.

Let C_{∗} be the direct sum of the entries (b), (d), (e) and (f) in Proposition 15.2. The omitted entries, together with the F_{p}[v_{1}]-free summands generated by 1 and h_{0}, are all concentrated in degrees 0, …, −3 modulo |v_{1}| and are finite in all degrees. We also have π_{∗}L̂S = E[ζ]⊗A_{∗} with |ζ| = −1. It will thus suffice to show that C_{∗} is finite in all integer degrees, but the set of degrees a congruent to −1, −2 or −3 mod |v_{1}| such that C_{a} is finite has measure zero. This will follow from Proposition 15.9 once we have determined the parameters λ, ϕ, ϕ′ and w for the families in entries (b), (d), (e) and (f). This is a straightforward calculation. It is easy to see that in each case the period λ and the width w tend to infinity as n does, but that p^{λ}/w ≤ p. Moreover, if we set ∗ = 0 we get a generator x whose degree satisfies 0 ≤ |x| < |v_{2}^{p^{λ}}|, which implies that ϕ = |x|. Given this, it is immediate that ϕ tends to infinity as n does. A little more calculation is required to find the numbers ϕ′. For example, in entry (b) we have ϕ = 2(pt + p − 1)p^{n−1}(p^{2} − 1) + 2p − 3 and w = p^{n} + p^{n−1} − 1 and λ = n + 1 so

ϕ′ = p^{λ}|v_{2}| − (w − 1)|v_{1}| − ϕ

= 2p^{n+1}(p^{2} − 1) − 2(p^{n} + p^{n−1} − 2)(p − 1) − 2(pt + p − 1)p^{n−1}(p^{2} − 1) − 2p + 3

= 2(p − 1 − t)p^{n−1}(p^{2} − 1) + 2p − 1.

As 0 ≤ t < p − 1 we see that this tends to infinity as well. The other cases are similar (although the case t = p − 1 in entry (f) needs to be done separately).

16. Questions and conjectures

In this section we assemble a long list of questions that we have not been able to answer. Some of them have a long history, but others seem not to have been asked before.

Problem 16.1. When X is a small spectrum in 𝒦, is the number len(π_{∗}X) defined in Definition 12.18 always an integer? If not, is there a universal constant M such that p^{M} len(π_{∗}X) is always an integer?

Problem 16.2. Is π_{k}L̂S always a finitely generated module over Z_{p} for all k ∈ Z? Theorem 15.1 suggests that this is true but subtle.

Problem 16.3. It would be pleasant to have a natural topology on π_{0}𝒟 making it a topological semiring. One idea would be to define U_{Z}(X) = {Y | Z ∧ Y ≃ Z ∧ X} and declare the sets U_{Z}(X) (as Z runs over ℱ) to be a basis of neighbourhoods of X. This has some good properties and some bad ones. For example, take n = 1 and p > 3. Then we can define a v_{1} self map v: S^{2p−2}/p ⟶ S/p, which becomes an isomorphism in 𝒦. Write X = ⋁_{i=0}^{p−2} S^{2i}/p, so that Σ^{2}X ≃ X ≄ ΣX. It seems that X cannot be separated from ΣX, so the topology is not Hausdorff, and the subspace π_{0}ℱ is not discrete. We hope nonetheless that some minor modifications will yield a satisfactory theory.

Problem 16.4. We can define a group K_{0}(ℱ) as the monoid of isomorphism classes of small spectra under the wedge operation, modulo relations [Y] = [X] + [Z] for every cofibre sequence X ⟶ Y ⟶ Z. There is a homomorphism χ: K_{0}(ℱ) ⟶ Z defined by

χ[X] = ∑_{0≤k<|v_{n}|}(−1)^{k} len(E_{k}X).

There is also a homomorphism ξ: K_{0}(ℱ) ⟶ Z[\frac{1}{p}], defined by

ξ[X] = len(π_{even}(X)) − len(π_{odd}(X)).

What is the relationship between χ and ξ? Is χ an isomorphism? If not, is Q ⊗ χ an isomorphism? Can one say anything about the higher Waldhausen K-theory of ℱ? There is a paper by Waldhausen about this [Wal84] but he assumed that the Telescope Conjecture would turn out to be true. Ravenel discusses this briefly in [Rav93] in the light of his disproof of the Telescope Conjecture [Rav92b].

Problem 16.5. Is Hopkins’ chromatic splitting conjecture true? The simplest form of this says that K(n − 1)_{∗}L_{K(n)}S has rank two over K(n − 1)_{∗}. A much more elaborate version is explained in [Hov95a], where some interesting consequences are deduced. Part of it can be rephrased as saying that

L_{n−1}L_{K(n)}S = L_{n−1}S_{p}^{∧} ∧ ⋀_{j=1}^{n}(S ∨ L_{n−j}S^{1−2j}).

One possible approach is to consider the functor v_{k}^{−1}((E/I_{k})^{∗}X) for k < n. This is a cohomology theory on the category of E-finite spectra; it would be interesting to know how it relates to K(k)^{∗}X.

Problem 16.6. A problem related to the previous one is as follows: in the language of Definition 6.7, what is the relationship between the support and cosupport of an E-local spectrum X? In particular, X is K-local if and only if cosupp(X) = {n}, and the examples suggest that in this case we have supp(X) = {m, m + 1, …, n} for some m. Is this always the case?

Problem 16.7. Is there a cofinal set of ideals I for which there is a generalised Moore spectrum of type S/I which is a commutative and associative ring spectrum? Given a map BP_{∗}/I ⟶ BP_{∗}/J of BP_{∗}BP -comodules, is there a map S/I ⟶ S/J inducing it? What about connecting maps for short exact sequences of comodules? How large a diagram of such maps can be made to commute on the nose? The first step here is clearly to find a convenient and conceptual formulation of the problem. A good answer would be helpful for the theory of unstable v_{n} periodicity, amongst other things.

Problem 16.8. What are the invariant prime ideals in E^{∗}? The obvious conjecture is that they are just the ideals I_{k} = (v_{j} | j < k) for k ≤ n. Many people believe, as we once did, that this is an easy consequence of Landweber’s classification of invariant prime ideals in BP_{∗}, but this does not seem to be the case. We also conjecture that every invariant radical ideal is prime. This would also help to prove that the categories 𝒟_{k} are the only nontrivial ideals in 𝒟.

Appendix A. Completion

In this appendix, we study the theory of (usually infinitely generated) modules over a regular local ring, that satisfy certain completeness properties. In particular, this applies to the modules E^{∗}X and E_{∗}^{∨}X over E_{∗}, for X ∈ 𝒦. Much of the theory presented here is obtained by specialising the results of [GM92] to our simpler context. Presumably a lot of it is known in some form to commutative algebraists.

Let R be a Noetherian graded ring with a unique maximal homogeneous ideal 𝔪, which is generated by a regular sequence of homogeneous elements (x_{0}, …, x_{n−1}). All modules, ideals and so on are required to be graded, and all maps and elements are required to be homogeneous.

A.1. Ordinary completion. Recall that the completion of a module M at 𝔪 is

M_{𝔪}^{∧} = lim_{←-}_{k} M/𝔪^{k}M.

Theorem A.1.

(a) (M_{𝔪}^{∧})_{𝔪}^{∧} = M_{𝔪}^{∧}.

(b) If M is finitely generated then M_{𝔪}^{∧} = R_{𝔪}^{∧} ⊗_{R} M, and this is an exact functor of M.

(c) R_{𝔪}^{∧} is flat over R.

(d) If f: M ⟶ N is epi, then so is the induced map M_{𝔪}^{∧} ⟶ N_{𝔪}^{∧}.

Proof. (a): Because M_{𝔪}^{∧} is defined as the inverse limit of {M/𝔪^{k}M}, there is a map π_{k}: M_{𝔪}^{∧} ⟶ M/𝔪^{k}M. Because the maps in the inverse system are epi, the map π_{k} is also epi. It is clear that π_{k} factors through an epimorphism M_{𝔪}^{∧}/𝔪^{k}M_{𝔪}^{∧} ⟶ M/𝔪^{k}M. On the other hand, the obvious map M ⟶ M_{𝔪}^{∧} induces a map M/𝔪^{k}M ⟶ M_{𝔪}^{∧}/𝔪^{k}M_{𝔪}^{∧}. One can check that these maps are mutually inverse, so that

(M_{𝔪}^{∧})_{𝔪}^{∧} = lim_{←-}_{k} M_{𝔪}^{∧}/𝔪^{k}M_{𝔪}^{∧} = lim_{←-} M/𝔪^{k}M = M_{𝔪}^{∧}

(b) and (c) are well-known consequences of the Artin-Rees lemma.

(d): It is easy to see that the induced maps M/𝔪^{n}M ⟶ N/𝔪^{n}N and 𝔪^{n}M ⟶ 𝔪^{n}N are epi, and thus that 𝔪^{n}M/𝔪^{n+1}M ⟶ 𝔪^{n}N/𝔪^{n+1}N is also epi. Let K_{n} be the kernel of the map M/𝔪^{n}M ⟶ N/𝔪^{n}N. We get a diagram as follows, in which the columns and the last two rows are exact.

L  𝔪^{n}M/𝔪^{n+1}M  𝔪^{n}N/𝔪^{n+1}N K_{n+1}  M/𝔪^{n+1}M  N/𝔪^{n+1}N K_{n}  M/𝔪^{n}M  N/𝔪^{n}N w v u  ww v u  v u v  w u  ww u u  u u v  w  ww

It follows from the snake lemma that the map K_{n+1} ⟶ K_{n} is epi, and thus that lim_{←-}_{n}^{1} K_{n} = 0. We therefore have a short exact sequence lim_{←-} K_{n} ⟶ lim_{←-} M/𝔪^{n}M ⟶

lim_{←-} N/𝔪^{n}N, so M_{𝔪}^{∧} ⟶ N_{𝔪}^{∧} is epi as claimed.

Note that completion does not preserve monomorphisms and is not right exact. For example, we can take R = Z_{(p)} and 𝔪 = (p). Then the completion of Z_{(p)} ⟶ Q is Z_{p} ⟶ 0, so completion does not preserve monomorphisms. Next, consider M = ⨁_{k} Z and f = ⨁_{k} p^{k} : M ⟶ M. One can show that the element (p, p^{2}, …) is nonzero in the cokernel of f : M_{𝔪}^{∧} ⟶ M_{𝔪}^{∧} but is zero in the completion of the cokernel. Thus, completion is not right exact.

A.2. Local homology. We next recall the theory of local (co)homology and the derived functors of the completion functor. Recall that one can define the leftderived functors of any additive functor F. One only needs F to be right exact in order to prove that the zeroth derived functor of F is F (which is false in the present context). Our response is to take the zeroth derived functor as the “correct” definition of completion.

For any x ∈ R we let K^{•}(x) denote the complex R ⟶ R[1/x], with R in degree 0 and R[1/x] in degree 1. We also write K^{•}(𝔪) = K^{•}(x_{0}) ⊗ … ⊗ K^{•}(x_{n−1}); there is then a natural map K^{•}(𝔪) ⟶ R. Note that K^{•}(𝔪) is a complex of flat modules. In fact, K^{•}(𝔪) is the finite acyclisation of R determined by R/𝔪 in the derived category of R, so it is determined by the ideal 𝔪 up to canonical quasiisomorphism

over R. To see this, write K^{•}(x) as the colimit of the complexes R ⟶^{x^{k}} R and tensor these complexes together. We also write P K^{•}(x) for the complex of projectives

R ⊕ R[t] ⟶^{(1,tx−1)} R[t], which is quasiisomorphic to K^{•}(x). Here the map tx − 1 : R[t] ⟶ R[t] is the R[t]-module map which takes 1 to tx − 1. We write P K^{•}(𝔪) = P K^{•}(x_{0}) ⊗ … ⊗ P K^{•}(x_{n−1}), which is quasiisomorphic to K^{•}(𝔪) by flatness. The local homology and cohomology groups of a module M are

L_{∗}M = H_{∗}^{𝔪}(M) = H_{∗}(Hom(P K^{•}(𝔪), M))
H_{𝔪}^{∗}(M) = H^{∗}(P K^{•}(I) ⊗ M) = H^{∗}(K^{•}(I) ⊗ M).

The last equality again uses the flatness of K^{•}(𝔪). Because R is a regular local ring, we have

H_{𝔪}^{k}(R) = \begin{cases}k < n & 0 \\ k = n & R/(x_{0}^{∞}, …, x_{n−1}^{∞}) \end{cases}

There are important questions to address about the extent to which this description of H_{𝔪}^{n}(R) is natural; however, we shall not address them here.

It follows from the above that K^{•}(𝔪) is actually quasiisomorphic to Σ^{n}H_{𝔪}^{n}(R) (in other words, the complex whose only nonzero term is H_{𝔪}^{n}(R), concentrated in degree n). Note that this is a complex of injectives.

It turns out that the functors L_{k}M are the left derived functors of completion, in the following sense.

Theorem A.2.

(a) There are natural maps M ⟶^{η_{M}} L_{0}M ⟶^{ϵ_{M}} M_{𝔪}^{∧}. Moreover, ϵ is an epimorphism, and the composite M ⟶ M_{𝔪}^{∧} is the obvious map.

(b) There is a short exact sequence

lim_{←-}_{k}^{1} Tor_{s+1}^{R}(R/𝔪^{k}, M) ↣ L_{s}M ↠ lim_{←-}_{k} Tor_{s}^{R}(R/𝔪^{k}, M).

In particular, there is a short exact sequence

lim_{←-}_{k}^{1} Tor_{1}^{R}(R/𝔪^{k}, M) ↣ L_{0}M ↠ M_{𝔪}^{∧}.

(c) For any short exact sequence M′ ⟶ M ⟶ M′′ there is a long exact sequence

L_{k+1}M′′ ⟶ L_{k}M′ ⟶ L_{k}M ⟶ L_{k}M′′ ⟶ L_{k−1}M′.

(d) There is a natural isomorphism

L_{s}M = Ext_{R}^{n−s}(H_{𝔪}^{n}(R), M).

Moreover, both sides vanish if s < 0 or s > n. (Thus L_{0} is right exact and L_{n} is left exact).

(e) If M is projective then ϵ_{M} is an isomorphism.

Proof. In [GM92] Greenlees and May construct functors L_{k} and a natural map ϵ : L_{0}M ⟶ M, satisfying (c) and (e). In Theorem 2.5, they prove that their groups L_{k}M are the same as H_{k}^{𝔪}(M) as defined above. They prove (b) as Proposition 1.1; it follows that ϵ_{M} is an epimorphism, as stated in (a). The map K^{•}(𝔪) ⟶ R induces a map

η: M = Hom(R, M) ⟶ H_{0}(Hom(K^{•}(𝔪), M)) = L_{0}M.

One can check that the composite M ⟶^{η} L_{0}M ⟶^{ϵ} M_{𝔪}^{∧} is the obvious map.

In our case, we know that P K^{•}(𝔪) is a cochain complex of projectives, whose only cohomology is H_{𝔪}^{n}(R), concentrated in degree n. Moreover, P K^{•}(𝔪) vanishes above degree n. It follows that P_{k} = P K^{n−k}(𝔪) defines a projective resolution of H_{𝔪}^{n}(R), and thus that L_{k}M = H_{k}(Hom(P K^{•}(𝔪), M)) = Ext_{R}^{n−k}(H_{𝔪}^{n}(R), M). This proves (d).

Proposition A.3. There is a natural map μ: L_{s}M ⊗L_{t}N ⟶ L_{s+t}(M ⊗N), which is commutative, associative and unital in the obvious sense.

Proof. The universal property of K^{•}(𝔪) as the finite acyclisation of R gives a quasiisomorphism ψ: P K^{•}(𝔪) ⟶ P K^{•}(𝔪) ⊗_{R} P K^{•}(𝔪) compatible with the maps P K^{•}(𝔪) ⟶ R ⟵ P K^{•}(𝔪) ⊗_{R} P K^{•}(𝔪), which is unique up to homotopy. This gives a map of chain complexes

Hom_{R}(P K^{•}(𝔪), M) ⊗_{R} Hom_{R}(P K^{•}(𝔪), N) ⟶ Hom_{R}(P K^{•}(𝔪), M ⊗ N).

By taking homology, we get the advertised pairing. Using the uniqueness of ψ (and the analogous map P K^{•}(𝔪) ⟶ P K^{•}(𝔪)^{⊗3}), we see that the pairing is commutative, associative, and unital.

Proposition A.4. If M is finitely generated then the map μ: M ⊗_{R} L_{0}N ⟶ L_{0}M ⊗_{R} L_{0}N ⟶ L_{0}(M ⊗_{R} N) is an isomorphism, and L_{0}M = M_{𝔪}^{∧}. In particular, R/𝔪^{k} ⊗ L_{0}N = N/𝔪^{k}N.

Proof. Because L_{0} is additive, it is clear that R^{k} ⊗_{R} L_{0}N = (L_{0}N)^{k} = L_{0}(N^{k}) = L_{0}(R^{k} ⊗ N). One can check that the map is just μ. For an arbitrary finitely generated module, choose an exact sequence R^{j} ⟶ R^{k} ⟶ M ⟶ 0, and use the use the right exactness of L_{0} to conclude that M ⊗_{R} L_{0}N = L_{0}(M ⊗_{R} N). In particular,

M_{𝔪}^{∧} = M ⊗_{R} R_{𝔪}^{∧} = M ⊗_{R} L_{0}R = L_{0}(M ⊗_{R} R) = L_{0}M.

A.3. L-complete modules.

Definition A.5. An R-module M is L-complete if η_{M}: M ⟶ L_{0}M is an isomorphism. We write ℳ for the category of R-modules, and ℳ̂ for the subcategory of L-complete modules.

Theorem A.6.

(a) For any M ∈ ℳ, the modules M_{𝔪}^{∧} and L_{k}M lie in ℳ̂. In particular, L_{0}^{2}M = L_{0}M.

(b) If M ∈ ℳ̂ then L_{k}M = 0 for k > 0.

(c) L_{0}M = 0 ⇔ M_{𝔪}^{∧} = 0 ⇔ M = 𝔪M.

(d) If M ∈ ℳ̂ and M = 𝔪M then M = 0.

(e) ℳ̂ is an Abelian subcategory of ℳ, which is closed under extensions.

(f) L_{0} : ℳ ⟶ ℳ̂ is left adjoint to the inclusion ℳ̂ ⟶ ℳ.

(g) If {M_{k}} is a collection of L-complete modules, then ∏_{k} M_{k} is L-complete. If they form an inverse system then lim_{←-}_{k} M_{k} is L-complete. If they form a tower

then lim_{←-}_{k}^{1} M_{k} is also L-complete.

Proof. It is proved as Theorem 4.1 in [GM92] that if N = M_{𝔪}^{∧} or N = L_{k}M then L_{0}N = N and L_{k}N = 0 for k > 0. It follows that M_{𝔪}^{∧} and L_{k}M are L-complete. It also follows that if M is L-complete and k > 0 then L_{k}M = L_{k}L_{0}M = 0. This proves (a) and (b).

(c): We have epimorphisms L_{0}M ⟶ M_{𝔪}^{∧} ⟶ M/𝔪M, so L_{0}M = 0 ⇒ M_{𝔪}^{∧} = 0 ⇒ M = 𝔪M. Suppose that M = 𝔪M. We shall prove that L_{0}M = 0, using an argument supplied by John Greenlees. It follows that M = 𝔪^{k}M, so that R/𝔪^{k} ⊗ M = 0 for all k. Using the short exact sequences

𝔪^{k}/𝔪^{k+1} ⟶ R/𝔪^{k+1} ⟶ R/𝔪^{k}

and the fact that 𝔪^{k}/𝔪^{k+1} is a free module over R/𝔪, we get an exact sequence

Tor_{1}^{R}(𝔪^{k}/𝔪^{k+1}, M) ⟶ Tor_{1}^{R}(R/𝔪^{k+1}, M) ⟶ Tor_{1}^{R}(R/𝔪^{k}, M) ⟶ 0.

From this we see that the maps in the tower

Tor_{1}^{R}(R/𝔪^{k}, M) ⟵ Tor_{1}^{R}(R/𝔪^{k+1}, M) ⟵ …

are surjective, so that lim_{←-}_{k}^{1} Tor_{1}^{R}(R/𝔪^{k}, M) = 0. Using part (b) of Theorem A.2, we see that L_{0}M = 0 as claimed.

(d): If M ∈ ℳ̂ and M = 𝔪M then M = L_{0}M = 0 by (c).

(e): First, we claim that the image of any map f: M′ ⟶ M′′ of L-complete modules is L-complete. To see this, factor f as a composite M′ ⟶^{q} M ⟶^{j} M′′, with q epi and j mono, so that M is the image of f. We have a diagram as follows:

M′  M  M′′ L_{0}M′  L_{0}M  L_{0}M′′^{q} ww_{u_{η′ ≃}}  v  w^{j}_{_{η} u}  _{u_{≃ η′′}} ww_{L_{0}q}  w_{L_{0}j}

Note that L_{0}q is epi because L_{0} is right exact. Because the left square commutes, we see that η is epi; because the right square commutes, we see that it is mono. Thus η is an isomorphism, and M is L-complete.

Next, suppose that N′ ⟶ N ⟶ N′′ is a short exact sequence, and that any two of the terms are L-complete. Part (c) of Theorem A.2 gives a long exact sequence relating the L_{k}-groups of N′, N and N′′, from which it is easy to see that the third of these is also L-complete. Thus ℳ̂ is closed under extensions, and under kernels and cokernels of epimorphisms and monomorphisms. For any map f: M′ ⟶ M′′ as in the previous paragraph, we have ker(f) = ker(q) and cok(f) = cok(j); it follows that ℳ̂ is closed under kernels and cokernels, and thus that it is Abelian.

(f): Suppose that N is L-complete and M is arbitrary. We need to show that η : M ⟶ L_{0}M induces an isomorphism η_{M}^{∗}: Hom(L_{0}M, N) ⟶ Hom(M, N). There is a map λ: Hom(M, N) ⟶ Hom(L_{0}M, N), defined by the commutativity of the following diagram:

M  N L_{0}M  L_{0}N^{f} w_{_{η_{M}} u}  _{u_{≃ η_{N}}} ^{ȷ ȷ ȷ ȷȷ fi}_{λ(f)} w_{L_{0}f}

It is clear that η_{M}^{∗} ∘ λ = 1, so that η_{M}^{∗} is epi. Suppose that f ∈ ker(η_{M}^{∗}), so that f can be factored as L_{0}M ⟶^{q} M′ ⟶^{g} N, where M′ is the cokernel of η_{M}. By the argument given above, we have an epimorphism η_{M′}^{∗} : Hom(L_{0}M′, N) ⟶ Hom(M′, N). However, because L_{0} is right exact and idempotent, we see that L_{0}M′ = 0. It follows that g = 0 and thus f = 0. Thus η_{M}^{∗}: Hom(L_{0}M, N) ⟶ Hom(M, N) is an isomorphism, as required.

(g): It is easy to see from the definitions that H_{s}^{𝔪}(∏_{k} M_{k}) = ∏_{k} H_{s}^{𝔪}(M_{k}). It follows that a product of L-complete modules is L-complete. If the modules {M_{k}} form an inverse system involving various maps u: M_{k} ⟶ M_{l} then lim_{←-}_{k} M_{k} is the

kernel of a map ∏_{k} M_{k} ⟶ ∏_{u} M_{l}, so it is L-complete by (d). If the inverse system is a tower, then lim_{←-}_{k}^{1} M_{k} is the cokernel of a map ∏_{k} M_{k} ⟶ ∏_{k} M_{k}, and thus is L-complete.

Corollary A.7. ℳ̂ is a symmetric monoidal category with tensor product M\overline{⊗}N = L_{0}(M ⊗ N).

Proof. It suffices to show that the map L_{0}μ: L_{0}(L_{0}M ⊗ L_{0}N) ⟶ L_{0}(M ⊗ N) induced by the pairing of Proposition A.3 is an isomorphism. Consider two modules M, N ∈ ℳ. Let M′ be the cokernel of M ⟶ L_{0}M. Because L_{0} is idempotent and right exact, we have L_{0}M′ = 0. It follows that M′ = 𝔪M′, or in other words that the natural map 𝔪 ⊗_{R} M′ ⟶ M′ is surjective. This implies that 𝔪 ⊗_{R} M′ ⊗_{R} N ⟶ M′ ⊗_{R} N is also surjective, so that L_{0}(M′ ⊗_{R} N) = 0. Using the right exactness of L_{0}, we see that the map L_{0}(η ⊗ 1): L_{0}(M ⊗_{R} N) ⟶ L_{0}(L_{0}M ⊗_{R} N) is surjective. Similarly, the map L_{0}(L_{0}M ⊗_{R} N) ⟶ L_{0}(L_{0}M ⊗_{R} L_{0}N) is surjective, and thus L_{0}(η ⊗ η): L_{0}(M ⊗_{R} N) ⟶ L_{0}(L_{0}M ⊗_{R} L_{0}N) is surjective. On the other hand, one sees from the definitions that L_{0}μ ∘ L_{0}(η ⊗ η) = 1. It follows that L_{0}μ is an isomorphism, with inverse L_{0}(η ⊗ η).

We next prove a useful criterion for modules to be L-complete.

Proposition A.8. If M is L-complete and a ∈ 𝔪 then lim_{←-}^{0} M = lim_{←-}^{1} M = 0, where the limits refer to the tower

M ⟵^{a} M ⟵^{a} M ⟵^{a} ….

Conversely, if (x_{0}, …, x_{n−1}) is a regular system of parameters and the above holds with a = x_{i} for each i, then M is L-complete.

Proof. Let C^{•} be the complex (of projectives)

R[t] ⟶^{at−1} R[t].

One checks directly that H^{s} Hom(C^{•}, M) = lim_{←-}^{s} M. Moreover, there is an obvious

short exact sequence C^{•} ⟶ P K^{•}(a) ⟶ R. It follows that lim_{←-}^{0} M = lim_{←-}^{1} M = 0 if and only if Hom(P K^{•}(a), M) is quasiisomorphic to M = Hom(R, M).

Suppose that M is L-complete, so that M ≃ Hom(P K^{•}(𝔪), M). The complex K^{•}(𝔪) is quasiisomorphic to K^{•}(a) ⊗ K^{•}(𝔪). Indeed, K^{•}(𝔪) ⊗ R[1/a] is acyclic, since K^{•}(𝔪) is finitely acyclic and R[1/a] is finitely local. Thus P K^{•}(𝔪) is homotopy equivalent to P K^{•}(a) ⊗ P K^{•}(𝔪). Therefore

Hom(P K^{•}(a), M) ≃ Hom(P K^{•}(a), Hom(P K^{•}(𝔪), M))
≃ Hom(P K^{•}(a) ⊗ P K^{•}(𝔪), M)
≃ M.

It follows that lim_{←-}^{0} M = lim_{←-}^{1} M = 0.

Conversely, let (x_{0}, …, x_{n−1}) be a regular system of parameters, and suppose that M ≃ Hom(P K^{•}(x_{i}), M) for each i. As P K^{•}(𝔪) = P K^{•}(x_{0})⊗…⊗P K^{•}(x_{n−1}), we see easily that M ≃ Hom(P K^{•}(𝔪), M), so M is L-complete.

A.4. Pro-free modules.

Theorem A.9. Let M be an L-complete R-module. The following are equivalent: 1. Every regular sequence of parameters \underline{x} is regular on M.

2. Some regular sequence of parameters \underline{x} is regular on M.

3. Tor_{s}^{R}(M, R/𝔪) = 0 for all s > 0.

4. Tor_{1}^{R}(M, R/𝔪) = 0.

5. M = L_{0}F = F_{𝔪}^{∧} for some free module F.

6. M is a projective object of ℳ̂.

Definition A.10. If M ∈ ℳ̂ satisfies the conditions of Theorem A.9, we say that M is pro-free.

Proof of Theorem A.9. (1)⇒(2): clear.

(2)⇒(3): Suppose that \underline{x} = (x_{0}, …, x_{n−1}) is a regular system of parameters for R, and that \underline{x} is also regular on M. Write I_{m} = (x_{0}, …, x_{m−1}), so that I_{n} = 𝔪. We therefore have short exact sequences

R/I_{m} ⟶^{x_{m}} R/I_{m} ⟶ R/I_{m+1}

and

M/I_{m}M ⟶^{x_{m}} M/I_{m}M ⟶ M/I_{m+1}M.

The first of these also gives a long exact sequence

Tor_{s+1}^{R}(R/I_{m+1}, M) ⟶ Tor_{s}^{R}(R/I_{m}, M) ⟶^{x_{m}} Tor_{s}^{R}(R/I_{m}, M) ⟶

Tor_{s}^{R}(R/I_{m+1}, M) ⟶ Tor_{s−1}^{R}(R/I_{m}, M).

By comparing this with the second short exact sequence, we see inductively that Tor_{s}^{R}(R/I_{m}, M) = 0 for all s > 0 and all m, in particular Tor_{s}^{R}(R/𝔪, M) = 0 for s > 0.

(3)⇒(4): clear.

(4)⇒(5): Suppose that Tor_{1}^{R}(M, R/𝔪) = 0. Choose a basis \{\overline{e}_{i} | i ∈ I\} for M/𝔪M over R/𝔪, and choose elements e_{i} ∈ M lifting \overline{e}_{i}. These give a map F = ⨁_{I} R ⟶ M, and thus a map f: L_{0}F ⟶ M. By proposition A.4, we see that R/𝔪 ⊗ L_{0}F = R/𝔪 ⊗ F ≃ R/𝔪 ⊗ M, so that R/𝔪 ⊗ cok(f) = 0. By part (d) of Theorem A.6, we see that cok(f) = 0, so f is epi. We therefore get an exact sequence

0 = Tor_{1}^{R}(R/𝔪, M) ⟶ R/𝔪 ⊗_{R} ker(f) ⟶ R/𝔪 ⊗_{R} L_{0}F ⟶^{≃} R/𝔪 ⊗_{R} M ⟶ 0.

It follows that R/𝔪 ⊗_{R} ker(f) = 0, so that ker(f) = 0. Thus f is an isomorphism, as required.

(5)⇒(1): Suppose that \underline{x} = (x_{0}, …, x_{n−1}) is a regular system of parameters for R. Write I_{m} = (x_{0}, …, x_{m−1}). Let S be a set, and a = (a_{s})_{s∈S} an element of ∏_{s∈S} R/I_{m}. We shall say that a converges to zero (or write a ⟶ 0) if for all k we have a_{s} ∈ 𝔪^{k}R/I_{m} for all but finitely many s. One can check directly that

(⨁_{k∈S} R/I_{m})_{𝔪}^{∧} = {a ∈ ∏_{s∈S} R/I_{m} | a ⟶ 0}.

We write F_{m} for this module. Using the above description, one can check directly that the sequence

F_{m} ⟶^{x_{m}} F_{m} ⟶ F_{m+1}

is short exact. It follows that \underline{x} is regular on F_{0} (which is the same as L_{0}(⨁_{S} R) by part (e) of Theorem A.2).

(5)⇒(6): Suppose that F is free and that M = L_{0}F. Then for N ∈ ℳ̂ we have ℳ̂(M, N) = ℳ(F, N) by part (f) of Theorem A.6. This is clearly an exact functor of N, as required.

(6)⇒(4): Suppose that P is a projective object of ℳ̂. Choose a free module F and an epimorphism F ⟶ P. This gives an epimorphism L_{0}F ⟶ L_{0}P = P, which must split because P is projective. It follows that Tor_{1}^{R}(R/𝔪, P) is a summand of Tor_{1}^{R}(R/𝔪, L_{0}F), which vanishes because (5)⇒(4).

Corollary A.11. The product of any family of pro-free modules is pro-free.

Proof. This follows easily from condition (1).

Corollary A.12. ℳ̂ has enough projectives.

Proof. Given M ∈ ℳ̂, choose a free module F and an epimorphism F ⟶ M. This gives an epimorphism L_{0}F ⟶ L_{0}M = M.

Proposition A.13. If f: P ⟶ Q is a map of pro-free modules such that the induced map P/𝔪P ⟶ Q/𝔪Q is a monomorphism, then f is a split monomorphism. In particular, if {P_{k}} is a family of pro-free modules, then the natural map L_{0} ⨁_{k} P_{k} ⟶ ∏_{k} P_{k} is a split monomorphism.

Proof. For the first claim, choose elements e_{s} ∈ P for s ∈ S giving a basis for P/𝔪P as an R/𝔪-vector space, and then choose elements e′_{t} ∈ Q for t ∈ T such that {fe_{s} | s ∈ S} ⨿ {e_{t} | t ∈ T } gives a basis for Q/𝔪Q. This gives an obvious diagram

⨁_{S} R  ⨁_{S⨿T} R P  Q u  w u w

As in the proof of (4)⇒(5) above, we see that the vertical maps are isomorphisms after applying L_{0}. The top horizontal map is a split monomorphism, so the bottom one is also.

Now consider a family {P_{k}} of pro-free modules. By proposition A.4, we know that

R/𝔪 ⊗_{R} L_{0} ⨁_{k} P_{k} = R/𝔪 ⊗_{R} ⨁_{k} P_{k} = ⨁_{k} R/𝔪 ⊗_{R} P_{k}.

It is easy to see that R/𝔪 ⊗_{R} ∏_{k} P_{k} = ∏_{k} R/𝔪 ⊗_{R} P_{k}. Thus, the first part of this proposition applies.

Appendix B. Small objects in other categories

In this appendix we investigate when various other categories of local spectra have no small objects. We assume that all spectra and Abelian groups are p-local. We write I for the Brown-Comenetz dual of the sphere, so that [X, I] = Hom(π_{0}X, Q/Z) and IX = F (X, I). Here Q/Z denotes Q/Z_{(p)} in keeping with our conventions. If L: 𝒮 ⟶ 𝒮 is a localisation functor, we say that a spectrum X is L-small if it is a small object in the category of L-local spectra. Localisation functors split naturally into two categories: we say that a localisation functor L has a finite local if there is a nonzero finite spectrum X such that LX = X. Similarly, we say that L has a finite acyclic if there is a nonzero finite X such that LX = 0. It is an old conjecture of the first author that every localisation functor either has a finite local or a finite acyclic: it certainly cannot have both.

Definition B.1. We say that a spectrum X is finite-dimensional if it admits a finite filtration 0 = X_{a} ≤ X_{a+1} ≤ … X_{b} = X for some integers a ≤ b, such that X_{k}/X_{k−1} is a wedge of copies of S^{k}. Clearly if Y is a finite-dimensional CW complex then Σ^{a}Σ^{∞}Y is a finite-dimensional spectrum. We define the width of X to be the minimum possible value of b − a.

Remark B.2. It is not hard to show that X has a filtration as above if and only if π_{k}X = 0 for k ≤ a and H_{b}X is free and H_{k}X = 0 for k > b. It follows that any retract of X admits a filtration of the same type.

Lemma B.3. Let L be a localisation functor on the category of spectra, and 𝒮_{L} the category of L-local spectra. If X is an L-small spectrum then it is a retract of Σ^{−N}LY for some integer N and some finite-dimensional suspension spectrum Y.

Proof. Let X_{k} denote the k’th space of X. We assume some foundational setting in which this is an infinite loop space with π_{j}X_{k} = π_{j−k}X for j ≥ 0. It is well-known that X is the telescope of the spectra Σ^{−k}Σ^{∞}X_{k}. It is not hard to conclude that X is also the telescope of the spectra Σ^{−k}Σ^{∞}Y_{k}, where Y_{k} is the 2k-skeleton of a CW complex weakly equivalent to X_{k}. As localisation functors preserve telescopes, we see that X = LX is the telescope of the spectra LΣ^{−k}Σ^{∞}Y_{k}. By smallness, it must be a retract of one of these terms.

Lemma B.4. For any set A, the spectrum (⋁_{i∈A} S)_{p}^{∧} is a retract of ∏_{i∈A} S_{p}^{∧}.

Proof. By Proposition A.13 (with R = Z_{p}) we see that there is a set B ⊇ A such that ∏_{i∈A} Z_{p} = (⨁_{j∈B} Z_{p})_{p}^{∧}. It follows that for any finitely generated Abelian group M we have ∏_{i∈A} M_{p}^{∧} = (⨁_{j∈B} M)_{p}^{∧}. We can use the set B to define a map (⋁_{j∈B} S_{p}^{∧})_{p}^{∧} ⟶ ∏_{i∈A} S_{p}^{∧}, and by applying the previous remark to M = π_{k}S we conclude that this map is an isomorphism. It follows that W_{p}^{∧} = (⋁_{i∈A} S_{p}^{∧})_{p}^{∧} is a retract of ∏_{i∈A} S_{p}^{∧} as claimed.

Lemma B.5. If X is a nontrivial finite spectrum then coloc⟨X⟩ = coloc⟨S_{p}^{∧}⟩, and this contains Z_{p}^{∧} for all finite-dimensional spectra Z. Moreover, coloc⟨X, HQ⟩ = coloc⟨S⟩, and this contains all finite-dimensional spectra.

Proof. Consider a generalised Moore spectrum S/I = S/(J, v). It is easy to see that S/(J, v^{k}) lies in the thick subcategory generated by S/I, and thus that (S/J)_{v}^{∧} = holim_{←-}_{k} S/(J, v^{k}) ∈ coloc⟨X⟩. If J ≠ 0 then |v| > 0 and S/J is (−1)-

connected and simple connectivity arguments show that (S/J)_{v}^{∧} = S/J. When J = 0 we know that v is just a power of p and (S/J)_{v}^{∧} = S_{p}^{∧}. By induction on the height we conclude easily that coloc⟨S/I⟩ = coloc⟨S_{p}^{∧}⟩. If X is an arbitrary nontrivial finite spectrum then X generates the same thick subcategory as some generalised Moore spectrum, so coloc⟨X⟩ = coloc⟨S_{p}^{∧}⟩. This part of the argument is essentially [Hov95a, Lemma 3.7].

Next, let X be a finite-dimensional spectrum, with filtration as in Definition B.1. Lemma B.4 tells us that (X_{k}/X_{k−1})_{p}^{∧} ∈ coloc⟨S_{p}^{∧}⟩, and it follows easily that X_{p}^{∧} ∈ coloc⟨S_{p}^{∧}⟩.

Next, recall that Z_{p}^{∧} = F (S^{−1}/p^{∞}, Z), so we have a fibration F (SQ, Z) ⟶ Z ⟶ Z_{p}^{∧}, in which the fibre is a rational spectrum. It is easy to see that whenever V is a nontrivial rational spectrum, we have coloc⟨V ⟩ = coloc⟨HQ⟩ = {all rational spectra}. By taking Z = S, we conclude that S ∈ coloc⟨S_{p}^{∧}, HQ⟩. We also conclude that HQ ∈ coloc⟨S, S_{p}^{∧}⟩ = coloc⟨S⟩. This shows that coloc⟨S⟩ = coloc⟨S_{p}^{∧}, HQ⟩ as claimed. Finally, we can take Z to be a finite-dimensional spectrum. We have seen that Z_{p}^{∧} ∈ coloc⟨S_{p}^{∧}⟩ and that F (SQ, Z) is rational and thus in coloc⟨HQ⟩, so it follows that Z ∈ coloc⟨S_{p}^{∧}, HQ⟩ = coloc⟨S⟩.

Theorem B.6. Suppose L is a localisation functor with a finite local.

(a) If LHQ ≠ 0, then L is the identity on finite-dimensional spectra. If LHQ = 0, then L is p-completion on finite-dimensional spectra.

(b) Every L-small spectrum is finite, and is torsion if LHQ = 0.

Proof. (a): Let Z be a finite-dimensional spectrum. The category 𝒮_{L} of L-local spectra is a colocalising category which contains a nontrivial finite spectrum, so by Lemma B.5 it contains Z_{p}^{∧}. If LHQ ≠ 0 then 𝒮_{L} contains a nontrivial rational spectrum so it contains all rational spectra, and in particular it contains F (SQ, Z). We have a fibration F (SQ, Z) ⟶ Z ⟶ Z_{p}^{∧} so we conclude that Z ∈ 𝒮_{L} and thus LZ = Z. On the other hand, suppose that LHQ = 0. If V is a rational spectrum we must have LV = 0, for otherwise 𝒮_{L} would contain a nontrivial rational spectrum and thus would contain HQ. Thus LF (SQ, Z) = 0 so our fibration shows that LZ = LZ_{p}^{∧} = Z_{p}^{∧}.

(b): We begin by showing that any L-small spectrum is finite-dimensional. In case LHQ ≠ 0, this follows from part (a) and Lemma B.3. If LHQ = 0, then the telescope holim_{-→}(p, X) of an L-small spectrum X computed in 𝒮_{L} is zero. Applying smallness to the canonical map X ⟶ holim_{-→}(p, X), we find that 0 = p^{m}: X ⟶ X

when m ≫ 0. Thus X is a retract of X/p^{m}, which is a retract of Σ^{−n}Y/p^{m} (because LY/p^{m} = Y/p^{m} by part (a)). It follows that X is finite-dimensional and torsion.

We next claim that we have [X, ⋁_{i} S] = ⨁_{i}[X, S] for any set of indices i. If LHQ ≠ 0 then this is clear, because X is small in 𝒮_{L} and the wedge in 𝒮 is finite-dimensional and thus L-local and thus the same as the coproduct in 𝒮_{L}. In the case LHQ = 0 we know that D(S/p^{m}) ∧ ⋁_{i} S = ⋁_{i} D(S/p^{m}) is L-local so [X ∧ S/p^{m}, ⋁_{i} S] = ⨁_{i}[X ∧ S/p^{m}, S] and we have seen that X is a retract of X ∧ S/p^{m} for some m so the claim follows.

Let 𝒞 be the full subcategory of finite-dimensional spectra Y such that

[Y, ⋁_{k=0}^{∞} S]_{∗} = ⨁_{k}[Y, S]_{∗}.

This is clearly thick, and contains both X and all finite spectra. We shall prove by induction on the width of Y that if Y ∈ 𝒞 then Y is finite. We may assume that Y has width greater than zero, so there is a cofibration Z ⟶ Y ⟶ W in which W is a wedge of spheres of the same dimension and Z has width strictly less than that of Y. By the defining property of 𝒞, there is a finite subwedge V of W such that the map Y ⟶ W factors through V. Let T be the cofibre of the map Y ⟶ V, so that T ∈ 𝒞. By applying the octahedral axiom to the maps Y ⟶ V ↣ W we get a cofibration T ↣ ΣZ ↠ U in which the first map is a split monomorphism, so T is a retract of Z and thus has width less than that of Y. It follows that T is finite, and thus that Y is finite. We conclude that our L-small spectrum X must be finite.

We now investigate when LF (n) is L-small. It is convenient to use the notation ⟨L⟩ for the collection of L-local spectra, and to think of ⟨L⟩ as a generalised Bousfield class. If L = L_{E}, then ⟨L⟩ = ⟨E⟩. We can then use notation such as ⟨L ∧ F (n)⟩, which is the local objects corresponding to the acyclics for the functor X ↦ LX ∧ F (n) = L(X ∧ F (n)).

Proposition B.7. Suppose L is a localisation functor on 𝒮. Then the following are equivalent.

(a) LF (n) is L-small.

(b) If {X_{α}} is a set of L-local spectra, then ⋁_{α} X_{α} ∧ F (n) is L-local.

(c) The natural map LS ∧ X ∧ F (n) ⟶ L(X ∧ F (n)) is an isomorphism for all X.

(d) ⟨LS ∧ F (n)⟩ = ⟨L ∧ F (n)⟩.

Furthermore, LF (n) is a small generator of 𝒮_{L} if and only if ⟨L⟩ = ⟨LS ∧ F (n)⟩.

Proof. (a)⇒(b): Suppose LF (n) is L-small, and {X_{α}} is a set of L-local spectra. Then we have

[F (n), L ⋁ X_{α}]_{∗} = [LF (n), L ⋁ X_{α}]_{∗}
= ⨁[LF (n), X_{α}]_{∗}
= ⨁[F (n), X_{α}]_{∗}
= [F (n), ⋁ X_{α}]_{∗}.

It follows using duality that the map ⋁ X_{α} ∧ F (n) ⟶ L(⋁ X_{α} ∧ F (n)) is an equivalence.

(b)⇒(c): Consider the collection of all X such that the map LS ∧ X ∧ F (n) ⟶ L(X ∧ F (n)) is an equivalence. This category is clearly thick and contains S. Part (b) implies that it is localising, so it is all of 𝒮.

(c)⇒(d): This is clear.

(d)⇒(a): We first show that if X_{α} is L-local for all α, then ⋁ X_{α} ∧ F (n) is also L-local. To see this, choose F (n) to be a ring spectrum (take F (n) = DY ∧ Y for a finite type n Y). Then ⋁ X_{α} ∧ F (n) is an LS ∧ F (n)-module spectrum, so in particular is LS ∧ F (n)-local. Since we are assuming part (d), we see that ⋁ X_{α} ∧ F (n) is L ∧ F (n)-local, and in particular is L-local.

We therefore have

[LF (n), L ⋁ X_{α}] = [F (n), L ⋁ X_{α}]
= [S, L(⋁ X_{α} ∧ DF (n))]
= [S, ⋁ X_{α} ∧ DF (n)]
= [F (n), ⋁ X_{α}]
= ⨁[F (n), X_{α}]
= ⨁[LF (n), X_{α}]

as required.

To complete the proof, note that LF (n) is a generator of 𝒮_{L} if and only if π_{∗}L(X∧ F (n)) = [DF (n), LX] = 0 implies that LX = 0. This is true if and only if ⟨L⟩ = ⟨L ∧ F (n)⟩.

Corollary B.8. Suppose L is a localisation functor with a finite local. Then 𝒮_{L} has a nonzero small object if and only if ⟨L⟩ ≥ ⟨F (n)⟩ for some n, and in this case the small objects in 𝒮_{L} are 𝒞_{n} for the smallest such n. The category 𝒮_{L} is an algebraic stable homotopy category if and only if L = L_{F (n)} for some n, and in this case F (n) is a small generator.

Proof. The small objects in 𝒮_{L} are all finite by Theorem B.6. Furthermore, LS is either S or the p-completion of S, and in either case we have ⟨LS⟩ = ⟨S⟩. The corollary then follows from Proposition B.7.

Proposition B.7 also applies to localisations with finite acyclics. For example, we get the following corollary.

Corollary B.9. Let E = K(m) ∨ K(m + 1) ∨ … K(n) for some 0 ≤ m ≤ n < ∞. Then 𝒮_{E} is an algebraic stable homotopy category with small generator L_{E}F (m).

It would be interesting to investigate these categories, as well as 𝒮_{F (n)}.

In order to apply Corollary B.8, we need a criterion to determine when ⟨L⟩ ≥ ⟨F (n)⟩ for some n.

Lemma B.10. Suppose ⟨L⟩ ≥ ⟨F (n)⟩ for some n. Then LI ≠ 0.

Proof. It suffices to show that DF (n) ∧ I ≠ 0 for all n. But DF (n) ∧ I = IF (n) and IX is never zero unless X is.

Lemma B.10 is more useful than it appears: in fact, we conjecture that its converse holds as well. There are very few spectra that detect I.

Lemma B.11. BP ∧ I = 0.

Proof. This follows from Ravenel’s results in [Rav84]. He proves in Lemma 3.2 that [P (1), S]_{∗} = 0, where P (1) = BP/p = BP ∧ S/p. As D(S/p) = S^{−1}/p we see that [BP, S/p]_{∗} = 0. As S/p has finite homotopy groups, we see that S/p = I^{2}(S/p) so [BP, S/p]_{∗} = Hom(π_{∗}(BP ∧ I(S/p)), Q/Z) = 0. Thus BP ∧ S/p ∧ I = 0. We also have S[\frac{1}{p}] ∧ I = 0 because π_{∗}I is a torsion group. As ⟨S⟩ = ⟨S/p⟩ ∨ ⟨S[\frac{1}{p}]⟩, we conclude that BP ∧ I = 0.

Corollary B.12. We have X ∧ I = 0 if X is a BP -module. In particular K(n), HZ, HF_{p} and HQ = SQ are I-acyclic. Furthermore, I itself is I-acyclic.

Proof. The only thing to prove is that I is I-acyclic. But the homotopy groups of I are concentrated in non-negative dimensions and are all finite except for π_{0}I = Q/Z. It follows that I is in the localising subcategory generated by HF_{p}. Since HF_{p} is I-acyclic, so is I.

Now recall that a spectrum is said to be harmonic if it is W -local, where W = ⋁_{0≤n<∞} K(n).

Corollary B.13. There are no nonzero small objects in the following categories.

(a) The BP -local category.

(b) The harmonic category.

(c) The HZ-local category.

(d) The HF_{p}-local category.

(e) The I-local category.

Proof. All of these localisation functors kill I, so it suffices to check that they do have a finite local. It is well-known that all finite spectra are BP -local and HZlocal, and that all finite torsion spectra are HF_{p}-local ( [Bou79, Theorem 3.1]). Similarly, all finite spectra are harmonic by [Rav84, Corollary 4.5]. To see that S/p is I-local, note that I^{2}(S/p) = S/p since the homotopy groups of S/p are finite. When Z is I-acyclic, we have

[Z, S/p] = [Z, F (I(S/p), I)] = [Z ∧ I(S/p), I] = [Z ∧ D(S/p) ∧ I, I] = 0

as required.

References

[Ada74] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, 1974.

[AK89] J. F. Adams and N. J. Kuhn, Atomic spaces and spectra, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 3, 473–481.

[And95] M. Ando, Isogenies of formal group laws and power operations in the cohomology theories E_{n}, Duke Math. J. 79 (1995), no. 2.

[Bak91] A. Baker, A_{∞} structures on some spectra related to Morava K-theory, Quart. J. Math. Oxf 42 (1991), 403–419.

[Bor53] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207.

[Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281.

[CS96] J. D. Christensen and N. P. Strickland, Phantom maps and homology theories, 25 pp., 1996, to appear in Topology.

[Dev92] E. S. Devinatz, Small ring spectra, J. Pure Appl. Alg. 81 (1992), 11–16.

[Dev96a] E. S. Devinatz, The generating hypothesis revisited, Preprint, 1996.

[Dev96b] E. S. Devinatz, Morava modules and Brown-Comenetz duality, Preprint, 1996.

[DHS88] E. S. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable homotopy theory, Ann. of Math. (2) 128 (1988), 207–241.

[EKMM96] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules and algebras in stable homotopy theory, Amer. Math. Soc. Surveys and Monographs, vol. 47, American Mathematical Society, 1996.

[Fra92] J. Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992), 43– 65.

[Fra96] J. Franke, Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence, Preprint, 1996.

[Fre66] P. Freyd, Stable homotopy, Proceedings of the Conference on Categorical Algebra (La Jolla, 1965) (S. Eilenberg, D. K. Harrison, S. Mac Lane, and H. Röhrl, eds.), 1966, pp. 121–172.

[GM92] J. P. C. Greenlees and J. P. May, Derived functors of I-adic completion and local homology, J. Alg. 149 (1992), no. 2, 438–453.

[GM95a] J. P. C. Greenlees and J. P. May, Completions in algebra and topology, Handbook of Algebraic Topology (Amsterdam) (Ioan M. James, ed.), Elsevier, Amsterdam, 1995.

[GM95b] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, vol. 113, Mem. Amer. Math. Soc., no. 543, American Mathematical Society, 1995.

[GS96a] J. P. C. Greenlees and H. Sadofsky, The Tate spectrum of v_{n}-periodic complex oriented theories, Math. Z. 222 (1996), 391–405.

[GS96b] J. P. C. Greenlees and N. P. Strickland, Varieties and local cohomology for chromatic group cohomology rings, 48 pp., Submitted to Topology, 1996.

[HG94] M. J. Hopkins and B. H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. 30 (1994), 76–86.

[HMS94] M. J. Hopkins, M. E. Mahowald, and H. Sadofsky, Constructions of elements in Picard groups, Topology and Representation Theory (E. M. Friedlander and M. E. Mahowald, eds.), Contemp. Math., vol. 158, Amer. Math. Soc., 1994, pp. 89–126.

[Hov95a] M. Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conjecture, The Čech Centennial (Mila Cenkl and Haynes Miller, eds.), Contemp. Math., vol. 181, Amer. Math. Soc.,1995, pp. 225–250.

[Hov95b] M. Hovey, Cohomological Bousfield classes, J. Pure Appl. Algebra 103 (1995), 45–59.

[HPS95] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, To appear in Mem. Amer. Math. Soc., 1995.

[HR92] M. J. Hopkins and D. C. Ravenel, Suspension spectra are harmonic, Bol. Soc. Mat. Mexicana 37 (1992), 271–280, This is a special volume in memory of José Adem, and is really a book. The editor is Enrique Ramírez de Arellano.

[HRW96] M. J. Hopkins, D. C. Ravenel, and W. S. Wilson, Morava Hopf algebras and spaces K(n) equivalent to finite Postnikov systems, Preprint, 1996.

[HS] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II, To appear in Ann. of Math.

[HS95] M. Hovey and H. Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, To appear in J. London Math. Soc., 1995.

[HS96] M. Hovey and H. Sadofsky, Tate cohomology lowers chromatic Bousfield classes, Proc. Amer. Math. Soc., 124 (1996), 3579–3585.

[Lan76] P. S. Landweber, Homological properties of comodules over MU_{∗}(MU) and BP_{∗}(BP), Amer. J. Math. 98 (1976), 591–610.

[LS69] R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94.

[Mar83] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland, 1983.

[Mil92] H. R. Miller, Finite localizations, Bol. Soc. Mat. Mexicana 37 (1992), 383–390, This is a special volume in memory of José Adem, and is really a book. The editor is Enrique Ramírez de Arellano.

[Mor85] J. Morava, Noetherian localizations of categories of cobordism comodules, Ann. of Math. (2) 121 (1985), 1–39.

[Mos68] R.M.F. Moss, On the composition pairing of Adams spectral sequences, Proc. London Math. Soc. 18 (1968), 179–192.

[Nas96] C. Nassau, On the structure of P (n)_{∗}(P (n)) for p = 2, preprint, 1996.

[Rav84] D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1) (1984), 351–414.

[Rav86] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986.

[Rav92a] D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, 1992.

[Rav92b] D. C. Ravenel, Progress report on the telescope conjecture, Adams Memorial Symposium on Algebraic Topology Volume 2 (Nigel Ray and Grant Walker, eds.), 1992, London Mathematical Society Lecture Notes 176, pp. 1–21.

[Rav93] D. C. Ravenel, Life after the telescope conjecture, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 205–222.

[Riv72] N. Saavedra Rivano, Catégories Tannakiennes, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 265.

[Rud86] Y. B. Rudyak, Exactness theorems for the cohomology theories MU,BP and P (n), Mat. Zametki 40 (1986), no. 1, 115–126,141. English translation: Math. Notes 40 (1986) 562–569.

[RW80] D. C. Ravenel and W. S. Wilson, The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691–748.

[Sad93] H. Sadofsky, Hopkins’ and Mahowald’s picture of Shimomura’s v_{1}-Bockstein spectral sequence calculation, Algebraic topology (Oaxtepec, 1991) (Martin Tangora, ed.), 1993, Contemp. Math. 146, Amer. Math. Soc., pp. 407–418.

[Shi86] K. Shimomura, On the Adams–Novikov spectral sequence and products of β–elements, Hiroshima Math. J. 16 (1986), 209–224.

[Spa66] E. H. Spanier, Algebraic topology, McGraw–Hill, 1966.

[Str92] N. P. Strickland, On the p-adic interpolation of stable homotopy groups, Adams Memorial Symposium on Algebraic Topology Volume 2 (Nigel Ray and Grant Walker, eds.), 1992, London Mathematical Society Lecture Notes 176, pp. 45–54.

[Str96] N. P. Strickland, Products on MU-modules, 34 pp., Submitted to Trans. Amer. Math. Soc., 1996.

[SY95] K. Shimomura and A. Yabe, The homotopy groups π_{∗}(L_{2}S^{0}), Topology 34 (1995), no. 2, 261–289.

[Wal84] F. Waldhausen, Algebraic K–theory of spaces, localization, and the chromatic filtration of stable homotopy theory, Algebraic Topology Aarhus 1982 (Ib Madsen and Robert Oliver, eds.), Lecture Notes in Mathematics, vol. 1051, Springer–Verlag, 1984, pp. 173–195.

Index

atomic, 24

BG, 48 Bousfield class, 32–33, 43 Brown-Comenetz duality, 52–55, 90 ℬ(X), 69

C^{f}, 44 Ĉ, 6 chromatic splitting, 82 cohomology functor, 50 coideal, 6 colocalising subcategory, 6, 38, 39, 43 completion, 82–90 cophantom, 51, 55 cosupport, 36, 82

𝒟, 46 𝒟_{E}, 5 deg, 68 𝒟_{MU}, 4, 22 dualisable, 7, 34, 46–48, 57–64

E, 4, 5, 7–9, 15–18, 45 E(n), 4 even finite spectrum, 9 evenly generated, 9–12 E_{∗}^{∨}X, 45

ℱ, 46 F (m), 6 F -small, 7, 34, 47 finite, 7, 34, 46, 47, 62 finite-dimensional, 90 ℱ_{n}, 30

generalised Moore spectrum, 91 good (P, u) self map, 70 good v_{n} element, 24 good v_{n} self map, 22, 27, 38, 57

Haar measure, 76, 79 harmonic, 94

holim_{-→}, 7

holim_{←-}, 7

homology functor, 50–52

ideal, 6, 42, 43, 61, 62 Î, 52, 54 I_{n}, 4 injective

comodule, 14

spectrum, 14, 19–21, 34 invariant prime, 82

K, 4 𝒦, 6 K(n), 4 K_{0}(ℱ), 81 Krull-Schmidt theorem, 60

ℒ, 6

L-complete, 8, 45, 85–90 Landweber exactness, 5, 9–14 len, 63, 81 L^{f}, 44 L̂, 6, 44–45 local homology, 83–85 localising subcategory, 6, 38, 39, 43

ℳ, 6, 40 MASS, 15 minimal weak colimit, 7, 10, 50 ℳ_{MU}, 4 monochromatic, 6, 40–41 monogenic, 7 Moore spectrum, 21, 26, 28, 53, 82 MU, 4, 10 μ-spectrum, 24 M̂U, 4

mwlim_{-→}, 7, 50

nilpotent map, 36, 57 spectrum, 6, 34, 39, 42, 64–67 topologically, 57–60

Noetherian, 15–18

𝒫, 67 𝒫^{0}, 69 phantom, 5, 9, 11, 30, 51, 55 π(P, u), 70 Pic, 67 Pic^{0}, 69 Picard group, 52, 63, 67–74 Pic′, 69 𝒫′, 69 Pro(ℱ), 29–31 pro-free, 88–90

representable, 51, 52

𝒮, 4, 5

semiring, 62–64

Seq(ℱ), 29 Σ^{∗}, 15 Σ(k)_{∗}, 16 Σ̂^{∗}, 15 S/I, 26, 28, 31 small, 7, 34, 46, 50, 62, 90–94 smashing, 32, 37 spectral sequence

Adams, 14, 15, 34, 35 universal coefficient, 14 stable homotopy category, 6–7 algebraic, 7, 41 monogenic, 7, 33

strongly central, 21, 22, 24, 25

element, 24 support, 36, 82

T (m), 6 telescope conjecture, 37 thick, 6, 36, 37 T_{n}, 30 Tow(ℱ), 29 tower, 29, 30, 32

width, 90 w_{k}, 4

MORAVA K-THEORIES AND LOCALISATION 99

MIT, Cambridge, MA

E-mail address: hovey@math.mit.edu

Trinity College, Cambridge CB2 1TQ, UK

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