Stable homotopy over the Steenrod algebra

John H. Palmieri

Box 354350, Department of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A.

E-mail address: palmieri@member.ams.org

1991 Mathematics Subject Classification. 55S10, 55U15, 18G35, 55U35, 55T15, 55P42, 55Q10, 55Q45, 18G15, 16W30, 18E30, 20J99

Key words and phrases. Steenrod algebra, Ext, stable homotopy, spectra, nilpotence theorem, Quillen stratification, chromatic convergence

Research partially supported by National Science Foundation grant DMS-9407459.

Abstract. We apply the tools of stable homotopy theory to the study of modules over the mod p Steenrod algebra A^{∗}. More precisely, let A be the dual of A^{∗}; then we study the category Stable(A) of unbounded cochain complexes of injective comodules over A, in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, Brown-Comenetz duality, and other homotopy-theoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of A, Ext_{A}^{∗∗}(F_{p}, F_{p}). We also have nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a number of other results.

Contents

List of Figures v

Preface vii

Chapter 0. Preliminaries 1 0.1. Grading and other conventions 1 0.2. Hopf algebras 2 0.3. Modules and comodules 4 0.4. Homological algebra 6 0.5. Two small examples 9

Chapter 1. Stable homotopy over a Hopf algebra 13 1.1. The category Stable(Γ) 14 1.2. The functor H 16 1.2.1. Remarks on Hopf algebra extensions 20 1.3. Some classical homotopy theory 22 1.4. The Adams spectral sequence 25 1.5. Bousfield classes and Brown-Comenetz duality 29 1.6. Further discussion 31

Chapter 2. Basic properties of the Steenrod algebra 35 2.1. Quotient Hopf algebras of A 35 2.1.1. Quasi-elementary quotients of A 41 2.2. P_{t}^{s}-homology 42 2.2.1. Miscellaneous results about P_{t}^{s}-homology 46 2.3. Vanishing lines for homotopy groups 48 2.3.1. Proof of Theorems 2.3.1 and 2.3.2 when p = 2 50 2.3.2. Changes necessary when p is odd 55 2.4. Self-maps via vanishing lines 57 2.5. Construction of spectra of specified type 60 2.6. Further discussion 63

Chapter 3. Chromatic structure 65 3.1. Margolis’ killing construction 65 3.2. A Tate version of the functor H 71 3.3. Chromatic convergence 75 3.4. Further discussion: work of Mahowald and Shick 76 3.5. Further discussion 77

Chapter 4. Computing Ext with elements inverted 79 4.1. The q_{n}-based Adams spectral sequence 80 4.2. The Q_{n}-based Adams spectral sequence 81 4.3. A(n) as an A-comodule 85 4.4. \frac{1}{2}A(n) satisfies the vanishing plane condition 88 4.5. \frac{1}{2}A(n) generates the expected thick subcategory 90 4.5.1. The proof of Proposition 4.5.7 95 4.6. Some computations and applications 99 4.6.1. Computation of (Q_{n})_{∗∗}(Q_{n}) 99 4.6.2. Eisen’s calculation 102 4.6.3. The v_{1}-inverted Ext of the mod 2 Moore spectrum 104

Chapter 5. Quillen stratification and nilpotence 107 5.1. Statements of theorems 108 5.1.1. Quillen stratification 108 5.1.2. Nilpotence 110 5.2. Nilpotence and F -isomorphism via the Hopf algebra D 112 5.2.1. Nilpotence: Proof of Theorem 5.1.5 116 5.2.2. F -isomorphism: Proof of Theorem 5.1.2 117 5.3. Nilpotence and F -isomorphism via quasi-elementary quotients 119 5.3.1. Nilpotence: Proof of Theorem 5.1.6 119 5.3.2. F -isomorphism: Proof of Theorem 5.1.3 121 5.4. Further discussion: nilpotence at odd primes 123 5.5. Further discussion: miscellany 124

Chapter 6. Periodicity and other applications of the nilpotence theorems 127 6.1. The periodicity theorem 127 6.2. y-maps and their properties 128 6.3. Properties of ideals 131 6.4. The proof of the periodicity theorem 132 6.5. Computation of some invariants in HD_{∗∗} 134 6.6. Computation of a few Bousfield classes 139 6.7. Ideals and thick subcategories 143 6.7.1. The thick subcategory conjecture 143 6.7.2. Rank varieties 146 6.8. Further discussion: slope supports 147 6.9. Further discussion: miscellany 150

Appendix A. An underlying model category 153

Appendix B. Steenrod operations and nilpotence in Ext_{Γ}^{∗∗}(k, k) 155 B.1. Steenrod operations in Hopf algebra cohomology 155 B.2. Nilpotence in Ext over quotients of A: p = 2 156 B.3. Nilpotence in Ext over quotients of A: p odd 157 B.3.1. Sketch of proof of Conjecture B.3.4, and other results 159

Appendix. Bibliography 165

Appendix. Index 169

List of Figures

2.1.A  Graphical representation of a quotient Hopf algebra of A.  37
2.1.B  Profile functions for A(n).  38

2.1.C Profile functions for maximal elementary quotients of A, p = 2. 40

2.3.A  Vanishing line at the prime 2.  49
3.1.A  Vanishing curve for π_{ij}(C_{n}^{f}S^{0}).  70
3.2.A  The coefficients of HA(1) and ĤA(1).  74
4.2.A  Vanishing plane in Theorem 4.2.6.  83
4.6.A  “Profile function” for H(A □_{E}F_{p}, Q_{n}).  100
4.6.B  Lightning flashes and v_{1}-towers.  105
5.1.A  Profile functions for D.  108
5.2.A  Profile function for D(n).  113
5.3.A  Profile functions for D_{r} and D_{r,q}.  120
6.5.A  Graphical depiction of coaction of A on lim HE_{∗∗}.  138
6.8.A  T (t, s) and T (m) as subsets of Slopes′.  149

Preface

The object of study for this book is the mod p Steenrod algebra A and its cohomology Ext_{A}^{∗∗}(F_{p}, F_{p}). Various people, including the author, have approached this subject by taking results in stable homotopy theory and then trying to prove analogous results for A-modules. This has proven to be successful, but the analogies were just that—there was no formal setting in which to do anything more precise.

In [HPS97], Hovey, Strickland, and the author developed “axiomatic stable homotopy theory.” In particular, we gave axioms for a stable homotopy category; in any such category, one has available many of the tools of classical and modern stable homotopy theory—tools like Brown representability and Bousfield localization. It turns out that a category Stable(A) (defined in the next paragraph) of modules over the Steenrod algebra is such a category; as one might expect, the trivial module F_{p} plays the role of the sphere spectrum S^{0}, and Ext_{A}^{∗∗}(−, −) plays the role of homotopy classes of maps. Since many of the tools of stable homotopy theory are focused on the study of the homotopy groups of S^{0} (and of other spectra), one should expect the corresponding tools in Stable(A) to help in the study of Ext_{A}^{∗∗}(F_{p}, F_{p}) (and related groups). In this book we apply some of these tools— Adams spectral sequences, nilpotence theorems, periodicity theorems, chromatic towers, etc.—to the study of Ext over the Steenrod algebra. It is our hope that this book will serve two purposes: first, to provide a reference source for a number of results about the Steenrod algebra and its cohomology, and second, to provide an example of an in-depth use of the language and tools of axiomatic stable homotopy theory in an algebraic setting.

Now we describe the category in which we work. We fix a prime p, let A^{∗} be the mod p Steenrod algebra, and let A = Hom_{F_{p}}(A^{∗}, F_{p}) be the (graded) dual of the Steenrod algebra. We let Stable(A) be the category whose objects are unbounded cochain complexes of injective left A-comodules, and whose morphisms are cochain homotopy classes of maps. This is a stable homotopy category. We prove a number of results about Stable(A); some of these are analogues of results in the ordinary stable homotopy category, and some are not. Some of these are new, and some already known, at least in the setting of A^{∗}-modules; the old results often need new proofs to apply in the more general setting we discuss here.

Note. This work arose from the study of the abelian category of left A^{∗}- modules; to apply stable homotopy theoretic techniques, though, it is most convenient to work in a triangulated category. One’s first guess for an appropriate category might be one with objects the chain complexes of projective A^{∗}-modules; it turns out that this category has some technical difficulties—see Remark 1.1.1(d). It is much more convenient to work with A-comodules instead of A^{∗}-modules, and fortunately, one does not lose much by doing this. Many A^{∗}-modules of interest can be viewed as A-comodules; the main effects of using comodules are things of the following sort: various arrows go the “wrong” way, Ext_{A}^{∗∗}(F_{p}, F_{p}) is covariant in A, and one studies A via its quotient Hopf algebras (because those are dual to the sub-Hopf algebras of A^{∗}).

Each chapter is divided into a number of sections; at the beginning of each chapter, we give a brief description of its contents, section by section. In this introduction, we give a brief overview of each chapter. We note that most chapters have at least one “Further discussion” section, in which we discuss issues auxiliary to the general discussion.

In Chapter 0, we provide some background material. We introduce some conventions related to grading; then we discuss Hopf algebras, modules, and comodules. We define what it means for a comodule to be injective, and we use this to define Ext in the category of comodules. We end the chapter by looking at some examples of small Hopf algebras; we classify the indecomposable comodules over these, and we compute the Ext groups for these indecomposables.

In Chapter 1, given a graded commutative Hopf algebra Γ over a field k, we define and examine the category Stable(Γ). The main examples we are concerned with are when Γ is the dual of a group algebra, the dual of an enveloping algebra, or the dual of the Steenrod algebra. Aside from setting up notation for use throughout this book, the main topics of this chapter include construction of cellular and Postnikov towers, an examination of generalized Adams spectral sequences in Stable(Γ), and some remarks on Bousfield classes and Brown-Comenetz duality.

In Chapter 2 we specialize to the case in which p is a prime, k = F_{p} is the field with p elements, and A is the dual of the mod p Steenrod algebra. Recall from [Mil58] that as algebras, we have

A ≅ \begin{cases}F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …], & if p = 2, \\ F_{p}[ξ_{1}, ξ_{2}, ξ_{3}, …] ⊗ Λ[τ_{0}, τ_{1}, τ_{2}, …], & if p is odd. \end{cases}

The coproduct Δ on A is determined by

Δ(ξ_{n}) = ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ ξ_{i},

Δ(τ_{n}) = ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ τ_{i} + τ_{n} ⊗ 1,

where ξ_{0} = 1. In this chapter, we discuss two tools with which to study Stable(A): quotient Hopf algebras of A and P_{t}^{s}-homology. We use these tools to prove several theorems. The first is a vanishing line theorem, which generalizes results of Anderson-Davis [AD73] and Miller-Wilkerson [MW81]: given conditions on the P_{t}^{s}-homology groups of X, then Ext_{A}^{i,j}(F_{p}, X) = 0 when i > mj − c, for some numbers m and c. The second is a “self-map” theorem, which first appeared for modules in [Pal92]: given a finite A-comodule M, we construct a non-nilpotent element of Ext_{A}^{∗∗}(M, M) satisfying certain properties. We use this second theorem to construct well-behaved objects with vanishing lines of specified slopes; these are analogues in the category Stable(A) of generalized Toda V (n)’s. We use these objects in Chapter 3.

In Chapter 3 we consider Steenrod algebra analogues of chromatic theory and the functors L_{n} and L_{n}^{f}. The latter functor L_{n}^{f} turns out be more tractable; in fact, it is a generalization (from the module setting) of Margolis’ killing construction [Mar83, Chapter 21]. We show that L_{n} ≠ L_{n}^{f} if n > 1, at least at the prime 2. We compute L_{n}^{f} on some particular ring spectra, and show that, at least for these rings, it turns “group cohomology” into “Tate cohomology.” We use this result to show that the chromatic tower constructed using the functors L_{n}^{f} converges for any finite object. (This is a variant of a theorem of Margolis [Mar83, Theorem 22.1].)

In Chapter 4, we discuss a spectral sequence for computing v_{n}^{−1} Ext_{A}^{∗∗}(F_{p}, M) for certain comodules M (those for which some power of v_{n} acts on Ext_{A}^{∗∗}(F_{p}, M)). As applications, we describe v_{0}^{−1} Ext_{A}^{∗∗}(F_{p}, M) in terms of the Q_{0}-homology of M, we reproduce Eisen’s calculation [Eis87], and we discuss Mahowald’s conjectured calculation in [Mah70] of the v_{1}-inverted Ext of the mod 2 Moore spectrum. The spectral sequence is constructed as a localization of an Adams spectral sequence, so convergence is an issue, but one which can be dealt with. As part of the discussion of convergence, we analyze the comodule structure on the Hopf algebras A(n), originally due to Mitchell [Mit85] and Smith [Smi]. Of particular interest is the subcomodule \frac{1}{2}A(n) of A(n).

Note. Although Chapters 3 and 4 can be read independently, they are philosophically linked, in that they both involve using nice objects which have vanishing P_{t}^{s}-homology groups for specified P_{t}^{s}’s. In Chapter 3, we use the analogues of generalized Toda V (n)’s to give a construction of the functor L_{n}^{f}. In Chapter 4, we study the comodules \frac{1}{2}A(n) in order to prove convergence of the localized Adams spectral sequence.

Let p = 2. In Chapter 5 we develop analogues in the category Stable(A) of the nilpotence theorem of Devinatz, Hopkins, and Smith [DHS88], as well as the stratification theorem of Quillen [Qui71]. In fact we give two nilpotence theorems: in one we describe a single ring object (like BP) that detects nilpotence; more precisely, there is a quotient Hopf algebra D of A so that, if M is a finite-dimensional A-comodule, an element z ∈ Ext_{A}^{∗∗}(M, M) is nilpotent under Yoneda composition if and only if its restriction to Ext_{D}^{∗∗}(M, M) is nilpotent. The second nilpotence theorem is similar, but uses a family of ring objects (somewhat like the Morava K-theories) to detect nilpotence. These are versions in Stable(A) of the nilpotence theorems of [DHS88] and [HS98]. We strengthen these results when studying Ext_{A}^{∗∗}(F_{2}, F_{2}), by “identifying” the image of Ext_{A}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{D}^{∗∗}(F_{2}, F_{2}) (and similarly for the other nilpotence theorem). One can view this as an analogue of Quillen’s theorem [Qui71, Theorem 6.2], which identifies the cohomology of a compact Lie group up to F -isomorphism.

Again, let p = 2. In Chapter 6, we discuss applications and illustrations of the theorems from the previous chapter. In ordinary stable homotopy theory, the nilpotence theorems lead to the periodicity theorem and the thick subcategory theorem (see [Hop87] and [HS98]); in our setting, things are a bit harder, so we get a weak version of a periodicity theorem, and only a conjecture as to a classification of the thick subcategories of finite objects in Stable(A). More precisely, if M is a finite-dimensional A-comodule, then we produce a number of central non-nilpotent elements in Ext_{A}^{∗∗}(M, M) by using the “ideal of M”: the kernel of Ext_{D}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{D}^{∗∗}(M, M). We study some properties of these ideals, and we conjecture that the thick subcategories of finite-dimensional comodules are determined by them. One of our analogues of Quillen’s theorem says that the elements of the kernel of Ext_{A}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{D}^{∗∗}(F_{2}, F_{2}) are nilpotent, and it identifies the image. This identification is not explicit, so we discuss some examples of elements known to be in the image. We also imitate [Rav84] to show that the objects that detect nilpotence have strictly smaller Bousfield classes than the sphere.

We also have two appendices: In Appendix A, we describe a model category whose associated homotopy category is Stable(Γ), for any commutative Hopf algebra Γ; the results here are due to Hovey [Hov99]. In Appendix B, we discuss the nilpotence of certain classes in Ext_{A}^{∗∗}(F_{p}, F_{p}) when A is the dual of the Steenrod algebra. The results depend on the prime; when p = 2, we get results that we use in Chapter 5 to prove our nilpotence theorems. When p is odd, we have more conjectures than results, and understanding those conjectures would be an important step in proving the nilpotence theorems at odd primes.

In this book we have a mix of results: some are extensions of older results to the cochain complex setting, and some are new. For each older result, if the proof in the literature extends easily to our setting, then we do not include a proof; otherwise, we at least give a sketch. It appears that when one uses the language of stable homotopy theory, one tends to change arguments with spectral sequences into simpler arguments with cofibration sequences (see Lemma 1.2.15, for example), so even though the setting is potentially more complicated, or at least less familiar, some of the proofs simplify. In such cases, we often give in to temptation and include the new proof in its entirety (as, for example, with the vanishing line theorem 2.3.1). Obviously, we include full proofs of all of the new results, and we give references for all of the old results.

Acknowledgments: I have had a number of entertaining and illuminating discussions with a number of people on this material, Mark Hovey perhaps more than most. Others include Haynes Miller, Mike Hopkins, Dan Christensen, Hal Sadofsky, Bill Dwyer, and Paul Shick.

The following chart indicates the interdependencies of the chapters. (This chart includes the primary connections but omits a few small ones; for instance, the proof of Theorem 6.8.3 uses a result from Chapter 4.)

[figure: Chapter 0; Chapter 1; Chapter 2; Chapter 3; Chapter 4; Chapter 5; Chapter 6]

CHAPTER 0 Preliminaries

In this chapter, we give preliminaries and background. Almost all of the material here is standard, but it may be helpful to readers who are more familiar with modules than comodules.

We start in Section 0.1 by giving our grading conventions. In Section 0.2 we define Hopf algebras and mention a few examples. We define modules and comodules in the next section; in this book, we focus primarily on comodules, so we also discuss the relationship between comodules over a coalgebra and modules over the dual algebra. In Section 0.4 we discuss homological algebra in the category of comodules, recalling the definitions of injective objects and of Ext. In the last section of the chapter, we discuss in some detail two examples of small Hopf algebras, comodules over them, and the computation of their Ext groups.

0.1. Grading and other conventions

Fix a field k. We work throughout with Z-graded vector spaces over k; every element from such a vector space is assumed to be homogeneous, unless otherwise indicated. If V is a graded vector space, we write the ith graded piece as V_{i}, so that V = ⨁_{i∈Z} V_{i}. V is of finite type if each V_{i} is finite-dimensional. V is bounded below if V_{i} = 0 for all sufficiently small i. Given a homogeneous element v of a graded vector space, we write |v| for the degree of v. We write s for the “shift” or “suspension” functor on graded vector spaces: for a graded vector space V = ⨁_{i∈Z} V_{i}, the ith graded piece of sV is (sV)_{i} = V_{i−1}. Of course, we may iterate s any number of times to define s^{n} for any n ≥ 0; s is also invertible, so s^{n} makes sense for any integer n. We also use s for the induced suspension functor on the categories of graded modules and graded comodules. (Note that in Section 1.1 we introduce another, doubly-indexed, suspension functor Σ^{i,j}.)

A degree zero map between graded vector spaces sends homogeneous elements to homogeneous elements of the same degree; we write Hom^{0}(V, W) for the collection of such maps. For any integer j, the set of degree j maps is the set

Hom^{j}(V, W) = Hom^{0}(s^{j}V, W).

Hom^{∗}(V, W) denotes the set of all graded maps from V to W, of all degrees.

Throughout, every map between graded vector spaces is assumed to be a graded map of possibly nonzero degree, and we will often omit the degree. In other words, if V and W are graded vector spaces, the notation f: V ⟶ W indicates a graded map, not necessarily of degree zero. If there is an explicit suspension, though, then the map is understood to have the specified degree: f: s^{n}V ⟶ W indicates a map of degree zero from s^{n}V to W, which is the same as a degree n map from V to W.

We will omit the word “graded” from this point on; all vector spaces and maps are understood to be graded.

There are several other important conventions: first, all unadorned tensor products are over the ground field k. Second, unless otherwise specified, “module” means “left module,” and similarly for “comodule.” Third, all algebras are assumed to be associative and unital; similarly, all coalgebras are coassociative and counital.

0.2. Hopf algebras

We start this section with the definition of a Hopf algebra. This is standard; two references are [MM65] and [Swe69]. Note that we insist that our Hopf algebras have antipodes.

Definition 0.2.1. Fix a field k. A Hopf algebra over k is a graded k-vector space Γ = ⨁_{i∈Z} Γ_{i} together with the following structure maps (all of which are graded maps of degree zero):

• a unit or coaugmentation map η: k ⟶ Γ,

• a multiplication map μ: Γ ⊗_{k} Γ ⟶ Γ,

• a counit or augmentation map ε: Γ ⟶ k,

• a comultiplication or diagonal map Δ: Γ ⟶ Γ ⊗_{k} Γ,

• and a conjugation or antipode map χ: Γ ⟶ Γ. The maps η and μ give Γ the structure of an associative unital k-algebra—i.e., the following two diagrams commute):

Γ ⊗ Γ ⊗ Γ ⟶^{1⊗μ} Γ ⊗ Γ
μ⊗1↓  ↓^{μ}
Γ ⊗ Γ ⟶^{μ} Γ,

and

Γ ⊗ k Γ ⊗ Γ  Γ. k ⊗ Γ_{_{1⊗η} u} _{[ [ [ []}_{=}_{^{μ} w} u_{η⊗1} ^{ȷ ȷ ȷȷ fi}_{=}

Dually, the maps ε and Δ make Γ into a coassociative counital k-coalgebra—the dual diagrams to those above (i.e., diagrams as above, but with all the arrows reversed) commute. We give Γ ⊗ Γ an algebra structure via the composite

Γ ⊗ Γ ⊗ Γ ⊗ Γ ⟶^{1⊗T ⊗1} Γ ⊗ Γ ⊗ Γ ⊗ Γ ⟶^{μ⊗μ} Γ ⊗ Γ,

where T: Γ ⊗ Γ ⟶ Γ ⊗ Γ is the twist map: T (a ⊗ b) = (−1)^{|a||b|}b ⊗ a. (Dually, we can give Γ ⊗ Γ the structure of a coalgebra.) We insist that the maps Δ and ε be algebra maps; equivalently, we insist that μ and η be coalgebra maps. The conjugation map χ makes the following diagram commute:

Γ  Γ ⊗ Γ  Γ ⊗ Γ k  Γ.^{Δ} w_{_{ε} u}  ^{1⊗χ} w _{u_{μ}}_{^{η}  w}

Lastly, the same diagram, except with χ ⊗ 1 replacing 1 ⊗ χ, also commutes.

Since the structure maps in this definition are of degree zero, the image of the unit η lies in Γ_{0}, and the kernel of the counit ε contains ⨁_{i≠0} Γ_{i}. We also note that it is well-known—see [Swe69, Proposition 4.0.1], for example—that χ is an “antihomomorphism” with respect to both the algebra and the coalgebra structures: χ ∘ μ = μ ∘ (χ ⊗ χ) ∘ T, and Δ ∘ χ = T ∘ (χ ⊗ χ) ∘ Δ.

Definition 0.2.2. Let Γ be a Hopf algebra over a field k. We say that Γ is commutative if it is commutative as a graded algebra—i.e., the following diagram commutes:

Γ ⊗ Γ Γ ⊗ Γ  Γ. _{[ [ [ []}_{μ}_{_{T} u}  _{^{μ} w}

Γ is cocommutative if the dual diagram commutes; Γ is bicommutative if it is both commutative and cocommutative. We say that Γ = ⨁_{i} Γ_{i} is connected if Γ_{i} = 0 when i < 0 and η: k ⟶ Γ_{0} is an isomorphism.

We work entirely with Hopf algebras that are either commutative (the usual situation) or cocommutative, or both. For such Hopf algebras, the antipode χ is an anti-automorphism of order 2: χ^{2} = 1. See [Swe69, Proposition 4.0.1].

Note that if Γ is a Hopf algebra for which each homogeneous piece Γ_{i} is finitedimensional (i.e., Γ is of finite type), then the graded dual Γ^{∗} of Γ has the structure of a Hopf algebra. (By “graded dual”, we mean that (Γ^{∗})_{i} = Hom_{k}^{i}(Γ, k) = Hom_{k}(Γ_{−i}, k).) In this case, quotient Hopf algebras of Γ correspond to sub-Hopf algebras of Γ^{∗}, Γ is commutative if and only if Γ^{∗} is cocommutative, etc. If some Γ_{i} is not finite-dimensional, then Γ^{∗} will have the structure of an augmented algebra, and the dual of the product on Γ will make Γ^{∗} a “completed” coalgebra rather than an honest coalgebra.

Example 0.2.3. Let k be a field.

(a) The homology of a topological group G with coefficients in k is a cocommutative Hopf algebra; it is connected if and only if G is connected.

(b) For any group G, the group algebra kG is a cocommutative Hopf algebra. It is commutative if and only if G is abelian. It is graded trivially: every element is homogeneous of degree zero; hence it is connected if and only if G is the trivial group. If G is finite, then the vector space dual of kG is a commutative Hopf algebra.

(c) For any Lie algebra L, its universal enveloping algebra U(L) is a cocommutative Hopf algebra. It is commutative if and only if L is abelian. As with kG, it is graded trivially (unless L is graded itself).

(d) Similarly, if k has characteristic p, then for any restricted Lie algebra L defined over k, its restricted universal enveloping algebra V (L) is a cocommutative Hopf algebra.

(e) Fix a prime p. The mod p Steenrod algebra A^{∗} is a cocommutative Hopf algebra over the field F_{p}; its dual A is a commutative Hopf algebra. It is conventional to grade each of these non-negatively, so that they are both connected; however, it is better to grade one positively and one negatively, so that they are graded duals of each other. We will grade A positively and use the induced negative grading on A^{∗}. Starting in Chapter 2, we will focus almost exclusively on this example.

Definition 0.2.4. An element γ of a Hopf algebra Γ is primitive if Δ(γ) = γ ⊗ 1 + 1 ⊗ γ. We write P Γ for the vector space of all primitives of Γ. An element γ is grouplike if Δ(γ) = γ ⊗ γ.

Note that a grouplike element must have degree zero, because the diagonal map Δ is a degree zero map.

0.3. Modules and comodules

We move on to a brief discussion of modules and comodules. Again, [MM65] is one of the standard references; [Boa] is also quite useful.

Definition 0.3.1. Let Γ be a coalgebra over a field k. A k-vector space M is a (left) Γ-comodule if there is a degree zero structure map ψ: M ⟶ Γ ⊗ M, called the coaction map, making the following diagrams commute:

M  Γ ⊗ M  M  Γ ⊗ M k ⊗ M,  Γ ⊗ M  Γ ⊗ Γ ⊗ M.^{ψ} w _{’ ’ ’ ’ ’’)}_{≅}  _{u_{ε⊗1}}  ^{ψ} w_{_{ψ} u}  _{u_{1⊗ψ}}_{^{Δ⊗1} w}

In other words, ψ defines a “coassociative” coaction. A (left) module over a kalgebra is defined dually, of course.

As mentioned in Section 0.1, we use left comodules and left modules throughout; from here on, we will omit the word “left.”

Given a coalgebra map ϕ: Γ ⟶ Λ, any Γ-comodule M inherits a Λ-comodule structure, via the coaction map

M ⟶ Γ ⊗ M ⟶^{ϕ⊗1} Λ ⊗ M.

We call the Λ-comodule structure on M the “corestriction,” the “push-forward,” or (abusing the language a bit) the “restriction” of the Γ-comodule structure.

Given two Γ-comodules M and N over a Hopf algebra Γ, then M ⊗ N is naturally a Γ-comodule, via the structure map

M ⊗ N ⟶^{ψ_{M} ⊗ψ_{N}} Γ ⊗ M ⊗ Γ ⊗ N ⟶^{1⊗T ⊗1} Γ ⊗ Γ ⊗ M ⊗ N ⟶^{μ⊗1⊗1} Γ ⊗ M ⊗ N.

We call this the diagonal coaction, although perhaps a better name would be the “codiagonal” coaction. Note that one can define it using the push-forward construction described in the previous paragraph: M ⊗ N has the structure of a Γ ⊗ Γ-comodule, and so inherits a Γ-comodule structure via the coalgebra map μ: Γ ⊗ Γ ⟶ Γ. We can also put the left coaction on M ⊗ N, in which the structure map is

M ⊗ N ⟶^{ψ_{M} ⊗1} Γ ⊗ M ⊗ N.

Note that this completely ignores the comodule structure on N. We rarely use the left comodule structure on the tensor product; when we do, we denote the comodule

by M ⊗^{L} N. If we want to explicitly distinguish the diagonal coaction from the left

coaction, we write M ⊗^{Δ} N for the tensor product with the diagonal coaction.

We will use the following lemma once or twice. It is fairly standard; see [Boa, Theorem 5.7], or [Mar83, Proposition 12.4] for the module version.

Lemma 0.3.2. Let M be a Γ-comodule. Then Γ ⊗^{Δ} M with the diagonal coaction

is naturally isomorphic, as a Γ-comodule, to Γ ⊗^{L} M with the left coaction. In

particular, Γ ⊗^{Δ} M is isomorphic to a direct sum of copies of Γ.

Proof. One can check that the following two composites are mutually inverse k-vector space maps:

Γ ⊗^{L} M ⟶^{1⊗ψ_{M}} Γ ⊗ Γ ⊗ M ⟶^{1⊗χ⊗1} Γ ⊗ Γ ⊗ M ⟶^{μ⊗1} Γ ⊗ M,

Γ ⊗ M ⟶^{1⊗ψ_{M}} Γ ⊗ Γ ⊗ M ⟶^{μ⊗1} Γ ⊗^{L} M.

Furthermore, the second of these is easily seen to be a Γ-comodule map, and hence an isomorphism; hence the first composite is also a Γ-comodule isomorphism. □

Starting in Chapter 2, we will work with comodules over A, the dual of the mod p Steenrod algebra. Since many readers may be more familiar with the study of modules over A^{∗}, the Steenrod algebra, we include a brief discussion of the relationship between comodules over a Hopf algebra Γ and modules over its dual Γ^{∗}.

Lemma 0.3.3. Let Γ be a coalgebra over k. Let Γ^{∗} denote the graded dual of Γ; this is a k-algebra. Every Γ-comodule has a natural Γ^{∗}-module structure, and if Γ is finite-dimensional, then the categories Γ-Comod and Γ^{∗}-Mod are equivalent.

Proof. Let M be a Γ-comodule. Then we make it a Γ^{∗}-module via the structure map

(0.3.4)  Γ^{∗} ⊗ M ⟶^{1⊗ψ} Γ^{∗} ⊗ Γ ⊗ M ⟶^{ev⊗1} k ⊗ M = M.

If Γ is finite-dimensional, one can easily dualize this: given dual bases (γ_{i}) and (g_{i}) for Γ and Γ^{∗}, then the dual of the evaluation map is

k ⟶^{ev^{∗}} Γ ⊗ Γ^{∗},
1 ⟼ ∑_{i} γ_{i} ⊗ χ(g_{i}).

Composing with the module structure map ϕ makes M into a Γ-comodule:

M ⟶^{ev^{∗} ⊗1} Γ ⊗ Γ^{∗} ⊗ M ⟶^{1⊗ϕ} Γ ⊗ M,
m ⟼ ∑_{j} γ_{j} ⊗ ϕ(χ(g_{j}) ⊗ m).

We leave the details of the proof to the reader; see also [HPS97, Lemma 9.5.3]. □

Regardless of whether Γ is finite-dimensional, the composite (0.3.4) defines a functor

J: Γ-Comod ⟶ Γ^{∗}-Mod.

Now, the functor J defines an equivalence between the category of Γ-comodules and the category of Γ^{∗}-modules satisfying a finiteness condition called “tameness”: a module M is tame, or locally finite, if every element m ∈ M generates a finite submodule of M. The functor J has a right adjoint R, where RM is the largest tame submodule of M—see [HPS97, Lemma 9.5.3]. One can use R to convert operations on modules into similar operations for comodules. For example, if Γ^{∗} is a Hopf algebra, given two Γ^{∗}-modules M and N, then Hom_{k}(M, N) is naturally a Γ^{∗}-module, via the “conjugation” action:

Γ ⊗ Hom_{k}(M, N) ⟶ Hom_{k}(M, N),
γ ⊗ f(−) ⟼ ∑ γ′f(χ(γ′′) · −),

where Δ(γ) = ∑ γ′ ⊗ γ′′. If Γ is a Hopf algebra and M and N are Γ-comodules, or equivalently tame Γ^{∗}-modules, then Hom_{k}(M, N) need not be tame. To fix this, one instead uses R Hom_{k}(M, N) as the internal Hom object in the comodule category. This is well-behaved; for instance, it is right adjoint to the tensor product. See [HPS97, Lemma 9.5.3] for more details.

Remark 0.3.5. The above discussion gives some idea of the relative advantages and disadvantages of the two categories Γ-Comod and Γ^{∗}-Mod. One has more flexibility when working with modules, whereas the finiteness restrictions on comodules may lead to some extra structure. In the module setting, for example, the “duality” functor d: M ⟼ Hom_{k}(M, k) satisfies ddM ≅ M for any M which is of finite type. This is not true for comodules, in general; for example, if Γ = A is the dual of the mod p Steenrod algebra, then dA = R Hom_{F_{p}}(A, F_{p}) is actually zero. On the other hand, “a comodule without generators is zero”: if M is a Γcomodule so that k □_{Γ}M = 0, then M = 0. The corresponding statement fails for modules, as Margolis points out in [Mar83, Proposition 13.9]: when Γ^{∗} = A^{∗}, then F_{p} ⊗_{A^{∗}} A = 0.

0.4. Homological algebra

Now we discuss a little homological “coalgebra.” [Boa] is a good reference for this material, and some of it may also be found in [HPS97, Section 9.5]. One can also dualize discussions of homological algebra for modules, as found in any number of places (such as [CE56, Wei94, Ben91a]).

Since we are working with comodules rather than modules, we work with the notions of cofree and injective comodules, which are dual to the notions of free and projective, respectively.

Definition 0.4.1. Let Γ be a k-coalgebra. A Γ-comodule M is injective if the functor Hom_{Γ}(−, M) is exact. A comodule M is projective if Hom_{Γ}(M, −) is exact. The forgetful functor U: Γ-Comod ⟶ k-Mod has a right adjoint, C:

Hom_{k}(UM, V) ≅ Hom_{Γ}(M, CV).

CV is called the cofree comodule on V; see Lemma 0.4.3 for a formula for C. See [Boa] or [HPS97, Lemma 9.5.4] for the following results.

Lemma 0.4.2. A comodule is injective if and only if it is a summand of a cofree comodule.

Lemma 0.4.3. For any vector space V, CV ≅ Γ ⊗ V as k-vector spaces, with Γ-coaction Δ ⊗ 1. In particular (with V = k), Γ is injective as a Γ-comodule.

As a consequence, every comodule map M ⟶ Γ is adjoint to a vector space map M ⟶ k.

Injective comodules are more important to us than projective comodules, because of the following.

Example 0.4.4. (a) On one hand, if Γ is a finite-dimensional commutative Hopf algebra, then injective comodules are the same as projective comodules: by a result of Larson and Sweedler [LS69, p. 85], finitedimensional cocommutative Hopf algebras are self-injective as modules, so in the module setting, projective and injective are the same. Applying Lemma 0.3.3 yields the result for comodules over a commutative Hopf algebra.

(b) On the other hand, if Γ = A is the dual of the mod p Steenrod algebra, then it seems likely that there are no nonzero projective comodules. As in the previous section, we can view any A-comodule as a module over the Steenrod algebra A^{∗}, and in the category of A^{∗}-modules, A is not projective—see [Mar83, Proposition 13.8]. This provides evidence that it is not projective as an A-comodule. Furthermore, we note that no finite-dimensional comodule has a projective comodule mapping onto it. To see this, we first observe that every finite-dimensional comodule maps surjectively to the trivial comodule F_{p}, so it suffices to show that F_{p} has no projective comodule mapping onto it. For each integer n ≥ 1, let M_{n} be the n-dimensional comodule which is spanned as a vector space by classes

{y_{0}, y_{1}, …, y_{n−1}},

where the degree of y_{i} is −|ξ_{1}|(1 + p + ⋯ + p^{i−1}). The comodule structure is given by

y_{i} ⟼ 1 ⊗ y_{i} + ∑_{j=i+1}^{n−1} ξ_{1}^{p^{i}+···+p^{j−1}} ⊗ y_{j}.

In other words, y_{i} ⟼ 1⊗y_{i} +ξ_{1}^{p^{i}} ⊗y_{i+1} +(other terms), where the “other terms” are of the form a_{j} ⊗ y_{j} where a_{j} is a non-primitive in A. For each n, there is a surjection M_{n} ↠ F_{p} which sends y_{0} to 1 and every other y_{i} to 0. Given any projective comodule P and any map f: P ⟶ F_{p}, suppose there were a class x ∈ P not in the kernel of f. For each n, consider the diagram

P M_{n}  F_{p}  0. ff ff ff ff fl _{u_{f}} w  w

Since P is projective, there must be a lift P ⟶ M_{n}. If x ∈ P maps nontrivially to F_{p}, then under the lift P ⟶ M_{n}, x must map to a nonzero scalar multiple of y_{0}. Since y_{0} has n terms in its diagonal, then x must have at least n terms in its diagonal; since n is arbitrary, then x must have infinitely many terms in its diagonal. This can’t happen, so every x ∈ P must be in the kernel of f: P ⟶ F_{p}. In other words, if P is a projective comodule, then zero is the only comodule map P ⟶ F_{p}.

For any Γ-comodule M, Lemma 0.3.2 says that Γ⊗M is isomorphic to Γ⊗UM = CUM, and hence is injective. The map

M = k ⊗ M ⟶^{η⊗1} Γ ⊗ M

which is adjoint to the identity on UM gives us the start of an injective resolution of M.

Definition 0.4.5. Given a comodule M, a sequence of injective comodules I_{•} = (I_{0} ⟶ I_{1} ⟶ I_{2} ⟶ ⋯) together with a map M ⟶ I_{0} is an injective resolution of M if the sequence

0 ⟶ M ⟶ I_{0} ⟶ I_{1} ⟶ I_{2} ⟶ ⋯

is exact.

It is well-known that injective resolutions are unique up to cochain homotopy equivalence. We are interested in these resolutions because they are used to compute derived functors, such as Ext.

Definition 0.4.6. Let Γ be a coalgebra over k, and let M and N be Γcomodules. Then Ext_{Γ}^{s}(M, N) is the sth derived functor of Hom_{Γ}(M, N). To compute it, one takes an injective resolution I_{•} of N, and defines Ext_{Γ}^{s}(M, N) to be the sth cohomology group of the cochain complex Hom_{Γ}(M, I_{•}).

Throughout this book, when we say Ext, we mean Ext in this setting: the category of comodules over a coalgebra.

Note that since Hom is graded, then Ext^{s} is graded for each s; in other words, Ext is bigraded. We write Ext^{s,t} for the degree t part of Ext^{s}, and we write Ext^{∗∗} for ⨁_{s,t} Ext^{s,t}.

The following two lemmas are standard, and easy, computations. We leave the details to the reader.

Lemma 0.4.7. Let Γ be a coalgebra over a field k. For any comodules M and N, the graded vector space Ext_{Γ}^{0}(M, N) is isomorphic to Hom_{Γ}^{∗}(M, N).

Lemma 0.4.8. Let Γ be a coalgebra over a field k. Then Ext_{Γ}^{1}(k, k) is isomorphic to P Γ, the space of primitives of Γ.

Remark 0.4.9. We have the following remarks, with which the reader is probably familiar.

(a) We point out that for any comodule, there is a canonical injective resolution, known as the cobar complex. We will do a few computations with it in Appendix B.3; we refer the reader to [Ada56] and [HMS74] for details.

(b) Ext_{Γ}^{∗∗}(M, N) has important naturality properties: of course, it is functorial in M and N, contravariant in M and covariant in N. One can use the cobar complex to show that it is also natural in the coalgebra Γ, covariantly.

(c) Finally, we note that Ext_{Γ}^{∗∗}(k, k) is a bigraded k-algebra, commutative (in the graded sense) if Γ is a commutative Hopf algebra. The algebra structure is given by the “Yoneda product”; see [Ben91a, Section 2.6] for a description. One can use naturality, together with a Künneth isomorphism, to explain the commutativity: the product map μ: Γ ⊗ Γ ⟶ Γ is a coalgebra map, so induces a map

μ_{∗}: Ext_{Γ⊗Γ}^{∗∗}(k, k) ⟶ Ext_{Γ}^{∗∗}(k, k).

Since we are tensoring over a field, then there is an isomorphism

Ext_{Γ⊗Γ}^{∗∗}(k, k) ≅ Ext_{Γ}^{∗∗}(k, k) ⊗ Ext_{Γ}^{∗∗}(k, k).

One can check that the composite map μ_{∗}: Ext_{Γ}^{∗∗}(k, k) ⊗ Ext_{Γ}^{∗∗}(k, k) ⟶ Ext_{Γ}^{∗∗}(k, k) agrees with the Yoneda product. Since the product μ is commutative, then so is μ_{∗}.

We use the following result several times, so we state it here for the readers’ convenience. This first appeared in [HS98], and is a generalization of results in [Wil81].

Theorem 0.4.10 (Theorem 4.13 in [HS98]). Suppose that Γ ↠ Λ is a surjection of finite-dimensional graded connected commutative Hopf algebras over a field k of characteristic p > 0. For any λ ∈ Ext_{Λ}^{∗∗}(k, k), there is a number m so that λ^{p^{m}} is in the image of the restriction map Ext_{Γ}^{∗∗}(k, k) ⟶ Ext_{Λ}^{∗∗}(k, k).

In [HS98], this was stated for Ext of modules, not comodules; we have dualized it.

0.5. Two small examples

In this section, we discuss two small coalgebras, their comodules, and homological algebra over them. One can understand these coalgebras pretty thoroughly, and it is common practice to try to reduce the study of more complicated Hopf algebras and coalgebras to these simpler ones.

We work over a field k of characteristic p > 0.

Notation 0.5.1. Given a homogeneous element x in a graded vector space, recall that |x| denotes its degree. Let E[x] denote the coalgebra with basis {1, x} with x primitive (Definition 0.2.4), where |x| is odd if p is odd. Let D[x] be the coalgebra with basis {x_{0} = 1, x_{1} = x, x_{2}, …, x_{p−1}}, where |x| is even if p is odd, where |x_{n}| = n|x|, and with coproduct given by this formula:

(0.5.2)  Δ(x_{n}) = ∑_{i=0}^{n} \binom{n}{i}x_{i} ⊗ x_{n−i}.

These coalgebras usually arise in this book as Hopf algebras with algebra structure given by E[x] ≅ k[x]/(x^{2}) and D[x] ≅ k[x]/(x^{p}). Another algebra structure arises by viewing D[x], for example, as being dual to the mod p group algebra of a cyclic group of order p. When we refer to “the Hopf algebra E[x],” we mean any Hopf algebra which is isomorphic as a coalgebra to E[x]; the same goes for D[x].

Note that we can choose a different basis {y_{0} = 1, y_{1} = x, y_{2}, …, y_{p−1}} for D[x] so that the coproduct formula is given by

(0.5.3)  Δ(y_{n}) = ∑_{i=0}^{n} y_{i} ⊗ y_{n−i}.

(The dual situation is perhaps more familiar: given a k-algebra with basis {z_{0} = 1, z_{1}, …, z_{p−1}\} and product z_{i}z_{j} = \binom{i+j}{i}z_{i+j} if i + j < p, then this is isomorphic to k[z]/(z^{p})—just let z = z_{1}.)

Now we classify the finite-type comodules over these coalgebras.

Proposition 0.5.4. Suppose that k is a finite field and m ≥ 1 is an integer. Let R be the k-algebra k[x]/(x^{m}), graded with |x| > 0.

(a) Every finite-type R-module may be written uniquely as a direct sum of indecomposable modules.

(b) Up to suspension and isomorphism, the modules k[x]/(x^{n}), with 1 ≤ n ≤ m, are the only nonzero indecomposable R-modules.

Proof. Part (a) is a special case of [Mar83, Theorem 11.21]; note that this result is for algebras defined over a finite field.

Part (b): we classify all indecomposable k[x]/(x^{m})-modules by induction on m. First we note that for any m, the algebra R is a Poincaré algebra (see [Mar83, p. 188]); hence every free R-module is both projective and injective.

When m = 1, then R = k; hence the result holds. Suppose that the result holds for modules over k[x]/(x^{m−1}), and let M be a nonzero indecomposable k[x]/(x^{m})- module. If there is an element y ∈ M so that x^{m−1}y is nonzero, then the submodule generated by y is free, hence injective; hence it splits off of M as a summand. Since M is indecomposable, this submodule must be all of M; in other words, M is isomorphic to k[x]/(x^{m}).

The other possibility is that x^{m−1}y = 0 for all y ∈ M. In this case, M is an indecomposable module over k[x]/(x^{m−1}). By induction, then, M is isomorphic to k[x]/(x^{n}) for some n with 1 ≤ n ≤ m − 1. □

The coproduct (0.5.3) in D[x] defines a comodule structure on the vector space Span(y_{0}, y_{1}, …, y_{n−1}) for each n ≤ p. We write M_{n} for this n-dimensional comodule.

Corollary 0.5.5. (a) Suppose that k is a finite field. Then every finitetype E[x]-comodule may be written uniquely as a direct sum of trivial comodules and cofree comodules.

(b) Suppose that k is a finite field. Then every finite-type D[x]-comodule may be written uniquely as a direct sum of suspensions of the comodules M_{n}, 1 ≤ n ≤ p.

Proof. By Lemma 0.3.3, the category of comodules over the coalgebra E[x] (or D[x]) is equivalent to the category of modules over the dual algebra k[x]/(x^{2}) (or k[x]/(x^{p})), so Proposition 0.5.4 applies to the coalgebras of interest. □

Next we work out some homological algebra over E[x] and D[x]. Both the methods and results will be useful later.

As in Section 0.1, we write s for the suspension functor on graded comodules. We note that there is a short exact sequence of E[x]-comodules

(0.5.6)  0 ⟶ k ⟶ E[x] ⟶ s^{|x|}k ⟶ 0.

Repeatedly splicing this together with itself (and removing the k from the start) yields this sequence, which is an injective resolution of k:

(0.5.7)  0 ⟶ E[x] ⟶ s^{|x|}E[x] ⟶ s^{2|x|}E[x] ⟶ ⋯ ⟶ s^{n|x|}E[x] ⟶ · · · .

This sequence is periodic: each map s^{n|x|}E[x] ⟶ s^{(n+1)|x|}E[x] sends x to 1 and sends 1 to 0. Note that one can extend this sequence to the left, as well, yielding an unbounded sequence which is exact:

⋯ ⟶ s^{(n−1)|x|}E[x] ⟶ s^{n|x|}E[x] ⟶ s^{(n+1)|x|}E[x] ⟶ · · · .

Now we work with D[x]. As above, we let M_{n} = Span(y_{0}, y_{1}, …, y_{n−1}), with comodule structure given by formula (0.5.3). For each n < p, there are two relevant short exact sequences of D[x]-comodules:

\begin{matrix}0 ⟶ M_{n} ⟶ D[x] ⟶ s^{n|x|}M_{p−n} ⟶ 0, \\ 0 ⟶ M_{p−n} ⟶ D[x] ⟶ s^{(p−n)|x|}M_{n} ⟶ 0. \end{matrix} ; (0.5.8)

Splicing these together alternately yields this injective resolution of M_{n}:

(0.5.9) 0 ⟶ D[x] ⟶ s^{n|x|}D[x] ⟶ s^{p|x|}D[x] ⟶ s^{(p+n)|x|}D[x] ⟶ s^{2p|x|}D[x] ⟶ · · · .

This sequence is periodic with period two. As with the E[x] sequence, one can extend this to the left to get an unbounded long exact sequence.

One can use these injective resolutions to compute Ext with coefficients in any indecomposable comodule.

Proposition 0.5.10. (a) As algebras, Ext_{E[x]}^{∗∗}(k, k) ≅ k[h], where the el-

ement h ∈ Ext_{E[x]}^{1,|x|}(k, k) corresponds to the short exact sequence (0.5.6).

(b) As algebras, Ext_{D[x]}^{∗∗}(k, k) ≅ Λ[h] ⊗ k[b], where h ∈ Ext_{D[x]}^{1,|x|}(k, k) corresponds to the extension

0 ⟶ k ⟶ M_{2} ⟶ s^{|x|}k ⟶ 0,

and b ∈ Ext_{D[x]}^{2,p|x|}(k, k) corresponds to the composite of the two extensions given in (0.5.8):

0 ⟶ k ⟶ D[x] ⟶ s^{|x|}D[x] ⟶ s^{p|x|}k ⟶ 0.

(c) For any n with 1 ≤ n ≤ p − 1 and for any i ≥ 0,

Ext_{D[x]}^{i}(k, M_{n}) ≅ \begin{cases}s^{jp|x|}k & if i = 2j, \\ s^{(jp+n)|x|}k & if i = 2j + 1. \end{cases}

This is a bigraded module over Ext_{D[x]}^{∗∗}(k, k). Multiplication by b gives an isomorphism between Ext_{D[x]}^{i}(k, M_{n}) and s^{−p|x|} Ext_{D[x]}^{i+2}(k, M_{n}). The algebra and module structures are not immediate from the resolutions, but

they are not hard to compute, and are also standard.

CHAPTER 1 Stable homotopy over a Hopf algebra

In this chapter we discuss stable homotopy theory over a graded commutative Hopf algebra Γ over a field k; a major focus of study is Ext_{Γ}^{∗∗}(k, k), where Ext denotes comodule Ext—derived functors of Hom in the category of left Γ-comodules. This material applies when Γ is the dual of a group algebra, the dual of an enveloping algebra, or the dual of the Steenrod algebra; in these cases, Ext_{Γ}^{∗∗}(k, k) is the ordinary cohomology of the dual Γ^{∗} with coefficients in k.

Our goals in this chapter are to establish some notation, make some basic definitions, and prove some general facts about the category Stable(Γ) of cochain complexes of injective Γ-comodules.

In more detail: we start in Section 1.1 by defining the setting for the rest of the book, the category Stable(Γ). We also set up some important notation; for instance, we explain the grading conventions on morphisms and (co)homology functors in Stable(Γ)—if X is an injective resolution of a left comodule M, then the (s, t)-homotopy group π_{s,t}X is equal to Ext_{Γ}^{s,t}(k, M). In Section 1.2 we construct some particular ring objects in our category, one object HB for every quotient Hopf algebra B of Γ. To be precise, HB is an injective resolution of Γ □_{B}k; this is a ring spectrum in Stable(Γ). So for instance, if we were working with Γ = kG^{∗}, we would have one such object H(kB^{∗}) for every subgroup B of G, and the object H(kB^{∗}) would have homotopy groups π_{∗}H(kB^{∗}) = H^{∗}(B; k). In Subsection 1.2.1 we establish some notation for Hopf algebra extensions, and we prove one or two useful results about extensions with small kernel. For example, given an extension of Hopf algebras of the form

E[x] ⟶ B ⟶ C,

the associated extension spectral sequence has only one possible differential; if B is a quotient Hopf algebra of Γ, then this manifests itself in the category Stable(Γ) as a cofibration sequence HB ⟶ HB ⟶ HC. In Section 1.3 we set up cellular towers and Postnikov towers in Stable(Γ), and we prove a Hurewicz theorem and a few useful lemmas. In Section 1.4 we discuss generalized Adams spectral sequences in Stable(Γ), focusing on the spectral sequence based on the homology theory associated to the ring spectrum HB, for B a “conormal” quotient of Γ. This turns out to be the same, up to a regrading, as the Lyndon-Hochschild-Serre spectral sequence associated to the Hopf algebra extension

Γ □_{B}k ⟶ Γ ⟶ B.

In Section 1.5 we define Bousfield classes and Brown-Comenetz duality, and we recall some results of Ravenel’s relating the two.

In Section 1.6 we apply some of this work to the study of stable homotopy over a group algebra. We point out, for example, that a corollary of work of Benson, Carlson, and Rickard is a classification of the Bousfield lattice in Stable(kG^{∗}), for G a p-group and k a field of characteristic p.

1.1. The category Stable(Γ)

In this section, we define the category in which we work, and we introduce notation which we will use for the rest of this book.

Let Γ be a graded commutative Hopf algebra over a field k; we work in the category Stable(Γ). The objects of this category are unbounded Z-graded cochain complexes of injective left Γ-comodules; we call these objects spectra. The morphisms are the cochain homotopy classes of bigraded maps; to be more precise, we first describe the bigrading on the category. One can shift each object in both the “cohomological direction” (changing the cochain complex grading) and the “internal direction” (since the cochain complexes consist of graded comodules). More precisely, given an object

X = (⋯ ⟶ I_{−1} ⟶ I_{0} ⟶ I_{1} ⟶ ⋯),

we let Σ^{i,j}X be the cochain complex which is s^{j}I_{n−i} in homological degree n, with the apparent differential; here s denotes the internal suspension functor on the category Γ-Comod, as described in Section 0.1. Σ^{i,j} is called the (i, j)-suspension functor.

For objects X and Y of Stable(Γ), we let [X, Y]_{0,0} denote the cochain homotopy classes of graded maps of bidegree (0, 0) from X to Y, and we let [X, Y]_{i,j} = [Σ^{i,j}X, Y]_{0,0}. The set of morphisms in Stable(Γ) from X to Y is the set

[X, Y]_{∗∗} = ⨁_{i,j}[X, Y]_{i,j}

of cochain homotopy classes of all graded maps. If we want to indicate the Hopf algebra Γ, we write the morphisms as [X, Y]_{∗∗}^{Γ}. As with maps of graded vector spaces, we often omit the degree of any given map, so the notation X ⟶ Y indicates an element in [X, Y]_{∗∗}, not necessarily an element in [X, Y]_{0,0}. We have chosen this grading so that it matches up well with Ext: see Remark 1.1.2.

The category Stable(Γ) is a stable homotopy category in the sense of [HPS97]; hence one can perform many standard stable homotopy theoretic constructions in it. (See [HPS97, Theorem 9.5.1] for a verification that Stable(Γ) is a stable homotopy category.) For instance, rather than having exact sequences, one has “exact triangles,” also known as “cofibrations” or “cofiber sequences.” We freely use other language and results from [HPS97], often without explicit citation. Stable(Γ) is (weakly) generated by the injective resolutions of the simple comodules—that is, if X is an object so that [J, X]_{∗∗} = 0 whenever J is an injective resolution of a simple comodule, then X is a contractible cochain complex. If the trivial comodule k and its suspensions are the only simples (say, if Γ is connected, or if Γ is the dual of the mod p group algebra of a p-group), then we say that Stable(Γ) is monogenic, at least in the graded sense. In this case, the stable homotopy constructions are even more familiar.

Let 𝒮 denote the set of simple Γ-comodules. If the set

⋃_{T,T′∈𝒮} Ext_{Γ}^{∗∗}(T, T′)

is countable, then Stable(Γ) is a Brown category, so that homology functors are representable. This is the case when Γ = A, the dual of the Steenrod algebra.

Remark 1.1.1. (a) As in any stable homotopy category, honest limits and colimits usually do not exist in Stable(Γ); rather, one has to work with homotopy limits and colimits. Hence the notation lim and colim stand for the homotopy versions. See [HPS97, Section 2.2] for information about limits and colimits in stable homotopy categories.

(b) Hovey [Hov99] has constructed a closed model category so that Stable(Γ) is equivalent to the associated homotopy category. A summary of the relevant results is given in Appendix A below.

(c) In the case when Γ = A is the dual of the mod p Steenrod algebra, Mahowald and Sadofsky studied the category Stable(Γ) in their paper [MS95].

(d) If Γ is finite-dimensional, then one could just as well work with the category of cochain complexes of injective Γ^{∗}-modules, because in the finitedimensional case, the categories of Γ-comodules and Γ^{∗}-modules are equivalent, with injective comodules corresponding to injective modules. When Γ^{∗} is not finite-dimensional, in particular when Γ^{∗} = A^{∗} is the Steenrod algebra, there are technical problems with the category of cochain complexes of injective A^{∗}-modules. For example, there are no maps from F_{p} to A^{∗}, so the “homotopy” of the injective module A^{∗} would be zero; therefore in the module setting, we would not have the implication π_{∗∗}X = 0 ⇒ X = 0. See the discussion at the end of Section 0.3 for more information.

(e) We also note that, regardless of the dimension of Γ, the category Stable(Γ) is rather different from the derived category of Γ-comodules, because homology isomorphisms are not necessarily invertible in Stable(Γ). For instance, if Γ = E[x] with x primitive, then the periodic cochain complex

⋯ ⟶ Γ ⟶ Γ ⟶ Γ ⟶ ⋯,

in which each map sends x to 1 and 1 to 0, has no homology, and hence is zero in the derived category. On the other hand, it is non-contractible in Stable(Γ); if we write Ext_{Γ}^{∗∗}(k, k) = k[v], then this complex is a ring spectrum with homotopy groups (as defined below) equal to k[v, v^{−1}].

The category Stable(Γ) has arbitrary coproducts; we use the symbol ∨ to denote the coproduct. We sometimes use the word wedge as a synonym for coproduct.

For objects X and Y of Stable(Γ), we write X ∧ Y for X ⊗_{k} Y, and we call this the smash product of X and Y. This operation is commutative, associative, and unital: if S is an injective resolution of the trivial comodule k, then S is the unit of the smash product. We call S the sphere spectrum. We write S^{i,j} for Σ^{i,j}S, so that Σ^{i,j}X = S^{i,j} ∧ X for any X.

Remark 1.1.2. The grading is the usual Ext grading: if X and Y are injective resolutions of comodules M and N, respectively, then [X, Y]_{i,j} = Ext_{Γ}^{i,j}(M, N). Hence if Γ is concentrated in degree 0 (e.g., if Γ = (kG)^{∗}), then one may as well work with Γ-comodules concentrated in degree 0, in which case [−, −]_{ij} = 0 if j ≠ 0, and [−, −]_{i0} = Ext_{Γ}^{i}(−, −). We will follow the (somewhat odd) tradition in homotopy theory of drawing pictures of Ext (and hence of [−, −]) using the Adams spectral sequence grading: Ext^{s,t} is drawn with s on the vertical axis and t − s on the horizontal axis.

Unfortunately, because of the form of long exact sequences in Ext, cofiber sequences look like this:

⋯ ⟶ Σ^{1,0}Z ⟶ X ⟶ Y ⟶ Z ⟶ Σ^{−1,0}X ⟶ · · · .

So one needs to take a little care when translating proofs from ordinary homotopy theory to this setting. Given a spectrum X and integers i and j, we define the homology functor associated to X, X_{ij}, by

X_{ij}: Stable(Γ) ⟶ Ab,
Y ⟼ [S, X ∧ Y]_{i,j},

and we define the cohomology functor associated to X, X^{ij}, by

X^{ij}: Stable(Γ)^{op} ⟶ Ab,
Y ⟼ [Y, X]_{−i,−j}.

When X = S, we have a special notation for X_{ij}: we define the (i, j)-homotopy group of Y to be π_{ij}Y = S_{ij}Y = [S, Y]_{ij}. As with morphisms, we write π_{∗∗}(−) for ⨁_{i,j} π_{ij}(−) (and similarly for other homology and cohomology functors). Also, given a spectrum X, we write X_{ij} for π_{ij}X = X_{ij}S, and X_{∗∗} for ⨁_{i,j} X_{ij}. Some people might refer to X_{∗∗} = π_{∗∗}X as the “hypercohomology” of the cochain complex X, but we will try to restrain ourselves. Note that if X ⟶ Y ⟶ Z is a cofiber sequence, then we have a long exact sequence

⋯ ⟶ π_{i−1,j}Z ⟶ π_{i,j}X ⟶ π_{i,j}Y ⟶ π_{i,j}Z ⟶ π_{i+1,j}X ⟶ ⋯

(and similarly for other homology functors).

We say that an object R in Stable(Γ) is a ring spectrum if there is a multiplication map μ: R ∧ R ⟶ R and a unit map η: S ⟶ R, making the appropriate diagrams commute.

We often abuse notation and let S^{0} = S^{0,0} = S. One of our main goals is to get as much information as possible about π_{∗∗}S^{0} = Ext_{Γ}^{∗∗}(k, k).

1.2. The functor H

We assume that Γ is a graded commutative Hopf algebra over a field k, and we work in the category Stable(Γ). The quotient coalgebras and Hopf algebras of Γ carry useful information; in this section we construct a spectrum HB in Stable(Γ) for each quotient coalgebra B of Γ, and we study the properties of the functor H.

Recall that if B is a quotient coalgebra of Γ and if M is a B-comodule, then the cotensor product Γ □_{B}M is defined to be the equalizer of the two maps

Γ ⊗ M ⟶^{1_{Γ}⊗ψ_{M}} Γ ⊗ B ⊗ M,

Γ ⊗ M ⟶^{ψ_{Γ}⊗1_{M}} Γ ⊗ B ⊗ M.

Here ψ_{Γ} is the right B-comodule structure map on Γ, and ψ_{M} is the left B-comodule structure map on M. (The tensor products are over k, as usual.)

Alternatively, taking the cotensor product with Γ is right adjoint to restriction; see Lemma 1.2.3 below.

Definition 1.2.1. We define a covariant functor H from quotient coalgebras of Γ to spectra by defining HB to be an injective resolution of Γ □_{B}k.

1.2. THE FUNCTOR H 17

For example, Hk = Γ, so that Hk_{∗∗}(X) is the homology of the cochain complex X; also, HΓ = S^{0}. H provides a useful source of (co)homology functors on Stable(Γ). The general philosophy is that if one has a quotient B of Γ, rather than studying B by working in Stable(B), one studies B by looking at HB in the category Stable(Γ). This is borne out by Corollary 1.2.7, as well as the other results in this section.

In order to study the objects HB, we need to examine the cotensor product. We start with its adjointness property.

Given a quotient coalgebra B of Γ, we write [−, −]^{B} for the set of cochain homotopy classes of B-comodule maps. We abuse notation and let Stable(B) denote the category with objects cochain complexes of injective B-comodules, and morphisms [−, −]^{B}. (This is an abuse of notation because the category Stable(B) need not be a stable homotopy category—it won’t have a smash product unless B is a quotient Hopf algebra.)

Definition 1.2.2. Fix a quotient coalgebra B of Γ. Given a Γ-comodule M, we let M↓_{B} denote its restriction to B; this is the B-comodule with structure map M ⟶ Γ ⊗ M ⟶ B ⊗ M. We occasionally write res_{Γ,B} for restriction. We use the same notations for related restriction functors, such as the one from Stable(Γ) to Stable(B).

Recall from [MM65] and [Rad77] that if B happens to be a quotient Hopf algebra of Γ, then Γ↓_{B} is injective as a right (and a left) B-comodule. When Γ↓_{B} is injective as a right B-comodule, the functor Γ □_{B}−: B-Comod ⟶ Γ-Comod is exact and takes injectives to injectives; hence it induces a functor Γ □_{B}−: Stable(B) ⟶ Stable(Γ), defined by applying Γ □_{B}− dimensionwise. Regardless of the context, we call Γ □_{B}− the induction functor, and we occasionally write ind_{B,Γ} instead of Γ □_{B}−.

Lemma 1.2.3. The functors −↓_{B} and Γ □_{B}− are adjoint:

(a) Let B be a quotient coalgebra of Γ. Given a Γ-comodule M and a Bcomodule N, there is an isomorphism

Hom_{B}(M↓_{B}, N) ≅ Hom_{Γ}(M, Γ □_{B}N),

natural in M and N.

(b) Let B be a quotient coalgebra of Γ over which Γ↓_{B} is injective as a right B-comodule. Then given objects X ∈ Stable(Γ) and Y ∈ Stable(B), the isomorphism of part (a) induces a natural isomorphism

[X↓_{B}, Y]^{B} ≅ [X, Γ □_{B}Y]^{Γ}.

Part (a) is dual to the classical statement for modules that induction M ⟼ Γ ⊗_{B} M is left adjoint to restriction; see [Ben91a, Proposition 2.8.3], for instance.

Proof. Part (a): Define

α: Hom_{B}(M↓_{B}, N) ⟶ Hom_{Γ}(M, Γ □_{B}N)

by α(f) = (1 ⊗ f) ∘ ψ_{M}, where ψ_{M} is the Γ-coaction map on M. One can verify that α(f) is a Γ-comodule map for any f ∈ Hom_{B}(M↓_{B}, N). Define

β: Hom_{Γ}(M, Γ □_{B}N) ⟶ Hom_{B}(M↓_{B}, N)

by β(g) = (ε ⊗ 1) ∘ g; one can verify that β(g) is a map of B-comodules. One can also check that α and β are inverses. We leave the details to the reader; the verifications are dual to the corresponding verifications for modules.

Part (b): One needs to check two things: that the maps α and β of part (a) induce well-defined maps on sets of cochain maps, and that they induce well-defined maps on cochain homotopy classes of maps. These are both straightforward verifications, which we leave to the reader. □

Example 1.2.4. If Γ↓_{B} is injective over B, M is a Γ-comodule and N is a B-comodule, then we have

Ext_{B}^{∗∗}(M↓_{B}, N) ≅ Ext_{Γ}^{∗∗}(M, Γ □_{B}N).

This result is often called a “change-of-rings isomorphism” or “Shapiro’s lemma”— see [Rav86, Theorem A1.3.12] and [Ben91a, Corollary 2.8.4], for instance. It follows from part (b) of the lemma by letting X be an injective resolution of M and Y an injective resolution of N.

Lemma 1.2.5. Suppose that B is a quotient coalgebra of Γ so that Γ ↓_{B} is injective as a right B-comodule. If M is a Γ-comodule, then there is an isomorphism of Γ-comodules

Γ □_{B}(M↓_{B}) ≅ (Γ □_{B}k) ⊗ M.

This result is given for modules in [HS98, p. 23]—it is called the “shearing isomorphism” there.

Proof. Note that Γ □_{B}(M↓_{B}) is a Γ-subcomodule of Γ⊗^{L}M (where ⊗^{L} indicates the tensor product with the left Γ-coaction), and (Γ □_{B}k) ⊗ M is a Γ-subcomodule of Γ ⊗ M (with the usual, diagonal, coaction). One only needs to check that the isomorphism

Γ ⊗^{L} M ≅ Γ ⊗ M

of Lemma 0.3.2 carries these subcomodules to each other. We leave the details to the reader. □

Applied dimensionwise to a cochain complex X in Stable(Γ), we have

Γ □_{B}(X↓_{B}) ≅ (Γ □_{B}k) ⊗ X.

Since HB is an injective resolution of Γ □_{B}k, we have the following.

Corollary 1.2.6. Suppose that B is a quotient coalgebra of Γ so that Γ↓_{B} is injective as a right B-comodule. For any object X ∈ Stable(Γ), there is a cochain homotopy equivalence Γ □_{B}(X↓_{B}) ≃ HB ∧ X.

Combining this with Lemma 1.2.3 yields the following.

Corollary 1.2.7. Suppose that B is a quotient coalgebra of Γ so that Γ↓_{B} is injective as a right B-comodule. Given objects X and Y of Stable(Γ), we have [X, HB ∧ Y]_{∗∗} ≅ [X↓_{B}, Y ↓_{B}]_{∗∗}^{B}. In particular, HB_{∗∗} is an algebra and HB_{∗∗}Y is a right module over it.

In summary, we have the following. Note that the result is stronger for quotient Hopf algebras than for quotient coalgebras.

Proposition 1.2.8. For any quotient coalgebra B of Γ over which Γ ↓_{B} is injective as a right B-comodule, there is an isomorphism HB_{ij} ≅ Ext_{B}^{ij}(k, k). If X is an injective resolution of a Γ-comodule M, then HB_{∗∗}X ≅ Ext_{B}^{∗∗}(k, M↓_{B}). Furthermore, we have the following.

(a) If B is a quotient Hopf algebra of Γ, then HB is a commutative associative ring spectrum.

(b) If B is a quotient coalgebra of Γ over which Γ↓_{B} is injective as a right comodule, then HB has many of the properties of a ring spectrum:

(i) HB_{∗∗} is a k-algebra, and for any spectrum Y, HB_{∗∗}Y is a right module over HB_{∗∗}.

(ii) There is a “unit map” S^{0} ⟶ HB which induces an algebra map

π_{∗∗}S^{0} ⟶ HB_{∗∗}. (iii) More generally, if B ↠ C are quotient coalgebras of Γ over which Γ is injective, then the induced map HB_{∗∗} ⟶ HC_{∗∗} is an algebra map.

Proof. That HB_{∗∗} ≅ Ext_{B}^{∗∗}(k, k) follows from Lemma 1.2.3.

Part (a) is clear—since Γ is commutative and associative as an algebra, then so is Γ □_{B}k, and the commutative associative product on Γ □_{B}k induces one on HB. The quotient map Γ ↠ B induces the unit map S^{0} = HΓ ⟶ HB, and hence a Hurewicz map π_{∗∗}(X) ⟶ HB_{∗∗}(X). This Hurewicz map is the same as the restriction map res_{Γ,B}: Ext_{Γ}^{∗∗}(k, k) ⟶ Ext_{B}^{∗∗}(k, k).

For part (b), part (i) is part of Corollary 1.2.7. The unit map is induced from the quotient Γ ↠ B, as in (a), and the induced map in (iii) is just the restriction map. □

Example 1.2.9. If B is a quotient coalgebra of Γ, not a quotient Hopf algebra, then HB need not be a ring spectrum. For example, suppose that k has characteristic p, and let Γ = k[x] with x primitive. Then B = k[x^{p}] is a quotient coalgebra of Γ, and HB is an injective resolution of M = k[x]/(x^{p}). A multiplication on HB would induce one on M (by taking homology), and it is easy to see that M is not a Γ-comodule algebra: the comodule structure on M is given by x ↦ 1 ⊗ x + x ⊗ 1 ∈ Γ ⊗ M, so if this coaction were multiplicative, we would have

0 = x^{p} ⟼ x^{p} ⊗ 1 ≠ 0.

Hence HB is not a ring spectrum.

The example Hk, which represents the homology of cochain complexes, is particularly important.

Corollary 1.2.10. The spectrum Hk is a field spectrum: it is a commutative ring spectrum, and any Hk-module spectrum is a wedge of suspensions of Hk. In particular, for any X, Hk ∧ X is a wedge of suspensions of Hk.

Proof. By Proposition 1.2.8(a), Hk = Γ is a commutative ring spectrum. By Lemma 0.3.2, for any Γ-comodule M, Γ ⊗ M is a direct sum of copies of Γ; one can check (by [Boa, Theorem 5.7], for instance) that given a comodule map M ⟶ N, the induced map Γ ⊗ M ⟶ Γ ⊗ N sends each summand of Γ either isomorphically to a summand, or to zero. So if X is any cochain complex of Γ-comodules, then Hk ∧ X = Γ ⊗ X splits into a direct sum of cochain complexes of the forms

0 ⟶ Γ ⟶^{=} Γ ⟶ 0

and

0 ⟶ Γ ⟶ 0.
□

Hence we have the following.

Corollary 1.2.11. Hk_{∗∗} satisfies a Künneth isomorphism.

1.2.1. Remarks on Hopf algebra extensions. Many results about group cohomology (and indeed about Hopf algebra cohomology in general) are proved using the spectral sequence associated to an extension. If one has a group extension in which the quotient is cyclic of prime order, the associated spectral sequence is particularly tractable, and hence quite useful. In this subsection we remind the reader of standard notation related to Hopf algebra extensions, and then we focus on extensions with small kernel. We observe that in this case, the spectral sequences degenerate to cofibrations. In Section 1.4 we discuss the spectral sequence associated to a general Hopf algebra extension.

Definition 1.2.12. (a) Suppose that C ↪ Γ is an inclusion of augmented algebras over a field k, and let IC be the augmentation ideal of C—the kernel of C ⟶ k. We say that C is normal in Γ if the left ideal of Γ generated by C (i.e., Γ · IC) is equal to the right ideal generated by C (i.e., IC · Γ). If C is normal in Γ, then we let Γ//C = Γ ⊗_{C} k = k ⊗_{C} Γ. In this case,

C ⟶ Γ ⟶ Γ//C

is an extension of augmented k-algebras.

(b) Dually, suppose that Γ ⟶ B is a surjective map of coaugmented coalgebras over k, and let JB denote the coaugmentation coideal of B—the cokernel of k ⟶ B. We say that B is a conormal quotient of Γ if Γ □_{B}k = k □_{B}Γ as sub-vector spaces of Γ. If B is a conormal quotient of Γ, then

Γ □_{B}k ⟶ Γ ⟶ B

is an extension of coaugmented k-coalgebras. As far as this definition goes, for us Γ will usually be a Hopf algebra with

commutative multiplication, so that every subalgebra of Γ will be normal. We will be more interested in quotients of Γ; if Γ ⟶ B is a surjective map of commutative Hopf algebras over k, then Γ □_{B}k is the algebra kernel. So B is conormal if the algebra kernel is a subcoalgebra of Γ and is closed under conjugation χ.

Remark 1.2.13. Suppose that Γ is a commutative Hopf algebra and B is a conormal quotient coalgebra of Γ over which Γ is injective. For any Γ-comodule M, there is a coaction of Γ □_{B}k on Ext_{B}^{∗∗}(k, M↓_{B}), which may be constructed as follows. Let I_{•} be a Γ-injective resolution of M; then this Ext group is the cohomology of the cochain complex with terms Hom_{B}(k, I_{n}). For each n, Hom_{B}(k, I_{n}) is a Γ □_{B}k-comodule, via the Γ-coaction on I_{n}, and the boundary maps in the cochain complex are Γ □_{B}k-comodule maps. Hence Ext_{B}^{n}(k, M↓_{B}) is a Γ □_{B}k-comodule for each integer n. More generally, the same argument produces a Γ □_{B}k-coaction on HB_{n∗}X for any object X in Stable(Γ), for each integer n.

For the remainder of this section, we assume that the ground field k has characteristic p > 0. We use the results of Section 0.5 here; in particular, see Notation 0.5.1 for the coalgebras E[x] and D[x].

Notation 1.2.14. Recall from Lemma 0.4.8 that if B is a Hopf algebra, then Ext_{B}^{1,∗}(k, k) = HB_{1,∗} is isomorphic to the vector space of primitives of B. If y ∈ B is primitive, we let [y] denote the associated element of HB_{1,∗}. Recall further that if p is odd, then there is a Steenrod operation

β𝒫̃^{0} : Ext_{B}^{s,t}(k, k) ⟶ Ext_{B}^{s+1,pt}(k, k).

See [May70, Wil81], as well as Appendix B.1. Given [y] ∈ HB_{1,∗}, the element β𝒫̃^{0} [y] is also the p-fold Massey product of [y] with itself, as mentioned in [May70, Remarks 11.11].

As mentioned above, when studying group cohomology, one often uses the spectral sequence for a group extension of the form

1 → H → G → Z/p → 1.

In our “dual” setting, we consider Hopf algebra extensions with small kernel. These give us cofibrations which we can use in place of spectral sequences.

Lemma 1.2.15. Fix a graded commutative Hopf algebra Γ over a field k of characteristic p > 0.

(a) Suppose that there is a Hopf algebra extension of the form

E[x] ⟶ B ⟶ C,

where B is a quotient Hopf algebra of Γ. Then x is primitive in B, so there is a nonzero element h = [x] in Ext_{B}^{1,|x|}(k, k) = HB_{1,|x|}. We also let h: Σ^{1,|x|}HB ⟶ HB denote the corresponding self-map of HB—i.e., the composite

Σ^{1,|x|}HB = S^{1,|x|} ∧ HB ⟶^{h∧1} HB ∧ HB ⟶^{μ} HB,

where μ is the multiplication map. Then there is a cofiber sequence

Σ^{1,|x|}HB ⟶^{h} HB ⟶ HC ⟶ Σ^{0,|x|}HB.

(b) Suppose that p is odd and there is a Hopf algebra extension of the form

D[x] ⟶ B ⟶ C,

where B is a quotient Hopf algebra of Γ. Then x is primitive in B; we let b = β𝒫̃^{0} [x] in Ext_{B}^{2,p|x|}(k, k) = HB_{2,p|x|}. We also let b: Σ^{2,p|x|}HB ⟶ HB denote the corresponding self-map of HB. Then there is a cofiber sequence

Σ^{2,p|x|}HB ⟶^{b} HB ⟶ H̃C ⟶ Σ^{1,p|x|}HB,

where H̃C is defined by a cofibration

Σ^{1,|x|}HC ⟶ H̃C ⟶ HC ⟶ Σ^{0,|x|}HC.

Proof. Part (a): It is clear that x is primitive, so we only need to discuss the putative cofibration. Equation (0.5.6) gives an exact sequence of E[x]-comodules:

0 ⟶ k ⟶ E[x] ⟶ s^{|x|}k ⟶ 0.

We apply the (exact) functor Γ □_{B}− to this, noting that E[x] = B □_{C}k:

0 ⟶ Γ □_{B}k ⟶ Γ □_{C}k ⟶ s^{|x|}Γ □_{B}k ⟶ 0.

Taking injective resolutions gives the desired cofibration.

Part (b): We apply the functor Γ □_{B}− to the two short exact sequences of D[x]-comodules in (0.5.8) and take injective resolutions; writing H̃B for an injective resolution of the comodule Γ □_{B}(k[x]/(x^{p−1})), we have the following cofiber sequences:

Σ^{1,|x|}H̃B ⟶^{η} HB ⟶ HC ⟶ Σ^{0,|x|}H̃B,

Σ^{1,(p−1)|x|}HB ⟶^{η′} H̃B ⟶ HC ⟶ Σ^{0,(p−1)|x|}HB.

It is standard (e.g., see [Ben91b, pp. 137–8]) that the map b is the composite η ∘η′, and the 3 × 3 lemma (or the octahedral axiom—see [HPS97, A.1.1–A.1.2]) allows us to identify the cofiber of η ∘ η′ in terms of the cofibers of η and η′. □

1.3. Some classical homotopy theory

We assume that Γ is a graded commutative Hopf algebra over a field k, and we work in the category Stable(Γ). For this section, we assume that Stable(Γ) is monogenic—i.e., the trivial comodule k and its suspensions are the only simple comodules. We also assume that Γ is non-negatively graded: Γ_{n} = 0 if n < 0.

Because π_{ij}S^{0} = Ext_{Γ}^{i,j}(k, k), then π_{∗∗}S^{0} is concentrated in the first quadrant. More precisely, π_{ij}S^{0} = 0 if j < 0 and (unless i = j = 0) if i ≤ 0; furthermore, π_{00}S^{0} = k. Since π_{∗∗}S^{0} is “connected” in this sense, we can construct cellular towers and Postnikov towers, as in the usual stable homotopy category (and indeed in any connective stable homotopy category—see [HPS97, Section 7]). We also have a Hurewicz theorem. Since we are working in a bigraded setting rather than the singly graded setting of [HPS97, Section 7], we state the relevant results. See [Mar83, Chapter 3] for the proofs in the ordinary stable homotopy category; the reader can modify them for this setting fairly easily.

Remark 1.3.1. A priori, since the boundary homomorphism in π_{∗∗} raises degrees rather than lowers them, there could be difficulties in translating the proofs in [Mar83] to the category Stable(Γ). On the other hand, we are “working over a field,” meaning that π_{00}S^{0} is a field k. This means that every homotopy group is a k-module; hence, given any object X and any bidegree (i, j), one can find a map ⋁ S^{i,j} ⟶ X which induces not just an epimorphism on π_{ij}, but an isomorphism. Indeed, because of the connectivity properties of π_{∗∗}S^{0}, for any numbers m ≤ 0 and n, there is a map from a wedge of spheres to X inducing an isomorphism on π_{ij} for all i and j with j = mi + n. This feature compensates for any possible problems with the boundary homomorphism.

Definition 1.3.2. Given a spectrum X and integers k and m, then the diagram

0 = X_{k} ⟶ X_{k+1} ⟶ X_{k+2} ⟶ ⋯

is a strong cellular tower of slope m for X if (a) X is the sequential colimit of the X_{n},

(b) the cofiber of each map X_{n} ⟶ X_{n+1} is a wedge of spheres S^{i,j} with j = mi + n.

In contrast, a cellular tower is a similar diagram in which the cofibers are wedges of spheres with no restrictions on their dimensions. According to [HPS97, Proposition 2.3.1], every object in Stable(Γ) has a cellular tower. Because of the connectivity properties of π_{ij}S^{0}, we have the following theorem, guaranteeing existence of strong cellular towers with and Postnikov towers.

Theorem 1.3.3. Let X be a spectrum and m a non-positive integer.

(a) Suppose that there is an integer k so that π_{ij}X = 0 whenever j ≤ mi + k. Then X has a strong cellular tower of slope m:

0 = X_{k} ⟶ X_{k+1} ⟶ X_{k+2} ⟶ · · · .

(b) For any integer n, there is a cofiber sequence

X[n, ∞]_{m} ⟶ X ⟶ X[−∞, n − 1]_{m},

where (i) π_{ij}(X[n, ∞]_{m}) = 0 if j < mi + n, (ii) π_{ij}(X[−∞, n − 1]_{m}) = 0 if j ≥ mi + n.

(c) If for some integer n we have spectra Y and Z with Y = Y [n, ∞]_{m} and Z = Z[−∞, n − 1]_{m}, then [Y, Z]_{∗∗} = 0.

(d) Hence we can construct a Postnikov tower of slope m:

⋯ ⟶ X[−∞, r]_{m} ⟶ X[−∞, r − 1]_{m} ⟶ X[−∞, r − 2]_{m} ⟶ ⋯
↑  ↑  ↑
X[r]_{m}  X[r − 1]_{m}  X[r − 2]_{m}

Here X[r]_{m} is a generalized Eilenberg-Mac Lane spectrum: a wedge of objects of the form Σ^{i,j}Hk, where j = mi + r. The sequential colimit of X[−∞, r]_{m} is 0, and the sequential limit of X[−∞, r]_{m} is X.

(e) Dually, we can construct a diagram

⋯ ⟶ X[r, ∞]_{m} ⟶ X[r − 1, ∞]_{m} ⟶ X[r − 2, ∞]_{m} ⟶ ⋯
↓  ↓  ↓
X[r]_{m}  X[r − 1]_{m}  X[r − 2]_{m}

The sequential colimit of the “connective covers” X[r, ∞]_{m} is X, and the sequential limit is 0.

Parts (d) and (e) are easy in our context: if k is the only simple comodule, then every injective is a direct sum of copies of Γ = Hk. So to construct X[r, ∞]_{m}, for example, one just truncates the cochain complex X at bidegrees (i, j) with j < mi + r.

Note that if Γ is a connected Hopf algebra (Definition 0.2.2), then π_{ij}S^{0} = 0 if j < i; since the Steenrod algebra is connected, we use this pattern for our definition of connectivity. On the other hand, if Γ is concentrated in degree zero, then π_{ij}S^{0} = 0 if j ≠ 0; this leads to a weaker notion of connectivity.

Definition 1.3.4. Given a spectrum X, if there exist numbers i_{0} and j_{0} so that π_{ij}X = 0 when i < i_{0} or j − i < j_{0}, then we say that X is (i_{0}, j_{0})-connective. We say that X is connective if X is (i_{0}, j_{0})-connective for some unspecified i_{0} and j_{0}. If for some i_{0} and j_{0}, we have π_{ij}X = 0 when i < i_{0} or j < j_{0}, we say that X is weakly (i_{0}, j_{0})-connective.

Here is the second main theorem of this section, a bigraded version of the Hurewicz theorem.

Theorem 1.3.5. If X is (i_{0}, j_{0})-connective, then the Hurewicz map π_{ij}X ⟶ Hk_{ij}X is an isomorphism when i < i_{0} or j − i ≤ j_{0}. Similarly, if X is weakly (i_{0}, j_{0})-connective, then π_{ij}X ⟶ Hk_{ij}X is an isomorphism when i < i_{0} or j ≤ j_{0}.

If Γ is connected and X is an injective resolution of a bounded below comodule M, then X is (0, 0)-connective, but it also satisfies a stronger property. This property occurs several times in this work, so we make it into a definition.

Definition 1.3.6. We say that a spectrum X is comodule-like, or CL, if X satisfies the following conditions:

(a) There exists an integer i_{0} such that π_{i∗}X = 0 if i < i_{0},

(b) There exists an integer j_{0} such that π_{ij}X = 0 if j − i < j_{0},

(c) There exists an integer i_{1} such that (Hk)_{i∗}X = 0 if i > i_{1}.

For example, when X is an injective resolution of a bounded below comodule M, we may take i_{0} = i_{1} = 0, while j_{0} is the degree of the bottom class of M. Note that in general, an object X is a CL-spectrum if and only if X has a cellular tower built of spheres S^{i,j} with i_{0} ≤ i ≤ i_{1} and j − i ≥ j_{0}.

We will need the following lemmas later. First we need to recall a few definitions from [HPS97, Definitions 1.4.3 and 2.1.1].

Definition 1.3.7. (a) A full subcategory 𝒟 of Stable(Γ) is localizing if it is “closed under cofibrations and coproducts”: if X ⟶ Y ⟶ Z is a cofibration and two of X, Y, and Z are in 𝒟, then so is the third; if {X_{α}} is a set of objects in 𝒟, then ⋁_{α} X_{α} is in 𝒟. Given an object Y, we let loc(Y) denote the localizing subcategory generated by Y, i.e., the intersection of all of the localizing subcategories containing Y.

(b) Similarly, a full subcategory 𝒟 of Stable(Γ) is thick if it is closed under cofibrations and retracts (if Y is in 𝒟 and there are maps X ⟶ Y ⟶ X so that the composite is an isomorphism, then X is in 𝒟); and thick(Y) denotes the thick subcategory generated by Y.

(c) A property P of spectra is generic if the full subcategory of spectra satisfying P is thick.

(d) An object X of Stable(Γ) is finite if and only if it is small (in the categorical sense), if and only if it is in thick(S^{0}).

If Γ is connected, then an object X is finite if and only if X is connective and has dim_{k} Hk_{∗∗}X < ∞. So if Γ is connected and B is a quotient Hopf algebra of Γ, then a connective object X of Stable(Γ) is finite if and only if its restriction X↓_{B} is finite in Stable(B).

Lemma 1.3.8. Suppose that X is a spectrum and that there is a line of nonpositive slope above which the homotopy of X is zero—i.e., for some m ≤ 0, there is an n so that if j ≥ mi + n, then π_{ij}X = 0. Then X is in the localizing subcategory generated by Γ = Hk.

Proof. The tower in Theorem 1.3.3(e) displays X as being a colimit of objects of loc(Γ); hence X is itself an object of loc(Γ). □

Lemma 1.3.9. If 𝒟 is a localizing subcategory of Stable(Γ) which contains a nonzero finite spectrum, then loc(Γ) ⊆ 𝒟.

Proof. It suffices to show that Γ ∈ ob 𝒟 if 𝒟 is as given. Let Y be a nonzero finite object of 𝒟; then Y ∧Γ is nonzero (by the Hurewicz theorem 1.3.5—remember that Γ = Hk) and is contained in ob 𝒟. On the other hand, Corollary 1.2.10 tells us that Y ∧Γ is a direct sum of suspensions of Γ, so by the Eilenberg swindle [HPS97, Lemma 1.4.9], Γ ∈ ob 𝒟. □

We will see in Corollary 6.6.8 that when Γ = A is dual of the mod p Steenrod algebra, the containment loc(A) ⊂ 𝒟 is strict (i.e., loc(A) contains no nonzero finite spectrum).

1.4. The Adams spectral sequence

As in the rest of this chapter, we assume that Γ is a graded commutative Hopf algebra over a field k. We discuss generalized Adams spectral sequences here; we focus to some extent on the spectral sequence associated to the homology theory HB_{∗∗}, when B is a conormal quotient Hopf algebra of Γ (see Definitions 1.2.1 and 1.2.12). In this case, we note that the Adams spectral sequence is the same as the spectral sequence associated to a Hopf algebra extension, and we mention a few consequences of this observation.

We start by giving a construction of the Adams spectral sequence, as can be found in [Mil81], [Ada74, Section III.15] or [Rav86, Section 2.2]. Let E be a ring spectrum satisfying the following (cf. [Ada74, pp. 317–318] and [Rav86, Assumptions 2.2.5]).

Condition 1.4.1. (a) E is a commutative associative ring spectrum. Let μ: E ∧ E ⟶ E be the multiplication map and η: S^{0} ⟶ E the unit map.

(b) E is a flat ring spectrum; i.e., E_{∗∗}E is flat as a left module over E_{∗∗}.

(c) E is weakly (0, 0)-connective (Definition 1.3.4), and the unit map η induces an isomorphism on π_{0,0}.

(d) The map μ_{∗}: π_{00}E ⊗ π_{00}E ⟶ π_{00}E induced by the multiplication map μ is an isomorphism.

(e) Hk_{ij}E is a finite-dimensional k-vector space for each i and j.

We let \overline{E} denote the fiber of the unit map S^{0} ⟶ E. For each integer s ≥ 0, we let

F_{s}X = \overline{E}^{∧s} ∧ X,
K_{s}X = E ∧ \overline{E}^{∧s} ∧ X.

The cofibration \overline{E} ⟶ S^{0} ⟶ E leads to the following diagram of cofibrations, which we call the (canonical) E-based Adams tower for X:

X  F_{0}X ⟵ F_{1}X ⟵ F_{2}X ⟵ · · · .
↓  ↓  ↓
K_{0}X  K_{1}X  K_{2}X

This construction satisfies the definition of an “E_{∗}-Adams resolution” for X, as given in [Rav86, Definition 2.2.1]—see [Rav86, Lemma 2.2.9]. Note also that F_{s}X = X ∧ F_{s}S^{0}, and the same holds for K_{s}X—the canonical Adams tower is functorial and exact.

If we apply π_{∗∗} to the Adams tower for X, we get an exact couple and hence a spectral sequence. This is called the E-based Adams spectral sequence.

Since we are assuming that E satisfies Condition 1.4.1, the pair (E_{∗∗}, E_{∗∗}E) is a Hopf algebroid and E_{∗∗}X is an E_{∗∗}E-comodule for any spectrum X—see [Rav86, Proposition 2.2.8]. As usual, we write Ext_{E_{∗∗}E} for comodule Ext; since E_{∗∗}E is a bigraded coalgebra, then this Ext is trigraded.

Theorem 1.4.2. Let E be a spectrum satisfying Condition 1.4.1. Then there is a spectral sequence, the E-based Adams spectral sequence, which has E_{2}-term

E_{2}^{s,t,u} = Ext_{E_{∗∗}E}^{s,t,u}(E_{∗∗}, E_{∗∗}X),

with differentials

d_{r}: E_{r}^{s,t,u} ⟶ E_{r}^{s+r,t+r−1,u−r+1}.

The spectral sequence abuts to π_{s+u,t+u}X.

This theorem does not address convergence. Convergence is related to the notion of “E-completeness”: a spectrum X is E-complete if the inverse limit of its Adams tower is contractible. If X is connective and E-complete, then the E-based Adams spectral sequence abutting to π_{∗∗}X will actually converge.

Proposition 1.4.3. Let E be a ring spectrum satisfying Condition 1.4.1. Then every weakly connective spectrum X is E-complete.

Proof. This is an easy connectivity result: our conditions on E ensure that \overline{E} is weakly (1, 0)-connective; hence if X is weakly (i, j)-connective, then by the Hurewicz theorem 1.3.5, F_{s}X is (i + s, j)-connective. □

Next we discuss the genericity of vanishing planes in Adams spectral sequences. Here is the relevant result; we do not use part (ii) of the following theorem, but we include the statement anyway.

We defined “generic” in Definition 1.3.7. If a spectrum W is (w_{1}, w_{2})-connective (Definition 1.3.4) but neither (w_{1} + 1, w_{2})-connective nor (w_{1}, w_{2} + 1)-connective, we write ∥W ∥ = (w_{1}, w_{2}). We also write ∥W ∥_{1} for w_{1} and ∥W ∥_{2} for w_{2}; in other words, ∥W ∥_{1} is the smallest number i so that π_{i,∗}W ≠ 0, and ∥W ∥_{2} is the smallest number ℓ so that π_{∗,∗+ℓ}W ≠ 0.

Theorem 1.4.4 ([HPS99]). Suppose that E is a spectrum satisfying Condition 1.4.1, and consider the E-based Adams spectral sequence

E_{∗}^{∗∗∗}(X) ⇒ π_{∗∗}(X).

Fix numbers M and N. The following properties of an E-complete spectrum X are each generic.

(i) There exist numbers r and b so that if s, t, and u satisfy

s ≥ M(s + u) + N(t + u) + b,

then E_{r}^{s,t,u}(X) = 0. (ii) There exist numbers r and b so that for all finite spectra W, if s, t, and u

satisfy

s ≥ M(s + u − ∥W ∥_{1}) + N(t + u + ∥W ∥_{2}) + b,

then E_{r}^{s,t,u}(X ∧ W) = 0. Note that the “slope” (M, N) of the vanishing plane is fixed, but the intercept

b and term r of the spectral sequence may vary in these generic conditions.

Remark 1.4.5. (a) We want E to be a nice ring spectrum so we can identify the E_{2}-term and so we have some convergence information. For the proof of this vanishing plane theorem, convergence is important, but the form of the E_{2}-term is not. Hence, we actually do not need the flatness hypothesis, Condition 1.4.1(b), and if we can guarantee convergence by some other means, then we can discard the other assumptions on E.

(b) Similarly, we assume that X is E-complete to ensure that the spectral sequence converges.

The proof in [HPS99] of the corresponding result in the ordinary stable homotopy category is formal and carries over easily to this setting.

For the remainder of this section, we let B be a conormal quotient Hopf algebra of Γ and focus on the HB-based Adams spectral sequence.

Proposition 1.4.6. Suppose that B is a conormal quotient Hopf algebra of Γ so that HB is weakly (0, 0)-connective and so that the graded vector space Γ □_{B}k is finite-dimensional in each degree. Then HB satisfies Condition 1.4.1. Furthermore, HB_{∗∗}HB is isomorphic to (Γ □_{B}k) ⊗ HB_{∗∗} as HB_{∗∗}-modules, and HB_{∗∗}HB is a Hopf algebra over HB_{∗∗}.

Note that if Γ is zero in negative dimensions, then HB will be weakly (0, 0)- connective for any quotient B of Γ. If Γ is of finite type, then Γ □_{B}k will be, also.

Proof. Proposition 1.2.8 says that HB is a commutative, associative ring spectrum. Flatness follows from the isomorphism HB_{∗∗}HB ≅ (Γ □_{B}k) ⊗ HB_{∗∗}, which we verify below. Almost all of the rest of Condition 1.4.1 follows from the fact that, as k-algebras, HB_{∗∗} ≅ Ext_{B}^{∗∗}(k, k); the only remaining piece is that Hk_{ij}HB is finite-dimensional for each i and j. Hk_{∗∗}HB is the homology of the cochain complex HB, and so is equal to Γ □_{B}k; hence our finite-dimensionality assumption on Γ □_{B}k finishes the verification of Condition 1.4.1.

To show that HB_{∗∗}HB is a Hopf algebra over HB_{∗∗}, we first note that since B is conormal, then Γ □_{B}k is trivial as a left B-comodule. Then by Proposition 1.2.8, we have

HB_{∗∗}HB = Ext_{B}^{∗∗}(k, Γ □_{B}k) ≅ (Γ □_{B}k) ⊗ Ext_{B}^{∗∗}(k, k) = (Γ □_{B}k) ⊗ HB_{∗∗}.

The left unit η_{L}: HB_{∗∗} ⟶ (Γ □_{B}k)⊗HB_{∗∗} is the map given by applying HB_{∗∗}(−) to k ⟶ HB. One can easily check that this is just the inclusion x ⟼ 1⊗x. Using the formula for the adjointness isomorphism between restriction and cotensor product in the proof of Lemma 1.2.3, one sees that η_{R} is the same map. Hence the Hopf algebroid (HB_{∗∗}, HB_{∗∗}HB) is a Hopf algebra. □

For the next two results, we need to know how the gradings behave in the above isomorphism:

Ext_{B}^{ℓ,m}(k, Γ □_{B}k) ≅ ⨁_{i+j=m}(Γ □_{B}k)_{i} ⊗ Ext_{B}^{ℓ,j}(k, k).

Since HB_{∗∗}X is an HB_{∗∗}HB-comodule for any X, then the proposition gives a coaction of Γ □_{B}k on HB_{∗∗}X. This is the coaction mentioned in Remark 1.2.13.

Corollary 1.4.7. Suppose that B is a conormal quotient of Γ as in Proposition 1.4.6. For any object X ∈ Stable(Γ), the E_{2}-term of the HB-based Adams spectral sequence abutting to π_{∗∗}X is isomorphic to Ext_{Γ □_{B}k}^{∗∗} (k, HB_{∗∗}X). More precisely,

E_{2}^{s,t,u}(X) ≅ Ext_{Γ □_{B}k}^{s,t+u} (k, HB_{u,∗}X).

Proof. This is an exercise in homological algebra, using the above computation of HB_{∗∗}HB. We leave the details to the reader. □

In the case when X is an injective resolution of a Γ-comodule M, this E_{2}-term is isomorphic to

Ext_{Γ □_{B}k}^{∗∗} (k, Ext_{B}^{∗∗}(k, M)),

which is the E_{2}-term of the change-of-rings spectral sequence associated to the extension

Γ □_{B}k ⟶ Γ ⟶ B.

More precisely, the change-of-rings spectral sequence has

(1.4.8)  ^{‵}E_{2}^{p,q,v} = Ext_{Γ □_{B}k}^{p,v} (k, Ext_{B}^{q,∗}(k, M))

and converges (strongly) to

Ext_{Γ}^{p+q,v}(k, M).

The differentials are indexed as follows:

d_{r}: ^{‵}E_{r}^{p,q,v} ⟶ ^{‵}E_{r}^{p+r,q−r+1,v}.

See [Sin73, II, §5], for more information; alternatively, one can dualize the construction of the Lyndon-Hochschild-Serre spectral sequence for the computation of group cohomology. See [Ben91b, Section 3.5] for a construction of this as the spectral sequence associated to a double complex, for instance.

Proposition 1.4.9. Suppose that B is a conormal quotient of Γ, and suppose that X is an injective resolution of a Γ-comodule M. Then the HB-based Adams spectral sequence abutting to π_{∗∗}X is isomorphic, up to a regrading, to the changeof-rings spectral sequence associated to the extension

Γ □_{B}k ⟶ Γ ⟶ B,

abutting to Ext_{Γ}^{∗∗}(k, M). The regrading is as follows: for all r ≥ 2, the E_{r}^{s,t,u}-term of the Adams spectral sequence is isomorphic to the ^{‵}E_{r}^{s,u,t+u}-term of the changeof-rings spectral sequence.

For example, the regrading gives this isomorphism of E_{2}-terms:

Ext_{Γ □_{B}k}^{p,v} (k, Ext_{B}^{q,∗}(k, M)) ≅ Ext_{HB_{∗∗}HB}^{p,v−q,q}(HB_{∗∗}, HB_{∗∗}X).

Proof. This is another exercise in homological algebra. If X is an injective resolution of a Γ-comodule M, then the Adams tower for X is a double complex; using the computation of HB_{∗∗}HB in Proposition 1.4.6, one can easily show that the resulting spectral sequence is the change-of-rings spectral sequence. □

Remark 1.4.10. Because of this proposition, we will occasionally use the terms “HB-based Adams spectral sequence” and “change-of-rings spectral sequence” interchangeably. Note the following, however.

(a) The Adams spectral sequence abutting to π_{∗∗}X is defined for any object X in Stable(Γ), not just when X is an injective resolution of a comodule.

(b) On the other hand, the change-of-rings spectral sequence may be generalized somewhat: for any Γ-comodules M_{1}, M_{2}, and M_{3}, there is a spectral sequence with

‵E_{2}^{p,q,v} = Ext_{Γ □_{B}k}^{p,v} (M_{1}, Ext_{B}^{q,∗}(M_{2}, M_{3})) ⇒ Ext_{Γ}^{p+q,v}(M_{1} ⊗ M_{2}, M_{3}).

See [Ben91b, Section 3.5] for a construction of such a spectral sequence in the setting of modules over a group algebra, including a proof of convergence.

As a particular example of this spectral sequence, we have the “hypercohomology” spectral sequence: when B = k, then HB = Hk = Γ, so Hk_{∗∗}Hk = Ext_{k}^{∗∗}(k, Γ) = Γ. Hence the spectral sequence has

E_{2} ≅ Ext_{Γ}^{∗∗}(k, Hk_{∗∗}X) ⇒ π_{∗∗}X.

In other words, it starts with Ext of the homology of the cochain complex X and computes the hypercohomology of X.

Because the HB-based Adams spectral sequence agrees with the change-of-rings spectral sequence, if X = S^{0} (or more generally if X is an injective resolution of a commutative Γ-comodule algebra), then one has Steenrod operations acting on the HB-based Adams spectral sequence, as described in [Sin73] (see also [Saw82]). We need the following result in Chapter 5.

Proposition 1.4.11. Suppose that Γ is a Hopf algebra over the field F_{p}, and suppose that B is a conormal quotient of Γ so that Γ □_{B}k is finite-dimensional in each degree. Consider the HB-based Adams spectral sequence converging to π_{∗∗}S^{0}. Given y ∈ E_{2}^{0,t,u}, with t and u even if p is odd, then for each n, y^{p^{n}} survives to E_{p^{n}+1}^{0,p^{n}t,p^{n}u}.

Note that the result is the same whether one is using the Adams grading— Theorem 1.4.2—or the change-of-rings grading—equation (1.4.8): the elements in ‵E_{2}^{0,q,v} in the change-of-rings grading correspond to elements in E_{2}^{0,v−q,q} = E_{2}^{0,t,u} in the Adams grading; those with q and v even correspond to those with t and u even.

Proof. This follows from properties of Steenrod operations on this spectral sequence, as discussed in [Sin73] and [Saw82]. Suppose we have a spectral sequence E_{r}^{s,t} which is a spectral sequence of algebras over the Steenrod algebra. Fix z ∈ E_{r}^{s,t}, and fix an integer k. If p = 2, then [Sin73, Proposition 1.4] tells us to which term of the spectral sequence Sq^{k} z survives (the result depends on r, s, t, and k). In particular, if z ∈ E_{r}^{0,t}, then Sq^{t}(z) = z^{2} survives to E_{2r−1}^{0,2t}. [Saw82, Proposition 2.5] is the corresponding result at odd primes. □

By the way, Singer’s results [Sin73] are stated in the case of an extension of commutative Hopf algebras

B ⟶ Γ ⟶ C

where C is also cocommutative (actually, he works in the dual situation). This cocommutativity condition is not, in fact, necessary, as forthcoming work of Singer shows [Sin99].

1.5. Bousfield classes and Brown-Comenetz duality

Again, we suppose that Γ is a graded commutative Hopf algebra over a field k. We assume that Stable(Γ) is monogenic. In this section we collect some useful results about Bousfield classes, Brown-Comenetz duality, and their interaction.

We remind the reader that the Bousfield class ⟨X⟩ of an object X in an arbitrary stable homotopy category is the collection of X-acyclic objects—all objects Z with X ∧ Z = 0. We order Bousfield classes by reverse inclusion, so ⟨X⟩ ≥ ⟨Y ⟩ means that X ∧ Z = 0 ⇒ Y ∧ Z = 0. X and Y are Bousfield equivalent if ⟨X⟩ = ⟨Y ⟩. We define operations ∨ and ∧ on Bousfield classes by ⟨X⟩ ∨ ⟨Y ⟩ = ⟨X ∨ Y ⟩ and ⟨X⟩ ∧ ⟨Y ⟩ = ⟨X ∧ Y ⟩; one can check that these are well-defined and that ∨ is the least-upper bound operation. A theorem of Ohkawa [Ohk89] (see also [DP]) says that there is a set of Bousfield classes; hence there is also a greatest-lower bound operation (take the least-upper bound of the set of classes less than or equal to both ⟨X⟩ and ⟨Y ⟩). For certain well-behaved classes of spectra, the greatest-lower bound is given by ∧, but that is not true in general.

These concepts were introduced (in the ordinary stable homotopy category) by Bousfield in [Bou79a] and [Bou79b]; Ravenel proved a number of fundamental results about them in [Rav84]. See [HPS97, Section 3.6] for a discussion of Bousfield classes in a general stable homotopy category.

For any spectrum X, we define its Brown-Comenetz dual IX to be the spectrum that represents the cohomology functor

Y ⟼ Hom_{k}^{∗∗}(π_{∗∗}(X ∧ Y), k).

Hence π_{∗∗}IX = Hom_{k}^{∗∗}(π_{∗∗}X, k). This is an analogue of the construction of Brown and Comenetz [BC76] in the ordinary stable homotopy category.

We say that a bigraded vector space V = ⨁ V_{i,j} is of finite type if V_{i,j} is finite-dimensional for each i and j.

Proposition 1.5.1. [Rav84] Let X and Y be spectra.

(a) Given a map f: X ⟶ X with cofiber X/f and telescope f^{−1}X, we have ⟨X/f⟩ ∨ ⟨f^{−1}X⟩ = ⟨X⟩ and ⟨X/f⟩ ∧ ⟨f^{−1}X⟩ = 0.

(b) If X is weakly connective (Definition 1.3.4), then IX is in the localizing subcategory generated by Hk = Γ.

(c) If X has finite type homotopy, then I^{2}X = X; hence if [Y, X]_{∗∗} = 0, then π_{∗∗}(Y ∧ IX) = 0.

(d) If X is a ring spectrum and Y is an X-module spectrum, then we have ⟨Y ⟩ = ⟨X ∧ Y ⟩ ≤ ⟨X⟩.

(e) If X is a ring spectrum, then IX is an X-module spectrum; hence, ⟨IX⟩ ≤ ⟨X⟩.

(f) Let X be a noncontractible ring spectrum with finite type homotopy and suppose that Y is an X-module spectrum. If [Y, X]_{∗∗} = 0, then ⟨Y ⟩ < ⟨X⟩.

See [Rav84] for the proofs. (See also Lemma 1.3.8 for (b).) Combining part (a) of this with Lemma 1.2.15 gives us a corollary. Corollary 1.5.2. (a) Suppose that there is a Hopf algebra extension of

the form

E[x] ⟶ B ⟶ C,

giving a cofiber sequence

Σ^{1,|x|}HB ⟶^{h} HB ⟶ HC ⟶ Σ^{0,|x|}HB.

Then ⟨HB⟩ = ⟨HC⟩ ∨ ⟨h^{−1}HB⟩ and HC ∧ h^{−1}HB = 0.

(b) Suppose that there is a Hopf algebra extension of the form

D[x] ⟶ B ⟶ C,

with |x| > 0, giving cofiber sequences

Σ^{2,p|x|}HB ⟶^{b} HB ⟶ H̃C ⟶ Σ^{1,p|x|}HB,

Σ^{1,0}HC ⟶^{f} Σ^{1,|x|}HC ⟶ H̃C ⟶ HC.

If the ring spectrum HC is weakly connective, then ⟨HB⟩ = ⟨HC⟩∨⟨b^{−1}HB⟩ and HC ∧ b^{−1}HB = 0.

Proof. Part (a) is immediate; part (b) follows once we show that ⟨HC⟩ = ⟨H̃C⟩. By the second cofibration above, it suffices to show that f^{−1}HC is contractible. For connectivity reasons, though, π_{∗∗}f^{−1}HC = 0. □

Note that π_{i∗}HC is zero when i < 0, so requiring HC to be weakly connective merely requires that there be a number j_{0} so that π_{ij}HC = Ext_{C}^{ij}(k, k) = 0 when j < j_{0}. This is immediate if the Hopf algebra C is zero in negative dimensions, for instance.

1.6. Further discussion

In this section, we note that we can apply the technology of this chapter to group algebras, and we give one or two examples.

Let k be a field of characteristic p > 0, and let G be a finite group. Then kG is a finite-dimensional cocommutative Hopf algebra, so we can use stable homotopy theory to study Stable(kG^{∗}). By Remark 1.1.1, we could just as well work with the category of cochain complexes of injective kG-modules; in either case, morphisms in the category relate to group cohomology, as mentioned in Remark 1.1.2. Since kG is concentrated in degree 0, then morphisms are only singly graded.

As a result, most of the results in this chapter apply to Stable(kG^{∗}). If G is p-group, then the trivial module k is the only simple, so all of the results of the chapter apply to Stable(kG^{∗}) for G a p-group. To illustrate, Lemma 1.2.15 translates to the following, when p = 2. When B is a subgroup of G, then kB^{∗} is a quotient Hopf algebra of kG^{∗}; note that the coefficient ring of H(kB^{∗}) is the same as the group cohomology H^{∗}(B; k) of B.

Corollary 1.6.1. Let k be a field of characteristic 2, and let G be a finite group. Any group extension

1 ⟶ B ⟶ G ⟶ Z/2 ⟶ 1

gives rise to a cofibration in Stable(kG^{∗}):

Σ^{1}H(kG^{∗}) ⟶^{z} H(kG^{∗}) ⟶ H(kB^{∗}),

where z ∈ H^{1}(G; k) is the inflation of the polynomial generator in H^{1}(Z/2; k).

As in Lemma 1.2.15, we are using the same letter z to denote a class in π_{∗}H(kG^{∗}) and the corresponding self map of H(kG^{∗}).

As mentioned in Subsection 1.2.1, when one has a group extension like

1 ⟶ B ⟶ G ⟶ Z/2 ⟶ 1,

one can often replace arguments involving the associated spectral sequence with simpler arguments involving the above cofibration. It is an amusing exercise to prove Chouinard’s theorem [Cho76] this way, for instance.

We should also discuss the Bousfield lattice in Stable(kG^{∗}). In any stable homotopy category, the partially ordered set of Bousfield classes, together with the meet and join operations, contains important structural information about the category. It turns out that when G is a finite p-group, there is a complete description of the Bousfield lattice, essentially due to Benson, Carlson, and Rickard [BCR96, BCR97].

Given a p-group G and an algebraically closed field k of characteristic p, they define V_{G}(k) to be the maximal ideal spectrum of the (graded) commutative Noetherian ring H^{∗}(G; k). Then for each closed homogeneous irreducible subvariety V of V_{G}(k), they define a kG-module κ(V) satisfying various properties, as outlined below. Throughout, they work in the “stable module category” StMod(kG)—see Section 3.2 for a definition of the stable comodule category, which is equivalent in this setting. This means that in their setting, κ(V) is only well-defined up to projective summands. One could just as easily work in Stable(kG^{∗}), in which case κ(V) is a cochain complex, well-defined up to cochain-homotopy equivalence. Naturally, we choose the latter course.

Theorem 1.6.2 ([BCR96]). The objects κ(V) satisfy the following:

(a) There is a Bousfield class decomposition of ⟨S^{0}⟩:

⟨S^{0}⟩ = ⋁_{V ⊆V_{G}(k)}⟨κ(V)⟩,

where the wedge is taken over all closed homogeneous irreducible V.

(b) If V ≠ W, then κ(V) ∧ κ(W) = 0.

(c) For any V ⊆ V_{G}(k) and objects X and Y, if κ(V)∧X ∧Y = 0, then either κ(V) ∧ X = 0 or κ(V) ∧ Y = 0.

By [HPS97, Corollary 5.2.3], this implies the following.

Corollary 1.6.3 ([BCR97]). When G is a p-group, the thick subcategories of finitely generated kG-modules are in one-to-one correspondence with collections of closed homogeneous subvarieties of V_{G}(k) which are closed under specialization.

Their result also has the following corollary.

Corollary 1.6.4. Each ⟨κ(V)⟩ is a minimal nonzero Bousfield class. Hence the Bousfield lattice is a Boolean algebra on the classes ⟨κ(V)⟩. Indeed, for every X,

⟨X⟩ = ⋁_{V : κ(V)∧X≠0}⟨κ(V)⟩.

Proof. To show that ⟨κ(V)⟩ is minimal, we let E = ⋁_{W ≠V} κ(W). Then ⟨S^{0}⟩ = ⟨κ(V)⟩ ∨ ⟨E⟩, and κ(V) ∧ E = 0—in other words, κ(V) is complemented. If X is any object with ⟨X⟩ < ⟨κ(V)⟩, then there is a spectrum Z with X ∧ Z = 0 but κ(V) ∧ Z ≠ 0. Hence κ(V) ∧ X ∧ Z = 0; by Theorem 1.6.2(c), this implies that κ(V) ∧ X = 0. Since ⟨S^{0}⟩ = ⟨κ(V)⟩ ∨ ⟨E⟩, then we have ⟨X⟩ = ⟨X ∧ E⟩. On the other hand, we have κ(V) ∧ E = 0 and ⟨κ(V)⟩ > ⟨X⟩, so X ∧ E = 0. Hence ⟨X⟩ = ⟨X ∧ E⟩ = ⟨0⟩, so X = 0.

Hence for any spectrum Y, if κ(V) ∧ Y ≠ 0, then ⟨κ(V) ∧ Y ⟩ = ⟨κ(V)⟩. Combined with Theorem 1.6.2(a), this gives the desired description of the Bousfield lattice. □

These results describe much of the global structure of the category Stable(kG^{∗}) (and also of StMod(kG)), but they leave open the question of classifying all localizing subcategories (Definition 1.3.7). We conjecture that every localizing subcategory is a Bousfield class, so Corollary 1.6.4 would classify them as well. This sort of general structure is discussed in [HPS97, Chapter 6].

One can construct the objects κ(V) in Stable(Γ) for any Γ whose cohomology ring is Noetherian—see [HPS97, Definition 6.0.8, Example 6.1.4]. This includes all finite-dimensional commutative Hopf algebras, by work of Friedlander and Suslin [FS97]. The analogues of parts (a) and (b) of Theorem 1.6.2 hold, but it is not clear whether part (c) does. It is natural to conjecture that a similar description of the Bousfield lattice is valid, as long as the Hopf algebra Γ is suitably well-behaved.

For instance, suppose that Γ is finite-dimensional and connected, and also that every quasi-elementary quotient of Γ is elementary (these terms are defined in Section 2.1). Hovey and the author (see [HPb] and [HPa]) were then able to imitate the work in [BCR96] and hence to prove Theorem 1.6.2 in this setting, giving a classification of thick subcategories and of Bousfield classes in Stable(Γ).

If the quasi-elementary quotients of Γ do not coincide with the elementaries, one still might be able to imitate Benson et al., if one had a good enough understanding of the quasi-elementary quotients of Γ. If Γ is a finite-dimensional quotient of the dual of the odd primary Steenrod algebra, the quasi-elementary quotients are not completely understood, and we do not know if Theorem 1.6.2 holds.

CHAPTER 2 Basic properties of the Steenrod algebra

The results and constructions in Chapter 1 hold as long as Γ is a graded commutative Hopf algebra over a field, occasionally with the assumption that Stable(Γ) is monogenic or that Γ is connected. Now we start to focus on the case when Γ is the dual of the Steenrod algebra.

Fix a prime p. In Section 2.1, we define the dual A of the mod p Steenrod algebra, and we give a classification of the quotient Hopf algebras of A. We also define several important families of quotient Hopf algebras of A: the A(n)’s, the elementary quotients, and the quasi-elementary quotients. We classify the latter two families, at least at the prime 2, and for each (quasi-)elementary E, we compute the homotopy groups of the ring spectrum HE. In Section 2.2 we introduce P_{t}^{s}- homology, a well-known tool for studying Ext over the Steenrod algebra. In our setting, P_{t}^{s}-homology is a homology theory, and hence is represented by an object P_{t}^{s} in Stable(A); we compute π_{∗∗}P_{t}^{s}, and we perform a few other computations.

Starting in Section 2.3, we begin to get to the main results for this chapter. We discuss a vanishing line theorem in Section 2.3: given conditions on the P_{t}^{s}- homology groups of an object X, then π_{ij}X = 0 when mi > j +c for some numbers m and c. (This is an extension to the cochain complex setting of theorems of Anderson-Davis [AD73] and Miller-Wilkerson [MW81].) In Section 2.4 we use the vanishing line theorem to construct “self-maps of finite objects” in Stable(A). For example, if M is an A-comodule and dim_{F_{p}} M is finite, and if X is an injective resolution of M, then we construct a cochain map Σ^{n_{1},n_{2}}X ⟶ X which is nonnilpotent under composition. We also establish that these self-maps have certain nice properties. (This is an extension to the cochain complex setting of a result of the author [Pal92].) In Section 2.5 we use these self-maps to construct finite objects whose P_{t}^{s}-homology groups vanish for certain specified P_{t}^{s}’s, and we study some properties of these objects. We use them to study chromatic phenomena in Chapter 3.

In Section 2.6 we mention a few topological applications of the vanishing line and self-map results, and one or two other issues.

2.1. Quotient Hopf algebras of A

In this section we define the dual A of the mod p Steenrod algebra, we give a classification of quotient Hopf algebras of A, and we discuss two important families of these quotient Hopf algebras: namely, the A(n)’s and the elementary quotients. Margolis’ book [Mar83] is a good reference for all of these topics. In a subsection, we also discuss the quasi-elementary quotients of A.

Fix a prime number p and let A be the dual of the mod p Steenrod algebra. Recall from [Mil58] Milnor’s description of A: as an algebra, we have

A = \begin{cases}F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …], & if p = 2, \\ F_{p}[ξ_{1}, ξ_{2}, ξ_{3}, …] ⊗ Λ[τ_{0}, τ_{1}, τ_{2}, …], & if p is odd, \end{cases}

where the degrees of the generators are |ξ_{n}| = 2^{n} − 1 if p = 2, |ξ_{n}| = 2(p^{n} − 1) and |τ_{n}| = 2p^{n} − 1 if p odd. The coproduct map Δ: A ⟶ A ⊗ A is determined by its values on the generators, and then by extending multiplicatively:

Δ: ξ_{n} ⟼ ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ ξ_{i},

Δ: τ_{n} ⟼ ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ τ_{i} + τ_{n} ⊗ 1.

(In both of these formulas, we take ξ_{0} to be 1.)

One can check that A is a graded connected commutative Hopf algebra over the field F_{p} (or see [Mil58]); as a result, the work in the previous chapter applies to the study of Stable(A).

The quotient Hopf algebras of A have been classified by Anderson-Davis (p = 2) and Adams-Margolis (p odd). We state the classification theorem here; see [Mar83] or the original papers for the proof. (Recall that we defined conormal quotient Hopf algebras in Definition 1.2.12.)

Theorem 2.1.1. [AD73, AM74]

(a) Every quotient Hopf algebra B of A is of the form

B = \begin{cases}A/(ξ_{1}^{2^{n_{1}}} , ξ_{2}^{2^{n_{2}}} , …), & p = 2, \\ A/(ξ_{1}^{p^{n_{1}}} , ξ_{2}^{p^{n_{2}}} , …; τ_{0}^{e_{0}}, τ_{1}^{e_{1}}, …), & p odd, \end{cases}

for some exponents n_{1}, n_{2}, … ∈ {0, 1, 2, … }∪{∞} and e_{0}, e_{1}, … ∈ {1, 2}. These exponents satisfy the following conditions:

(i) For all primes p: for each i and r with 0 < i < r, either n_{r} ≥ n_{r−i}−i or n_{r} ≥ n_{i}.

(ii) For p odd: if e_{r} = 1, then for each i and j with 1 ≤ i ≤ r and

i + j = r, either n_{i} < j or e_{j} = 1.

(b) Conversely, any set of exponents {n_{i}} and {e_{i}} satisfying conditions (i)- (ii) above determines a quotient Hopf algebra of A.

(c) Let B be a quotient Hopf algebra of A as in (a). Then B is a conormal

quotient Hopf algebra of A if and only if, for p = 2,

(i) n_{1} ≤ n_{2} ≤ n_{3} ≤ ⋯, and for p odd,

(i) n_{1} ≤ n_{2} ≤ n_{3} ≤ ⋯, (ii) e_{0} ≤ e_{1} ≤ e_{2} ≤ ⋯, (iii) e_{k} = 1 ⇒ n_{k} = 0.

For part (a), if some n_{i} = ∞, that means that one does not divide out by any power of ξ_{i}. Similarly, if some e_{i} = 2, then one does not mod out by τ_{i}.

Part (a) says that there is a monomorphic function from the set of quotient Hopf algebras of A to the set of sequences either of the form (n_{1}, n_{2}, …), or of the form (n_{1}, n_{2}, …; e_{0}, e_{1}, …); part (b) gives the image of this function. For p = 2,

2.1. QUOTIENT HOPF ALGEBRAS OF A 37

p = 2 0 1⋮^{r−1} ⋮ 1 2 3 . . .. . . n… ξ_{1} ξ_{1}^{2} ξ_{2}^{2} ξ_{3}^{2} ξ_{2}^{4} ξ_{3}^{4}  ξ_{n}^{2^{r−1}} ξ_{n}^{2^{r}} ⋮. . .. . . ⋮…  ^{p odd ξ_{1} ξ_{1}^{p} ξ_{1}^{p^{2}}… ⋮  … τ_{n} τ_{n+1}… ⋮…}

Figure 2.1.A. Graphical representation of a quotient Hopf algebra of A. For p = 2, this is a bar chart; the nth column is height r − 1 if one is dividing out by ξ_{n}^{2^{r}}. For p odd, this is a similar bar chart, together with an extra row at the bottom; in this row, one marks which τ_{n}’s are nonzero in the quotient.

given a Hopf algebra B, one can view the sequence of exponents n_{1}, n_{2}, … as a function

{1, 2, … } ⟶ {0, 1, 2, … } ∪ {∞},
i ⟼ n_{i}.

We refer to this as the profile function of B. There is, of course, a similar function when p is odd. We will occasionally give graphical representations of quotient Hopf algebras via their profile functions, as in [Mar83, p. 234–5]. See Figure 2.1.A, for example.

Here is a simple, but useful, result. See Notation 0.5.1 for the definition of D[x] and E[x].

Lemma 2.1.2. (a) Suppose that B is a quotient Hopf algebra of A, and

that for some s and t we have • ξ_{t}^{p^{s}} ≠ 0 in B,

• ξ_{t}^{p^{s+1}} = 0 in B,
• ξ_{j}^{p^{s}} = 0 in B for all j < t.

Then there is a Hopf algebra extension

D[ξ_{t}^{p^{s}}] ⟶ B ⟶ C.

(b) Fix p odd. Suppose that B is a quotient Hopf algebra of A, and that for

some n we have • τ_{n} ≠ 0 in B, and either

• τ_{j} = 0 in B for all j < n, or
• ξ_{j} = 0 in B for all j ≤ n.

Then there is a Hopf algebra extension

E[τ_{n}] ⟶ B ⟶ C.

Proof. One only has to check that given the conditions on B, then ξ_{t}^{p^{s}} (respectively, τ_{n}) is primitive in B. This check is straightforward. □

[figure: p = 2; n; n+1; n; n+1; p odd; n−1; n; n; τ_{0}; …; τ_{n}]

Figure 2.1.B. Profile functions for A(n). The actual profile functions have a staircase shape, which we abbreviate as lines with slope −1.

Remark 2.1.3. (a) Hence the results of Lemma 1.2.15 apply. The usual notation is:

h_{ts} = [ξ_{t}^{p^{s}}] ∈ HB_{1,|ξ_{t}^{p^{s}} |} = Ext_{B}^{1,|ξ_{t}^{p^{s}} |}(F_{p}, F_{p}),
b_{ts} = β𝒫̃^{0} (h_{ts}) ∈ HB_{2,|pξ_{t}^{p^{s}} |} = Ext_{B}^{2,p|ξ_{t}^{p^{s}} |}(F_{p}, F_{p}),
v_{n} = [τ_{n}] ∈ HB_{1,|τ_{n}|} = Ext_{B}^{1,|τ_{n}|}(F_{p}, F_{p}).

(b) Also, note that if B is a finite-dimensional quotient of A, then one can always find an integer n or a pair (s, t) so that the hypotheses of Lemma 2.1.2 hold. For instance, one can take s and t to be as follows:

t = min{n | ξ_{n} ≠ 0 in B},
s = max{i | ξ_{t}^{p^{i}} ≠ 0 in B}.

This provides an inductive procedure for studying finite quotients of A. We need to use several different families of quotient Hopf algebras of A. The

A(n)’s form the first family. These quotients are quite well-known; see [Mar83, p. 235], for example.

Example 2.1.4. We define A(n) as follows; see also Figure 2.1.B:

A(n) = \begin{cases}A/(ξ_{1}^{2^{n+1}}, ξ_{2}^{2^{n}}, …, ξ_{n+1}^{2}, ξ_{n+2}, ξ_{n+3}, …), & p = 2, \\ A/(ξ_{1}^{p^{n}}, ξ_{2}^{p^{n−1}}, …, ξ_{n}^{p}, ξ_{n+1}, ξ_{n+2}, …; τ_{n+1}, τ_{n+2}, …), & p odd. \end{cases}

Then A(n) is a quotient Hopf algebra of A, and the map A ⟶ A(n) is an isomorphism below degree

|ξ_{1}^{2^{n+1}}| = 2^{n+1},  if p = 2,
|ξ_{1}^{p^{n}}| = 2(p − 1)p^{n}, if p is odd.

One important property of the A(n)’s is that the dual A^{∗} of A is the union of the duals of the A(n)’s, and this gives A^{∗} the structure of a “P -algebra” (see [Mar83, Chapter 13] for the precise definition). In our setting, this translates into the following (cf. [Mar83, Proposition 13.4]).

Proposition 2.1.5. Suppose that Y is a spectrum so that for each i, π_{ij}Y = 0 when j ≪ 0. Let B be a quotient Hopf algebra of A, and define B(n) by the following pushout diagram of Hopf algebras:

A  B A(n)  B(n). ww u u  u u ww

Then HB ∧ Y is the sequential limit of

⋯ ⟶ HB(3) ∧ Y ⟶ HB(2) ∧ Y ⟶ HB(1) ∧ Y ⟶ HB(0) ∧ Y.

Hence for any spectrum X, there is a Milnor exact sequence

0 ⟶ \underleftarrow{lim}^{1}[X, HB(n) ∧ Y]_{i−1,j} ⟶ [X, HB ∧ Y]_{i,j} ⟶ \underleftarrow{lim}[X, HB(n) ∧ Y]_{i,j} ⟶ 0.

Here are two alternate descriptions of B(n). First, the profile function for B(n) is given by intersecting the profile functions of B and A(n). Second, in the setting of sub-Hopf algebras of the dual of A, the dual of B(n) is B(n)^{∗} = B^{∗} ∩ A(n)^{∗}.

Proof. We write the cochain complex Y as ⋯ ⟶ Y_{j} ⟶ Y_{j+1} ⟶ · · ·. We may assume that for each j, the injective comodule Y_{j} is bounded below. Because A ↠ A(n) is an isomorphism in a range of dimensions increasing with n, so is the map B ↠ B(n); hence the inverse system of comodules

⋯ ⟶ (A □_{B(n)}F_{p}) ⊗ Y_{j} ⟶ (A □_{B(n−1)}F_{p}) ⊗ Y_{j} ⟶ (A □_{B(n−2)}F_{p}) ⊗ Y_{j} ⟶ ⋯

stabilizes in any given degree, and the inverse limit is (A □_{B}F_{p}) ⊗ Y_{j}. We let (A □_{B(n)}F_{p}) ⊗ Y denote the cochain complex which is (A □_{B(n)}F_{p}) ⊗ Y_{j} in degree j; then the inverse system of cochain complexes stabilizes in each bidegree, and the inverse limit is HB ∧ Y. This finishes the proof. (Note that (A □_{B(n)}F_{p}) ⊗ Y is isomorphic to HB(n) ∧ Y in Stable(A).) □

For example, with Y as in the proposition and B = A, Y is the sequential limit of

⋯ ⟶ HA(3) ∧ Y ⟶ HA(2) ∧ Y ⟶ HB(1) ∧ Y ⟶ HA(0) ∧ Y.

We consider one other family of quotient Hopf algebras. See [Wil81] for some results related to these.

Definition 2.1.6. We say that a connected commutative Hopf algebra B over a field k of characteristic p is elementary if it is isomorphic (as a Hopf algebra) to a tensor product of Hopf algebras of the forms k[x]/(x^{p^{n}}) with x primitive (and |x| even, if p is odd), and E[y] with y primitive (and |y| odd, if p is odd). In other words, B is bicommutative and its dual B^{∗} has z^{p} = 0 for all z in the augmentation ideal.

See [Mar83], [Lin77a] and [Wil81] for the following.

Proposition 2.1.7. (a) Suppose that p = 2. A quotient Hopf algebra

B = A/(ξ_{1}^{2^{n_{1}}} , ξ_{2}^{2^{n_{2}}} , …)

of A is elementary if and only if for some r, (i) if i < r, then n_{i} = 0, (ii) if i ≥ r, then n_{i} ≤ r.

p = 2 0 1 2 3 4 1 2 3 4 5 6 E(0)  E(4) . . .. . .. . . ⋮ ⋮ ⋮  ⋮…

Figure 2.1.C. Profile functions for maximal elementary quotients of A at the prime 2. For reference, we have included a staircase above which are the elements ξ_{t}^{2^{s}} with s ≥ t.

(b) Suppose that p is odd. A quotient Hopf algebra

B = A/(ξ_{1}^{p^{n_{1}}} , ξ_{2}^{p^{n_{2}}} , …; τ_{0}^{e_{0}}, τ_{1}^{e_{1}}, …),

of A is elementary if and only if either n_{i} = 1 for all i, or for some r, (i) if i < r, then n_{i} = 0, (ii) if i ≥ r, then n_{i} ≤ r, (iii) if i < r, then e_{i} = 1.

We also describe the maximal elementary quotient Hopf algebras of A; see Figure 2.1.C for the profile function at the prime 2. Every elementary quotient Hopf algebra of A is a quotient of one of these.

Corollary 2.1.8. (a) Suppose that p = 2. The maximal elementary quotient Hopf algebras of A are

E(m) = A/(ξ_{1}, …, ξ_{m}, ξ_{m+1}^{2^{m+1}}, ξ_{m+2}^{2^{m+1}}, ξ_{m+3}^{2^{m+1}}, …), m ≥ 0.

(b) Suppose that p is odd. The maximal elementary quotient Hopf algebras of A are

E(−1) = A/(ξ_{1}, ξ_{2}, …) ≅ Λ[τ_{0}, τ_{1}, …],

E(m) = A/(ξ_{1}, …, ξ_{m}, ξ_{m+1}^{p^{m+1}}, ξ_{m+2}^{p^{m+1}}, ξ_{m+3}^{p^{m+1}}, …; τ_{0}, …, τ_{m}), m ≥ 0.

Note that the quotient Hopf algebras E(m) are conormal for all m. We use the elementary quotient Hopf algebras of A to prove the vanishing line

theorem of Section 2.3; we need to know their coefficient rings. The following is standard; it also follows from Proposition 0.5.10 and the Künneth theorem.

Proposition 2.1.9. (a) Let p = 2, and assume that E is an elementary quotient Hopf algebra of A. Then

HE_{∗∗} = F_{2}[h_{ts} | ξ_{t}^{2^{s}} ≠ 0 in E].

Here, |h_{ts}| = (1, |ξ_{t}^{2^{s}}|).

(b) Let p be odd, and assume that E is an elementary quotient Hopf algebra of A. Then

HE_{∗∗} = Λ[h_{ts} | ξ_{t}^{p^{s}} ≠ 0 in E] ⊗ F_{p}[b_{ts} | ξ_{t}^{p^{s}} ≠ 0 in E]
⊗ F_{p}[v_{n} | τ_{n} ≠ 0 in E].

Here, |h_{ts}| = (1, |ξ_{t}^{p^{s}}|), |b_{ts}| = (2, p|ξ_{t}^{p^{s}}|), and |v_{n}| = (1, |τ_{n}|).

Indeed, if E is an elementary quotient of A with ξ_{t}^{p^{s}} ≠ 0, then ξ_{t}^{p^{s}} is primitive in E. Following Notation 1.2.14 and Remark 2.1.3(a), we have h_{ts} = [ξ_{t}^{p^{s}}]; similarly, if τ_{n} is non-zero in E, then it is primitive and v_{n} = [t_{n}].

2.1.1. Quasi-elementary quotients of A. We also need to consider one other family of quotient Hopf algebras, the “quasi-elementary” Hopf algebras. It turns out that when p = 2, these coincide with elementary Hopf algebras; when p is odd, there are quasi-elementary Hopf algebras which are not elementary, but we do not have a complete classification.

The quasi-elementary quotients of A arise in the nilpotence theorem 5.1.6 of Section 5.1, and indeed in many of the results of Chapters 5 and 6; the lack of a classification at odd primes is one obstacle to proving those theorems when p is odd. So when p is 2, we have already studied these; when p is odd, we have no immediate use for them. Hence the contents of this subsection may be safely ignored, except for the term “quasi-elementary.”

Definition 2.1.10. Let B be a connected commutative Hopf algebra B over a field k.

(a) Suppose that k has characteristic 2. Then B is quasi-elementary if for every finite set S ⊂ Ext_{B}^{1,∗}(k, k)−{0}, the product ∏_{w∈S} w is non-nilpotent.

(b) Suppose that k has characteristic p > 2. Recall from Notation 1.2.14 that there is a Steenrod operation β𝒫̃^{0} on Ext_{B}^{∗∗}(k, k). B is quasi-elementary if for all finite sets

S_{odd} ⊂ ⨁_{n∈Z} Ext_{B}^{1,2n+1}(k, k) − {0},

S_{even} ⊂ ⨁_{n∈Z} Ext_{B}^{1,2n}(k, k) − {0},

the product

( ∏_{u∈S_{odd}} u)( ∏_{v∈S_{even}} β𝒫̃^{0} v)

is non-nilpotent. Every elementary Hopf algebra is quasi-elementary; Wilkerson [Wil81, Section

6] gives several examples of quasi-elementary Hopf algebras which are not elementary. In particular, one of his examples is a quotient of A, when p is odd. See [Wil81, Counterexample 6.3] for the following.

Example 2.1.11. Suppose that p is odd, and let B be this quotient Hopf algebra of A:

B = F_{p}[ξ_{1}, ξ_{2}, ξ_{3}]/(ξ_{1}^{p}, ξ_{2}^{p^{2}}, ξ_{3}^{p}).

Then B is quasi-elementary, but not elementary.

When p is 2, things are a bit nicer. See [Wil81, Theorem 6.4] for the following. Proposition 2.1.12. Suppose that p = 2. A quotient Hopf algebra B of A is

elementary if and only if it is quasi-elementary.

Note that Lin [Lin77a] has classified the elementary quotients of A—see Proposition 2.1.7 and Corollary 2.1.8 above. We have also computed the coefficient rings of these Hopf algebras in Proposition 2.1.9.

At odd primes, we do not have a classification of the quasi-elementary Hopf algebras, so we content ourselves with the following. This would follow from Conjecture 5.4.1.

Conjecture 2.1.13. Suppose that p is odd. Then every quasi-elementary quotient Hopf algebra of A is a quotient of

D = A/(ξ_{1}^{p}, ξ_{2}^{p^{2}}, ξ_{3}^{p^{3}}, …, ξ_{n}^{p^{n}}, …).

See [Pal97] for some results relating to quasi-elementary Hopf algebras. We should also point out that there is a classification of finite-dimensional quasielementary quotients of A at odd primes in [NP98]; however, the proof given there has a gap.

2.2. P_{t}^{s}-homology

In this section we discuss P_{t}^{s}-homology; this tool has been used by many authors to study Ext_{A}, so it is reasonable to expect it to be useful in the present setting. [Mar83, Section 19.1] is a good reference for basic results on P_{t}^{s}-homology of modules; in a subsection, we prove analogues of some of those results.

Let A^{∗} be the dual of A. Note that an A-comodule M with structure map ψ is naturally an A^{∗}-module, via the composite

A^{∗} ⊗ M ⟶^{1⊗ψ} A^{∗} ⊗ A ⊗ M ⟶^{ev⊗1} M.

We remind the reader that if one dualizes with respect to the monomial basis for A, then P_{t}^{s} is the element of A^{∗} dual to ξ_{t}^{p^{s}}, and (when p is odd) Q_{n} is the element dual to τ_{n}. If s < t, then (P_{t}^{s})^{p} = 0, so in this case one may define the P_{t}^{s}-homology of an A-comodule M by

H(M, P_{t}^{s}) = \frac{ker(M ⟶^{(P_{t}^{s})^{p−1}} M)}{im(M ⟶^{P_{t}^{s}} M)}.

For all n we have Q_{n}^{2} = 0, so one may define H(M, Q_{n}) similarly. One may do these things, but that does not mean that one should do them when working in the context of cochain complexes of injective A-comodules.

We defined the coalgebras E[x] and D[x] in Notation 0.5.1.

Definition 2.2.1. Given integers s and t with 0 ≤ s < t, we define the connective P_{t}^{s}-homology spectrum, p_{t}^{s}, to be the cochain complex HD[ξ_{t}^{p^{s}}]. In other words, it is an injective resolution of A □_{D[ξ_{t}^{p^{s}}]}F_{p}. Hence it is chain homotopy equivalent to the complex obtained by applying A □_{D[ξ_{t}^{p^{s}}]}− to the injective resolution of the

D[ξ_{t}^{p^{s}}]-comodule F_{p} as given in the diagram (0.5.9). So for j ≥ 0, we may take its jth term to be

(p_{t}^{s})_{j} = \begin{cases}s^{kp|ξ_{t}^{p^{s}} |}A, & j = 2k, \\ s^{(kp+1)|ξ_{t}^{p^{s}} |}A, & j = 2k + 1, \end{cases}

where s denotes the “internal” suspension functor on graded comodules. The differentials are given as in the following diagram:

\begin{aligned}⋯ & ⟶ & A & ⟶ & A & ⟶ & A & ⟶ & A & ⟶ & · · · . \\ & & ξ_{t}^{p^{s}} & ⟼ & 1, ξ_{t}^{(p−1)p^{s}} & ⟼ & 1, ξ_{t}^{p^{s}} & ⟼ & 1 \end{aligned}

2.2. P_{t}^{s}-HOMOLOGY 43

(A is a rank 1 cofree comodule, so a comodule map to A is determined by specifying a vector space map to F_{p}—see Lemma 0.4.3. In this diagram, “x ↦ 1” indicates the vector space map in which x goes to 1 ∈ F_{p}, and the other elements in the monomial basis for A go to 0. One can also describe this map as “right multiplication by x^{∗},” where x^{∗} ∈ A^{∗} is the element dual to x ∈ A.) Note that this cochain complex is periodic with period 2 if p is odd, period 1 if p = 2. We define the (periodic) P_{t}^{s}-homology spectrum, P_{t}^{s}, to be the cochain complex obtained by extending the periodicity to negative dimensions; in other words, for all integers j,

(P_{t}^{s})_{j} = \begin{cases}s^{kp|ξ_{t}^{p^{s}} |}A, & j = 2k, \\ s^{(kp+1)|ξ_{t}^{p^{s}} |}A, & j = 2k + 1, \end{cases}

with differentials given as above. In other words, this cochain complex is obtained by applying A □_{D[ξ_{t}^{p^{s}}]}− to the long exact sequence of D[ξ_{t}^{p^{s}}]-comodules

⋯ ⟶ s^{kp|ξ_{t}^{p^{s}} |}D[ξ_{t}^{p^{s}}] ⟶ s^{(kp+1)|ξ_{t}^{p^{s}} |}D[ξ_{t}^{p^{s}}] ⟶ s^{(k+1)p|ξ_{t}^{p^{s}} |}D[ξ_{t}^{p^{s}}] ⟶ · · · .

Similarly, when p is odd, we define the connective Q_{n}-homology spectrum, q_{n}, to be the cochain complex HE[τ_{n}]. Hence it may be obtained by applying A □_{E[τ_{n}]}− to the long exact sequence (0.5.7), in which case its jth term is given by

(q_{n})_{j} = \begin{cases}0 & if j < 0, \\ s^{j|τ_{n}|}A & if j ≥ 0, \end{cases}

with differentials

\begin{aligned}⋯ & ⟶ & A & ⟶ & A & ⟶ & A & ⟶ & A & ⟶ & · · · . \\ & & τ_{n} & ⟼ & 1, τ_{n} & ⟼ & 1, τ_{n} & ⟼ & 1 \end{aligned}

We define the (periodic) Q_{n}-homology spectrum, Q_{n}, to be the periodic version of q_{n}; in other words, for all integers j, (Q_{n})_{j} = s^{j|τ_{n}|}A, with differentials as above. For any spectrum X, we define its P_{t}^{s}-homology to be (P_{t}^{s})_{∗∗}X, and we define its Q_{n}-homology to be (Q_{n})_{∗∗}X.

From the construction, we can compute the homology (HF_{p})_{∗∗} of these spectra. Lemma 2.2.2. (a) For all s < t,

(HF_{p})_{i,∗}p_{t}^{s} = \begin{cases}A □_{D[ξ_{t}^{p^{s}}]}F_{p}, & if i = 0, \\ 0, & if i ≠ 0. \end{cases}

(b) For all s < t, (HF_{p})_{∗∗}P_{t}^{s} = 0.
(c) For all n,

(HF_{p})_{i,∗}q_{n} = \begin{cases}A □_{E[τ_{n}]}F_{p}, & if i = 0, \\ 0, & if i ≠ 0. \end{cases}

(d) For all n, (HF_{p})_{∗∗}Q_{n} = 0.

Proof. p_{t}^{s} is an injective resolution of A □_{D[ξ_{t}^{p^{s}}]}F_{p} so it only has homology in degree zero. P_{t}^{s} is defined by extending the periodicity of the p_{t}^{s} cochain complex into negative dimensions, so it has no homology at all. The same arguments work for q_{n} and Q_{n}. □

In the remainder of this section, we compute the coefficient rings of these spectra and we prove a few results for later use.

Proposition 2.2.3. (a) If s < t, then D[ξ_{t}^{p^{s}}] is a quotient coalgebra of A over which A is injective as a right comodule. For any n ≥ 0, E[τ_{n}] is a quotient Hopf algebra of A, and hence A is injective as a right comodule over it.

(b) Hence,

(p_{t}^{s})_{∗∗} ≅ \begin{cases}F_{2}[h_{ts}], & p = 2, \\ F_{p}[b_{ts}] ⊗ Λ[h_{ts}], & p odd, \end{cases}

where |h_{ts}| = (1, |ξ_{t}^{p^{s}}|) and |b_{ts}| = (2, p|ξ_{t}^{p^{s}}|). Also,
(q_{n})_{∗∗} ≅ F_{p}[v_{n}],
where |v_{n}| = (1, |τ_{n}|).

Proof. Part (a) is well-known. It is dual to the statement that A^{∗}, the dual of A, is free over the subalgebra F_{p}[P_{t}^{s}]/(P_{t}^{s})^{p}; i.e., H(A^{∗}, P_{t}^{s}) = 0. (And similarly, A^{∗} is free over E[Q_{n}]; i.e., H(A^{∗}, Q_{n}) = 0.) See [AM71], [MP72], or [Mar83, Proposition 19.1].

Part (b) is standard, trivial, or both, using Propositions 0.5.10 and 1.2.8, for instance. □

By construction, the periodicity of the cochain complex p_{t}^{s} yields a self-map of one of the following forms:

h_{ts}: Σ^{1,|ξ_{t}^{2^{s}} |}p_{t}^{s} ⟶ p_{t}^{s},  if p = 2,
b_{ts}: Σ^{2,p|ξ_{t}^{p^{s}} |}p_{t}^{s} ⟶ p_{t}^{s}, if p is odd.

On homotopy groups, the first of these induces multiplication by the element h_{ts} ∈ π_{1,|ξ_{t}^{2^{s}} |}(p_{t}^{s}), and similarly for the second; this explains our names for the self-maps at the cochain complex level. At odd primes, q_{n} has a self-map v_{n}: Σ^{1,|τ_{n}|}q_{n} ⟶ q_{n}. Taking the mapping telescopes of these self-maps gives the periodic spectra P_{t}^{s} and Q_{n}, respectively; indeed, one can take this to be the definition of those spectra.

Lemma 2.2.4. (a) Let p = 2. For all s < t, P_{t}^{s} is the mapping telescope of the self-map h_{ts}: p_{t}^{s} ⟶ Σ^{−1,−|ξ_{t}^{2^{s}} |}p_{t}^{s}. In symbols, P_{t}^{s} = h_{ts}^{−1}p_{t}^{s}.

(b) Let p be an odd prime. For all s < t, P_{t}^{s} = b_{ts}^{−1}p_{t}^{s}.

(c) Let p be an odd prime. For all n, Q_{n} = v_{n}^{−1}q_{n}.

Proof. This follows from the definition of P_{t}^{s} and Q_{n} in terms of p_{t}^{s} and q_{n}. □

See [Mar83, Propositions 19.2–3] for an alternate formulation of the following. Corollary 2.2.5. Hence

(P_{t}^{s})_{∗∗} = \begin{cases}F_{2}[h_{ts}^{±1}], & p = 2, \\ F_{p}[b_{ts}^{±1}] ⊗ Λ[h_{ts}], & p odd, \end{cases}

(Q_{n})_{∗∗} = F_{p}[v_{n}^{±1}],

and for any connective X,

(P_{t}^{s})_{∗∗}X = \begin{cases}h_{ts}^{−1}(p_{t}^{s})_{∗∗}X, & p = 2, \\ b_{ts}^{−1}(p_{t}^{s})_{∗∗}X, & p odd, \end{cases}

(Q_{n})_{∗∗}X = v_{n}^{−1}(q_{n})_{∗∗}X.

By the way, we note that for all primes, the spectrum p_{t}^{0} = HD[ξ_{t}] is a ring spectrum, while P_{t}^{0} is a field spectrum for p = 2 because P_{t}^{s} is commutative and every homogeneous element in π_{∗∗}P_{t}^{s} is invertible—see [HPS97, Proposition 3.7.2]. (Perhaps P_{t}^{0} is an example of what one might call an “Artinian ring spectrum” when p is odd.) Similarly, when p is odd, q_{n} = HE[τ_{n}] is a ring spectrum and Q_{n} is a field spectrum. The spectra p_{t}^{s} for s > 0 are not ring spectra, but are examples of the sort discussed in Proposition 1.2.8(b).

Remark 2.2.6. As one might expect, there is a relationship between the module definition of P_{t}^{s}-homology (given just before Definition 2.2.1) and the homology functor represented by the P_{t}^{s}-homology spectrum. This relationship is straightforward when p = 2, more complicated when p is odd. If X is an injective resolution of a comodule M of finite-type, then the change-of-rings isomorphism of Example 1.2.4 gives

(p_{t}^{s})_{∗∗}X ≅ Ext_{D[ξ_{t}^{p^{s}}]}^{∗∗}(F_{p}, M).

Corollary 0.5.5 gives a classification of finite-type D[ξ_{t}^{p^{s}}]-comodules, so we just need to know these Ext groups for each indecomposable summand. These are easily computed by taking minimal resolutions; the answers are given in Proposition 0.5.10. When p = 2, the indecomposable comodules are the trivial comodule F_{p} and the cofree comodule. We have the following isomorphism for each of these, and hence for every comodule M: when i ≥ 1,

(2.2.7)  Ext_{D[ξ_{t}^{p^{s}}]}^{i,∗}(F_{p}, M) ≅ H_{∗}(M, P_{t}^{s}) ⊗ Ext_{D[ξ_{t}^{p^{s}}]}^{i,∗}(F_{p}, F_{p}).

Hence the periodic P_{t}^{s}-homology of the injective resolution X of M is given by

(P_{t}^{s})_{∗∗}X = H_{∗}(M, P_{t}^{s}) ⊗ (P_{t}^{s})_{∗∗}.

(Here we are viewing the singly-graded vector space H_{∗}(M, P_{t}^{s}) as doubly-graded by putting H_{j}(M, P_{t}^{s}) in bidegree (0, j).) In particular, (P_{t}^{s})_{∗∗}X = 0 if and only if H_{∗}(M, P_{t}^{s}) = 0.

Similarly, we have

(Q_{n})_{∗∗}X = H_{∗}(M, Q_{n}) ⊗ (Q_{n})_{∗∗}.

In the P_{t}^{s} case when p is odd, the isomorphism (2.2.7) is not quite valid for every indecomposable comodule M; rather, one finds that

Ext_{D[ξ_{t}^{p^{s}}]}^{i,∗}(F_{p}, M) ≅ \begin{cases}s^{e(M)}H_{∗}(M, P_{t}^{s}) ⊗ Ext_{D[ξ_{t}^{p^{s}}]}^{i,∗}(F_{p}, F_{p}), & if i is even, \\ s^{o(M)}H_{∗}(M, P_{t}^{s}) ⊗ Ext_{D[ξ_{t}^{p^{s}}]}^{i,∗}(F_{p}, F_{p}), & if i is odd, \end{cases}

where s is the suspension functor in the category of graded vector spaces, and e(M) and o(M) are numbers that depend on the indecomposable comodule M. In particular, we still see that if M is any finite-type comodule and X is its injective resolution, (P_{t}^{s})_{∗∗}X = 0 if and only if H_{∗}(M, P_{t}^{s}) = 0.

It is convenient to have alternate notation for the spectra P_{t}^{s} and Q_{n}, based on the “slopes” of the polynomial generators in their coefficient rings. We use this notation for much of the remainder of the book.

Notation 2.2.8. We define the slope of P_{t}^{s} to be ^{\frac{p|ξ_{t}^{p^{s}} |}{2}}, and the slope of Q_{n} to be |τ_{n}|. Note that these spectra all have distinct slopes. The set

\begin{matrix}\{|ξ_{t}^{2^{s}}| : s < t\}, & p = 2, \\ \{^{\frac{p|ξ_{t}^{p^{s}} |}{2}} : s < t\} ∪ \{|τ_{n}| : n ≥ 0\}, & p odd, \end{matrix}

is called the set of slopes of A; the phrase “fix a slope m” means “fix an element m of this set.” Given a slope m, we let Z(m) denote the corresponding P_{t}^{s} or Q_{n} spectrum. For example, when p = 2, we have

Z(1) = P_{1}^{0}, Z(3) = P_{2}^{0}, Z(6) = P_{2}^{1}, Z(7) = P_{3}^{0}, …, Z(|ξ_{t}^{2^{s}}|) = P_{t}^{s}, ….

When p is odd, we have

\begin{aligned}Z(1) = Q_{0}, & Z(p − 1) = Q_{1}, Z(p^{2} − p) = P_{1}^{0}, Z(p^{2} − 1) = Q_{2}, …, \\ & Z(\frac{p|ξ_{t}^{p^{s}}|}{2}) = P_{t}^{s}, …, Z(|τ_{n}|) = Q_{n}, …. \end{aligned}

Note that Z(4), for instance, is meaningless when p = 2, because 4 is not a slope at that prime. We let z(m) be the connective version of Z(m), either p_{t}^{s} or q_{n}. We let y_{m} denote the corresponding element of A, either ξ_{t}^{p^{s}} or τ_{n}.

2.2.1. Miscellaneous results about P_{t}^{s}-homology. In this subsection, we present a few results about P_{t}^{s}-homology, either for later use or for comparison with work of other authors, especially Margolis.

See [Mar83, Theorem 19.21] for the following at the prime 2.

Proposition 2.2.9. Let B be a quotient Hopf algebra of A. Then (P_{t}^{s})_{∗∗}HB ≠ 0 if and only if ξ_{t}^{p^{s}} ≠ 0 in B. For p odd, (Q_{n})_{∗∗}HB ≠ 0 if and only if τ_{n} ≠ 0 in B.

We will use this result in Section 3.3.

Proof. We will give the proof for P_{t}^{s} and leave the Q_{n} proof for the reader. By Remark 2.2.6, we are interested in H(A □_{B}F_{p}, P_{t}^{s}). If ξ_{t}^{p^{s}} ≠ 0 in B, then 1 ∈ A □_{B}F_{p} generates a nonzero homology class.

To prove the converse, first we reduce to the case when B is finite. We define B(n) to be the quotient Hopf algebra of A defined as in Proposition 2.1.5 by the following pushout diagram of Hopf algebras:

A ⟶ B
↓  ↓
A(n) ⟶ B(n).

(A(n) is defined in Example 2.1.4.) In other words, B(n) is the quotient of B induced by the map A ↠ A(n). Note that each B(n) is finite, and if ξ_{t}^{p^{s}} = 0 in B, then the same is true in B(n) for all n. So if we assume the result for finite quotients of A, then we know that (P_{t}^{s})_{∗∗}HB(n) = 0 for all n. Hence the Milnor short exact sequence in Proposition 2.1.5

0 ⟶ \underleftarrow{lim}^{1}[S^{0}, HB(n) ∧ P_{t}^{s}]_{i−1,j} ⟶ [S^{0}, HB ∧ P_{t}^{s}]_{i,j} ⟶ \underleftarrow{lim}[S^{0}, HB(n) ∧ P_{t}^{s}]_{i,j} ⟶ 0

has first and last terms zero; therefore the middle term is also zero.

So it suffices to show that if B is a finite quotient of A in which ξ_{t}^{p^{s}} = 0, then (P_{t}^{s})_{∗∗}HB = 0. We do this by induction on dim_{F_{p}} B. The induction starts with B = F_{p}, in which case HF_{p} = A. Then

(P_{t}^{s})_{∗∗}HF_{p} = (HF_{p})_{∗∗}P_{t}^{s},

the homology of the cochain complex P_{t}^{s}. This complex is acyclic, so its homology is zero—see Lemma 2.2.2. (Alternatively, see the remark following this proof.) This starts the induction.

Now fix a finite-dimensional B with ξ_{t}^{p^{s}} = 0 in B, and assume that for every proper quotient C of B, we have (P_{t}^{s})_{∗∗}HC = 0. As noted in Remark 2.1.3, there is a Hopf algebra quotient C of B with Hopf algebra kernel either F_{p}[ξ_{r}^{p^{q}}]/(ξ_{r}^{p^{q+1}}) or E[τ_{m}]. So by Lemma 1.2.15 this leads to a cofiber sequence (in which we neglect suspensions)

HB ⟶^{f} HB ⟶ Z,

where either Z = HC or Z is the cofiber of a self-map of HC. In either case, (P_{t}^{s})_{∗∗}HC = 0 ⇒ (P_{t}^{s})_{∗∗}Z = 0, so (P_{t}^{s})_{∗∗}f is an isomorphism.

Now we argue essentially as in the proof of Lemma 2.3.12 to see that because comodule A □_{B}F_{p} is bounded below, then (P_{t}^{s})_{∗∗}HB = 0. To be precise, we first have to specify the degree of the map f. There are three cases:

(1) p = 2: then f has bidegree (1, |ξ_{r}^{2^{q}}|)—i.e., f is a map Σ^{1,|ξ_{r}^{2^{q}} |}HB ⟶ HB. (f is also known as h_{rq}.)

(2) p odd, Hopf algebra kernel E[τ_{m}]: then f has bidegree (1, |τ_{m}|). (f is also known as v_{m}.)

(3) p odd, Hopf algebra kernel D[ξ_{r}^{p^{q}}] = F_{p}[ξ_{r}^{p^{q}}]/(ξ_{r}^{p^{q+1}}): then f has bidegree

(2, p|ξ_{r}^{p^{q}}|). (f is also known as b_{rq}.) We deal with cases (1) and (3); (2) is handled the same way. By our computations in Corollary 2.2.5, we know that for all i and j, we have an isomorphism

(P_{t}^{s})_{ij}HB ≅ (P_{t}^{s})_{i+2,j+p|ξ_{t}^{p^{s}} |}HB.

(This is given by (h_{ts})^{2} when p = 2, and by b_{ts} when p is odd.) In cases (1) and (3), f induces an isomorphism

(P_{t}^{s})_{ij}HB ≅ (P_{t}^{s})_{i+2,j+p|ξ_{r}^{p^{q}} |}HB.

(In other words, f^{2} when p = 2, and f when p is odd.) Combining these, for each integer k we get an isomorphism

(P_{t}^{s})_{ij}HB ≅ (P_{t}^{s})_{i,j+pk(|ξ_{t}^{p^{s}} |−|ξ_{r}^{p^{q}} |)}HB.

Now, ξ_{r}^{p^{q}} ≠ 0 in B and ξ_{t}^{p^{s}} = 0, so ξ_{r}^{p^{q}} ≠ ξ_{t}^{p^{s}}. In particular, these two elements have different degrees. By Remark 2.2.6, for fixed i, (P_{t}^{s})_{i,∗}HB is equal (up to suspension) to H_{∗}(A □_{B}F_{p}, P_{t}^{s}); since A □_{B}F_{p} is a bounded below comodule, then this homology must be zero in small enough degrees. By the above isomorphism, we may then conclude that it is zero in all degrees. □

We remark that it is easy to show that (HF_{p})_{∗∗}P_{t}^{s} = (P_{t}^{s})_{∗∗}A = 0—this follows by a result of Milnor-Moore [MM65], as in [Mar83, Proposition 19.1]. It seems as though there should be a similar proof of Proposition 2.2.9, but we have not been able to find one.

Recall from [HPS97, Definition A.2.4] that if X is any object in Stable(A), then DX denotes the Spanier-Whitehead dual of X. DX is defined to be the object representing the cohomology theory Y ⟼ [X ∧ Y, S^{0}]; hence for any Y,

[X ∧ Y, S^{0}] = [Y, DX].

We mention the following easy fact.

Proposition 2.2.10. Let E and X be spectra with X finite. Then E_{∗∗}X = 0 if and only if E_{∗∗}DX = 0.

Proof. If X is E-acyclic, then so is X ∧ Y for any Y. In particular, DX ∧ X ∧ DX is E-acyclic. Since DX is a retract of this when X is finite, we are done. □

Corollary 2.2.11. Let X be a finite spectrum. Then (P_{t}^{s})_{∗∗}X = 0 if and only if (P_{t}^{s})_{∗∗}DX = 0; and for p odd, (Q_{n})_{∗∗}X = 0 if and only if (Q_{n})_{∗∗}DX = 0.

One should be able to get more precise information about the relationship between (P_{t}^{s})_{∗∗}X and (P_{t}^{s})_{∗∗}DX, as in [Mar83, Proposition 19.12], but we do not need it.

We end this section with a note on operations on P_{t}^{s}-homology.

Proposition 2.2.12. (a) When p = 2,

(P_{t}^{s})_{∗∗}P_{t}^{s} = (P_{t}^{s})_{∗∗}p_{t}^{s} = H_{∗}(A □_{D[ξ_{t}^{p^{s}}]}F_{p}, P_{t}^{s}) ⊗ (P_{t}^{s})_{∗∗}.

(b) When p is odd,

(Q_{n})_{∗∗}Q_{n} = (Q_{n})_{∗∗}q_{n} = H_{∗}(A □_{E[τ_{n}]}F_{p}, Q_{n}) ⊗ (Q_{n})_{∗∗}.

(c) When p is odd,

(P_{t}^{s})_{∗∗}P_{t}^{s} = (P_{t}^{s})_{∗∗}p_{t}^{s}.

Proof. To fix notation, we do part (a); the other cases are virtually identical. By Lemma 2.2.4, P_{t}^{s} = b_{ts}^{−1}p_{t}^{s}, so

P_{t}^{s} ∧ P_{t}^{s} = b_{ts}^{−1}p_{t}^{s} ∧ b_{ts}^{−1}p_{t}^{s} = b_{ts}^{−1}p_{t}^{s} ∧ p_{t}^{s} = P_{t}^{s} ∧ p_{t}^{s}.

This proves the first equality. The rest follows from Corollary 2.2.5 and Remark 2.2.6. □

Margolis has calculated H_{∗}(A □_{D[ξ_{t}^{p^{s}}]}F_{p}, P_{t}^{s}) for s = 0 at the prime 2 [Mar83, Corollary 19.26]; similar calculations work at odd primes for H_{∗}(A □_{E[τ_{n}]}F_{p}, Q_{n}). We give details in Proposition 4.6.1.

2.3. Vanishing lines for homotopy groups

In this section we prove several theorems relating the vanishing of P_{t}^{s}- and Q_{n}- homology groups of an object X to the homotopy groups of X; these are versions in the cochain complex category of well-known theorems about modules over the Steenrod algebra. These results are at the heart of many of the other results of the book.

Here are our main theorems. In the setting of modules over the Steenrod algebra, the first is due to Anderson-Davis [AD73] (p = 2) and Miller-Wilkerson [MW81] (p odd), and the second to Adams-Margolis [AM71] (p = 2) and Moore-Peterson [MP72] (p odd). Both of the results are proved in [MW81]; we follow those proofs.

[figure: π_{ij}X; j − i; i; i_{0}; i_{1}; j_{0} − α; j_{0}; slope \frac{1}{d−1}]

Figure 2.3.A. Vanishing line at the prime 2. In this picture, π_{ij}X = 0 above the slanted line (the vanishing line), to the left of the vertical line j − i = j_{0}, and below the horizontal line i = i_{0}. α is a number depending only on the slope d.

Recall that if X is an injective resolution of a comodule M, then π_{∗∗}(X) = Ext_{A}^{∗∗}(F_{p}, M). We make heavy use of Notation 2.2.8 in this section; CL-spectra are defined in Definition 1.3.6.

Theorem 2.3.1 (Vanishing line theorem). Let X be a CL-spectrum. Suppose that there is a number d so that Z(n)_{∗∗}X = 0 for all slopes n with n < d. Then π_{∗∗}X has a vanishing line of slope d: for some c, π_{ij}X = 0 when j < di − c.

Of course in (j − i, i)-coordinates, this vanishing line has slope \frac{1}{d−1}, and the vanishing occurs above this line. See Figure 2.3.A for a picture; the numbers i_{0}, i_{1}, and j_{0} there come from the definition of CL-spectra, Definition 1.3.6.

Theorem 2.3.2. Let X be a CL-spectrum, and assume that Z(n)_{∗∗}X = 0 for all slopes n; then π_{∗∗}X has a “horizontal” vanishing line: for some c, π_{ij}X = 0 when i > c.

Corollary 2.3.3. Under the hypotheses of Theorem 2.3.2, X is in the localizing subcategory generated by A = HF_{p}.

Proof. This follows from Lemma 1.3.8. □

For comparison purposes, we mention the following result, which we do not prove. See Definition 0.4.1 for the definition of “cofree.”

Theorem 2.3.4 ([AM71], [MP72]). Suppose that M is a bounded below Acomodule and X is its injective resolution. Then Z(n)_{∗∗}X = 0 for all slopes n if and only if M is cofree as an A-comodule.

Remark 2.3.5. (a) These theorems are false without the assumption that X is CL. For example, the (0, 0)-connective spectrum

X = ⋁_{i≥0, j≥0} Σ^{i,i+j}HF_{p}

is Z(n)-acyclic for all n, since Z(n)_{∗∗}HF_{p} = 0, but its homotopy π_{∗∗}X does not have a horizontal vanishing line.

(b) When we speak of phenomena “above” a vanishing line, we mean “above” in the grading of Figure 2.3.A—i.e., points (i, j−i) so that i > \frac{1}{d−1}(j−i)+b for some b.

(c) Suppose that X is CL with associated constants i_{0}, i_{1}, and j_{0}. We will see in the proof that the “intercept” c of the vanishing line in Theorem 2.3.1 may be given by c = (d − 1)i_{1} − j_{0} + α, where α = α(d) is a number depending only on d. In the grading of Figure 2.3.A, π_{ij}X = 0 above the line of slope \frac{1}{d−1} through the point (j_{0} − α, i_{1}). To compute α, we find the smallest integer n so that the map A ⟶ A(n) is an isomorphism through degree d (i.e., so that 2^{n+1} > d when p = 2, or 2(p − 1)p^{n} > d when p is odd); then

α = \begin{cases}∑_{\begin{matrix}s+t≤n+1 \\ |ξ_{t}^{p^{s}} |≤d \end{matrix}}|ξ_{t}^{p^{s}}|, & if p = 2, \\ ∑_{\begin{matrix}s+t≤n \\ |ξ_{t}^{p^{s}} |≤d \end{matrix}}(d + (p − 1)|ξ_{t}^{p^{s}}|) + ∑_{\begin{matrix}i≤n \\ |τ_{i}|≤d \end{matrix}}|τ_{i}|, & if p is odd. \end{cases}

For Theorem 2.3.2, the intercept is i_{1}—the homotopy π_{ij}X is zero if i > i_{1}.

(d) For both of these theorems (in fact, for all of the results of this section), we can weaken condition (b) of Definition 1.3.6 slightly—it suffices to assume that for all i, there is a j_{0} = j_{0}(i) so that π_{ij}X = 0 if j − i < j_{0}. With this assumption, one replaces j_{0} in the above formulas for c with min{j_{0}(i) | i_{0} ≤ i ≤ i_{1}}.

One can generalize these results somewhat. The same proofs carry over essentially unchanged, so to keep the notation simple we prove Theorems 2.3.1 and 2.3.2, rather than the following.

Theorem 2.3.6. Let X be a CL-spectrum, and fix a quotient Hopf algebra A′ of A.

(a) Suppose that there is a number d so that Z(n)_{∗∗}X = 0 for all slopes n with n < d and y_{n} ≠ 0 in A′. Then HA′_{∗∗}X has a vanishing line of slope d: for some c, HA′_{ij}X = 0 when j < di − c.

(b) If Z(n)_{∗∗}X = 0 for all slopes n with y_{n} ≠ 0 in A′, then HA′_{∗∗}X has a “horizontal” vanishing line; hence HA′ ∧ X is in the localizing subcategory generated by A = HF_{p}.

(c) Suppose that X is an injective resolution of a bounded below A-comodule M. Then Z(n)_{∗∗}X = 0 for all slopes n with y_{n} ≠ 0 in A′ if and only if M is a cofree A′-comodule.

Of course, HA′_{∗∗} relates to Ext_{A′} the same way π_{∗∗} relates to Ext_{A}, so one should view this result as a vanishing line theorem over A′. (One could also work in Stable(A′) and prove a result about π_{∗∗} in that category, but it seems better to work in Stable(A) whenever possible.)

2.3.1. Proof of Theorems 2.3.1 and 2.3.2 when p = 2. We give the proofs of Theorems 2.3.1 and 2.3.2 when p = 2. In the next subsection we indicate the changes necessary when working at odd primes.

The proof has two parts: in Lemma 2.3.11, we show that it suffices to check that HB_{∗∗}X has a vanishing line for all finite-dimensional quotient Hopf algebras B of A. In the rest of the lemmas, we verify the presence of the vanishing line for finite-dimensional quotients by induction on dimension. We give the full proofs of the theorems at the end of the subsection.

As noted in Remark 2.1.3, the classification theorem 2.1.1 of quotient Hopf algebras B of A tells us that if B is finite-dimensional, then for some s and t we have the following conditions:

• ξ_{t}^{2^{s}} ≠ 0 in B,

• ξ_{t}^{2^{s+1}} = 0 in B,
• ξ_{j}^{2^{s}} = 0 in B for all j < t.

Hence by Lemma 2.1.2, there is a Hopf algebra extension

E[ξ_{t}^{2^{s}}] ⟶ B ⟶ C,

and so by Lemma 1.2.15 an element h_{ts} ∈ HB_{1,|ξ_{t}^{2^{s}} |} and a cofibration

Σ^{1,|ξ_{t}^{2^{s}} |}HB ⟶^{h_{ts}} HB ⟶ HC.

Given this situation, we have the following result. (Given a spectrum Y and a self-map v: Σ^{ij}Y ⟶ Y, we say that v acts nilpotently on π_{∗∗}Y if for all y ∈ π_{∗∗}Y, some power of π_{∗∗}(v) annihilates y.)

Lemma 2.3.7. Assume that B is a finite-dimensional quotient Hopf algebra of A, and fix s and t as above.

(a) If s ≥ t, then h_{ts} ∈ HB_{∗∗} is nilpotent; hence the self-map h_{ts}: HB ⟶ HB is nilpotent.

(b) Given an object X in Stable(A), consider the cofibration

Σ^{1,|ξ_{t}^{2^{s}} |}HB ∧ X ⟶^{h_{ts}∧1_{X}} HB ∧ X ⟶ HC ∧ X.

If h_{ts} ∧ 1_{X} acts nilpotently on HB_{∗∗}X, and if HC_{∗∗}X has a vanishing line of slope d, then HB ∧ X has a vanishing line of slope d. The difference in intercepts depends only on d and |ξ_{t}^{2^{s}}|—it is independent of X. Remark 2.3.8. In fact, the nilpotence of h_{ts} in part (a) does not depend on B

being finite-dimensional. See Theorem B.2.1 for a generalization, due to Lin.

Proof. Lin proved part (a) in [Lin77a, Corollary 3.2]; see also Anderson-Davis [AD73] and Miller-Wilkerson [MW81, Proposition 4.1].

For part (b), we look at the long exact sequence in homotopy coming from the given cofibration (we write h for π_{∗∗}(h_{ts} ∧ 1_{X})):

⋯ ⟶ HC_{i−1,j}X ⟶ HB_{i−1,j−|ξ_{t}^{2^{s}} |}X ⟶^{h} HB_{ij}X ⟶ HC_{ij}X ⟶ · · · .

Fix a bidegree (i, j) above the vanishing line for HC_{∗∗}X. There are two cases: suppose first that |ξ_{t}^{2^{s}}| > d. The exact sequence tells us that

HB_{i−1,j−|ξ_{t}^{2^{s}} |}X ⟶^{h} HB_{ij}X

is an epimorphism. Since |ξ_{t}^{2^{s}}| > d, then (i − 1, j − |ξ_{t}^{2^{s}}|) is above the vanishing line for HC_{∗∗}X. Hence

HB_{i−k,j−k|ξ_{t}^{2^{s}} |}X ⟶^{h} HB_{i−(k−1),j−(k−1)|ξ_{t}^{2^{s}} |}X

is an epimorphism for all k > 0. Also since |ξ_{t}^{2^{s}}| > d, then one can see that for k ≫ 0, (i − k, j − (k − 1)|ξ_{t}^{2^{s}}|) is a bidegree above the vanishing line for HC_{∗∗}X, so

HB_{i−k,j−k|ξ_{t}^{2^{s}} |}X ⟶^{h} HB_{i−(k−1),j−(k−1)|ξ_{t}^{2} |}X

is an isomorphism. Since h acts nilpotently on HB_{∗∗}X, though, these groups must be zero for k large; since they surject onto HB_{ij}X, then HB_{ij}X = 0. Note that in this case, the vanishing line for HC_{∗∗}X is the same as that for HB_{∗∗}X.

Suppose, on the other hand, that |ξ_{t}^{2^{s}}| ≤ d. If (i, j) is a bidegree above the vanishing line, then so is (i + 1, j), so the map

HB_{i,j−|ξ_{t}^{2^{s}} |}X ⟶^{h} HB_{i+1,j}X

is an isomorphism. Arguing as above, we see that the group HB_{i,j−|ξ_{t}^{2^{s}} |}X must be zero. In this case, the intercept of the HB vanishing line changes by |ξ_{t}^{2^{s}}|: HB_{ij}X = 0 when j < di − c − |ξ_{t}^{2^{s}}|. □

Lemma 2.3.9. Fix a spectrum X satisfying condition (a) of Definition 1.3.6.

(a) Given an extension

E[ξ_{t}^{2^{s}}] ⟶ B ⟶ C

where B and C are quotient Hopf algebras of A, if |ξ_{t}^{2^{s}}| ≥ d and if HC_{∗∗}X has a vanishing line of slope d, then HB_{∗∗}X has a vanishing line of slope d. In fact, HB_{∗∗}X has the same vanishing line as HC_{∗∗}X.

(b) Given finite-dimensional quotient Hopf algebras B ↠ C of A, if HC_{∗∗}X has a vanishing line of slope d and if the map B ↠ C is an isomorphism in dimensions less than d, then HB_{∗∗}X has a vanishing line of slope d. In fact, HB_{∗∗}X has the same vanishing line as HC_{∗∗}X.

Proof. Part (a): (This proof is based on that of [MW81, Proposition 3.2].) We assume that HC_{i,j}X = 0 when j < di−c, and we want to show that HB_{i,j}X = 0 when j < di − c. We prove this by induction on i. By Lemma 1.2.15, we have a cofibration

Σ^{1,|ξ_{t}^{2^{s}} |}HB ∧ X ⟶^{h_{ts}} HB ∧ X ⟶ HC ∧ X.

This gives us a long exact sequence in homotopy:

(2.3.10)  ⋯ ⟶ HB_{i−1,j−|ξ_{t}^{2^{s}} |}X ⟶ HB_{i,j}X ⟶ HC_{i,j}X ⟶ · · · .

Since X satisfies condition (a) of Definition 1.3.6, then for i sufficiently small and for all j, we have

π_{i,j}(X) = 0 = HB_{i,j}X = HC_{i,j}X.

This starts the induction: if i_{0} is the smallest value of i for which HB_{i,j}X is nonzero, then we have an inclusion HB_{i_{0},j}X ↪ HC_{i_{0},j}X (for all j).

The inductive step is also easy: if we have i and j with j < di − c, then j − |ξ_{t}^{2^{s}}| < d(i − 1) − c; hence both HC_{i,j}X and HB_{i−i,j−|ξ_{t}^{2^{s}} |}X are zero. So we apply exactness in the long exact sequence (2.3.10).

Part (b): Given B ↠ C with B finite-dimensional (or more generally, with C of finite index in B), then there is a sequence of extensions

E[ξ_{t_{1}}^{2^{s_{1}}} ] ⟶ B ⟶ B_{1},
E[ξ_{t_{2}}^{2^{s_{2}}} ] ⟶ B_{1} ⟶ B_{2},
⋮
E[ξ_{t_{n}}^{2^{sn}}] ⟶ B_{n−1} ⟶ C.

(See [HS98, Lemma A.15] or [MW81, Lemmas 3.4–3.5], for example.) If, furthermore, B ↠ C is an isomorphism through degree d − 1, then each ξ_{t_{i}}^{2^{s_{i}}} has degree at least d. So apply induction and part (a). □

Lemma 2.3.9 is a special case of the following; we have stated and proved them separately because at odd primes we need a variant of the above result, while the following holds as stated.

Lemma 2.3.11. Fix an object X satisfying condition (a) of Definition 1.3.6. Given a surjection of coalgebras f: A ↠ B, if f is an isomorphism below degree d, and if HB_{∗∗}X has a vanishing line of slope d, then HA_{∗∗}X has a vanishing line of slope d. In fact, they have the same vanishing line.

Proof. We have an injection of comodules F_{2} = A □_{A}F_{2} ⟶ A □_{B}F_{2}, and the cokernel is zero below dimension d. Taking injective resolutions gives a cofibration

Σ^{1,0}Z ⟶ HA ⟶ HB ⟶ Z,

where Z is (0, d)-connective (Definition 1.3.4). So to show that HA_{ij}X = π_{ij}X = 0 if j < di − c, one argues by induction on i just as in Lemma 2.3.9(a). □

Lemma 2.3.12. Let X be a spectrum satisfying conditions (b) and (c) of Definition 1.3.6. Suppose that E is a finite-dimensional elementary quotient Hopf algebra of A. If (P_{t}^{s})_{∗∗}X = 0 whenever ξ_{t}^{2^{s}} ≠ 0 in E, then HE_{∗∗}X has a horizontal vanishing line. In fact, given i_{1} as in condition (c), then HE_{i∗}X = 0 if i > i_{1}.

Proof. Since the dual of E is an exterior algebra, we abuse notation and write E = E[x_{1}, x_{2}, …, x_{n}] where each x_{k} is primitive; in other words, each x_{k} is equal to ξ_{t}^{2^{s}} for some s and t. We point out that the x_{k}’s have distinct degrees.

Now HE_{∗∗} is a polynomial algebra on n generators; we write v_{k} ∈ HE_{1,|x_{k}|} for the generator corresponding to x_{k}. We also use v_{k} to denote the corresponding selfmap of HE, and we let HE/(v_{k}) be its cofiber. Note that the self-map v_{k} induces multiplication by v_{k} on homotopy, and since v_{k} ∈ HE_{∗∗} is a nonzero divisor,

π_{∗∗}(HE/(v_{k})) = (π_{∗∗}HE)/(v_{k}).

We define HE/(v_{k_{1}}, …, v_{k_{m}}) similarly, assuming that the numbers k_{1}, …, k_{m} are distinct. Hence if ℓ is an integer with 1 ≤ ℓ ≤ n and ℓ ∉ {k_{1}, …, k_{m}}, then v_{ℓ}: HE ⟶ HE induces a self-map of HE/(v_{k_{1}}, …, v_{k_{m}}).

Also note that if x_{k} = ξ_{t}^{2^{s}}, then by a change-of-coalgebras, we have

(P_{t}^{s})_{∗∗}X = v_{k}^{−1}(HE/(v_{1}, …, v̂_{k}, …, v_{n}))_{∗∗}X.

We refer to this as the x_{k}-homology of X.

We claim for any set {k_{1}, …, k_{m}} ⊆ {1, …, n},

(HE/(v_{k_{1}}, …, v_{k_{m}}))_{∗∗}(X)

has a horizontal vanishing line. We prove this by induction on n − m. (We will have proved the lemma when m = 0, i.e., when n − m = n.)

We have to deal with the first few cases before we can apply the inductive step. If n − m = 0, then {k_{1}, …, k_{m}} = {1, …, n}, so HE/(v_{k_{1}}, …, v_{k_{m}}) = HF_{2}, and by assumption, HF_{2}_{∗∗}X has a horizontal vanishing line with intercept i_{1}. If n−m = 1, then we let k be the integer so that

{k} ∪ {k_{1}, …, k_{m}} = {1, 2, …, n}.

To simplify the notation, we let HE_{m} = HE/(v_{k_{1}}, …, v_{k_{m}}). Then we have a cofibration

HE_{m} ∧ X ⟶^{v_{k}} HE_{m} ∧ X ⟶ HF_{2} ∧ X.

By hypothesis, HF_{2}_{∗∗}X has a horizontal vanishing line, so the map v_{k} induces an isomorphism in π_{ij} for i > i_{1}. On the other hand, the x_{k}-homology of X is zero, and the x_{k}-homology of X is equal to v_{k}^{−1}(HE_{m})_{∗∗}X. Since v_{k} induces an isomorphism for i > i_{1}, and since this localization is zero, then v_{k} must be zero when i > i_{1}. So (HE_{m})_{∗∗}X has a horizontal vanishing line with intercept i_{1}.

Now fix m with n − m ≥ 2, and assume that

(HE/(v_{ℓ_{1}}, …, v_{ℓ_{t}}))_{∗∗}X

has a horizontal vanishing line whenever n − t < n − m. As above, let HE_{m} = HE/(v_{k_{1}}, …, v_{k_{m}}). Pick distinct integers k, ℓ ≤ n which are not in {k_{1}, …, k_{m}}, and consider the following diagram, in which each row and column is a cofibration:

Σ^{2,|x_{k}|+|x_{ℓ}|}HE_{m} ⟶^{v_{k}} Σ^{1,|x_{ℓ}|}HE_{m} ⟶ Σ^{1,|x_{ℓ}|}HE_{m}/(v_{k})
v_{ℓ}↓  ^{v_{ℓ}}↓  ↓
Σ^{1,|x_{k}|}HE_{m}  ⟶^{v_{k}} HE_{m} ⟶ HE_{m}/(v_{k})
↓  ↓  ↓
Σ^{1,|x_{k}|}HE_{m}/(v_{ℓ}) ⟶ HE_{m}/(v_{ℓ}) ⟶ HE_{m}/(v_{k}, v_{ℓ}).

Now smash this diagram with X. By induction, all of the spectra HE_{m}/(v_{k}, v_{ℓ})∧X, HE_{m}/(v_{k})∧X, and HE_{m}/(v_{ℓ})∧X have horizontal vanishing lines with intercept i_{1}. Now we apply π_{ij}(−); for i > i_{1}, the maps labeled v_{k} and v_{ℓ} induce isomorphisms on π_{ij}, so we have

π_{i,j}HE_{m} ∧ X ⟶_{≅}^{v_{ℓ}^{−1}∘v_{k}} π_{i,j+|x_{k}|−|x_{ℓ}|}HE_{m} ∧ X.

This isomorphism, combined with the facts that |x_{k}| ≠ |x_{ℓ}| and that π_{ij}(HE_{m} ∧X) is zero when j ≪ 0, implies that π_{ij}(HE_{m} ∧ X) = 0 for all j; i.e., π_{∗∗}(HE_{m} ∧ X) has the predicted horizontal vanishing line. This completes the inductive step, and hence the proof. □

Proof of Theorem 2.3.1. By Lemma 2.3.11, if we know that HB_{∗∗}X has a vanishing line of slope d for all finite-dimensional quotient Hopf algebras B of A, then π_{∗∗}(X) will, also. (For example, the map A ⟶ A(n) is an isomorphism through degree 2^{n+1} − 1, so apply the lemma to the case B = A(n) where n is chosen large enough so that 2^{n+1} − 1 ≥ d.)

Now we show that HB_{∗∗}X has a vanishing line of slope d for all quotients B of A with dim_{F_{2}} B < ∞, by induction on dim_{F_{2}} B: The case where dim_{F_{2}} B = 1 (i.e., B = F_{2}) is taken care of by condition (c) of the definition of CL, Definition 1.3.6, so we move on to the inductive step. By Remark 2.1.3, there is a Hopf algebra extension

E[ξ_{t}^{2^{s}}] ⟶ B ⟶ C.

By hypothesis, HC_{∗∗}X has a vanishing line of slope d, and we want to produce a vanishing line for HB_{∗∗}X. There are several cases:

(1) If s ≥ t, then we are done by Lemma 2.3.7.

(2) If s < t and |ξ_{t}^{2^{s}}| > d, then we are done by Lemma 2.3.9(a).

(3) If s < t and |ξ_{t}^{2^{s}}| ≤ d, then we have to prove something. We may assume that ξ_{j} = 0 in B for j < t. Let

B = F_{2}[ξ_{t}, ξ_{t+1}, ξ_{t+2}, …]/(ξ_{t}^{2^{s+1}}, ξ_{t+1}^{2^{n_{1}}}, ξ_{t+2}^{2^{n_{2}}}, ξ_{t+3}^{2^{n_{3}}}, …),
E = F_{2}[ξ_{t}, ξ_{t+1}, ξ_{t+2}, …]/(ξ_{t}^{2^{s+1}}, ξ_{t+1}^{2^{m_{1}}} , ξ_{t+2}^{2^{m_{2}}} , ξ_{t+3}^{2^{m_{3}}} , …),

where m_{i} = max(min(n_{i}, s − i), 0). By Proposition 2.1.7, E is an elementary quotient Hopf algebra of A (and of B). Furthermore, if ξ_{v}^{2^{u}} ≠ 0 in E, then |ξ_{v}^{2^{u}}| ≤ |ξ_{t}^{2^{s}}|. By hypothesis on X, then, (P_{v}^{u})_{∗∗}X = 0 when ξ_{v}^{2^{u}} ≠ 0 in E. So we apply Lemma 2.3.12 to conclude that HE_{∗∗}X has a horizontal vanishing line with intercept i_{1}. Since X satisfies condition (b) of Definition 1.3.6, then HE_{∗∗}X also has a vanishing line of slope |ξ_{t}^{2^{s}}| + 1 (the line with this slope through the point (i_{1}, j_{0})). We then apply Lemma 2.3.9(b) to the case B ↠ E to conclude that HB_{∗∗}X has a vanishing line of slope |ξ_{t}^{2^{s}}| + 1 (and in fact the same vanishing line).

Now, the map h_{ts} acts at slope |ξ_{t}^{2^{s}}|, and hence acts nilpotently on HB_{∗∗}X. By Lemma 2.3.7(b), then, the vanishing line for HC_{∗∗}X, which has slope d, gives one for HB_{∗∗}X. □

Proof of Theorem 2.3.2. By Proposition 2.1.5, it suffices to show that there is a uniform horizontal vanishing line for all of the groups HA(n)_{∗∗}X. Lemma 2.3.12 and Lemma 2.3.9 together show that each HA(n)_{∗∗}X has a horizontal vanishing line, and in fact these results identify the vanishing line: HA(n)_{i∗}X = 0 if i > i_{1}, with i_{1} as in condition (c) of Definition 1.3.6. □

2.3.2. Changes necessary when p is odd. The proofs of Theorems 2.3.1 and 2.3.2 given above go through with a few changes when the prime p is odd; we indicate those changes in this subsection.

We start with the same set-up as when p = 2: we have an extension of Hopf algebras in one of the following forms:

(E) E[τ_{n}] ⟶ B ⟶ C,
(D) D[ξ_{t}^{p^{s}}] ⟶ B ⟶ C.

If case (E) arises, then Lemma 2.3.7(b) carries over as stated (writing v_{n} for the homotopy element associated to the element τ_{n}, rather than h_{ts}).

Otherwise extension (D) arises, and we may assume that we have • ξ_{t}^{p^{s}} ≠ 0 in B,

• ξ_{t}^{p^{s+1}} = 0 in B,

• ξ_{j}^{p^{s}} = 0 in B for all j < t. Lemma 1.2.15 then gives an element b_{ts} ∈ HB_{2,p|ξ_{t}^{p^{s}} |}, and hence a cofibration

(2.3.13)  Σ^{2,p|ξ_{t}^{p^{s}} |}HB ⟶^{b_{ts}} HB ⟶ H̃C ⟶ Σ^{1,p|ξ_{t}^{p^{s}} |}HB,

where H̃C is the cofiber of a self-map of HC, as in Lemma 1.2.15. Hence if HC_{∗∗}X has a vanishing line of slope d, then so does H̃C_{∗∗}X. More precisely, if HC_{ij}X = 0 when j < di − c, then H̃C_{ij}X = 0 when j < di − c + min(0, |ξ_{t}^{p^{s}}| − d).

Here is the odd prime analogue of Lemma 2.3.7. Lemma 2.3.14. (a) Given s and t as above so that we have extension (D),

if s ≥ t, then b_{ts} ∈ HB_{∗∗} is nilpotent; hence the self-map b_{ts}: HB ⟶ HB is nilpotent.

(b) Given extension (D) and an object X, consider the cofibration

Σ^{2,p|ξ_{t}^{p^{s}} |}HB ∧ X ⟶^{b_{ts}∧1_{X}} HB ∧ X ⟶ H̃C ∧ X.

If b_{ts} ∧ 1_{X} acts nilpotently on HB_{∗∗}X, and if H̃C_{∗∗}X has a vanishing line of slope d, then HB ∧ X has a vanishing line of slope d. The difference in intercepts depends only on d and |ξ_{t}^{p^{s}}|—it is independent of X.

Proof. For part (a), see [MW81, Proposition 4.1]. Part (b) is proved just as in the p = 2 case. □

The odd prime version of Lemma 2.3.9 is as follows. Lemma 2.3.15. Fix an object X satisfying condition (a) of Definition 1.3.6.

(a) Given an extension

D[ξ_{t}^{p^{s}}] ⟶ B ⟶ C,

if ^{\frac{p|ξ_{t}^{2^{s}} |}{2}} ≥ d and if HC_{∗∗}X has a vanishing line of slope d, then HB_{∗∗}X has a vanishing line of slope d. In fact, HB_{∗∗}X has the same vanishing line as H̃C_{∗∗}X.

(b) Given an extension

E[τ_{n}] ⟶ B ⟶ C,

if |τ_{n}| ≥ d and if HC_{∗∗}X has a vanishing line of slope d, then HB_{∗∗}X has a vanishing line of slope d. In fact, HB_{∗∗}X has the same vanishing line as HC_{∗∗}X.

(c) Given finite-dimensional quotients B ↠ C of A, if HC_{∗∗}X has a vanishing line of slope d and if the map B ↠ C is an isomorphism in odd dimensions less than d and in even dimensions less than \frac{2d}{p}, then HB_{∗∗}X has a vanishing line of slope d. Furthermore, the difference in intercept between the two vanishing lines is independent of X.

Proof. Part (a) is proved just as is Lemma 2.3.9(a), but based on the cofibration (2.3.13). Part (b) is the same as Lemma 2.3.9(a), except for the characteristic of the ground field, which is not relevant. Part (c) is proved, as in Lemma 2.3.9(b), by induction and parts (a) and (b). Since in (a), the vanishing line for H̃C_{∗∗}X may have a different intercept than that for HC_{∗∗}X, the intercept for HB_{∗∗}X will change as the induction proceeds, but it will change by amounts dependent only on d. □

Lemma 2.3.11 holds as stated, regardless of the prime involved. Here is the analogue of Lemma 2.3.12.

Lemma 2.3.16. Let X be an object satisfying conditions (b) and (c) of Definition 1.3.6. Suppose that E is a finite-dimensional elementary quotient Hopf algebra of A. If (P_{t}^{s})_{∗∗}X = 0 whenever ξ_{t}^{2^{s}} ≠ 0 in E, and (Q_{n})_{∗∗}X = 0 whenever τ_{n} ≠ 0 in E, then HE_{∗∗}X has a horizontal vanishing line. In fact, given i_{1} as in condition (c), then HE_{i∗}X = 0 if i > i_{1}.

The proof is the same as that for Lemma 2.3.12, using cofibrations of the form

(2.3.13) repeatedly. We also need to point out that the slopes ^{\frac{p|ξ_{t}^{p^{s}} |}{2}} and |τ_{n}| are all distinct. (Note also that in this case, we may choose d as large as we like, so that a horizontal vanishing line for HC_{∗∗}X induces the same one for H̃C_{∗∗}X.)

Finally, given these modified tools, the proofs of the main theorems go through essentially unchanged.

2.4. Self-maps via vanishing lines

In this section we use vanishing lines to construct self-maps of finite spectra. For each finite X, we construct one self-map; we give many others in Theorem 6.1.3. We make heavy use of Notation 2.2.8 in this section.

Our approach for this section is based on some recent work in ordinary stable homotopy theory, as described in work of Ravenel [Rav86] and Hopkins and Smith [HS98]. For now, we view the objects P_{t}^{s} and Q_{n} (i.e., the Z(n)’s) as the analogues of Morava K-theories. This is not a perfect analogy, because the Morava K-theories detect nilpotence, while the Z(n)’s do not.

The following definition is based on the definition of “v_{n}-map” in ordinary stable homotopy theory; see [HS98] and [Rav92]. We will investigate and generalize this definition in Section 6.2.

Notation 2.4.1. Fix a slope n. The ring z(n)_{∗∗} is either polynomial on one generator or has such a polynomial algebra as a tensor factor; we call the polynomial generator u_{n}. (In the notation of Remark 2.1.3 and Proposition 2.2.3, u_{n} is one of h_{ts}, b_{ts}, or v_{n}.) We write η_{Z(n)} for the composite S^{0} ⟶^{η} z(n) ⟶ Z(n), where η is the “unit map” of the spectrum z(n)—i.e., the map given by Proposition 1.2.8(b).

Definition 2.4.2. Fix a spectrum X and a slope n.

(a) A self-map f ∈ [X, X]_{∗∗} is a u_{n}-map if for some j,

η_{Z(n)} ∧ f = u_{n}^{j} ∧ 1_{X}
as elements of [X, Z(n) ∧ X]_{∗∗}.

(b) We say that X is of type n if Z(d)_{∗∗}X = 0 for d < n, and Z(n)_{∗∗}X ≠ 0. Notice that if Z(n)_{∗∗}X = 0, then the zero map of X is a u_{n}-map. We point out that in the ordinary stable homotopy category, Ravenel showed

in [Rav84, Theorem 2.11] that for any finite spectrum X, K(n)_{∗}X ≠ 0 ⇒ K(n + 1)_{∗}X ≠ 0. The analogous statement here, with Z(n)_{∗∗} rather than K(n)_{∗}, does not hold. See Proposition 6.8.2 (and also [Pal96b, Proposition 3.10 and Theorem A.1]) for the correct statement when p = 2, and for a guess in the odd prime case. We do know that if X is a nonzero finite spectrum, then by Theorem 2.3.2 and Corollary 6.6.8, for some n we have Z(n)_{∗∗}X ≠ 0. (One could also use the Atiyah-Hirzebruch spectral sequence to show this.)

The following first appeared for modules in [Pal92]. It is a slight generalization of a result of Hopkins and Smith [HS98]. See Theorems 5.1.2 and 6.1.3 for stronger results when p = 2.

Theorem 2.4.3. Fix a finite spectrum X of type n. Then for some k, there is a non-nilpotent u_{n}-map

v: Σ^{k,kn}X ⟶ X.

To prove this, we need a “relative vanishing line” result; this is a generalization of a standard result—see [Rav86, Lemma 3.4.9], for instance.

Lemma 2.4.4. Fix a slope n. Suppose that X is a CL-spectrum, and suppose that X is of type at least n; hence π_{∗∗}X has a vanishing line of slope n. Given m ≥ 0, let M = M(m) be the number below which degree A ↠ A(m) is an isomorphism. Then the Hurewicz map h: π_{∗∗}X ⟶ HA(m)_{∗∗}X is an isomorphism above a line of slope n: for some c independent of m, h is an isomorphism on π_{ij} when j < ni + M − c.

Proof. Let W denote the fiber of the map S^{0} ⟶ HA(m). Since the kernel of the map A ⟶ A(m) is zero below degree M, if we choose numbers i_{0}, i_{1}, and j_{0} so that X satisfies Definition 1.3.6, then W ∧ X satisfies the conditions with the numbers i_{0}, i_{1}, and j_{0} + M. Hence for some number c, the vanishing line theorem 2.3.1 implies that π_{ij}(W ∧ X) = 0 when j < ni + M − c. By Remark 2.3.5, c depends only on n, i_{1}, and j_{0}. □

We have the following result based on work of Lin [Lin77a] and Wilkerson [Wil81]; this follows immediately from [HS98, Theorem 4.13], stated above as Theorem 0.4.10.

Proposition 2.4.5. Fix m and consider the quotient Hopf algebra A(m) of A.

(a) Let p = 2. Fix integers s < t with ξ_{t}^{2^{s}} nonzero in A(m) (i.e., with s + t ≤ m + 1). Then the restriction map

Ext_{A(m)}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{F_{2}[ξ_{t}]/(ξ_{t}^{2^{s+1}} )}^{∗∗}(F_{2}, F_{2}) ≅ F_{2}[h_{t0}, h_{t1}, …, h_{ts}]

is “surjective modulo nilpotence” (i.e., for every element in the codomain, some power of it is in the image). Hence for some i = i(m), there is a non-nilpotent element

w ∈ Ext_{A(m)}^{i,i|ξ_{t}^{2^{s}} |}(F_{2}, F_{2}) = HA(m)_{i,i|ξ_{t}^{2^{s}} |}

which restricts to h_{ts}^{i}.

(b) Let p be odd. Fix integers s < t with ξ_{t}^{p^{s}} nonzero in A(m) (i.e., with

s + t ≤ m). Then the restriction map

Ext_{A(m)}^{∗∗}(F_{p}, F_{p}) ⟶  Ext_{F_{p}[ξ_{t}]/(ξ_{t}^{p^{s+1}} )}^{∗∗}(F_{p}, F_{p})
∥
F_{p}[b_{t0}, …, b_{ts}] ⊗ Λ[h_{t0}, …, h_{ts}]

is surjective modulo nilpotence. Hence for some j = j(m), there is a non-nilpotent element

w ∈ Ext_{A(m)}^{2j,jp|ξ_{t}^{p^{s}} |}(F_{p}, F_{p}) = HA(m)_{2j,jp|ξ_{t}^{p^{s}} |}

which restricts to b_{ts}^{j}.

(c) Let p be odd. Fix an integer t with τ_{t} nonzero in A(m) (i.e., with t ≤ m). Then the restriction map

Ext_{A(m)}^{∗∗}(F_{p}, F_{p}) ⟶ Ext_{E[τ_{t}]}^{∗∗}(F_{p}, F_{p}) ≅ F_{p}[v_{t}]

is surjective mod nilpotence. Hence for some k = k(m), there is a nonnilpotent element

w ∈ Ext_{A(m)}^{k,k|τ_{t}|}(F_{p}, F_{p}) = HA(m)_{k,k|τ_{t}|}

which restricts to v_{t}^{k}. We need the following lemma. Recall from Notation 0.5.1 that D[y] denotes

the Hopf algebra F_{p}[y]/(y^{p}) with y primitive.

Lemma 2.4.6. Fix a slope n, and choose an integer m large enough that y_{n} is nonzero in A(m) (and hence so that D[y_{n}] or E[y_{n}] is a quotient coalgebra of A(m) over which A(m) is injective).

(a) Then the map HA(m)_{∗∗} ⟶ z(n)_{∗∗} is an algebra map, and some power of the polynomial generator u_{n} of z(n)_{∗∗} is in the image.

(b) For any object X, the following diagram commutes:

[X, HA(m) ∧ X]_{∗∗} ⟶ [X, z(n) ∧ X]_{∗∗}
−∧X↑  ↑^{−∧X}
HA(m)_{∗∗}  ⟶  z(n)_{∗∗}.

(c) For any object X with Z(n)_{∗∗}X ≠ 0, the map z(n)_{∗∗} ⟶ [X, z(n) ∧ X]_{∗∗} is an injection.

Proof. Part (a) follows from Proposition 1.2.8(b) and Proposition 2.4.5. Part (b) follows from the fact that [X, z(n) ∧ X]_{∗∗} is isomorphic to cochain

homotopy classes of D[y_{n}]-linear (respectively, E[y_{n}]-linear) self-maps of X, and the horizontal maps are just restriction.

For part (c), we merely note that if Z(n)_{∗∗}X ≠ 0, then the identity map on X is not cochain homotopic to zero over D[y_{n}] (respectively, over E[y_{n}]). □

Proof of Theorem 2.4.3. Since X is of type n, so is X ∧ DX, where DX is the Spanier-Whitehead dual of X; hence Lemma 2.4.4 applies to [X, X]_{∗∗} = π_{∗∗}(X∧ DX). By that lemma, if we choose m large enough, then the map [X, X]_{∗∗} ⟶ [X, HA(m) ∧ X]_{∗∗} is an isomorphism in the bidegrees (k, kn), for all integers k. Now we apply Lemma 2.4.6 to find a non-nilpotent element w ∈ HA(m)_{k,kn} (for some k) which maps nontrivially to z(n)_{∗∗} and to [X, z(n)∧X]_{∗∗}. The lift of w∧1_{X} to [X, X]_{k,kn} has the desired properties. □

We will need the following lemma in Section 2.5.

Lemma 2.4.7. Fix a finite type n spectrum X. Then for some k, the u_{n}- map v ∈ [X, X]_{∗∗} constructed in Theorem 2.4.3 is central in a band parallel to the vanishing line. This band includes the origin.

Proof. Consider the element w ∈ HA(m)_{k,kn}, as given by Proposition 2.4.5. Since HA(m)_{∗∗} is a commutative ring and [X, HA(m) ∧ X] is an algebra over it, w maps to a central element in [X, HA(m) ∧ X]_{∗∗}. By Lemma 2.4.4, [X, X]_{∗∗} ⟶ [X, HA(m) ∧ X]_{∗∗} is an isomorphism in a band parallel to the vanishing line; by choice of m, that band includes the origin. Hence the lift of w to [X, X]_{∗∗} is central in that band. □

2.5. Construction of spectra of specified type

In this section we construct certain objects for later use; these are analogues of the “generalized Toda V (n) spectra,” as used by Mahowald and Sadofsky [MS95], among others. Some of the basic ideas are standard; most of the rest are due to them. A few of the details are different in our setting.

We start by constructing the relevant spectra. See Notations 2.2.8 and 2.4.1 for the terminology used here.

Warning 2.5.1. Recall that u_{n} is the polynomial generator of z(n)_{∗∗}, and its powers lie on a line of slope n; u_{n} is an element of either z(n)_{1,n} or z(n)_{2,2n}, depending on the prime and whether z(n) is p_{t}^{s} or q_{n}—see Proposition 2.2.3. Recall also that Z(n)_{∗∗} = u_{n}^{−1}z(n)_{∗∗}. So as to avoid dividing all of our arguments into cases, we will abuse notation and write u_{n}^{i} for the power of u_{n} in Z(n)_{i,in} (and similarly for u_{n}-maps: a self-map u_{n}^{i}: X ⟶ X has bidegree (i, in)). Hence when Z(n) = P_{t}^{s} and p is odd, only even powers of u_{n} make sense.

Proposition 2.5.2. Let p be a prime, and fix a slope n. Let 1 = d_{1} < d_{2} < ⋯ < d_{m} be the slopes less than n. For any integers k_{1}, …, k_{m}, there are integers j_{1}, …, j_{m} with k_{i} ≤ j_{i} for each i, so that there is a finite spectrum F = F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) satisfying the following.

(a) When n = 1, F = S^{0}.

(b) F is Z(d)_{∗∗}-acyclic for d < n, and Z(n)_{∗∗}F ≠ 0.

(c) Hence by Theorem 2.4.3, F has a u_{n}-map; call it u. If Z(n)_{∗∗}(u) has bidegree (j_{m+1}, j_{m+1}n), then F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}, u_{n}^{j_{m+1}}) is the fiber of u. More precisely, there is a fiber sequence

F (u_{d_{1}}^{j_{1}}, …, u_{n}^{j_{m+1}}) ⟶ F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) ⟶^{u_{n}^{j_{m+1}}} Σ^{−j_{m+1},−nj_{m+1}}F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}).

(d) If u and v are two u_{n}-maps of F, then u^{i} = v^{j} for some numbers i and j.

(e) F is self-dual, and the dual of u is a u_{n}-map. That is, for some numbers q and r, the Spanier-Whitehead dual DF of F is isomorphic to Σ^{q,r}F, and u maps to a u_{n}-map under the chain of isomorphisms

[F, F] ≅ [DF, DF] ≅ [Σ^{q,r}F, Σ^{q,r}F] ≅ [F, F].

Proof. The statements of (a)-(c) indicate how the spectra are constructed. Starting with S^{0}, one applies Theorem 2.4.3 to find a u_{1}-map u_{1}^{j_{1}} of it, and one lets F (u_{1}^{j_{1}}) be the fiber. By definition of u_{1}-map, u_{1}^{j_{1}} induces an isomorphism on Z(1)_{∗∗}, so F (u_{1}^{j_{1}}) is Z(1)_{∗∗}-acyclic; u_{1}^{j_{1}} induces zero on Z(d)_{∗∗} for d > 1, so F (u_{1}^{j_{1}}) is not Z(d)_{∗∗}-acyclic for any larger value of d. One proceeds inductively. This proves (a)-(c).

Part (d): Since u and v are u_{n}-maps, then after raising them to suitable powers, we may assume that they agree when restricted to [F, z(n) ∧ F]_{∗∗}. For any m ≥ 0, we can raise them to large enough powers so that they agree when restricted to [F, HA(m) ∧ F]_{∗∗}. But when m is large enough, this ring is isomorphic to [F, F]_{∗∗} in the bidegree of interest; hence large enough powers of our self-maps agree in [F, F]_{∗∗}.

Part (e): By the definition 2.4.2 of a u_{n}-map, the dual of a u_{n}-map of a finite object X is a u_{n}-map of the dual DX. □

Note that the fiber sequences in part (c) give a sequence of degree zero maps

F = F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) ⟶ F (u_{d_{1}}^{j_{1}}, …, u_{d_{m−1}}^{j_{m−1}}) ⟶ ⋯ ⟶ F (u_{d_{1}}^{j_{1}}) ⟶ S^{0}.

We refer to the composite F ⟶ S^{0} as “projection to the top cell.”

Up to suspension, the object F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) is the analogue in Stable(A) of Mahowald and Sadofsky’s spectrum (in the usual stable homotopy category) M(p^{j_{0}}, v_{1}^{j_{1}}, …, v_{n−1}^{j_{n−1}}). We have used the letter F rather than M since our spectra are iterated fibers rather than cofibers, as they are in [MS95]. Also, the letter M can get somewhat overworked; in particular, we want to avoid confusion with the functor M_{n}^{f} of Section 3.3.

Now we establish the main properties of the spectra F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}). Thick subcategories are defined in Definition 1.3.7, as is the notation thick(Y). Bousfield classes are defined in Section 1.5.

Theorem 2.5.3. Fix a prime p, a slope n, and other notation as in Proposition 2.5.2, and let F = F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) be as in that result. Then F satisfies the following properties.

(a) For any finite spectrum W which is Z(d)_{∗∗}-acyclic for all d < n, W is in thick(F).

(b) Hence the thick subcategory generated by F and the Bousfield class of F are both independent of the choice of exponents j_{i}.

(c) Let u denote the u_{n}-map of F. Suppose that ℓ_{1}, …, ℓ_{m} are integers so that W = F (u_{d_{1}}^{ℓ_{1}}, …, u_{d_{m}}^{ℓ_{m}}) exists, and let v denote the u_{n}-map of W as given by Proposition 2.5.2(c). Then there are integers i and j so that u^{i} ∧ 1_{W} = 1_{F} ∧ v^{j} as self-maps of F ∧ W.

(d) Suppose that ℓ_{1}, …, ℓ_{m} are integers so that F (u_{d_{1}}^{ℓ_{1}}, …, u_{d_{m}}^{ℓ_{m}}) exists. If ℓ_{i} ≫ j_{i} for each i, then there is a map

F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) ⟶ F (u_{d_{1}}^{ℓ_{1}}, …, u_{d_{m}}^{ℓ_{m}})

commuting with projection to the top cell—i.e., with the map F (⋯) ⟶ S^{0}.

Proof. Part (a): For each i ≤ m, let F (U_{i}) = F (u_{d_{1}}^{j_{1}}, …, u_{d_{i}}^{j_{i}}). We have cofiber sequences

F (U_{i}) ⟶ F (U_{i−1}) ⟶^{u_{d_{i}}^{j_{i}}} F (U_{i−1}).

We claim that 1_{W} ∧u_{d_{i}}^{j_{i}} is a nilpotent self map of W ∧F (U_{i−1}). Once we know this, then Lemma 5.2.5 tells us that W ∧ F (U_{i−1}) is in the thick subcategory generated by W ∧F (U_{i}) for each i. By induction, W ∧F (U_{0}) = W is in the thick subcategory generated by W ∧F (U_{m}), which is a subcategory of the thick subcategory generated by F (U_{m}) = F.

The claim that 1_{W} ∧u_{d_{i}}^{j_{i}} is nilpotent follows by application of the vanishing line theorem 2.3.1: the group

[W ∧ F (U_{i−1}), W ∧ F (U_{i−1})]_{∗∗} = π_{∗∗}(W ∧ DW ∧ F (U_{i−1}) ∧ DF (U_{i−1}))

has a vanishing line of slope n, and 1_{W} ∧ u_{d_{i}}^{j_{i}} acts along a line of slope d_{i} < n. Part (b): This follows immediately from part (a).

Part (c): The proof of Proposition 2.5.2(d) works here, too. By Proposition 2.5.2(b), F ∧ W is Z(d)_{∗∗}-acyclic for d < n; hence [F ∧ W, F ∧ W]_{∗∗} has a vanishing line of slope at least n. If this slope is larger than n, then the powers of both u ∧ 1_{W} and 1_{F} ∧ v would eventually lie above the vanishing line, and hence would both be zero. So we may assume that the vanishing line has slope equal to n. Now, as in the proof of Proposition 2.5.2(d), we know that for some i and j, the self-maps u^{i} ∧ 1_{W} and 1_{F} ∧ v^{j} agree when restricted to

[F ∧ W, HA(m) ∧ F ∧ W]_{∗∗},

if m is large enough. Since this ring is isomorphic to [F ∧ W, F ∧ W]_{∗∗} in the bidegree of interest, we are done.

Part (d): This follows from (c). We prove it by induction on m. When m = 1, then the following diagram commutes:

F (u_{1}^{j_{1}}) ⟶ S^{0} ⟶^{u_{1}^{j_{1}}} S^{0}
∥  ↓_{u_{1}^{ℓ_{1}−j_{1}}}
F (u_{1}^{ℓ_{1}}) ⟶ S^{0} ⟶^{u_{1}^{ℓ_{1}}} S^{0}.

Hence there is an induced map F (u_{1}^{j_{1}}) ⟶ F (u_{1}^{ℓ_{1}}).

Assume that we have a map

F (u_{d_{1}}^{j_{1}}, …, u_{d_{m−1}}^{j_{m−1}}) ⟶^{f} F (u_{d_{1}}^{ℓ_{1}}, …, u_{d_{m−1}}^{ℓ_{m−1}}).
∥  ∥
F  W

We abbreviate these spectra as F and W, as indicated. Consider the following diagram, in which the rows are fiber sequences:

F (u_{d_{m}}^{j_{m}}) ⟶ F ⟶^{u_{dm}^{jm}} F
∥  ↓^{u_{dm}^{ℓm−jm}}
F (u_{d_{m}}^{ℓ_{m}}) ⟶ F ⟶^{u_{dm}^{ℓm}} F
↓^{f}  ↓^{f}
W (u_{d_{m}}^{ℓ_{m}}) ⟶ W ⟶^{u_{dm}^{ℓm}} W.

Clearly the top right square commutes. We claim that part (c) implies that the lower right square commutes, when ℓ_{m} is large enough. Given this, we get maps

F (u_{d_{m}}^{j_{m}}) ⟶ F (u_{d_{m}}^{ℓ_{m}}) ⟶ W (u_{d_{m}}^{ℓ_{m}}).

The composite is the desired map.

To see that the lower right square commutes, we imitate the proof of Corollary 6.2.11. The key is to use Spanier-Whitehead duality so that the question is whether 1∧u_{d_{m}}^{ℓ_{m}} and Du_{d_{m}}^{ℓ_{m}} ∧1 agree as self-maps of the spectrum DF ∧W. By part (c), it suffices to show that F is self-dual (up to suspension), and that the dual of u_{d_{m}}^{ℓ_{m}} is a u_{n}-map. But this is Proposition 2.5.2(e). Hence if ℓ_{m} is sufficiently large, the square commutes, and we are done. □

We point out that since F = F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) is well-defined up to Bousfield class, and since its u_{n}-map is essentially unique, then the telescope u_{n}^{−1}F is welldefined up to Bousfield class. But the telescope of a u_{n}-map of an arbitrary finite spectrum of type n could have a different Bousfield class. For example, at the prime 2, there is a non-nilpotent self-map of the sphere called d_{0}:

d_{0} : S^{4,18} ⟶ S^{0}.

(In the May spectral sequence, this element would be called h_{20}^{2}h_{21}^{2}—see Example 6.5.7(b).) Although S^{0} and S^{0}/d_{0} are both type 0, they are not Bousfieldequivalent—S^{0} sees d_{0}^{−1}S^{0}, while S^{0}/d_{0} does not. Hence the telescopes h_{0}^{−1}S^{0} and h_{0}^{−1}(S^{0}/d_{0}) probably have distinct Bousfield classes.

2.6. Further discussion

As mentioned in Subsection 2.1.1, one of the main gaps in this theory for odd p is the lack of a classification of the quasi-elementary quotient Hopf algebras of A. See Appendix B.3 for a discussion of conjectures and results related to this issue.

We note that the vanishing line theorem 2.3.1 has been used many times in many papers; it provides a convenient way to get valuable information about the Adams E_{2}-term. Combined with newer results, such as the genericity of vanishing lines in Adams spectral sequences (Theorem 1.4.4), it is even more powerful.

Theorem 2.4.3, the result that ensures the existence of a non-nilpotent self-map of any finite object, also has been used in topological applications. For example, Hopkins and Smith used it to prove the periodicity theorem [HS98, Theorem 9]: they had constructed a particular spectrum X, and they used the theorem to find a v_{n}-map of X at the E_{2}-term of the Adams spectral sequence. Later, Theorem 2.4.3 was used by Sadofsky and the author in [PS94] to give a new proof of the periodicity theorem: we used it not only to produce a v_{n}-map, but also to construct the spectrum X in question. This was done by taking iterated cofibers, as in Section 2.5: when p = 2, the cofiber of u_{1} : S^{0} ⟶ S^{0} has a u_{3}-map (3 is the next slope after 1); the cofiber of this map has a u_{6}-map, etc. This was done with modules over the Steenrod algebra, and then realized at the spectrum level in the ordinary stable homotopy category.

CHAPTER 3 Chromatic structure

In this chapter we discuss “chromatic” results in Stable(A). We start in Section 3.1 by discussing Margolis’ killing construction [Mar83, Chapter 21]. This is the analogue, in our setting, of the functor L_{n}^{f} in the ordinary stable homotopy category, so we use the same notation. We give several different constructions of the functor, and we prove various properties (e.g., for X with nice connectivity properties, if X has finite type homotopy, then so does L_{n}^{f}X). We also define an analogue of the functor L_{n}, and we show that L_{n} ≠ L_{n}^{f} if n > 1, at least at the prime 2.

We have been using the functor H heavily throughout this book; the homotopy groups of HB, for B a quotient of A, are the cohomology groups of the Hopf algebra B. In Section 3.2 we construct Ĥ(−), a version of this functor whose homotopy groups are the Tate cohomology groups of B. The spectrum ĤA(m) turns out to be equal to L_{n}^{f}HA(m), for n sufficiently large compared to m; we use this result in Section 3.3 to prove that the “chromatic tower,” the tower

L_{0}^{f}X ⟵ L_{1}^{f}X ⟵ L_{2}^{f}X ⟵ L_{3}^{f}X ⟵ ⋯,

converges to X, if X is finite. This is an extension of a theorem of Margolis [Mar83, Theorem 22.1].

In Section 3.4 we try to compare our chromatic filtration of Ext_{A}^{∗∗}(F_{p}, F_{p}) to that of Mahowald and Shick in [MS87] and [Shi88]; we have more questions than answers, though.

In Section 3.5 we discuss some other issues related to chromatic phenomena, such as constructing chromatic towers in different orders, and relating the chromatic tower construction to the multiple complex construction of Benson and Carlson.

Warning 2.5.1 applies throughout this chapter.

3.1. Margolis’ killing construction

In this section we present Margolis’ killing construction. This is a (smashing) localization functor that kills off P_{t}^{s}- and Q_{n}-homology for ξ_{t}^{p^{s}} and τ_{n} of large slope.

We make use of Notation 2.2.8 and 2.4.1 in this section; as remarked above, Warning 2.5.1 applies here. We defined the notions of “thick” and “localizing” subcategories in Definition 1.3.7.

Definition 3.1.1. Recall from [Mil92] and [HPS97, Theorem 3.3.3] that given any thick subcategory 𝒞 of finite spectra, there is a functor L_{𝒞}^{f} : Stable(A) ⟶ Stable(A), called finite localization away from 𝒞, with these properties:

(a) L_{𝒞}^{f} is exact. When viewed as a functor L_{𝒞}^{f} : Stable(A) ⟶ L_{𝒞}^{f} Stable(A) to the category of L_{𝒞}^{f} -local spectra (i.e., spectra of the form L_{𝒞}^{f} X), it is left adjoint to the inclusion functor.

(b) There is a natural transformation 1 ⟶ L_{𝒞}^{f} .

(c) L_{𝒞}^{f} is idempotent—for any X, the map L_{𝒞}^{f} X ⟶ L_{𝒞}^{f} L_{𝒞}^{f} X induced by the natural transformation in (b) is an equivalence.

(d) L_{𝒞}^{f} is Bousfield localization with respect to the spectrum L_{𝒞}^{f} S^{0}.

(e) For any X, L_{𝒞}^{f} X = X ∧ L_{𝒞}^{f} S^{0}.

(f) For any finite X, L_{𝒞}^{f} X = 0 if and only if X ∈ 𝒞. For any X, L_{𝒞}^{f} X = 0 if and only if X is in the localizing subcategory generated by 𝒞.

Properties (a)–(c) say that L_{𝒞}^{f} is a localization functor [HPS97, Definition 3.1.1]; property (e) says that L_{𝒞}^{f} is smashing [HPS97, Definition 3.3.2]. We write C_{𝒞}^{f} for the corresponding acyclization functor; that is, C_{𝒞}^{f} X is the fiber of X ⟶ L_{𝒞}^{f} X. Since L_{𝒞}^{f} is smashing, then C_{𝒞}^{f} X = X ∧ C_{𝒞}^{f} S^{0}.

Definition 3.1.2. For any slope n, let L_{n}^{f} denote finite localization away from the thick subcategory of finite spectra which are ⋁_{d≤n} Z(d)-acyclic, and write C_{n}^{f} for the corresponding acyclization.

In Section 2.5, we constructed a finite spectrum F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) for an appropriately chosen set of exponents i_{1}, …, i_{n}, indexed by slopes. We will also write F (U_{n}) for this spectrum. If d is the next largest slope after n, then by construction, F (U_{n}) has a u_{d}-map. Let Tel(d) denote its telescope. Theorem 2.5.3 tells us that the Bousfield class ⟨F (U_{n})⟩ is independent of the exponents i_{1}, …, i_{n}; as noted after that theorem, the same is true of ⟨Tel(d)⟩.

The following is based on Mahowald and Sadofsky’s analysis in [MS95] of the functor L_{n}^{f} in the ordinary stable homotopy category.

Proposition 3.1.3. The functor L_{n}^{f} can be described in any of the following ways:

(a) as finite localization away from the thick subcategory generated by F (U_{n});

(b) as Bousfield localization with respect to ⋁_{d≤n} Tel(d);

(c) and as a colimit: if we let \overline{F} (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) be the cofiber of projection to the top cell F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) ⟶ S^{0}, then the map X ⟶ L_{n}^{f}X is given by

X ⟶ \underrightarrow{lim}_{i_{1},...,i_{n}} X ∧ \overline{F} (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}).

(The maps in the direct system will be defined in the proof.) Note also that for n large, L_{n}^{f} is not Bousfield localization with respect to

⋁_{d≤n} Z(d). See Proposition 3.1.10.

Proof. By Theorem 2.5.3(a), the thick subcategory of finite ⋁_{d≤n} Z(d)-acyclic spectra is equal to the thick subcategory generated by F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) for any choice of exponents i_{1}, …, i_{n}. This proves part (a).

Part (b) is essentially a Bousfield class computation. Repeated application of Proposition 1.5.1(a) gives us

(3.1.4)  ⟨S^{0}⟩ = ⟨F (U_{n})⟩ ∨ ⟨Tel(1)⟩ ∨ ⋯ ∨ ⟨Tel(n)⟩,

as well as pairwise orthogonality of the Bousfield classes on the right. We need to show that a spectrum X is ⋁_{d≤n} Tel(d)-acyclic if and only if it is L_{n}^{f}-acyclic. Assume that ⋁_{d≤n} Tel(d)_{∗∗}X = 0. By the decomposition of Bousfield classes (3.1.4), we see that ⟨X⟩ = ⟨X ∧ F (U_{n})⟩, so L_{n}^{f}X = 0 if and only if L_{n}^{f}X ∧ F (U_{n}) = 0. But certainly L_{n}^{f}F (U_{n}) = 0, so since L_{n}^{f} is smashing, then L_{n}^{f}F (U_{n}) ∧ X = 0. To show the converse, it suffices to show that F (U_{n}) is ⋁_{d≤n} Tel(d)-acyclic; this follows by the orthogonality of the above Bousfield classes.

To prove (c), we first need to construct the direct system. By “projection to the top cell,” we mean the composite F (U_{n}) ⟶ ⋯ ⟶ F (U_{1}) ⟶ F (U_{0}) = S^{0}. Using properties of the spectra F (U_{m}) and their u_{j}-maps (Theorem 2.5.3), we can choose the u_{j}-maps compatibly; i.e., if we have exponents i_{j} and ℓ_{j} so that the objects F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) and F (u_{1}^{ℓ_{1}}, …, u_{n}^{ℓ_{n}}) are defined and with ℓ_{j} ≫ i_{j}, then there is a map

F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) ⟶ F (u_{1}^{ℓ_{1}}, …, u_{n}^{ℓ_{n}})

which commutes with projection to the top cell. We let \overline{F} (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) denote the cofiber of F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) ⟶ S^{0}, and we define the spectrum L′_{n}X to be the following colimit:

X ⟶ \underrightarrow{lim}_{i_{1},...,i_{n}}\overline{F} (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) ∧ X
∥  ∥
X ⟶^{g′}  L′_{n}X.

For example, if n = 1, then we have S^{0} ⟶^{g′} u_{1}^{−1}S^{0}, where u_{1}^{−1}S^{0} is the mapping telescope of u_{1} : S^{0} ⟶ S^{0}, and fiber(g′) = F (u_{1}^{∞}) is the analogue of the mod p^{∞} Moore spectrum.

We need to verify that L′_{n} agrees with L_{n}^{f}. To do this, we note that L′_{n} is smashing, we show that g′: S^{0} ⟶ L′_{n}S^{0} is an L_{n}^{f}S^{0}-equivalence, and we show that L′_{n}S^{0} is L_{n}^{f}S^{0}-local. By construction, the fiber of g′ is in the localizing subcategory generated by the spectra F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}), and hence g′ is an L_{n}^{f}S^{0}-equivalence.

To show that L′_{n}S^{0} is local, we have to show that [W, L′_{n}S^{0}]_{∗∗} = 0 for any L_{n}^{f}- acyclic W. For any finite localization, the acyclics are the localizing subcategory generated by the finite acyclics, so it suffices to show this when W is a finite acyclic. Note that if W is a finite acyclic, then so is DW, the Spanier-Whitehead dual of W. Since

[W, L′_{n}S^{0}]_{∗∗} = [S^{0}, DW ∧ L′_{n}S^{0}]_{∗∗} = π_{∗∗}L′_{n}DW,

it suffices to show that L′_{n}W = 0 for any finite acyclic W, so let W be an acyclic. For each slope j ≤ n, if k is the next largest slope after j, then by vanishing lines, every u_{k}-map of F (u_{1}^{i_{1}}, …, u_{j}^{i_{j}}) ∧ W is nilpotent. Thus for all large enough exponents, F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) ∧ W is equivalent to a wedge of suspensions of W. One can then argue as in [MS95, Proposition 3.8] to conclude that

\underrightarrow{lim}_{i_{1},...,i_{n}}F (u_{1}^{i_{1}}, …, u_{n}^{i_{n}}) ∧ W ⟶ W

is an equivalence; hence the cofiber L′_{n}W of this map is zero. □

Recall from Definition 2.4.2 that a spectrum is of type n if it is ⋁_{d<n} Z(d)_{∗∗}- acyclic, but not Z(n)_{∗∗}-acyclic. By Theorem 2.4.3, any finite type n spectrum has a u_{n}-map.

Proposition 3.1.5. Suppose that X is a finite type n spectrum with u_{n}-map u. Then L_{n}^{f}X = u^{−1}X.

Proof. As in the proof of Proposition 3.1.3(c), we have to check that the map X ⟶ u^{−1}X is an L_{n}^{f}-equivalence, and that u^{−1}X is L_{n}^{f}-local. For the former, note that we have a diagram of cofibrations

X/(u) ⟶ X ⟶^{u} X
↓  ∥  ↓^{u}
X/(u^{2}) ⟶ X ⟶^{u^{2}} X
↓  ∥  ↓^{u}
X/(u^{3}) ⟶ X ⟶^{u^{3}} X
↓  ∥  ↓^{u}
⋮  ⋮  ⋮

The sequential colimit is the cofibration

X/(u^{∞}) ⟶ X ⟶ u^{−1}X.

Note that each spectrum X/(u^{n}) is finite of type greater than n, and so is in the thick subcategory generated by F (U_{n}) by Theorem 2.5.3(a). Hence X/(u^{∞}) is in loc(F (U_{n})); hence X ⟶ u^{−1}X is an L_{n}^{f}-equivalence.

To show that u^{−1}X is L_{n}^{f}-local, we have to show that [F (U_{n}), u^{−1}X]_{∗∗} = 0. We compute:

[F (U_{n}), u^{−1}X]_{∗∗} = π_{∗∗}(DF (U_{n}) ∧ u^{−1}X)
= π_{∗∗}(u^{−1}DF (U_{n}) ∧ X)
= u^{−1}π_{∗∗}(DF (U_{n}) ∧ X).

Since F (U_{n}) is of type greater than n, it has a vanishing line of slope greater than n; the same then goes for DF (U_{n}) ∧ X. Since u acts at slope n, it acts nilpotently on π_{∗∗}(DF (U_{n}) ∧ X). □

Here is the main theorem of this section. It tells us that L_{n}^{f} “kills off” certain P_{t}^{s}-homology groups; we also have a connectivity result.

Theorem 3.1.6 (Margolis’ killing construction). Fix a slope n and let X be a spectrum.

(a) In the cofiber sequence C_{n}^{f}X ⟶^{f} X ⟶^{g} L_{n}^{f}X,

(i) Z(d)_{∗∗}g is an isomorphism if d ≤ n, and (ii) Z(d)_{∗∗}f is an isomorphism if d > n.

In other words, (i)′ Z(d)_{∗∗}C_{n}^{f}X = 0 if d ≤ n, and (ii)′ Z(d)_{∗∗}L_{n}^{f}X = 0 if d > n.

(b) Suppose that X is a CL-spectrum (Definition 1.3.6). Then the homotopy of L_{n}^{f}X is “bounded to the left”: for each i, π_{ij}L_{n}^{f} = 0 for j ≪ 0. If, in addition, X has finite type homotopy, then so does L_{n}^{f}X.

Proof. Part (a): By the theory of finite localization in [HPS97], the spectrum C_{n}^{f}X is in the localizing subcategory generated by the finite ⋁_{d≤n} Z(d)-acyclics, and hence is ⋁_{d≤n} Z(d)-acyclic itself. This proves (i).

Since homology commutes with direct limits, we see that Z(d)_{∗∗} Tel(k) is zero if d ≠ k; in particular, ⋁_{k≤n} Tel(k) is ⋁_{d>n} Z(d)-acyclic. Since L_{n}^{f} is Bousfield localization with respect to ⋁_{k≤n} Tel(k), then we conclude that L_{n}^{f}S^{0} is ⋁_{d>n} Z(d)- acyclic. Since L_{n}^{f} is smashing, this proves (ii).

Part (b): Given X as in the statement, we show that the maps in the direct system defining L_{n}^{f}X (Proposition 3.1.3(c)) are isomorphisms on π_{∗∗} in a range increasing with the exponents.

We let F (U_{k}) = F (u_{1}^{i_{1}}, …, u_{k}^{i_{k}}) and we assume that d is the next largest slope after k. We write F (U_{k})(u_{d}^{i_{d}}) = F (u_{1}^{i_{1}}, …, u_{d}^{i_{d}}) for the fiber of the u_{d}-map on F (U_{k}). Consider the following commutative diagram:

⋯ ⟶ Σ^{−i_{d}+1,−i_{d}d}F (U_{k}) ⟶ F (U_{k})(u_{d}^{i_{d}}) ⟶ F (U_{k}) ⟶^{u_{d}^{i_{d}}} ⋯
↓_{u_{d}^{i_{d}}}  ↓^{f_{1}}  ∥
⋯ ⟶ Σ^{−2i_{d}+1,−2i_{d}d}F (U_{k}) ⟶ F (U_{k})(u_{d}^{2i_{d}}) ⟶ F (U_{k}) ⟶^{u_{d}^{2i_{d}}} ⋯
↓_{u_{d}^{i_{d}}}  ↓^{f_{2}}  ∥
⋯ ⟶ Σ^{−3i_{d}+1,−3i_{d}d}F (U_{k}) ⟶ F (U_{k})(u_{d}^{3i_{d}}) ⟶ F (U_{k}) ⟶^{u_{d}^{3i_{d}}} ⋯
↓_{u_{d}^{i_{d}}}  ↓^{f_{3}}  ∥
⋮  ⋮  ⋮

It suffices to show that the maps f_{r} ∧ 1_{X} are isomorphisms in a range increasing with r. Well, f_{r} ∧ 1_{X} is an isomorphism whenever the map

(3.1.7)  Σ^{−ri_{d}+1,−ri_{d}d}F (U_{k}) ∧ X ⟶^{u_{d}^{i_{d}} ∧1_{X}} Σ^{−(r+1)i_{d}+1,−(r+1)i_{d}d}F (U_{k}) ∧ X

from the left column is. The fiber is Σ^{−ri_{d}+1,−ri_{d}d}F (U_{k})(u_{d}^{i_{d}}) ∧ X; this spectrum has a vanishing line with some slope ℓ > d, and we can use Remark 2.3.5 to compute its intercept. To do this, we compute (HF_{p})_{∗∗}F (U_{k})(u_{d}^{i_{d}})—each u_{j}-map induces the zero map on (HF_{p})_{∗∗}, so this computation is easy. We find that

(i) (HF_{p})_{i∗}F (U_{k})(u_{d}^{i_{d}}) = 0 for i ≪ 0,

(ii) (HF_{p})_{i∗}F (U_{k})(u_{d}^{i_{d}}) = 0 for i > 0, and
(iii) (HF_{p})_{∗j}F (U_{k})(u_{d}^{i_{d}}) = 0 for j < N for some number N.

By the Hurewicz Theorem 1.3.5, (i) and (iii) also hold for π_{∗∗}; hence we can compute the equation of the vanishing line for F (U_{k})(u_{d}^{i_{d}}). Smashing with X moves the intercept a bit, so we find that π_{ij}F (U_{k})(u_{d}^{i_{d}})∧X = 0 when j < ℓi+N′ for some N′. Therefore the map (3.1.7) is an isomorphism on π_{ij} when j+ri_{d}d < ℓ(i+ri_{d}−1)+N′, i.e., when j < ℓi + ri_{d}(ℓ − d) + (N′ − ℓ). Since ℓ > d, then as r increases, this range increases.

So we see that for each s, then the graded groups π_{s∗} of the direct system have a uniform bound to the left; hence the same is true of the direct limit. If X also has finite type homotopy, then since the homotopy of each F (U_{k}) is of finite type, the same goes for F (U_{k}) ∧ X; since the homotopy of the direct system stabilizes at each bidegree, we are done. □

[figure: j − i; i; slope \frac{1}{n−1}; slope \frac{1}{d−1}]

Figure 3.1.A. “Vanishing curve” for π_{ij}(C_{n}^{f}S^{0}): π_{ij}(C_{n}^{f}S^{0}) is zero above the two indicated lines. Here, d is the slope preceding n, and slopes are labeled in (i, j − i)-coordinates. These lines are drawn through the origin for convenience, but their intercepts may be nonzero.

Remark 3.1.8. (a) Much of this was already known in the module setting. For p = 2, this is due to Margolis [Mar83, Theorem 21.1]; for arbitrary primes, part (a) and the connectivity in part (b) can be found in [Pal92, Theorem 3.1]. Given an A-comodule M and a slope n, then the module-version of the killing construction (dualized to the category of comodules) gives comodules M⟨n + 1, ∞⟩ and M⟨1, n⟩, well-defined up to injective summands, and an injective comodule J so that the following is short exact:

0 ⟶ M⟨n + 1, ∞⟩ ⟶ M ⊕ J ⟶ M⟨1, n⟩ ⟶ 0.

Being well-defined up to injective summands translates in our setting to being well-defined up to an object of loc(A) (cf. Lemma 3.1.9). So if X is an injective resolution of M, then an injective resolution for M⟨1, n⟩ is a connective cover of L_{n}^{f}X.

(b) By studying the proof of Theorem 3.1.6(b), one can show that C_{n}^{f}S^{0} has a nice vanishing curve: for i ≥ 0, π_{ij}C_{n}^{f}S^{0} has a vanishing line of slope n, and if d is the slope preceding n, then for i < 0, π_{ij}C_{n}^{f}S^{0} has a vanishing line of slope d. For example, when n = 1, we have

C_{1}^{f}S^{0} ⟶ S^{0} ⟶ L_{1}^{f}S^{0}
∥  ∥  ∥
F (u_{1}^{∞}) ⟶ S^{0} ⟶ u_{1}^{−1}S^{0}.

So

π_{∗∗}L_{1}^{f}S^{0} = u_{1}^{−1}π_{∗∗}S^{0} = h_{10}^{−1}π_{∗∗}S^{0} = F_{2}[h_{10}^{±1}].

Hence π_{ij}C_{1}^{f}S^{0} has a vanishing line of slope 2 for positive i, and a vanishing line of slope 1 (a vertical line, in (i, j − i)-coordinates) for i negative. See also Figure 3.1.A.

We need the following property of L_{n}^{f} in the next section. Lemma 3.1.9. For any slope n, L_{n}^{f}A = 0. Hence any spectrum X in the

localizing subcategory generated by A is L_{n}^{f}-acyclic.

3.2. A TATE VERSION OF THE FUNCTOR H 71

Proof. Since the category of L_{n}^{f}-acyclics contains a finite spectrum F (U_{n}), then it contains A by Lemma 1.3.9. □

We close this section with a few remarks about the telescope conjecture (see [Rav84] and [HPS97, Definition 3.3.8]). Let L_{n} denote Bousfield localization with respect to ⋁_{d≤n} Z(d).

Proposition 3.1.10. Let p = 2. We have L_{1} = L_{1}^{f}. If n > 1, then L_{n} ≠ L_{n}^{f}. One can interpret the statement “L_{n} = L_{n}^{f}” as an analogue of the telescope

conjecture, so this result would say that the telescope conjecture fails except when n = 1. However, it seems more proper to refer to the statement “every smashing localization functor is a finite localization” as the telescope conjecture—see [HPS97, Definition 3.3.8]. Since we do not know whether L_{n} is smashing, this result does not necessarily present a counterexample to this version of the telescope conjecture.

Proof. We know that the sphere has a u_{1}-map; a simple computation shows that u_{1}^{−1}S^{0} = Z(1), so that Bousfield localization with respect to L_{1}^{f}S^{0} = u_{1}^{−1}S^{0} is the same as that with respect to Z(1).

There is a non-nilpotent element d_{0} ∈ π_{4,18}S^{0} = Ext_{A}^{4,18}(F_{2}, F_{2}). We claim that d_{0}^{−1}S^{0} is L_{n}-acyclic for all n, and L_{n}^{f}-local for n ≥ 3. Since 3 is the next slope after 1, this covers all of the possibilities.

For degree reasons, d_{0} induces the zero map on Z(d)_{∗∗} for all d. (If an element α ∈ π_{ij}S^{0} is nonzero on Z(d)_{∗∗}S^{0} = Z(d)_{∗∗}, then j/i must be a multiple of d.) Hence Z(d)_{∗∗}(d_{0}^{−1}S^{0}) = 0 for all d, and d_{0}^{−1}S^{0} is L_{n}-acyclic for all n. To show that d_{0}^{−1}S^{0} is L_{n}^{f}-local, we show that [F, d_{0}^{−1}S^{0}]_{∗∗} = 0 for any finite ⋁_{d≤n} Z(d)-acyclic F. We compute:

[F, d_{0}^{−1}S^{0}]_{∗∗} = [S^{0}, DF ∧ d_{0}^{−1}S^{0}]_{∗∗}
= [S^{0}, d_{0}^{−1}DF]_{∗∗}
= d_{0}^{−1}π_{∗∗}DF.

Since F is a finite ⋁_{d≤n} Z(d)_{∗∗}-acyclic, so is DF (by Proposition 2.2.10). By Theorem 2.3.1, π_{∗∗}DF has a vanishing line of some slope m > n. Since n ≥ 3, we know that m ≥ 6, so the slope m is larger than \frac{9}{2}, the slope of d_{0}. Hence d_{0} acts nilpotently on π_{∗∗}DF; i.e., d_{0}^{−1}π_{∗∗}DF = 0. □

When p is odd, one can again show that L_{1} = L_{1}^{f}. There is every reason to expect that there are non-nilpotent elements in Ext_{A}^{∗∗}(F_{p}, F_{p}) which are not detected by any single Z(d); given this, one could conclude that L_{n} ≠ L_{n}^{f} for n large.

3.2. A Tate version of the functor H

In this section we introduce a “Tate” version of the functor H, defined for finite-dimensional quotients of A, and we show that when n is large enough, then L_{n}^{f}HB = ĤB. We will use our computations here to prove chromatic convergence in Section 3.3.

Let B be a quotient Hopf algebra of A. Note that practically all of our results hold in the category Stable(B); the only exceptions are computational results like the strict inequalities in Theorem 6.6.1. Let res_{A,B}: Stable(A) ⟶ Stable(B) denote the forgetful functor (also known as restriction); res_{A,B} has a right adjoint, induction, written ind_{B,A}, and defined by X ↦ A □_{B}X (cf. Section 1.2—in this section, we use the notations res_{A,B} and ind_{B,A} rather than ↓_{B} and A □_{B}−). These functors are exact (i.e., they take cofibrations to cofibrations), and restriction preserves the smash product and the sphere object. In the language of [HPS97, Section 3.4], the restriction functor is a stable morphism.

When B is a finite-dimensional Hopf algebra, we may consider another stable homotopy category associated to it, the stable category of B-comodules, written StComod(B). We provide a brief review of the relevant results here; see [HPS97, Section 9.6] for a few more details. The objects of StComod(B) are B-comodules, and the morphisms \underline{Hom}_{B}^{∗} are defined as follows: define the morphisms of degree zero to be \underline{Hom}_{B}(X, Y) = Hom_{B}(X, Y)/ ≃, where f ≃ g: X ⟶ Y if f − g factors through an injective comodule. Hence two comodules are equivalent in StComod(B) if they differ by injective summands. Define the desuspension functor Σ^{−1} = Σ^{−1,0} by the short exact sequence

0 ⟶ X ⟶^{η∧1_{X}} B ⊗ X ⟶ Σ^{−1}X ⟶ 0,

where B ⊗ X is the cofree comodule on X (Definition 0.4.1). This functor is invertible: ΣX is any comodule which fits into a short exact sequence

0 ⟶ ΣX ⟶ I ⟶ X ⟶ 0,

where I is injective. (Since B is finite, then injectives and projectives are the same, and there are enough projectives; hence there is always a surjection I ↠ X.) We let

\underline{Hom}_{B}^{i}(X, Y) = \underline{Hom}_{B}(Σ^{i}X, Y).

As with the category Stable(B), this category is bigraded; thus the suspension functor has a second, “internal,” index, as do sets of morphisms.

StComod(B) is a stable homotopy category. Indeed, on Stable(B) one has finite localization (Definition 3.1.1) away from the thick subcategory generated by the finite spectrum B = HF_{p}; StComod(B) is equivalent to the full subcategory of Stable(B) of L_{B}^{f}-local objects—see [HPS97, Theorems 9.6.3–9.6.4]. So we have a functor L_{B}^{f}: Stable(B) ⟶ StComod(B) with a right adjoint J: StComod(B) ⟶ Stable(B). (Every localization functor has a right adjoint, namely inclusion of the local objects into the category.)

Definition 3.2.1. Let B be a quotient Hopf algebra of A. We note that the object HB in Stable(A) is ind_{B,A}(S^{0}), and when B is finite-dimensional, we define the object ĤB in Stable(A) to be ind_{B,A}(J(S^{0})).

Fix a slope n. On each of the categories Stable(A), Stable(B), and StComod(B), we can define the functor L_{n}^{f} to be smashing with L_{n}^{f}S^{0}, or with its image under res or L_{B}^{f} ∘ res. This gives us a commuting diagram of functors

Stable(A) ⟶^{res} Stable(B) ⟶^{L_{B}^{f}} StComod(B)
↓^{L_{n}^{f}}  ↓^{L_{n}^{f}}  ↓^{L_{n}^{f}}
L_{n}^{f}Stable(A) ⟶^{res} L_{n}^{f}Stable(B) ⟶^{L_{B}^{f}} L_{n}^{f}StComod(B),

along with the commuting diagram of their right adjoints

Stable(A) ⟵^{ind} Stable(B) ⟵^{J} StComod(B)
↑^{R_{Stable(A)}}  ↑^{R_{Stable(B)}}  ↑^{R_{St(B)}}
L_{n}^{f}Stable(A) ⟵^{ind} L_{n}^{f}Stable(B) ⟵^{J} L_{n}^{f}StComod(B).

(The first diagram commutes by the definition of the vertical functors, and the second diagram commutes as a result—given a commuting square of left adjoints, their right adjoints also commute.) Recall from Notation 2.2.8 that y_{d} is the element of A—either ξ_{t}^{p^{s}} or τ_{n}—“with slope d.”

Proposition 3.2.2. Fix a finite-dimensional quotient Hopf algebra B of A, and let

N = max{d | y_{d} ≠ 0 in B}.

If n is a slope larger than N, then L_{n}^{f}HB = ĤB.

Proof. We claim that the two maps

α: HB ⟶ L_{n}^{f}HB,
β: HB ⟶ ĤB,

are the same. To show this, we show that the spectrum ĤB is L_{n}^{f}-local, and that the map β: HB ⟶ ĤB is an L_{n}^{f}-equivalence. By Lemma 3.2.4 below, the fiber of β has homotopy concentrated in the third quadrant, so by Lemma 1.3.8, it is in loc(A). In particular, it is L_{n}^{f}-acyclic by Lemma 3.1.9; hence β is an L_{n}^{f}-equivalence.

To show that ĤB is local, we show that it is in the image of the right adjoint R_{Stable(A)} of L_{n}^{f}: Stable(A) ⟶ L_{n}^{f}Stable(A). We claim, in fact, that R_{St(B)} is the identity functor. Given this claim, then we use the commutativity of the diagram of right adjoints: we have

ĤB = ind(JS^{0})
= ind(J(R_{St(B)}S^{0}))
= R_{Stable(A)}(ind(JS^{0})).

Hence ĤB is in the image of R_{Stable(A)}, and so it is local.

It remains to verify the claim that R_{St(B)} is the identity functor, or what is the same, that

L_{n}^{f}: StComod(B) ⟶ StComod(B)

is the identity functor. So it suffices to show that the cofiber of S^{0} ⟶ L_{n}^{f}S^{0} is zero in the category StComod(B). This cofibration is obtained by applying the functor L_{B}^{f} ∘ res to the following cofibration in Stable(A):

C_{n}^{f}S^{0} ⟶ S^{0} ⟶ L_{n}^{f}S^{0}.

So the cofiber under consideration is L_{B}^{f}(res C_{n}^{f}S^{0}). By Proposition 3.1.3, L_{n}^{f} is finite-localization away from the thick subcategory generated by F (U_{n}); but F (U_{n}) has no Z(d)-homology for any d with y_{d} ∈ B, so res(F (U_{n})) is in loc(B). Since res(F (U_{n})) is finite, it is in fact in thick(B). By fundamental properties of finite localizations (Definition 3.1.1), res(C_{n}^{f}S^{0}) is in loc(F (U_{n})), and hence in loc(B). Hence L_{B}^{f}(res C_{n}^{f}S^{0}) = 0, as desired. □

[figure: HA(1)_{st}; t − s; s; ĤA(1)_{st}; t − s; s]

Figure 3.2.A. The coefficients of HA(1) and ĤA(1) when p = 2. Since the top cell of A(1) is in degree 6, the third quadrant of ĤA(1)_{∗∗} is the same as the first quadrant, reflected across the origin and then translated by (−1, −6).

We are particularly interested in the case B = A(m), the quotient of A defined in Example 2.1.4.

Corollary 3.2.3. Fix an integer m ≥ 0. For n sufficiently large, we have

L_{n}^{f}HA(m) = ĤA(m).

In particular, n should be larger than

max\{d | y_{d} ≠ 0 in A(m)\} = \begin{cases}|ξ_{m+1}| = 2^{m+1} − 1, & if p = 2, \\ \frac{p}{2}|ξ_{m}| = p^{m+1} − p, & if p is odd. \end{cases}

Now we “compute” the homotopy groups of ĤB. See Figure 3.2.A for an example.

Lemma 3.2.4. Let B be a finite-dimensional quotient Hopf algebra of A, and consider π_{ij}ĤB.

(a) When ij < 0, then π_{ij}ĤB = 0. (In other words, the homotopy is concentrated in the first and third quadrants.)

(b) When i and j are nonnegative, the map HB ⟶ ĤB induces an isomorphism π_{ij}ĤB ≅ π_{ij}HB.

(c) Let d be the maximal degree in which B is nonzero. When i and j are negative, we have an isomorphism π_{ij}ĤB ≅ π_{−1−i,−d−j}HB.

Proof. We work in the category Stable(B). We have adjoint functors

L: Stable(B) ⟶ StComod(B),

J: StComod(B) ⟶ Stable(B),

and we want to compute the homotopy of JS^{0} by comparing it to that of S^{0} ∈ Stable(B). To do this, we need to recall the description of the functor J from [HPS97, Lemma 9.6.7]. Let I denote an injective resolution of the B-comodule F_{p} (i.e., I ≅ S^{0} in Stable(B)); then Hom_{F_{p}}(I, F_{p}) is a projective resolution of F_{p}. We splice these resolutions together to get the “Tate complex” t_{B}(F_{p}):

⋯ ⟶ Hom(I_{1}, F_{p}) ⟶ Hom(I_{0}, F_{p}) ⟶ I_{0} ⟶ I_{1} ⟶ · · · .

Then the functor J is defined by J(M) = t_{B}(F_{p}) ⊗ M. It is clear that the map S^{0} ⟶ J(S^{0}) induces an isomorphism on π_{i∗} for i ≥ 0. If we take I to be a minimal injective resolution, then Hom(I, F_{p}) is a minimal projective resolution; by minimality, [S^{0}, JS^{0}]_{i∗} = [F_{p}, JS^{0}]_{i∗} gives the primitives in JS^{0} in degree i, and there is one primitive for each summand isomorphic to B. Since Hom(B, F_{p}) ≅ Σ^{−d}B, we get the reflection and translation as described in (c). □

3.3. Chromatic convergence

It is easy to see that given a spectrum X and slopes n_{1} and n_{2} with n_{1} < n_{2}, then the map X ⟶ L_{n_{1}}^{f}X factors through X ⟶ L_{n_{2}}^{f}X. Hence we get a tower of cofibrations:

⋮  ⋮  ⋮
↓  ∥  ↓
C_{3}^{f}X ⟶ X ⟶ L_{3}^{f}X
↓  ∥  ↓
C_{2}^{f}X ⟶ X ⟶ L_{2}^{f}X
↓  ∥  ↓
C_{1}^{f}X ⟶ X ⟶ L_{1}^{f}X.

(Strictly speaking, we have defined L_{n}^{f} only when n is a slope. For a general n, we define L_{n}^{f}, as in Definition 3.1.2, to be finite localization away from the finite spectra which are ⋁_{d≤n} Z(d)-acyclic. So if n is not a slope, then L_{n}^{f} = L_{n−1}^{f}.) We may as well just focus on the right hand column, giving the following diagram:

0 ⟵ L_{1}^{f}X ⟵ L_{2}^{f}X ⟵ L_{3}^{f}X ⟵ · · · .
=↑  ↑  ↑
M_{1}^{f}X  M_{2}^{f}X  M_{3}^{f}X

Here, M_{n}^{f}X = X ∧ M_{n}^{f}S^{0} is defined to be the fiber of L_{n}^{f}X ⟶ L_{n−1}^{f}X. We call this diagram the chromatic tower for X. (Note that if n is not a slope, then M_{n}^{f}X = 0.)

Margolis [Mar83, Theorem 22.4] proved the following theorem for A^{∗}-modules when p = 2; see also [Pal94].

Theorem 3.3.1 (Chromatic convergence). If X is finite, then X = \underleftarrow{lim}_{n} L_{n}^{f}X.

Indeed, the proof shows that the tower of groups π_{∗∗}L_{n}^{f}X is pro-constant. In order to prove Theorem 3.3.1, we need the following observation.

Corollary 3.3.2. For any m, the chromatic tower for HA(m) stabilizes to ĤA(m).

Proof. This follows from Corollary 3.2.3. Actually, that result merely says that L_{n}^{f}HA(m) = ĤA(m) for all sufficiently large n; it does not say that the map L_{n+1}^{f}HA(m) ⟶ L_{n}^{f}HA(m) is an equivalence when n ≫ 0. This follows from the proof of Proposition 3.2.2, though. □

Note that the chromatic tower for HA(m) converges to ĤA(m), not to HA(m)— we do not have chromatic convergence in this case.

Proof of Theorem 3.3.1. If X is finite, then smashing with X commutes with inverse limits. Since L_{n}^{f}X = X ∧ L_{n}^{f}S^{0}, then chromatic convergence for S^{0} implies chromatic convergence for all finite spectra.

By Theorem 3.1.6(b), L_{n}^{f}S^{0} has finite type homotopy bounded to the left, so by Proposition 2.1.5, we see that

L_{n}^{f}S^{0} = \underleftarrow{lim}_{m} HA(m) ∧ L_{n}^{f}S^{0} = \underleftarrow{lim}_{m} L_{n}^{f}HA(m).

Hence we have

\underleftarrow{lim}_{n} L_{n}^{f}S^{0} = \underleftarrow{lim}_{n}(\underleftarrow{lim}_{m} L_{n}^{f}HA(m))

= \underleftarrow{lim}_{m}(\underleftarrow{lim}_{n} L_{n}^{f}HA(m))

= \underleftarrow{lim}_{m} ĤA(m).

Now we use Lemma 3.2.4 to compute the homotopy groups of \underleftarrow{lim} ĤA(m): in the first quadrant, the inverse limit stabilizes in each bidegree to the homotopy of \underleftarrow{lim} HA(m) = S^{0}, by Proposition 2.1.5. In the second and fourth quadrants, there is no homotopy at any stage in the inverse system. In the third quadrant, for each bidegree (i, j) there is an m_{0} so that π_{ij}ĤA(m) = 0 for all m ≥ m_{0}; so the inverse limit has no homotopy in the third quadrant, either. Hence the map S^{0} ⟶ \underleftarrow{lim} ĤA(m) is an isomorphism in homotopy. □

One could probably get another proof of chromatic convergence by making precise, and then using, the vanishing curve result mentioned in Remark 3.1.8(b).

By the way, Margolis gets chromatic convergence in his setting for all bounded below modules; since we do not have convergence for HA(n) (which is an injective resolution of a bounded below comodule), we cannot expect our result to generalize, precisely as stated, to other spectra. On the other hand, perhaps one could modify the proof of Theorem 3.3.1 to show that for any CL-spectrum X, then X ⟶ \underleftarrow{lim} L_{n}^{f}X is a connective cover. For this, the alternate formulation of CL after Definition 1.3.6 might be useful—X is CL if and only if there are numbers i_{0}, i_{1}, and j_{0} so that X has a cellular tower built of spheres S^{i,j} with i_{0} ≤ i ≤ i_{1} and j − i ≥ j_{0}.

3.4. Further discussion: work of Mahowald and Shick

Chromatic convergence for the sphere gives a convergent filtration of π_{∗∗}S^{0} = Ext_{A}^{∗∗}(F_{p}, F_{p}). When p = 2, Mahowald and Shick [MS87] give another convergent filtration; are these filtrations the same? If p = 2, the slope of the element ξ_{n+1} is 2^{n+1} − 1; this element corresponds to the periodicity operator v_{n} = h_{n+1,0}. Mahowald and Shick also construct something they call v_{n}^{−1} Ext_{A}^{∗∗}(F_{2}, F_{2}). How does this compare to π_{∗∗}(L_{2^{n+1}−1}^{f}S^{0})? (Shick [Shi88] has done similar work at odd primes, and one can naturally ask the same questions about that.) We describe their work. When p = 2, we write v_{n} for h_{n+1,0}, and we write E for the quotient F_{2}[ξ_{n+1}]/(ξ_{n+1}^{2}); when p is odd, we write E for E[τ_{n}]. They show that for every k ≥ n, the restriction map

Ext_{A(k)}^{∗∗}(F_{p}, F_{p}) ⟶ Ext_{E}^{∗∗}(F_{p}, F_{p}) ≅ F_{p}[v_{n}]

is nontrivial, so some power of v_{n} lifts to Ext_{A(k)}^{∗∗}(F_{p}, F_{p}) (this is a special case of Proposition 2.4.5); furthermore, one can choose the lifts compatibly as k varies. Hence the map

Ext_{A(k+1)}^{∗∗}(F_{p}, F_{p}) ⟶ Ext_{A(k)}^{∗∗}(F_{p}, F_{p})

induces a map after inverting v_{n}. They define

v_{n}^{−1} Ext_{A}^{∗∗}(F_{p}, F_{p}) = \underleftarrow{lim}_{k}(v_{n}^{−1} Ext_{A(k)}^{∗∗}(F_{p}, F_{p})).

From our point of view, the lift of v_{n} to Ext_{A(k)}^{∗∗}(F_{p}, F_{p}) is an element in π_{∗∗}HA(k). Given a homotopy class α in a ring spectrum R, one can turn it into a self-map by the composite

R ⟶^{η∧1} S^{0} ∧ R ⟶^{α∧1} R ∧ R ⟶^{μ} R.

Therefore we can invert the self-map v_{n} of HA(k); Mahowald and Shick’s result says that we can do this compatibly, giving maps of spectra

v_{n}^{−1}HA(k + 1) ⟶ v_{n}^{−1}HA(k).

Since the inverse limit of the rings HA(k + 1) is the sphere, by Proposition 2.1.5, then it seems plausible to take the inverse limit of the objects v_{n}^{−1}HA(k) to define v_{n}^{−1}S^{0}.

More generally, if X is any spectrum, one has (at least) two choices for the definition of v_{n}^{−1}X: either

X ∧ v_{n}^{−1}S^{0} = X ∧ \underleftarrow{lim}_{k}(v_{n}^{−1}HA(k)),

or

\underleftarrow{lim}_{k}(X ∧ v_{n}^{−1}HA(k)).

If X is finite, then these are the same; we write v_{n}^{−1}X to denote either of them. Otherwise, though, they need not be the same, and it is not clear which is the better choice. The first makes the functor v_{n}^{−1}(−) smashing, which could be an advantage.

Let a_{n} be the slope of v_{n}, either 2^{n+1} − 1 or 2p^{n} − 1, depending on whether p is even or odd. When X is finite, one can see that v_{n}^{−1}X is L_{a_{n}}^{f}-local, so the map X ⟶ v_{n}^{−1}X factors through L_{a_{n}}^{f}X, giving a map

L_{a_{n}}^{f}X ⟶ v_{n}^{−1}X.

Is this map an equivalence?

3.5. Further discussion

It seems most convenient to consider the chromatic tower as above, in which we kill off the P_{t}^{s}-homology groups of X in order of slope. This ordering is what allows us to prove chromatic convergence, via the “connectivity result” of Theorem 3.1.6(b). But the results of Chapters 5 and 6 suggest another approach, at least at the prime 2. There is a quotient Hopf algebra D of A, and the corresponding ring HD_{∗∗} seems to play a very important role in the category Stable(A); instead of viewing the chromatic construction as killing off P_{t}^{s}-homology groups, one can view it as killing off generators of the ring HD_{∗∗} in a particular order. As we describe in Section 6.9, though, when p = 2 we can kill off these generators in many different orders, and indeed, we can choose different generators entirely. With a different choice of generators and/or a different ordering, can one still prove convergence? Does other structure reveal itself when one works with other generators of HD_{∗∗} or with other orderings of the same generators?

As noted after the statement of Proposition 3.1.10, the validity of the strong form of the telescope conjecture is not known in Stable(A). If one could show that Bousfield localization with respect to ⋁_{d≤n} Z(d) were smashing, that would disprove it; we would be interested in any progress along these lines.

It is conceivable that the finite-type result of Theorem 3.1.6(b) could have bearing on the convergence (in the ordinary stable homotopy category) of the tower

⋯ ⟶ L_{2}^{f}X ⟶ L_{1}^{f}X ⟶ L_{0}^{f}X.

After all, the v_{n}-localized Adams spectral sequence in [MS95] has an E_{2}-term with some relation to what we call π_{∗∗}L_{m}^{f}S^{0} for a suitable m, so computations and other information about π_{∗∗}L_{m}^{f}S^{0} could have applications to the finite localization functor L_{n}^{f} in ordinary stable homotopy theory.

Work of Mahowald and others has led to calculations similar to that of ĤA(m) in Lemma 3.2.4; see [MR], for example. This may provide more connections between our functors L_{n}^{f} and topology.

In parallel with the study of chromatic phenomena in the ordinary stable homotopy category, one should try to understand the filtration pieces in the chromatic tower, and “monochromatic” objects in general: these are objects X so that M_{n}^{f}X = X. Other than the trivial case of M_{1}^{f}X = h_{10}^{−1}X, we have no information about these objects. See [Pal94] for some information about chromatic structure in the module setting.

Lastly, it seems possible that the chromatic tower constructed here, as well as those with other orderings, are the Steenrod algebra analogues of the multiple complexes of Benson and Carlson [BC87]. Is this a good analogy? If so, does it give any new insight into Stable(A), Stable(kG), or Stable(Γ) for an arbitrary commutative Hopf algebra Γ? Evens and Siegel [ES96] have extended the multiple complex construction to modules over finite-dimensional cocommutative Hopf algebras, or equivalently to comodules over finite-dimensional commutative Hopf algebras, so their work is relevant here.

CHAPTER 4 Computing Ext with elements inverted

In this chapter we present a spectral sequence for computing v_{n}^{−1} Ext_{A}^{∗∗}(M, N) for appropriate comodules M and N, and we give several applications. The spectral sequence is a localized Adams spectral sequence; we prove convergence using vanishing planes and a genericity argument.

In this chapter, we use the following notation: when p = 2, then q_{n} = p_{n+1}^{0}, Q_{n} = P_{n+1}^{0}, and v_{n} = h_{n+1,0}. When p is odd, then q_{n}, Q_{n}, and v_{n} are as usual—see Section 2.2.

We start the chapter in Section 4.1 with a discussion of the q_{n}-based Adams spectral sequence. Since q_{n} is connective, convergence for this spectral sequence is tractable. q_{n} is a nice ring spectrum, and its coefficient ring is polynomial on a class v_{n}. In Section 4.2 we try to invert some power of this class in the q_{n}-based spectral sequence converging to π_{∗∗}X. Depending on X, this may yield a new spectral sequence. When it does, we give a condition on the unlocalized spectral sequence— the existence of a vanishing plane with specified “slope”—that ensures convergence of the localized one. By appealing to Theorem 1.4.4, we see that convergence is generic in X.

Next, in Section 4.3 we present a spectrum X for which we can show that the E_{2}-term has an appropriate vanishing plane. It turns out that X can be built out of the sphere by a series of self-maps, similarly to the spectra F (u_{d_{1}}^{j_{1}}, …, u_{d_{n}}^{j_{n}}) of Section 2.5. This allows us to show that all finite spectra which are L_{a_{n}}^{f}-acyclic satisfy the convergence condition for the localized spectral sequence, where a_{n} is the slope of v_{n}. We remark that X is the injective resolution of the comodule known as \frac{1}{2}A(n), so our discussion of X involves an investigation of the comodule structure on A(n) which was constructed by Mitchell [Mit85] and Smith [Smi]. This discussion spans Sections 4.3–4.5.

We close the chapter with Section 4.6, in which we do some computations and discuss some applications. First we compute (Q_{n})_{∗∗}(Q_{n}) for all n, and from this Ext_{(Q_{0})_{∗∗}(Q_{0})}^{∗∗}((Q_{0})_{∗∗}, (Q_{0})_{∗∗}X) for every connective X. This lets us compute v_{0}^{−1}π_{∗∗}X for such X. Then we reproduce Eisen’s calculation [Eis87] of v_{n−1}^{−1} Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}), where Y (n) is the quotient Hopf algebra

Y (n) = F_{2}[ξ_{n}, ξ_{n+1}, …].

Finally, we discuss Mahowald’s conjectured calculation [Mah70] at the prime 2 of v_{1}^{−1}π_{∗∗}M, where M is an injective resolution of the homology of the mod 2 Moore spectrum. We note that the spectral sequence presented here will converge, and we have guesses for the differentials, but we do not know how to verify the guesses.

The main result of this chapter, the convergence of the localized spectral sequence, was originally proved by Margolis at the prime 2 in unpublished notes [Mar]; his approach was also presented in the author’s Ph.D. thesis [Pal91], as was much of the analysis of the comodule structure of A(n) in Sections 4.3–4.5. We have taken a different approach to convergence here, which was suggested by Hopkins.

4.1. The q_{n}-based Adams spectral sequence

We will use the following notation throughout this chapter.

Notation 4.1.1. We first introduced the spectra P_{t}^{s} and Q_{n}, as well as their connective covers p_{t}^{s} and q_{n}, in Section 2.2.

(a) When p = 2, we follow convention and write Q_{n} for the field spectrum P_{n+1}^{0}. Similarly, we write q_{n} for the connective version, the ring spectrum p_{n+1}^{0}. We computed the coefficients of these spectra in Proposition 2.2.3 and Corollary 2.2.5 and found that (q_{n})_{∗∗} = F_{2}[v_{n}] and (Q_{n})_{∗∗} = F_{2}[v_{n}^{±1}]; again, we are following convention and writing v_{n} for h_{n+1,0}. Recall that |v_{n}| = (1, 2^{n+1} − 1).

(b) At odd primes, q_{n}, Q_{n}, and v_{n} are as usual. These have the same coefficient rings as at the prime 2, with |v_{n}| = (1, 2p^{n} − 1).

(c) Let a_{n} denote the slope of Q_{n}, as defined in Notation 2.2.8:

a_{n} = \begin{cases}|ξ_{n+1}| = 2^{n+1} − 1, & when p = 2, \\ |τ_{n}| = 2p^{n} − 1, & when p is odd. \end{cases}

Regardless of the prime, the element v_{n} has bidegree |v_{n}| = (1, a_{n}).

(d) Define the Hopf algebra E to be

E = \begin{cases}F_{2}[ξ_{n+1}]/(ξ_{n+1}^{2}), & when p = 2, \\ E[τ_{n}], & when p is odd. \end{cases}

E is a quotient Hopf algebra of A, and indeed of any quotient A′ of A in which ξ_{n+1} (respectively, τ_{n}) is nonzero. Regardless of the prime, q_{n} = HE.

We consider the q_{n}-based Adams spectral sequence. Ideally, we would like q_{n} to satisfy Condition 1.4.1: q_{n} should be a nice ring spectrum, q_{n} should be flat (i.e., (q_{n})_{∗∗}(q_{n}) should be flat as a left module over (q_{n})_{∗∗}), and q_{n} should have good connectivity properties. q_{n} is defined to be HE, so it is a commutative and associative ring spectrum, by Proposition 1.2.8. We computed its coefficient ring in Proposition 2.2.3 (see also Notation 4.1.1), so we know that it satisfies the proper connectivity requirements. Because of this, the Adams spectral sequence converges to the homotopy of the “q_{n}-completion” of X. By Proposition 1.4.3, every connective spectrum is q_{n}-complete.

Proposition 4.1.2. Let X be a connective spectrum. Then the q_{n}-based Adams spectral sequence abutting to π_{∗∗}X converges.

The next issue is whether q_{n} is flat, which would allow us to identify the E_{2}- term of the spectral sequence as an Ext group. It turns out that it isn’t—see Corollary 4.6.2—so we do not have a formula for the E_{2}-term. Since we only use the q_{n}-based Adams spectral sequence as a tool to study the Q_{n}-based one, we do not need to know the E_{2}-term; hence this is not a big problem.

4.2. THE Q_{n}-BASED ADAMS SPECTRAL SEQUENCE 81

Remark 4.1.3. If one works over a quotient A′ of A in which ξ_{n+1} (respectively, τ_{n}) is nonzero, then the analogue of Proposition 4.1.2 holds. By “working over a quotient A′ of A,” we mean replacing the spectrum X with HA′ ∧ X, as in Theorem 2.3.6 (and see Remarks 5.1.4 and 5.1.7 for other examples). The resulting spectral sequence will converge to π_{∗∗}(HA′ ∧ X) = HA′_{∗∗}X. If X is connective, so is HA′ ∧ X, so Proposition 4.1.2 holds exactly as stated. As we progress through the chapter, we will make similar comments, most of which will have a bit more substance.

4.2. The Q_{n}-based Adams spectral sequence

In this section, we invert the element v_{n} in the q_{n}-based Adams spectral sequence converging to π_{∗∗}X, in the hopes of computing v_{n}^{−1}π_{∗∗}X. Of course, for v_{n}^{−1}π_{∗∗}X to make sense, some power of v_{n} must act on π_{∗∗}X. The easiest way to arrange this is for the spectrum X under consideration to have an appropriate sort of self-map. Here is a special case of Definition 2.4.2; see also Definition 6.2.1.

Definition 4.2.1. Fix a spectrum X and an integer n ≥ 0. A self-map f ∈ [X, X]_{∗∗} is a v_{n}-map if for some j,

η_{Q_{n}} ∧ f = v_{n}^{j} ∧ 1_{X}

as maps X ⟶ Q_{n} ∧ X. We write v_{n}^{−1}X for the telescope

\underrightarrow{lim}(X ⟶^{f} X ⟶^{f} X ⟶^{f} ⋯).

This is equivalent to requiring that 1_{Q_{n}} ∧f = v_{n}^{j}∧1_{X} in [Q_{n}∧X, Q_{n}∧X]_{∗∗}—see Remark 6.2.2. If X is finite, it also implies that η_{q_{n}} ∧ f^{i} = v_{n}^{ij} ∧ 1_{X} when i ≫ 0.

Remark 4.2.2. (a) v_{n}: q_{n} ⟶ q_{n} is an example of a v_{n}-map, and in this

case, Q_{n} = v_{n}^{−1}q_{n}.

(b) In general, the telescope \underrightarrow{lim}(X ⟶^{f} X ⟶^{f} ⋯) may depend on the map f, so the notation v_{n}^{−1}X is perhaps not ideal.

(c) One can use Theorem 2.4.3 to produce a v_{n}-map on any finite spectrum of type a_{n}, and any two such v_{n}-maps agree up to a power—see the statement and proof of Proposition 2.5.2(d). Hence in this case, v_{n}^{−1}X is independent of the v_{n}-map. We focus on this situation below.

(d) Since Q_{n} is a field spectrum, then the Künneth formula says that if f: X ⟶ X is a v_{n}-map, then so is 1_{Z} ∧ f for any spectrum Z.

Suppose that X is a spectrum with a v_{n}-map. Then we claim that inverting v_{n} in the q_{n}-based Adams spectral sequence for π_{∗∗}X gives the Q_{n}-based Adams spectral sequence abutting to π_{∗∗}(v_{n}^{−1}X). Q_{n} is a commutative associative ring spectrum, and in fact a field spectrum, hence flat, so we can compute the E_{2}- term of this spectral sequence. Convergence is an issue, though, because Q_{n} is not connective. To summarize:

Theorem 4.2.3. Suppose that X is a connective spectrum with a v_{n}-map.

(a) The Q_{n}-based Adams spectral sequence abutting to π_{∗∗}(v_{n}^{−1}X) may be obtained from the q_{n}-based Adams spectral sequence converging to π_{∗∗}X by inverting some power of v_{n} at the E_{2}-term.

(b) The E_{2}-term of the Q_{n}-based Adams spectral sequence for π_{∗∗}(v_{n}^{−1}X) is

E_{2}^{∗∗∗} ≅ Ext_{(Q_{n})_{∗∗}(Q_{n})}^{∗∗∗}((Q_{n})_{∗∗}, (Q_{n})_{∗∗}X).

(c) In this spectral sequence, multiplication by v_{n} increases degree by (0, a_{n} − 1, 1).

(d) The differential d_{r} increases degree by (r, r − 1, 1 − r).

Proof. Part (a): It suffices to show that given a q_{n}-Adams tower for X, inverting v_{n} turns it into a Q_{n}-Adams tower for v_{n}^{−1}X. The first section of Miller’s paper [Mil81] is a good reference for the construction of Adams spectral sequences; it tells us that a q_{n}-Adams tower for X is a diagram of the form

(4.2.4)  X ⟶ K_{0}X ⟶ Σ^{−1,0}K_{1}X ⟶ Σ^{−2,0}K_{2}X ⟶ ⋯,

in which

(i) Each K_{s}X is q_{n}-injective, meaning that K_{s}X is a retract of a spectrum of the form q_{n} ∧ Y for some Y, and

(ii) The sequence (4.2.4) is q_{n}-exact, meaning that applying [−, J] gives an

exact sequence for every q_{n}-injective spectrum J. In fact, according to the construction of the Adams spectral sequence given in Section 1.4, we may assume that K_{s}X is of the form K_{s}X = q_{n} ∧ Z_{s} ∧ X for some spectrum Z_{s}. (Z_{s} is actually \overline{q_{n}}^{∧s}, where \overline{q_{n}} is the fiber of the unit map of q_{n}, but we do not use this here.)

Let f: X ⟶ X denote the v_{n}-map of X; then 1_{Z_{s}} ∧ f is a v_{n}-map on Z_{s} ∧ X. By the remarks following Definition 4.2.1, after replacing f by a suitable power, then the two self-maps v_{n}^{j} ∧ 1_{Z_{s}} ∧ 1_{X} and 1_{q_{n}} ∧ 1_{Z_{s}} ∧ f of q_{n} ∧ Z_{s} ∧ X agree on homotopy; hence they have the same telescope. Thus if we invert 1 ∧ 1 ∧ f in each term of the tower, we get

v_{n}^{−1}X ⟶ q_{n} ∧ Z_{0} ∧ v_{n}^{−1}X ⟶ Σ^{−1,0}q_{n} ∧ Z_{1} ∧ v_{n}^{−1}X ⟶ · · · .
∥  ∥
Q_{n} ∧ Z_{0} ∧ v_{n}^{−1}X  Σ^{−1,0}Q_{n} ∧ Z_{1} ∧ v_{n}^{−1}X
∥  ∥
Q_{n} ∧ Z_{0} ∧ X  Σ^{−1,0}Q_{n} ∧ Z_{1} ∧ X

Certainly, every term in this tower is Q_{n}-injective. To show that the tower is Q_{n}-exact, we apply [−, Q_{n} ∧ Y]. We claim that

(4.2.5) [Q_{n} ∧ Z_{s} ∧ X, Q_{n} ∧ Y] = [q_{n} ∧ Z_{s} ∧ X, Q_{n} ∧ Y] = v_{n}^{−1}[q_{n} ∧ Z_{s} ∧ X, q_{n} ∧ Y].

Given this, since localization is exact, then Q_{n}-exactness for the localized tower follows from q_{n}-exactness of the original tower. The first equality in Equation (4.2.5) follows because the fiber of

q_{n} ∧ Z_{s} ∧ X ⟶ Q_{n} ∧ Z_{s} ∧ X

is killed by inverting v_{n}, so there are no maps from that fiber to Q_{n}, on which v_{n} is an equivalence. The second equality follows because Q_{n} = v_{n}^{−1}q_{n}.

Part (b): Since Q_{n} is flat, we can compute the E_{2}-term as an Ext group. Since X is connective, then (Q_{n})_{∗∗}X = (Q_{n})_{∗∗}(v_{n}^{−1}X), so the E_{2}-term for the spectral sequence abutting to π_{∗∗}(v_{n}^{−1}X) is as given.

Part (c): We check this for the q_{n}-based Adams spectral sequence. Since v_{n} is in the coefficient ring (q_{n})_{∗∗}, then it has filtration zero in the spectral sequence. For it to act correctly in the abutment π_{∗∗}X, then it must act with the indicated degree.

Part (d): This is true for any Adams spectral sequence. See Theorem 1.4.2. □

[figure: s; s + u; (t + u) − (s + u); = t − s; s = 0; s = 1; s = 2; slope ε; v_{n}]

Figure 4.2.A. Vanishing plane in Theorem 4.2.6, with b = 0. The E_{r}-term is zero above the plane containing the lines labeled “s = 0,” “s = 1,” etc. The tridegrees with fixed s+u and t−s coordinates, with s increasing, form a composition series for π_{s+u,t+u}; we have chosen axes so that the bidegrees with the filtration degree s fixed are drawn with the usual Adams spectral sequence grading—see Remark 1.1.2. As indicated, multiplication by v_{n} moves parallel to the (s + u, t − s)-plane, at a slope of \frac{1}{a_{n}−1}. With this choice for the axes, d_{r} goes up one, left one, and increases s by r.

As mentioned above, convergence of the Q_{n}-based Adams spectral sequence is not guaranteed. Here is our first result about convergence; we use it to prove Theorem 4.3.1 below. Recall from Notation 4.1.1 that a_{n} is the slope of Q_{n}.

Theorem 4.2.6. Fix an integer n ≥ 0. Let X be a connective spectrum with a v_{n}-map. Consider the Q_{n}-based Adams spectral sequence abutting to v_{n}^{−1}π_{∗∗}X. If for some numbers r, b, and ε > 0, the E_{r}-term of the q_{n}-based Adams spectral sequence for X has a vanishing plane of the form

(4.2.7)  E_{r}^{s,t,u} = 0 when s ≥ \frac{−a_{n}}{ε} (s + u) + \frac{1}{ε}(t + u) + \frac{b}{ε},

then the Q_{n}-based spectral sequence converges.

We will only apply this result with ε = 1, but the theorem is just as easy to prove with a general ε.

See Figure 4.2.A for a picture of the vanishing plane.

Proof. Consider the q_{n}-based Adams spectral sequence for π_{∗∗}X, which converges because X is connective. Since X has a v_{n}-map, we can invert this map to get the Q_{n}-based Adams spectral abutting to v_{n}^{−1}π_{∗∗}X. Since the v_{n}-map corresponds to multiplication by some power of v_{n}, say v_{n}^{ℓ}, then the q_{n}-based spectral sequence is one of F_{p}[v_{n}^{ℓ}]-modules, and the Q_{n}-based spectral sequence is obtained by inverting v_{n}^{ℓ}. When we have inverted v_{n}^{ℓ}, there are two ways the resulting spectral sequence can fail to converge to v_{n}^{−1}π_{∗∗}X: differentials and extensions.

We will call a class x in the q_{n}-spectral sequence v_{n}-periodic if (v_{n}^{ℓ})^{j}x ≠ 0 for all j. If a class is not v_{n}-periodic, we say that it is v_{n}-torsion.

Since the q_{n}-spectral sequence is one of F_{p}[v_{n}^{ℓ}]-modules, the only allowable differentials go from v_{n}-periodic classes to other v_{n}-periodic classes, or from v_{n}- periodic classes to v_{n}-torsion classes. The first sort of differentials don’t cause any problems with convergence; the second sort only cause problems if an infinite sequence of them conspires to kill off a v_{n}-tower; none of these differentials would be present in the localized spectral sequence, so at E_{∞} of the localized spectral sequence, there would be a superfluous v_{n}-tower.

The second possible problem is extensions: if, at E_{∞} of the q_{n}-spectral sequence, an infinite collection of v_{n}-torsion families is connected by extensions to form a single v_{n}-tower, this tower will not be present in the localized spectral sequence, so the localized spectral sequence will give the wrong answer: it won’t have enough v_{n}-towers.

We claim that the vanishing plane in (4.2.7) will prevent both of these problems. Suppose that we have the vanishing plane, as given, at the E_{r_{0}}-term, and hence at the E_{r}-term for all r ≥ r_{0}. Note that since |v_{n}| = (0, a_{n} − 1, 1) in the spectral sequence, the plane is parallel to the action by v_{n}. Given a class y ∈ E_{r}^{i,j,k}, then d_{r}y has degree

|d_{r}y| = (i + r, j + r − 1, k + 1 − r).

We check the vanishing plane condition: after multiplying through by ε > 0, we have

ε(i + r) ≥^{?} −a_{n}(i + k + 1) + (j + k) + b.

Since ε is positive, then for all sufficiently large r this inequality will hold; hence d_{r}y is above the vanishing plane for all r ≫ 0, and hence zero. In other words, each class in the spectral sequence can only support a differential d_{r} if r is small enough. Since the vanishing plane is parallel to the v_{n}-action, then the class y and all the classes {v_{n}^{d}y} are affected the same way by the plane’s presence.

Similarly, along any given line in the abutment π_{s+u,t+u} of slope a_{n}, the E_{∞}- term is nonzero in only finitely many filtrations above that line; hence there can not be infinitely many extensions, so it is impossible to produce a v_{n}-tower from v_{n}-torsion families. □

Notice that if one can find a spectrum X which satisfies the vanishing plane condition at some E_{r}, then every spectrum in the thick subcategory generated by X also satisfies it, perhaps at a different term of the spectral sequence, by Theorem 1.4.4.

4.3. A(n) AS AN A-COMODULE 85

Remark 4.2.8. If one works over a quotient A′ of A in which ξ_{n+1} (respectively, τ_{n}) is nonzero, one has similar results: the Q_{n}-based spectral sequence is the v_{n}-localization of the q_{n}-based spectral sequence, and the vanishing plane condition (4.2.7) guarantees convergence.

4.3. A(n) as an A-comodule

Here is our goal for the next three sections:

Theorem 4.3.1. Fix n ≥ 0. The Q_{n}-based Adams spectral sequence abutting to v_{n}^{−1}π_{∗∗}X converges for all finite X which are ⋁_{d<a_{n}} Z(d)_{∗∗}-acyclic.

Notice that by Theorem 2.4.3, every finite ⋁_{d<a_{n}} Z(d)_{∗∗}-acyclic spectrum has a v_{n}-map, so it makes sense to discuss v_{n}^{−1}π_{∗∗}X.

To prove the theorem, we will use a finite spectrum called \frac{1}{2}A(n). Recall that we defined the quotient Hopf algebras A(n) of A in Example 2.1.4:

A(n) = \begin{cases}A/(ξ_{1}^{2^{n+1}}, ξ_{2}^{2^{n}}, …, ξ_{n+1}^{2}, ξ_{n+2}, ξ_{n+3}, …), & p = 2, \\ A/(ξ_{1}^{p^{n}}, ξ_{2}^{p^{n−1}}, …, ξ_{n}^{p}, ξ_{n+1}, ξ_{n+2}, …; τ_{n+1}, τ_{n+2}, …), & p odd. \end{cases}

By a theorem of Mitchell [Mit85], and independently, Smith [Smi], A(n) admits a self-dual A-comodule structure extending its left A(n)-comodule structure. Let \frac{1}{2}A(n) denote the sub-vector space of A(n) defined by

\frac{1}{2}A(n) = A(n) □_{E}F_{p},

with E as in Notation 4.1.1. \frac{1}{2}A(n) is certainly an A(n)-subcomodule of A(n); we note in Corollary 4.3.8 below that it in fact inherits an A-comodule structure from A(n). We write \frac{1}{2}A(n) for both the comodule and its injective resolution, as long as it is clear from context which we intend.

We can now prove the main theorem, modulo a few details.

Proof of Theorem 4.3.1. Theorem 4.4.1 below says that \frac{1}{2}A(n) satisfies the vanishing plane condition (4.2.7) at the E_{2}-term of the q_{n}-based Adams spectral sequence. Theorem 4.5.1 says that \frac{1}{2}A(n) generates the thick subcategory of all finite ⋁_{d<a_{n}} Z(d)_{∗∗}-acyclics. By the genericity of vanishing planes, Theorem 1.4.4, we conclude that every ⋁_{d<a_{n}} Z(d)_{∗∗}-acyclic satisfies the vanishing plane condition. By Theorem 4.2.6, the Q_{n}-based Adams spectral sequence converges for these spectra, as desired. □

Remark 4.3.2. If one works over a quotient A′ of A in which ξ_{n+1} (respectively, τ_{n}) is nonzero, then HA′ ∧ \frac{1}{2}A(n) has the same properties as \frac{1}{2}A(n), hence the analogue of Theorem 4.3.1 holds in that context. Note that verifying this does not require reproving all of the results in the next few sections, but rather smashing those results with HA′.

For the rest of this section, we discuss some of the basic properties of the object \frac{1}{2}A(n). In Section 4.4 we prove Theorem 4.4.1: the q_{n}-based Adams spectral sequence for \frac{1}{2}A(n) has a nice vanishing plane at the E_{2}-term. This requires the construction of a specific q_{n}-Adams tower for \frac{1}{2}A(n). In Section 4.5 we prove Theorem 4.5.1: \frac{1}{2}A(n) generates the thick subcategory of all finite ⋁_{d<a_{n}} Z(d)_{∗∗}- acyclics. Parts of this are straightforward, but one piece requires a careful analysis of Mitchell’s and Smith’s A-comodule structure on A(n). We do this careful analysis in Subsection 4.5.1.

\frac{1}{2}A(n) is a sub-vector space, and we claim an A-subcomodule, of A(n); to verify this, we need to examine some features of the A-comodule structure on A(n). First, we establish some notation. The main difference between A(n) at the prime 2 and A(n) at odd primes is, obviously, the τ_{i}’s. It is sometimes useful to deal with the Hopf algebra generated by just the ξ_{i}’s at all primes simultaneously; the following gives us notation to do that.

Definition 4.3.3. Let P (n) be this quotient Hopf algebra of A:

P (n) = F_{p}[ξ_{1}, ξ_{2}, ξ_{3}, …]/(ξ_{1}^{p^{n+1}}, ξ_{2}^{p^{n}}, …, ξ_{n}^{p^{2}}, ξ_{n+1}^{p}, ξ_{n+2}, ξ_{n+3}, …).

So when p = 2, A(n) = P (n). Mitchell [Mit85] and Smith [Smi] also produce an A-comodule structure on P (n) extending its P (n)-comodule structure.

Remark 4.3.4. Mitchell’s and Smith’s constructions of comodule structures on A(n) and P (n) are somewhat different—Mitchell uses GL_{m}(F_{2})-invariants, and Smith uses idempotents in the group algebra F_{2}[Σ_{r}], for appropriate numbers m and r—but the main result of [Mit86] is that these two constructions result in the same A-comodule structure. Note also that [Mit86] is a good reference for Smith’s construction; the construction is also described in [Rav92].

There may be many possible A-comodule structures on A(n) and P (n)—at the prime 2, one can see that there are four different comodule structures on P (1) = A(1) (depending on the possible coactions of ξ_{1}^{4}), and hundreds of different comodule structures on P (2) = A(2) (see [Rot77]). We will use Mitchell’s and Smith’s comodule structure on P (n); when p is odd, we will use another comodule structure on A(n), which we now describe.

Definition 4.3.5. For p an odd prime and n ≥ 0, let V (n) denote the Hopf algebra

V (n) = Λ[τ_{0}, …, τ_{n}].

Let V (−1) = F_{p}.

Then V (n) is a quotient Hopf algebra of A and of A(n). V (−1) is obviously an A-comodule, and it turns out that the same is true for all of the V (n)’s.

Lemma 4.3.6. Fix an odd prime p and n ≥ 0.

(a) V (n) has an A-comodule structure extending its V (n)-comodule structure.

(b) There is a short exact sequence of A-comodules

0 ⟶ V (n − 1) ⟶ V (n) ⟶ s^{|τ_{n}|}V (n − 1) ⟶ 0.

Here, s is the internal suspension functor on the comodule category—see Section 0.1.

(c) If M is an A-comodule which is cofree of rank one as a P (n−1)-comodule, and if N is an A-comodule which is cofree of rank one as a V (n)-comodule, then M ⊗ N is an A-comodule which is cofree of rank one as an A(n)- comodule.

Proof. Part (a): The coproduct on τ_{k},

Δτ_{k} = ∑_{i=0}^{k} ξ_{k−i}^{p^{i}} ⊗ τ_{i} + τ_{k} ⊗ 1,

extended multiplicatively, defines an A-comodule structure map V (n) ⟶ A ⊗ V (n). Part (b): This is clear, given part (a).

Part (c): An A(n)-comodule L is cofree over A(n) if and only if it is cofree over P (n − 1) and V (n). An appeal to Lemma 0.4.3 finishes the proof. □

Notation 4.3.7. It is important which A-comodule structure we use on these Hopf algebras, so we fix Mitchell’s and Smith’s A-comodule structures on P (n), the above comodule structure on V (n), and the structure P (n − 1) ⊗ V (n) on A(n). We will use A(n) and P (n) to denote the Hopf algebras, the comodules with the given structure, and their injective resolutions, depending on the context. For the remainder of this section, we will use the notation A(n) only at odd primes.

Corollary 4.3.8. There is an A-comodule structure on \frac{1}{2}A(n) which extends its A(n)-comodule structure.

Proof. When p = 2, this is part of [Mit85, Corollary 3.14(b)]. When p is odd, then P (n − 1) ⊗ V (n − 1) is an A-comodule which is isomorphic, as an A(n)-comodule, to \frac{1}{2}A(n). □

Remark 4.3.9. We should point out that the quotient Hopf algebra structure and the A-comodule structure on A(n) do not interact as well as they might; the same goes for P (n). To start with, while A(n) is a quotient Hopf algebra of A, the quotient map π: A ↠ A(n) is not a map of A-comodules. For example, chase the primitive element ξ_{1}^{p^{n+1}} around the following diagram:

A ⟶^{π} A(n)
Δ↓  ^{ψ}↓
A ⊗ A ⟶^{1⊗π} A ⊗ A(n).

Going one way, ψ∘π(ξ_{1}^{p^{n+1}}) = 0; going the other way, (1⊗π)∘Δ(ξ_{1}^{p^{n+1}}) = ξ_{1}^{p^{n+1}} ⊗1. Of course, since A is cofree of rank one (Definition 0.4.1), then there is a unique A-comodule map i: A(n) ⟶ A which sends 1 to 1. As an A(n)-comodule, A(n) is cofree of rank one, so there is a unique A(n)-comodule map A(n) ⟶ A(n) so that 1 ↦ 1, namely the identity map. Since the A-comodule structure on A(n) extends its A(n)-comodule structure, the following diagram commutes:

A(n)  A A(n).^{i} w _{[ [ [[]}_{=}  _{u_{π}}

Hence the A-comodule map i: A(n) ⟶ A is one-to-one. Similar comments apply to P (n), so there is a one-to-one A-comodule map j: P (n) ⟶ A. Unfortunately, the inclusions i and j are not, in general, coalgebra maps. Consider this diagram:

P (n)  ⟶^{i} A
Δ↓  ↓^{Δ}
P (n) ⊗ P (n) ⟶^{i⊗i} A ⊗ A.

Let p = 2 and n = 1 and consider the element ξ_{1}^{3}ξ_{2} ∈ P (1). We have i(ξ_{1}^{3}ξ_{2}) = ξ_{1}^{3}ξ_{2} + cξ_{1}^{6} for some scalar c which depends on the choice of A-comodule structure on P (1). Regardless of what c is, one summand in Δ ∘ i(ξ_{1}^{3}ξ_{2}) is ξ_{1}^{5} ⊗ ξ_{1}, whereas (i⊗i)∘Δ(ξ_{1}^{3}ξ_{2}) has no such term. (In module language, this corresponds to the fact that no matter which A^{∗}-module structure one puts on P (1)^{∗}, the Adem relations force a nonzero action by Sq^{4}; therefore the A^{∗}-module quotient map A^{∗} ↠ P (1)^{∗} is not an algebra map.)

4.4. \frac{1}{2}A(n) satisfies the vanishing plane condition

Recall that the number a_{n}, defined in Notation 4.1.1, is the slope of Q_{n}. Our goal here is to prove the following.

Theorem 4.4.1. Fix n ≥ 0. The q_{n}-based Adams spectral spectral sequence converging to π_{s+u,t+u}(\frac{1}{2}A(n)) satisfies:

E_{2}^{s,t,u} = 0 when s ≥ −a_{n}(s + u) + (t + u).

In other words, it satisfies the vanishing plane condition (4.2.7) with ε = 1 and b = 0.

The philosophy behind the proof is that there is a nice relationship between (Q_{n})_{∗∗}(Q_{n}) and \frac{1}{2}A(n): we show in Proposition 4.6.3 below that

(Q_{n})_{∗∗}(Q_{n}) ≅ F_{p}[v_{n}^{±1}] ⊗ H(A □_{E}F_{p}, Q_{n}).

Here, E is as in Notation 4.1.1. We compute the Hopf algebra Γ = H(A □_{E}F_{p}, Q_{n}) in Proposition 4.6.1; from that computation, we can see that \frac{1}{2}A(n) is a quotient Hopf algebra of Γ, and the two agree in dimensions less than a_{n} + 1. \frac{1}{2}A(n) also has a Γ-comodule structure. As a result, \frac{1}{2}A(n) has a Γ-resolution in which the terms increase rapidly in connectivity; in fact, they increase by at least a_{n} + 1 at each stage.

The statements in the above paragraph can be translated directly into a vanishing plane for the E_{2}-term of the Q_{n}-based Adams spectral sequence. We want to look at the q_{n}-based spectral sequence, though, so we give the same argument, but in somewhat different language. The following proof does not use any results from Section 4.6.

Proof. We would like to show the spectrum \frac{1}{2}A(n) satisfies the vanishing plane condition (4.2.7) at the E_{2}-term of the q_{n}-based Adams spectral sequence. Since we do not have a formula for that E_{2}-term in general, we analyze it here by using a specific Adams tower; for this specific Adams tower, we in fact show that we have the desired vanishing plane at E_{1}.

As A(n)-comodules, we have \frac{1}{2}A(n) = A(n) □_{E}F_{p}; therefore, as A-comodules we have (A □_{A(n)}F_{p}) ⊗ \frac{1}{2}A(n) ≅ A □_{E}F_{p}, using the shearing isomorphism, Lemma 1.2.5. As spectra, this translates into the equality

HA(n) ∧ \frac{1}{2}A(n) = HE = q_{n}.

We use this fact to build a q_{n}-Adams tower for \frac{1}{2}A(n).

Lemma 4.4.2. The canonical HA(n)-Adams tower for \frac{1}{2}A(n) is a q_{n}-Adams tower for \frac{1}{2}A(n).

We prove this below.

4.4. \frac{1}{2} A(n) SATISFIES THE VANISHING PLANE CONDITION 89

Recall from Section 1.4 that the canonical HA(n)-based Adams tower for X = \frac{1}{2}A(n) is of the form

(4.4.3)  \begin{aligned}X & F_{0}X & ⟵ & F_{1}X & ⟵ & F_{2}X & ⟵ & ⋯, \\ & ↓ & & ↓ & & ↓ \\ & K_{0}X & & K_{1}X & & K_{2}X \end{aligned}

where

F_{s}X = \overline{HA(n)}^{∧s} ∧ X,
\overline{HA(n)} = fiber (S^{0} ⟶ HA(n)),
K_{s}X = HA(n) ∧ \overline{HA(n)}^{∧s} ∧ X.

The comodule A □_{A(n)}F_{p} is an F_{p}-algebra; let \overline{A □_{A(n)}F_{p}} denote the cokernel of the unit map η: F_{p} ⟶ A □_{A(n)}F_{p}. This unit map is an A-comodule map, so \overline{A □_{A(n)}F_{p}} is an A-comodule.

Lemma 4.4.2 says that the spectral sequence obtained by applying π_{s+u,t+u} to the tower in (4.4.3) agrees with the q_{n}-based Adams spectral sequence starting with the E_{2}-term. Our next goal is to compute the E_{1}-term obtained from (4.4.3). We use the following lemma here, and also to prove Lemma 4.4.2.

Lemma 4.4.4. Let X = \frac{1}{2}A(n).

(a) K_{s}X = q_{n} ∧ \overline{HA(n)}^{∧s}.

(b) Σ^{−1,0}\overline{HA(n)} is an injective resolution of the comodule \overline{A □_{A(n)}F_{p}}.

We prove this below. By the lemma, we have

E_{1}^{s,t,u} = π_{s+u,t+u}(K_{s}X)
= π_{s+u,t+u}(q_{n} ∧ \overline{HA(n)}^{∧s})
= (q_{n})_{s+u,t+u}(\overline{HA(n)}^{∧s})
= (q_{n})_{u,t+u}(Σ^{−s,0}\overline{HA(n)}^{∧s})
= Ext_{E}^{u,t+u}(F_{p}, \overline{A □_{A(n)}F_{p}}^{⊗s}).

For any E-comodule M, Ext_{E}^{u,t+u}(F_{p}, M) is a direct sum of copies of F_{p}, which occur when u = 0, and copies of F_{p}[v_{n}], which start when u = 0—see Proposition 0.5.10. Recall that in this setting, the (s, t, u)-degree of v_{n} is (0, a_{n} − 1, 1); hence if the comodule M has bottom cell in degree m, this Ext group is zero when

a_{n}u ≥ (t + u) + m.

Since A ↠ A(n) is an isomorphism in degrees less than a_{n} + 1, then the bottom class in \overline{A □_{A(n)}F_{p}} is in degree at least a_{n} + 1, so \overline{A □_{A(n)}F_{p}}^{⊗s} starts in degree at least (a_{n} + 1)s. Hence

Ext_{E}^{u,t+u}(F_{p}, \overline{A □_{A(n)}F_{p}}^{⊗s}) = 0

when (a_{n} + 1)s + a_{n}u ≥ (t + u), or equivalently, when

s ≥ −a_{n}(s + u) + (t + u),

as claimed. □

Proof of Lemma 4.4.2. Turning the tower (4.4.3) around, we have

(4.4.5)  X ⟶ K_{0}X ⟶ Σ^{−1,0}K_{1}X ⟶ Σ^{−2,0}K_{2}X ⟶ · · · .

We use the criteria from [Mil81], reproduced in the proof of Theorem 4.2.3(a), to check that (4.4.3) is a q_{n}-based Adams tower for X = \frac{1}{2}A(n). It is clear that every term in the tower is q_{n}-injective, since each K_{s}X is already of the form q_{n} ∧ Y. It suffices to check q_{n}-exactness for spectra J of the form J = q_{n} ∧ Y. In other words, we want to prove exactness for the sequence

[Σ^{−1,0}K_{s+1}X, J] ⟶^{f_{s}^{∗}} [K_{s}X, J] ⟶^{f_{s−1}^{∗}} [Σ^{1,0}K_{s−1}X, J].

By Lemma 4.4.4, K_{s}X is the (s, 0)-suspension of an injective resolution of the comodule

(A □_{E}F_{p}) ⊗ \overline{A □_{A(n)}F_{p}}^{⊗s}.

Furthermore, the map K_{s}X ⟶ Σ^{−1,0}K_{s+1}X comes from tensoring the following composition of maps of comodules with \frac{1}{2}A(n) ⊗ \overline{A □_{A(n)}F_{p}}^{⊗s}:

A □_{A(n)}F_{p} ⟶^{g} \overline{A □_{A(n)}F_{p}} ⟶^{η⊗1} (A □_{A(n)}F_{p}) ⊗ (\overline{A □_{A(n)}F_{p}}),

where g: A □_{A(n)}F_{p} ⟶ \overline{A □_{A(n)}F_{p}} is the quotient map and η: F_{p} ⟶ A □_{A(n)}F_{p} is the unit map. Hence the entire sequence (4.4.5) may be obtained by taking injective resolutions of a sequence of comodules.

We point out that as an E-comodule, A □_{A(n)}F_{p} splits via the maps g and η as A □_{A(n)}F_{p} ≅_{E} F_{p} ⊕ \overline{A □_{A(n)}F_{p}}. One can see this by first recalling from Corollary 0.5.5 that every finite-type E-comodule breaks into a direct sum of cofree comodules and trivial comodules, and then noting that A □_{A(n)}F_{p} is zero in dimension a_{n}, so the class in dimension 0 must give a trivial summand, not a cofree one.

Given all of this, it is easy to prove q_{n}-exactness of (4.4.5) at [K_{s}X, HE ∧ Y]. By Corollary 1.2.7, this is the same as maps over E from K_{s}X to Y:

[K_{s}X, HE ∧ Y] = [K_{s}X, Y]^{E}.

Over E, though, the map K_{s−1}X ⟶ K_{s}X is the composite of first projecting K_{s−1}X onto a summand, and then including that as a summand of K_{s}X. The image of this map is clearly the kernel of K_{s}X ⟶ K_{s+1}X. It follows that (4.4.5) is q_{n}-exact. □

Proof of Lemma 4.4.4. Part (a): Since X ∧ HA(n) = q_{n}, then K_{s}X = q_{n} ∧ \overline{HA(n)}^{∧s}, as desired.

Part (b): S^{0} is an injective resolution of F_{p}, HA(n) is an injective resolution of A □_{A(n)}F_{p}, and the unit map S^{0} ⟶ HA(n) is induced by the monomorphism F_{p} ↪ A □_{A(n)}F_{p}. Hence \overline{HA(n)} is an injective resolution of the cokernel \overline{A □_{A(n)}F_{p}} = A □_{A(n)}F_{p}/F_{p}, suspended: \overline{HA(n)} = Σ^{1,0}I, where I is an injective resolution of \overline{A □_{A(n)}F_{p}}. □

4.5. \frac{1}{2}A(n) generates the expected thick subcategory

In this section, we examine the thick subcategory generated by the object \frac{1}{2}A(n). (Throughout, we will use \frac{1}{2}A(n) to denote both the comodule and its injective resolution, as long as the context makes it clear which we mean.) In particular, we prove the following.

4.5. \frac{1}{2} A(n) GENERATES THE EXPECTED THICK SUBCATEGORY 91

Theorem 4.5.1. (a) Z(d)_{∗∗} \frac{1}{2}A(n) is zero if and only if d < a_{n}.

(b) For any finite spectrum W which is Z(d)_{∗∗}-acyclic for all d < a_{n}, W is in thick(\frac{1}{2}A(n)).

Let d_{m} be the slope preceding a_{n},

d_{m} = a_{n} − 1 = \begin{cases}|ξ_{n}^{2}| & when p = 2, \\ |ξ_{n}| & when p is odd. \end{cases}

Then Theorem 2.5.3(a) says that the spectrum F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) has the same properties. We will prove Theorem 4.5.1 the same way we proved that one: we will show that \frac{1}{2}A(n) can be built as an iterated cofiber of well-understood self-maps of finite spectra.

We need some definitions and preliminary results before we start the proof. The Hopf algebra V (k) was defined in Definition 4.3.5; we produced an A-comodule structure on it in Lemma 4.3.6.

Definition 4.5.2. (a) Suppose that p is odd. For any k ≥ 0, define the

vector space A(n)_{k} to be A(n)_{k} = P (n − 1) ⊗ V (k − 1).

(b) For any prime p, for any k ≥ 1, let B(k) be this quotient Hopf algebra of

A:

B(k) = F_{p}[ξ_{k}, ξ_{k+1}, ξ_{k+2}, …]/(ξ_{i}^{p} : i ≥ k),

and let

P (n)_{k} = ker (P (n) ⟶^{\overline{ψ}} JA ⊗ P (n) ⟶ JB(k) ⊗ P (n)) ,

where JA = cok(F_{p} → A) is the coaugmentation coideal of A, JB(k) is the coaugmentation coideal of B(k), and \overline{ψ} is the reduced coaction map on P (n). (See Definition 1.2.12(b) for “coaugmentation coideal.”) For example, A(n)_{n+1} = A(n) and A(n)_{0} = P (n − 1); more generally, if k ≤

n + 1, then each A(n)_{k} is an A-subcomodule of A(n). We show in Proposition 4.5.6 that each P (n)_{k} is an A-subcomodule of P (n). To start with, we have the following.

Proposition 4.5.3. Fix k > n + 1. For all primes, we have P (n)_{k} = P (n). In particular, if k > n + 1, then P (n)_{k} is an A-comodule.

Proof. This is due to Mitchell: when p = 2, [Mit85, Corollary 3.14(a)] is exactly the result that P (n)_{k} = P (n) when k > n + 1. When p is odd and k > n + 1, the proof of Mitchell’s result can be imitated to show that P (n)_{k} = P (n). (Alternatively, this result says that ξ_{k} “coacts trivially” on P (n) if k > n + 1— there are no terms ξ_{k} ⊗ x in the A-coaction map on elements of P (n). We describe Smith’s construction of the A-comodule structure on P (n) in Subsection 4.5.1; P (n) is constructed as a summand of a tensor product of comodules on which ξ_{k} coacts trivially; as a result, ξ_{k} coacts trivially on the entire tensor product, and hence on P (n).) □

Corollary 4.5.4. As a sub-vector space of P (n) when p = 2 (respectively, of A(n) when p is odd), we have

\frac{1}{2}A(n) = \begin{cases}P (n)_{n+1} & when p = 2, \\ A(n)_{n} & when p is odd. \end{cases}

Proof. At the prime 2, \frac{1}{2}A(n) is defined to be P (n) □_{E}F_{2}, with E as in Notation 4.1.1. Since P (n)_{n+2} = P (n), then

P (n)_{n+1} = ker (P (n) ⟶ JE ⊗ P (n))
= F_{2} □_{E}P (n)
= P (n) □_{E}F_{2},

the last equality holding because E is a conormal quotient of P (n).

The p odd case is even easier. In this case, \frac{1}{2}A(n) = A(n) □_{E}F_{p}, and one can see from our comodule structure on A(n) = P (n − 1) ⊗ V (n) that \frac{1}{2}A(n) = P (n − 1) ⊗ V (n − 1), as desired. □

In order to prove Theorem 4.5.1, we will need the following results. The Ext elements v_{k}, h_{k0}, and b_{k0} are defined in Remark 2.1.3.

Proposition 4.5.5. For each k with 0 ≤ k ≤ n, there is a short exact sequence of A-comodules

0 → A(n)_{k} → A(n)_{k+1} → s^{|τ_{k}|}A(n)_{k} → 0.

The short exact sequence corresponds to v_{k} ∈ Ext_{A}^{1}(A(n)_{k}, A(n)_{k}).

Proof. This follows from Lemma 4.3.6(b). □

We have a similar result for P (n). The proof is more involved, so it will help to have a more precise statement.

Proposition 4.5.6. Fix k with 1 ≤ k ≤ n + 1.

(a) P (n)_{k} is an A-subcomodule of P (n)_{k+1}, and it is spanned by the monomials in A(n) in which each ξ_{i}, k ≤ i ≤ n + 1, occurs to a power divisible by p.

(b) When p = 2, there is a short exact sequence of A-comodules

0 → P (n)_{k} → P (n)_{k+1} → s^{|ξ_{k}|}P (n)_{k} → 0,

corresponding to h_{k0} ∈ Ext_{A}^{1}(P (n)_{k}, P (n)_{k}).

(c) When p is odd, there is an A-comodule M and two short exact sequences of A-comodules

0 → P (n)_{k} → P (n)_{k+1} → s^{|ξ_{k}|}M → 0,
0 → M → P (n)_{k+1} → s^{|ξ_{k}^{p−1}|}P (n)_{k} → 0,

corresponding to elements α ∈ Ext_{A}^{1}(M, P (n)_{k}) and β ∈ Ext_{A}^{1}(P (n)_{k}, M), respectively, such that βα = b_{k0} in Ext_{A}^{2}(P (n)_{k}, P (n)_{k}).

Proof. Part (a): We prove this by downward induction on k. The induction starts with P (n)_{n+2} = P (n), which we know is an A-comodule with the advertised monomial basis. Given that P (n)_{k+1} has the right basis, we will show that P (n)_{k} is an A-subcomodule and that its basis is of the correct form.

By the definition 4.5.2 of P (n)_{k}, we have

P (n)_{k} = ker (P (n)_{k+1} ⟶ JD[ξ_{k}] ⊗ P (n)_{k+1}) ,

where the map is induced by the coaction map. The coaugmentation coideal JD[ξ_{k}] is equal to Span_{F_{p}}(ξ_{k}, …, ξ_{k}^{p−1}), and by coassociativity, an element x has a term ξ_{k}^{i} ⊗ y in its coproduct for some i with 0 < i < p if and only if it has a term ξ_{k} ⊗ z. Hence

P (n)_{k} = ker (P (n)_{k+1} ⟶ Span(ξ_{k}) ⊗ P (n)_{k+1})
= ker (P (n)_{k+1} ⟶ s^{|ξ_{k}|}P (n)_{k+1}) ,

where the map is defined by first applying the coaction map on P (n)_{k}, then projecting from A to Span(ξ_{k}). In other words, the map is multiplication by P_{k}^{0}, the element of A^{∗} dual to ξ_{k}. We claim that this is an A-comodule map, and hence its kernel, P (n)_{k}, inherits an A-comodule structure from P (n)_{k+1}. We have to check that this diagram commutes:

P (n)_{k+1}  ⟶^{P_{k}^{0}} s^{|ξ_{k}|}P (n)_{k+1}
ψ↓  ↓^{ψ}
A ⊗ P (n)_{k+1} ⟶^{1⊗P_{k}^{0}} A ⊗ s^{|ξ_{k}|}P (n)_{k+1}.

Since P (n)_{k+1} is an A-subcomodule of P (n), and since we have formulas for the P (n)-coaction on P (n), then we understand the P (n)-coaction on P (n)_{k+1}. In particular, the only way for a monomial y ∈ P (n)_{k+1} to have a summand ξ_{k} ⊗ z in its coproduct is if ξ_{k}^{j} is a factor in y, where j is not divisible by p. In this case, multiplication by P_{k}^{0} takes ξ_{k}^{j}z to ξ_{k}^{j−1}z. Hence the kernel of P_{k}^{0} is spanned by the monomials in P (n)_{k+1} in which ξ_{k} occurs to a power divisible by p. Furthermore, we can see that if x is in the kernel of P_{k}^{0}, then it goes to zero each way around the diagram. Given a monomial y which is not in the kernel, then y is of the form y = ξ_{k}z for some z, and P_{k}^{0}(y) = σz, where we have written σz for the element

σz = ξ_{k} ⊗ z ∈ Span(ξ_{k}) ⊗ P (n)_{k+1}.

If the comultiplication map applied to z gives ∑_{i} a_{i} ⊗ z_{i}, then by coassociativity, going each way around the diagram yields ∑_{i} a_{i} ⊗ (σz_{i}). In other words, the diagram commutes for all monomials in P (n)_{k+1}. Since the monomials form a basis, we conclude that the diagram commutes.

We have just shown that P_{k}^{0} : P (n)_{k+1} ⟶ P (n)_{k+1} is an A-comodule map; hence its kernel P (n)_{k} is an A-subcomodule of P (n)_{k+1}. We also verified the monomial basis for P (n)_{k}. This completes the inductive step, and hence the proof of part (a).

Part (b): By part (a), P (n)_{k} is an A-subcomodule of P (n)_{k+1}, so we have to check that the cokernel of the inclusion map is isomorphic to the appropriate suspension of P (n)_{k}. But we understand exactly what that cokernel is, by our description of the map P_{k}^{0}: for x ∈ P (n)_{k}, we have

P (n)_{k+1} ⟶^{P_{k}^{0}} s^{|ξ_{k}|}P (n)_{k+1}.
x ⟼ 0
ξ_{k}x ⟼ σx

Since multiplication by P_{k}^{0} is an A-comodule map, then the A-coaction on σx is σ applied to the A-coaction on x; in other words, the cokernel of P_{k}^{0} is s^{|ξ_{k}|}P (n)_{k}, as an A-comodule. This finishes the proof of part (b).

Part (c): This is similar to (b); see also the proof of Lemma 1.2.15(b). By the proof of (a), multiplication by the element P_{k}^{0} ∈ A^{∗} is a map of A-comodules P (n)_{k+1} ⟶ P (n)_{k+1}; therefore, so is multiplication by (P_{k}^{0})^{p−1}. The kernel of P_{k}^{0} is P (n)_{k}; the kernel of (P_{k}^{0})^{p−1} is another A-comodule, M. As in part (b), one can identify the cokernels of P_{k}^{0} and (P_{k}^{0})^{p−1} as suspensions of M and P (n)_{k}, respectively, yielding the advertised short exact sequences. As in the proof of Lemma 1.2.15(b), one can identify the Yoneda product βα as b_{k0}. □

We need to use the “pth power” functor Φ on the category of A-comodules. When p is odd, we assume that M is an evenly graded comodule; when p = 2, M may be arbitrary. ΦM is defined to be

(ΦM)_{n} = \begin{cases}M_{r} & n = pr, \\ 0 & n not divisible by p, \end{cases}

with comodule structure ψ_{ΦM}: ΦM ⟶ A ⊗ ΦM given by

ψ_{ΦM}(x) = (F ⊗ i) ∘ ψ_{M}(i^{−1}x).

Here, F is the Frobenius map on A defined by F (a) = a^{p}, ψ_{M} is the comodule structure map on M, and i: M ⟶ ΦM is the apparent vector space isomorphism (which multiplies degrees by p).

Proposition 4.5.7. For any prime p, P (n)_{1} ≅ ΦP (n − 1) as A-comodules. This proposition is the main technical result here; its proof requires a careful

analysis of the A-comodule structure on P (n). We prove this in Subsection 4.5.1. We can now prove the main theorem of this section.

Proof of Theorem 4.5.1. Part (a): Corollary 4.5.4 lets us describe \frac{1}{2}A(n) as either P (n)_{n+1} (when p = 2) or A(n)_{n} (otherwise).

Suppose that p is odd. To identify the Z(d)-homology groups of \frac{1}{2}A(n), we start with the computation of the Z(d)-homology groups of A(n)_{n+1} = A(n) in Theorem 2.3.6(c): when ξ_{t}^{p^{s}} is nonzero in A(n), then (P_{t}^{s})_{∗∗}A(n) = 0; and when τ_{i} is nonzero in A(n), then (Q_{i})_{∗∗}A(n) = 0.

On the comodule \frac{1}{2}A(n) = A(n) □_{E}F_{p}, the element Q_{n}—the dual of τ_{n}—acts trivially. Hence H(\frac{1}{2}A(n), Q_{n}) ≠ 0. Now assume that d ≠ a_{n}. The short exact sequence

0 → \frac{1}{2}A(n) → A(n) → s^{a_{n}} \frac{1}{2}A(n) → 0

gives a long exact sequence in Z(d)-homology; since A(n) is Z(d)_{∗∗}-acyclic, there is an isomorphism

Z(d)_{∗∗}Σ^{0,a_{n}} \frac{1}{2}A(n) ⟶_{≅}^{Z(d)_{∗∗}v_{n}} Z(d)_{∗∗}Σ^{−1,0} \frac{1}{2}A(n).

Composing this map with itself and using the periodicity in Z(d)_{∗∗}, we end up with

Z(d)_{i,j} \frac{1}{2}A(n) ≅ Z(d)_{i+2,j+2a_{n}} \frac{1}{2}A(n)

≅ Z(d)_{i,j+2a_{n}−2d} \frac{1}{2}A(n)

≅ Z(d)_{i,j+2ℓ(a_{n}−d)} \frac{1}{2}A(n) ∀ℓ ∈ Z.

Since \frac{1}{2}A(n) is connective and a_{n}−d is nonzero, we conclude that Z(d)_{∗∗} \frac{1}{2}A(n) = 0. (See Proposition 2.2.9 for a similar argument, including a few more details.)

As far as showing that Z(d)_{∗∗} \frac{1}{2}A(n) is nonzero for various d’s, if ξ_{t}^{p^{s}} = 0 in A(n), then a Poincaré series argument shows that at the comodule level, H(\frac{1}{2}A(n), P_{t}^{s}) ≠ 0. Similarly, H(\frac{1}{2}A(n), Q_{i}) ≠ 0 when τ_{i} = 0 in A(n).

The same argument works at the prime 2, after replacing A(n) by P (n).

Part (b): Writing A(n)_{k} for both the comodule and its injective resolution, we have a sequence of spectra

A(n)_{n+1} = A(n), A(n)_{n}, A(n)_{n−1}, A(n)_{n−2}, …, A(n)_{1},
A(n)_{0} = P (n − 1), P (n − 1)_{n+1}, P (n − 1)_{n}, …, P (n − 1)_{2},
P (n − 1)_{1} = ΦP (n − 2), ΦP (n − 2)_{n}, ΦP (n − 1)_{n−1}, …, ΦP (n − 2)_{2},
ΦP (n − 2)_{1} = Φ^{2}P (n − 3), …,
…
Φ^{n}P (0), Φ^{n}P (0)_{1} = S^{0},

together with cofibrations connecting them, of the following forms when p is odd:

A(n)_{k} → A(n)_{k+1} → Σ^{0,|τ_{k}|}A(n)_{k} ⟶^{v_{k}} Σ^{−1,0}A(n)_{k},

Φ^{i}P (n − i)_{k} → X̃_{k+1} → Σ^{1,2p|ξ_{k}^{p^{i}} |}Φ^{i}P (n − i)_{k} ⟶^{b_{ki}} Σ^{−1,0}Φ^{i}P (n − i)_{k},

where the spectrum X̃_{k+1} is the cofiber of a nilpotent self-map of Φ^{i}P (n−i)_{k+1} (cf. Lemma 1.2.15(b) and Corollary 1.5.2(b)). We should point out that Φ is a functor on A-comodules, not on spectra, so Φ^{i}P (n − i)_{k} means an injective resolution of the comodule P (n − i)_{k}, not some functor Φ^{i} applied to the spectrum P (n − i)_{k}.

When p = 2, we have cofibrations

Φ^{i}P (n − i)_{k} → Φ^{i}P (n − i)_{k+1} → Σ^{0,|ξ_{k}^{2^{i}} |}Φ^{i}P (n − i)_{k} ⟶^{h_{ki}} Σ^{−1,0}Φ^{i}P (n − i)_{k}.

For any prime p, we can use these cofibrations to imitate the proof of Theorem 2.5.3(a). □

We still owe the proof of Proposition 4.5.7, which we have put into its own subsection.

4.5.1. The proof of Proposition 4.5.7. Recall from Definition 4.3.3 that we have Hopf algebras

P (n) = F_{p}[ξ_{1}, ξ_{2}, ξ_{3}, …]/(ξ_{1}^{p^{n+1}}, ξ_{2}^{p^{n}}, …, ξ_{n}^{p^{2}}, ξ_{n+1}^{p}, ξ_{n+2}, ξ_{n+3}, …),
B(1) = F_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …]/(ξ_{i}^{p} : i ≥ 1).

Recall also that P (n) is an A-comodule, and that

P (n)_{1} = ker (P (n) ⟶ JA ⊗ P (n) ⟶ JB(1) ⊗ P (n)) .

Our goal in this subsection is to show that P (n)_{1} is isomorphic, as an A-comodule, to ΦP (n − 1), the “Frobenius” operator applied to P (n − 1).

By Proposition 4.5.6(a), we know that P (n)_{1} and ΦP (n − 1) are isomorphic as vector spaces, and indeed as ΦP (n − 1)-comodules. As noted after Remark 4.3.4, this does not guarantee that they are isomorphic as A-comodules. Thus we want to show that the A-comodule structure on P (n)_{1} is Φ applied to Mitchell’s and Smith’s structure on P (n − 1).

We start by making some reductions. We want to show that there is a short exact sequence of A-comodules

0 → ΦP (n − 1) → P (n) → P (n)/P (n)_{1} → 0.

It is enough to show that there is an A-comodule monomorphism i: ΦP (n − 1) → P (n): since ΦP (n − 1) is nonzero only in degrees divisible by p, then the cokernel of such a map must contain every ξ_{k}, 1 ≤ k ≤ n + 1, and hence there will be an induced surjection P (n)/ cok i ↠ P (n)/P (n)_{1}. Since P (n)_{1} and ΦP (n − 1) have the same dimension, this map must be an isomorphism.

To construct i: ΦP (n − 1) ⟶ P (n), we use Jeff Smith’s description of the Amodule structure on P (n) as given by Mitchell in [Mit86]: for each integer n ≥ 1, we let Z_{n} denote the n-dimensional A-comodule with vector space basis

(y_{1}, y_{p}, y_{p^{2}}, …, y_{p^{n−1}}),

with |y_{p^{i}}| equal to p^{i} when p = 2, or 2p^{i} when p is odd, and with comodule structure given by

y_{p^{i}} ↦ ∑_{j=0}^{i} ξ_{i−j}^{p^{j}} ⊗ y_{p^{j}}.

At the prime 2, for example, if we let Z = Span(y_{1}, y_{2}, y_{4}, …) be the quotient comodule of H_{∗}(RP^{∞}; F_{2}) “generated” by the class in degree 1, then Z_{n} is an ndimensional subcomodule of Z. For any non-negative integer k, we write Σ_{k} for the symmetric group on k letters, and we let e_{n} be a certain idempotent (defined below) in the group algebra F_{p}[Σ_{(p−1)\binom{n}{2}}].

Note that Z_{n+2} is an A-comodule, and the A-coaction commutes with the sym-

metric group action on Z_{n+2}^{⊗(p−1)\binom{n+2}{2}}, so Z_{n+2}^{⊗(p−1)\binom{n+2}{2}}e_{n+2} inherits an A-comodule structure. By restriction, it has a P (n)-comodule structure, as well.

We have the following.

Theorem 4.5.8 (Smith, [Mit86]). Z_{n+2}^{⊗(p−1)\binom{n+2}{2}}e_{n+2} is cofree of rank one as

a P (n)-comodule. It is self-dual as an A-comodule. Z_{n}^{⊗(p−1)\binom{n}{2}}e_{n} has bottom class in degree 2(\frac{p^{n}−1}{p−1} − n) when p is odd, and in degree 2^{n} − 1 − n when p = 2.

(The self-duality and the identification of the degree of the bottom class come from Mitchell’s construction, [Mit85, Theorem 3.1], combined with Mitchell’s isomorphism between the two A-comodule structures, [Mit86, Theorem 4.2].)

Since the A-comodule structure on P (n) is self-dual, we have the following. Corollary 4.5.9. Up to suspension, there is an isomorphism of A-comodules:

Z_{n+2}^{⊗(p−1)\binom{n+2}{2}}e_{n+2} ≅ (Z_{n+2}^{∗})^{⊗(p−1)\binom{n+2}{2}}e_{n+2}.

Hence (Z_{n+2}^{∗})^{⊗(p−1)\binom{n+2}{2}}e_{n+2} is cofree of rank one as a P (n)-comodule.

We note that ΦZ_{n+1}^{∗} is a subcomodule of Z_{n+2}^{∗}. In order to construct the map i: ΦP (n − 1) ↪ P (n), we prove in Proposition 4.5.11 that there is an isomorphism of A-comodules (neglecting suspensions)

Z_{n+1}^{⊗(p−1)\binom{n+1}{2}}e_{n+1} ≅ Z_{n+1}^{⊗(p−1)\binom{n+2}{2}}e_{n+2}.

Hence the inclusion ΦZ_{n+1}^{∗} ↪ Z_{n+2}^{∗} induces an injective map

(ΦZ_{n+1}^{∗})^{⊗(p−1)\binom{n+2}{2}}e_{n+2} ↪ (Z_{n+2}^{∗})^{⊗(p−1)\binom{n+2}{2}}e_{n+2},

which is the map i that we want. (We remark that one can check that the bottom class in the domain is in the same degree as the bottom class in codomain, but one can see more from general principles: the map is P (n)-linear, and any nonzero P (n)- comodule map into P (n) must have the bottom class in the image, by cofreeness. Since the map is injective, we conclude that it must send 1 ∈ ΦP (n−1) to 1 ∈ P (n).)

Now we discuss the details of the construction, starting with the description of the idempotent e_{n} ∈ F_{p}[Σ_{(p−1)\binom{n}{2}}]. James and Kerber [JK81] have written one of the standard references for representations of the symmetric group; in [Mit86], Mitchell provides most of the results needed to study Smith’s construction.

In general, idempotents for the symmetric group Σ_{k} come from a partition of k; in our case, we consider the partition

λ_{n} = ((n − 1)(p − 1), (n − 2)(p − 1), …, 2(p − 1), (p − 1))

of (p − 1)\binom{n}{2}. Associated to the partition λ_{n} is a Young diagram of shape λ_{n}, as described in [JK81, Section 4]. Given a vector space V together with a basis for V, we can view a standard basis element of V^{⊗(p−1)\binom{n}{2}} as a Young tableau—a Young diagram of shape λ_{n} in which each box is filled in with a basis element of V. For

example, if p = 2, n = 3, and {u, v, w} is a basis for V, then the tableau u v ; w

represents the element u ⊗ v ⊗ w in V^{⊗3}. An arbitrary element of V^{⊗(p−1)\binom{n}{2}} is a sum of such tableaux.

The idempotent element e_{n} ∈ F_{p}[Σ_{(p−1)\binom{n}{2}}] is defined as follows: we let R = R_{n} be the row-stabilizer subgroup of Σ_{(p−1)\binom{n}{2}} associated to the Young diagram of

shape λ_{n}—in other words, the subgroup of permutations of (p − 1)\binom{n}{2} that send each row of λ to itself—and we let \overline{R} be the sum of the elements in R. Similarly, we let C = C_{n} be the column stabilizer subgroup and C̃ the signed sum of the elements in C. We let f_{n} = \overline{R} C̃. One can show (e.g., by applying [JK81, Theorem 3.1.10] and noting that the hook lengths of the diagram are all prime to p) that f_{n}^{2} = αf_{n} for some nonzero scalar α ∈ F_{p}, so e_{n} = \frac{1}{α}f_{n} is an idempotent in F_{p}[Σ_{(p−1)\binom{n}{2}}].

Results from the theory of representations of the symmetric groups give the following (cf. [Mit86, Lemma 1.2]):

Lemma 4.5.10. Let V be a vector space over a field k of characteristic p. Then

dim_{k}(V^{⊗(p−1)\binom{n}{2}}e_{n}) = \begin{cases}0 & if dim_{k} V < n − 1, \\ p^{\binom{n−1}{2}} & if dim_{k} V = n − 1, \\ p^{\binom{n}{2}} & if dim_{k} V = n. \end{cases}

Sketch of proof. Let W_{n} = V^{⊗(p−1)\binom{n}{2}}e_{n}. A formula for the dimension of W_{n} in terms of hook lengths of the partition λ_{n} is given in [JK81, Theorem 5.2.14]. It is clear that if dim V < n − 1, then dim W_{n} = 0; see [JK81, Theorem 5.2.15], for instance. If dim V = n − 1 or dim V = n, then a reasonably straightforward computation using this formula gives the advertised dimension for W_{n}. (When dim V > n, the dimension formula is more complicated.) □

Thus we have

dim Z_{n−1}^{⊗(p−1)\binom{n−1}{2}}e_{n−1} = dim Z_{n−1}^{⊗(p−1)\binom{n}{2}}e_{n} = dim P (n − 3).

As remarked after Corollary 4.5.9, Proposition 4.5.7 follows if we know not just that these vector spaces have the same dimension, but that they are isomorphic as A-comodules. Reindexing, we have to prove this result.

Proposition 4.5.11. As A-comodules, Z_{n}^{⊗(p−1)\binom{n}{2}}e_{n} ≅ Z_{n}^{⊗(p−1)\binom{n+1}{2}}e_{n+1}, up to suspension.

Corollary 4.5.12. (a) There is an A-linear injection P (n − 1) ↪ P (n); dually, there is an A-linear surjection P (n) ↠ s^{c_{n}}P (n − 1), where c_{n} is chosen so that the top class of P (n) gets sent to the top class of P (n − 1).

(b) There is an A-linear injection ΦP (n − 1) ↪ P (n); dually, there is an A-linear surjection P (n) ↠ s^{b_{n}}ΦP (n − 1).

Part (a) of this corollary is originally due to Mitchell [Mit85, Corollary 3.13].

Proof of Proposition 4.5.11. Define a map φ: Z_{n}^{⊗(p−1)\binom{n}{2}} → Z_{n}^{⊗(p−1)\binom{n+1}{2}} by

φ((a_{1,1}⊗ ⋯ ⊗ a_{1,(p−1)(n−1)}) ⊗ ⋯ ⊗ (a_{n−1,1} ⊗ ⋯ ⊗ a_{n−1,p−1})) =

∑_{σ∈Σ_{n}} sgn(σ)(y_{p^{σ(0)}}^{⊗p−1} ⊗ a_{1,1} ⊗ ⋯ ⊗ a_{1,(p−1)(n−1)})

⊗ ⋯ ⊗ (y_{p^{σ(n−1)}}^{⊗p−1} ⊗ a_{n−1,1} ⊗ ⋯ ⊗ a_{n−1,p−1}).

(Here Σ_{n} is permuting the set {0, 1, …, n − 1}.) In other words, φ takes a tableau T of shape λ_{n} to the signed sum of n! tableaux of shape λ_{n+1}, one for each σ ∈ Σ_{n}. The new tableau corresponding to σ is obtained by adding p−1 columns of height n to the left of T, by putting y_{p^{σ(i)}} in each new box in the ith row, and by multiplying that new tableau by sgn(σ). For example, when p = 2 and n = 3, φ is given by

a b ; c ⟼ 1 a b ; 2 c ; 4 + 1 a b ; 4 c ; 2 + 2 a b ; 1 c ; 4 + 2 a b ; 4 c ; 1 + 4 a b ; 1 c ; 2 + 4 a b ; 2 c ; 1 .

Here we are writing k for the basis element y_{k} ∈ Z_{n}. φ is clearly F_{p}-linear and monomorphic. φ is also a map of A-comodules: to see this, let

b = ∑_{σ∈Σ_{n}} sgn(σ)y_{p^{σ(0)}}^{⊗p−1} ⊗ ⋯ ⊗ y_{p^{σ(n−1)}}^{⊗p−1}.

By commutativity of A, φ is A-linear if and only if the map

ψ: Z_{n}^{⊗(p−1)\binom{n}{2}} ⟶ Z_{n}^{⊗(p−1)\binom{n+1}{2}}
T ⟼ b ⊗ T

is A-linear. Moreover, ψ is A-linear if and only if b is primitive in Z_{n}^{⊗(p−1)n}. By commutativity of A, though, it is: one can easily see that under the A-coaction, b goes to 1 ⊗ b.

Hence we have a monomorphism of A-comodules,

φ: Z_{n}^{⊗(p−1)\binom{n}{2}} → Z_{n}^{⊗(p−1)\binom{n+1}{2}}.

The idempotents e_{n} and e_{n+1} split each into summands as A-comodules, so we have an induced map

\overline{φ}: Z_{n}^{⊗(p−1)\binom{n}{2}}e_{n} → Z_{n}^{⊗(p−1)\binom{n+1}{2}}e_{n+1}.

Since these have the same vector space dimension and since φ is monomorphic, it suffices to check that \overline{φ} is nonzero, so it suffices to find one element in the image of e_{n} which gets sent to a nonzero element in the image of e_{n+1}.

This last step is essentially done by Mitchell in [Mit86, Section 2]; at least, Mitchell identifies elements in the image of e_{n}. One can check by induction that if T is the tableau of shape λ_{n} with rows of the form

y_{p^{i}}^{⊗p−1} ⊗ y_{p^{i−1}}^{⊗p−1} ⊗ ⋯ ⊗ y_{p}^{⊗p−1} ⊗ y_{1}^{⊗p−1},

then T ·C̃_{n} is a nonzero element of Z_{n}^{⊗(p−1)\binom{n}{2}}e_{n}, and \overline{φ}(T ·C̃_{n}) is a nonzero element

of Z_{n}^{⊗(p−1)\binom{n+1}{2}}e_{n+1}. □

(Indeed, the bottom class of Z_{n}^{⊗(p−1)\binom{n}{2}}e_{n} is the tableau T · C̃_{n}, and hence

similarly for Z_{n}^{⊗(p−1)\binom{n+1}{2}}e_{n+1} and \overline{φ}(T · C̃_{n}).)

4.6. Some computations and applications

In this section, we compute (Q_{n})_{∗∗}(Q_{n}) and give several applications: we show that v_{0}^{−1}π_{∗∗}X ≅ (Q_{0})_{∗∗}X for every connective X, we reproduce Eisen’s calculation of a particular localized Ext group, and we examine Mahowald’s conjectured calculation of the v_{1}-inverted Ext of the Moore spectrum.

4.6.1. Computation of (Q_{n})_{∗∗}(Q_{n}). In this subsection, we compute the Hopf algebra (Q_{n})_{∗∗}(Q_{n}) of (Q_{n})_{∗∗}-operations for all n, and we use this to compute v_{0}^{−1}π_{∗∗}X when X is connective.

We start by computing (q_{n})_{∗∗}(q_{n}). Margolis computed this when p = 2; see [Mar83, Corollary 19.26] for his version. We extend his work to all primes in the following proposition.

For an A-comodule M, we write H(M, Q_{n}) for the conventional Q_{n}-homology of M,

H(M, Q_{n}) = \frac{ker(M ⟶^{Q_{n}} M)}{im(M ⟶^{Q_{n}} M)},

as discussed at the beginning of Section 2.2. Recall from Notation 4.1.1 that the Hopf algebra E is defined by

E = \begin{cases}F_{2}[ξ_{n+1}]/(ξ_{n+1}^{2}), & when p = 2, \\ E[τ_{n}], & when p is odd. \end{cases}

Proposition 4.6.1. Fix n ≥ 0.

(a) (q_{n})_{∗∗}(q_{n}) = HE_{∗∗}HE ≅ Ext_{E}^{∗∗}(F_{p}, A □_{E}F_{p}).

(b) As an E-comodule, A □_{E}F_{p} is a direct sum of trivial comodules and cofree comodules. The trivial summands are indexed by H(A □_{E}F_{p}, Q_{n}), and the cofree summands are indexed by im(Q_{n}: A □_{E}F_{p} → A □_{E}F_{p}).

(c) Hence (q_{n})_{∗∗}(q_{n}) is a direct sum of copies of F_{p}[v_{n}] and F_{p}. The F_{p}[v_{n}] summands are indexed by H(A □_{E}F_{p}, Q_{n}), and the F_{p} summands are indexed by im Q_{n}.

(d) Let C_{n} be the quotient Hopf algebra of A defined by C_{n} = A/(ξ_{k}^{p^{n}} : k ≥ 1).

p = 2 0 1⋮n⋮ 1 2 3 … n… ξ_{1} ⋮ ξ_{1}^{2^{n}} …  ξ_{n} …  ^{p odd 0 1⋮^{n−1} ⋮ ξ_{1} ⋮ ξ_{1}^{p^{n−1}} . . . . . . ξ_{n} τ_{0}  … τ_{n−1}}

Figure 4.6.A. “Profile function” for H(A □_{E}F_{p}, Q_{n}). Although H(A □_{E}F_{p}, Q_{n}) is not a quotient of A, it still makes sense to show which ξ_{i}^{p^{j}}’s and which τ_{i}’s are nonzero in it.

(i) When p = 2, H(A □_{E}F_{2}, Q_{n}) is isomorphic to the sub-Hopf algebra C_{n+1} □_{B(n+1)}F_{2} of C_{n+1}, where B(n+1) is the quotient Hopf algebra

B(n + 1) = F_{2}[ξ_{n+1}, ξ_{n+2}, …]/(ξ_{k}^{2} : k ≥ n + 1).

(ii) When p is odd, H(A □_{E}F_{p}, Q_{n}) is isomorphic to the sub-Hopf algebra C_{n} □_{Λ(n)}F_{p} of C_{n}, where Λ(n) is the quotient Hopf algebra

Λ(n) = Λ[τ_{n}, τ_{n+1}, …].

In both cases, im Q_{n} is nonzero. For example, ξ_{1}^{p^{n+1}} is in im Q_{n} for all primes p.

See Figure 4.6.A for the profile function of H(A □_{E}F_{p}, Q_{n}).

Proof. Part (a): This is an application of Shapiro’s Lemma—see Example 1.2.4, and also Proposition 2.2.3.

Part (b): This is a basic fact about comodules over an exterior coalgebra, applied to the particular comodule A □_{E}F_{p}, using the fact that the element Q_{n} of A^{∗} is dual to either ξ_{n+1} (p = 2) or τ_{n} (p odd). See Corollary 0.5.5.

Part (c): This is a basic fact about Ext over an exterior coalgebra. See Proposition 0.5.10.

Part (d): This is the interesting part. Since Q_{n} is primitive in A^{∗}, if we know how it acts on the algebra generators of A □_{E}F_{p}, then we can understand the action on the whole algebra. Indeed, if we know how Q_{n} acts on the generators of A itself, then we can understand how it acts on A □_{E}F_{p}.

Let p = 2. Since Q_{n} is dual to ξ_{n+1}, then multiplication by Q_{n} does the following on the generators of A:

ξ_{k} ↦ 0,  if k < n + 1,
ξ_{n+1} ↦ 1,
ξ_{k} ↦ ξ_{k−n−1}^{2^{n+1}}, if k > n + 1.

Therefore, given any monomial in A, if it contains an odd power of any ξ_{k} with k ≥ n + 1, then it is not in A □_{E}F_{p}. The cotensor product is spanned by all other monomials. The description of the action of Q_{n}, together with the primitivity of Q_{n}, allows us to identify H(A □_{E}F_{p}, Q_{n}).

Similarly, when p is odd, multiplication by Q_{n} does this:

τ_{k} ↦ 0,  if k < n,
τ_{n} ↦ 1,
τ_{k} ↦ ξ_{k−n}^{p^{n}}, if k > n,
ξ_{k} ↦ 0,  for all k.
□

See Example 4.6.4 for some computations for small values of n. Corollary 4.6.2. q_{n} is not a flat ring spectrum.

Proof. The trivial module F_{p} is not flat over the polynomial ring (q_{n})_{∗∗} = F_{p}[v_{n}], so one only has to check that im Q_{n} is not zero. This is part (d) of the proposition. □

We can use Proposition 4.6.1 to compute (Q_{n})_{∗∗}(Q_{n}). Proposition 4.6.3. Fix n ≥ 0. Then

(Q_{n})_{∗∗}(Q_{n}) ≅ v_{n}^{−1}(q_{n})_{∗∗}(q_{n})
≅ v_{n}^{−1} Ext_{E}^{∗∗}(F_{p}, A □_{E}F_{p})
≅ F_{p}[v_{n}^{±1}] ⊗ H(A □_{E}F_{p}, Q_{n}).

Proof. We only need to verify that (Q_{n})_{∗∗}(Q_{n}) ≅ v_{n}^{−1}(q_{n})_{∗∗}(q_{n}), but this is clear, once we note that Q_{n} = v_{n}^{−1}q_{n}. The rest follows from Proposition 4.6.1. □

Example 4.6.4. Here are some computations of the Q_{n}-homology of A □_{E}F_{p}, along with the corresponding coalgebras of operations.

(a) Consider Q_{0}. For all primes p, H(A □_{E}F_{p}, Q_{0}) = F_{p}, so (Q_{0})_{∗∗}(Q_{0}) = F_{p}[v_{0}^{±1}]. This is a graded field, so Ext_{(Q_{0})_{∗∗}(Q_{0})}^{s}(−, −) = 0 when s > 0.

(b) Consider Q_{1}. If p = 2, then C_{2} = A/(ξ_{k}^{4}) and B(2) = F_{2}[ξ_{2}, ξ_{3}, …]/(ξ_{k}^{2}),

so

H(A □_{E}F_{2}, Q_{1}) ≅ F_{2}[ξ_{1}, ξ_{2}^{2}, ξ_{3}^{2}, …, ξ_{i}^{2}, …]/(ξ_{k}^{4}).

When p is odd,

H(A □_{E}F_{p}, Q_{1}) ≅ E[τ_{0}] ⊗ F_{p}[ξ_{1}, ξ_{2}, …]/(ξ_{k}^{p}).

In either case, one obtains (Q_{1})_{∗∗}(Q_{1}) from this by tensoring with F_{p}[v_{1}^{±1}]. It is often easy to compute Ext over these coalgebras. For example, if M is the homology of the mod p Moore spectrum, M = F_{2}[ξ_{1}]/(ξ_{1}^{2}) or M = E[τ_{0}], then Ext_{(Q_{1})_{∗∗}(Q_{1})}^{∗∗}((Q_{1})_{∗∗}, (Q_{1})_{∗∗}M) is given by

F_{2}[v_{1}^{±1}] ⊗ F_{2}[h_{11}, h_{21}, h_{31}, …],  if p = 2,
F_{p}[v_{1}^{±1}] ⊗ Λ[h_{10}, h_{20}, h_{30}, …] ⊗ F_{p}[b_{10}, b_{20}, b_{30}, …], if p is odd.

(c) For all n, one can give a similar description of (Q_{n})_{∗∗}(Q_{n}); however, the larger n gets, the more difficult it gets to compute its cohomology.

Combining the computation of (Q_{0})_{∗∗}(Q_{0}) with the vanishing plane condition (4.2.7) yields the following.

Corollary 4.6.5. For every connective spectrum X, the Q_{0}-based Adams spectral sequence abutting to v_{0}^{−1}π_{∗∗}X converges. Furthermore, the spectral sequence collapses at E_{2}, and v_{0}^{−1}π_{∗∗}X = (Q_{0})_{∗∗}X.

Proof. We claim that the vanishing plane condition (4.2.7), with ε = 1, is satisfied at the E_{1}-term of the spectral sequence. Note that the slope of Q_{0} is 1, so the condition is that for some number b,

E_{1}^{s,t,u} = 0 when 2s ≥ t + b.

We give a construction of the Adams spectral sequence in Section 1.4; from there, we recall that

E_{1}^{s,t,u} = π_{s+u,t+u}(q_{0} ∧ \overline{q_{0}}^{∧s} ∧ X).

Since S^{0} → q_{0} is an epimorphism on π_{∗∗}, and an isomorphism on π_{ij} when j −i ≤ 0 and also when i ≤ 0, then π_{ij}(\overline{q_{0}}) is zero when j − i ≤ 0 and when i ≤ 0. By the Hurewicz theorem 1.3.5 and the Künneth theorem 1.2.11 for mod p homology HF_{p}, we find that if X is (i_{0}, j_{0})-connective (Definition 1.3.4), then

π_{s+u,t+u}(q_{0} ∧ \overline{q_{0}}^{∧s} ∧ X) = 0 when (t + u) − (s + u) < j_{0} + s.

In other words, E_{1}^{s,t,u} = 0 when 2s > t − j_{0}: the vanishing plane condition holds at E_{1} with ε = 1 and b = −j_{0}. So the spectral sequence converges.

We know the E_{2}-term of the Q_{0}-based spectral sequence by Theorem 4.2.3; using our computation in Example 4.6.4 that (Q_{0})_{∗∗}(Q_{0}) ≅ (Q_{0})_{∗∗} ≅ F_{p}[v_{0}^{±1}], we have

E_{2}^{s,t,u} ≅ Ext_{F_{p}[v_{0}^{±1}]}^{s,t,u}(F_{p}[v_{0}^{±1}], (Q_{0})_{∗∗}X)
≅ Ext_{F_{p}}^{s,t,u}(F_{p}, (Q_{0})_{∗∗}X)
≅ \begin{cases}(Q_{0})_{∗∗}X, & when s = 0, \\ 0, & when s ≠ 0. \end{cases}

Since the spectral sequence lies completely in the plane s = 0, there is no room for any differentials. □

Remark 4.6.6. If one works over a quotient A′ of A in which ξ_{n+1} (respectively, τ_{n}) is nonzero, one can make similar computations and draw similar conclusions: (q_{n})_{∗∗}(q_{n}) is computed via H(A′ □_{E}F_{p}, Q_{n}), the action of the element Q_{n} on A′ □_{E}F_{p} can be computed by the same formulas, and (Q_{n})_{∗∗}(Q_{n}) = v_{n}^{−1}(q_{n})_{∗∗}(q_{n}).

As a consequence, if no ξ_{i} in A′ is nilpotent, then the computation of the homology group H(A′ □_{E}F_{p}, Q_{n}) is much like the computations above. If there is torsion, though, things can be more complicated. For example, when p = 2 and n = 0, then in A □_{E}F_{2}, we have Q_{0}(ξ_{1}^{2}ξ_{2}) = ξ_{1}^{4}. On the other hand, ξ_{1}^{4} = 0 in A(1), so in A(1) □_{E}F_{2}, we have Q_{0}(ξ_{1}^{2}ξ_{2}) = 0. Hence over A(1), (Q_{0})_{∗∗}(Q_{0}) ≅ F_{2}[v_{1}^{±1}] ⊗ E[x_{5}], where |x_{5}| = 5. Hence v_{0}^{−1} Ext_{A(1)}^{∗∗}(F_{2}, F_{2}) ≅ F_{2}[v_{0}^{±1}, y_{4}], where y_{4} is “in the 4-stem”: y_{4} has bidegree (1, 5).

4.6.2. Eisen’s calculation. Let Y (n) be the quotient Hopf algebra of A defined by

Y (n) = \begin{cases}F_{2}[ξ_{n+1}, ξ_{n+2}, ξ_{n+3}, …], & if p = 2, \\ F_{p}[ξ_{n+1}, ξ_{n+2}, ξ_{n+3}, …] ⊗ Λ[τ_{n}, τ_{n+1}, …], & if p is odd. \end{cases}

By Proposition 1.2.8, HY (n)_{∗∗} ≅ Ext_{Y (n)}^{∗∗}(F_{p}, F_{p}). Since ξ_{n+1} (respectively, τ_{n}) is primitive in Y (n), then v_{n} is a class in HY (n)_{1,a_{n}}. This class is non-nilpotent because it is detected by the restriction map associated to the quotient map Y (n) → E.

Eisen proved the following in [Eis87].

Theorem 4.6.7. Fix n ≥ 0.

(a) When p = 2,

v_{n}^{−1} Ext_{Y (n)}^{∗∗}(F_{2}, F_{2}) ≅ F_{2}[v_{n}^{±1}] ⊗ F_{2}[h_{t0} : n + 1 < t < 2n + 2]
⊗ F_{2}[h_{ts} : 0 < s < n + 1, t ≥ n + 1].

(b) When p is odd,

v_{n}^{−1} Ext_{Y (n)}^{∗∗}(F_{p}, F_{p}) ≅ F_{p}[v_{n}^{±1}] ⊗ F_{p}[v_{i} : n + 1 ≤ i ≤ 2n]
⊗ Λ[h_{ts} : 0 ≤ s ≤ n − 1, t ≥ n + 1]
⊗ F_{p}[b_{ts} : 0 ≤ s ≤ n − 1, t ≥ n + 1].

Remarks 4.1.3, 4.2.8, 4.3.2, and 4.6.6 come into play now: we work over the quotient Hopf algebra Y (n) via HY (n). Since v_{n} is an element in HY (n)_{∗∗}, then S^{0} ∧ HY (n) has a v_{n}-map, so by Remarks 4.1.3 and 4.2.8, we can consider the Q_{n}-based Adams spectral sequence abutting to v_{n}^{−1}HY (n)_{∗∗}.

Remark 4.6.6 tells us how to compute the coalgebra of operations, and hence the E_{2}-term.

Proposition 4.6.8. Consider the Q_{n}-based Adams spectral sequence abutting to v_{n}^{−1}HY (n)_{∗∗}.

(a) Suppose that p = 2. Let

B = F_{2}[ξ_{n+1}, …, ξ_{2n+1}, ξ_{2n+2}^{2}, ξ_{2n+3}^{2}, …]/(ξ_{k}^{2^{n+1}} : k ≥ n + 1).

Then the E_{2}-term is

E_{2} ≅ F_{2}[v_{n}^{±1}] ⊗ Ext_{B}^{∗∗}(F_{2}, F_{2})
≅ F_{2}[v_{n}^{±1}] ⊗ F_{2}[h_{t0} : n + 1 < t < 2n + 2]
⊗ F_{2}[h_{ts} : 0 < s ≤ n, t ≥ n + 1].

(b) Suppose that p is odd. Let

C = Λ[τ_{n}, …, τ_{2n}] ⊗ F_{p}[ξ_{n+1}, ξ_{n+2}, …]/(ξ_{k}^{p^{n}} : k ≥ n + 1).

Then the E_{2}-term is

E_{2} ≅ F_{p}[v_{n}^{±1}] ⊗ Ext_{C}^{∗∗}(F_{p}, F_{p})
≅ F_{p}[v_{n}^{±1}] ⊗ F_{p}[v_{i} : n + 1 ≤ i ≤ 2n]
⊗ Λ[h_{ts} : 0 ≤ s ≤ n − 1, t ≥ n + 1]
⊗ F_{p}[b_{ts} : 0 ≤ s ≤ n − 1, t ≥ n + 1].

Remark 4.3.2 tells us about convergence. Since Q_{n} is the first differential in Y (n), then for every finite spectrum X, HY (n) ∧ X is Z(d)_{∗∗}-acyclic for all d < a_{n}; hence the spectral sequence converges for every finite spectrum. (Alternatively, one can check that, as in the proof of Corollary 4.6.5, the vanishing plane condition holds at the E_{2}-term of the spectral sequence for the sphere.)

Corollary 4.6.9. For any finite spectrum X, the Q_{n}-based Adams spectral sequence abutting to v_{n}^{−1}HY (n)_{∗∗}X converges.

Eisen’s theorem 4.6.7 follows from the next lemma.

Lemma 4.6.10. The spectral sequence converging to v_{n}^{−1}HY (n)_{∗∗} collapses at E_{2}.

Sketch of proof. Let E(n) be the quotient Hopf algebra of Y (n) defined by

E(n) = \begin{cases}Y (n)/(ξ_{k}^{2^{n+1}} : k ≥ n + 1), & if p = 2, \\ Y (n)/(ξ_{k}^{p^{n}} : k ≥ n + 1), & if p is odd. \end{cases}

Just as for Y (n), one can show that the Q_{n}-based Adams spectral sequence abutting to v_{n}^{−1}HE(n)_{∗∗} converges. One can also compute HE(n)_{∗∗} directly, by observing that E(n) is a tensor product of Hopf algebras of the forms F_{p}[ξ_{k}]/(ξ_{k}^{p^{r}}) with ξ_{k} primitive, and E[τ_{k}] with τ_{k} primitive. Hence:

HE(n)_{∗∗} = \begin{cases}F_{2}[h_{ts} : 0 ≤ s ≤ n − 1, t ≥ n + 1], & if p = 2, \\ F_{p}[v_{i} : i ≥ n] ⊗ Λ[h_{ts} : 0 ≤ s ≤ n, t ≥ n + 1] \\ ⊗ F_{p}[b_{ts} : 0 ≤ s ≤ n, t ≥ n + 1], & if p is odd. \end{cases}

Of course, one can compute v_{n}^{−1}HE(n)_{∗∗} immediately from this.

One can conclude three things: first, the restriction map HY (n) ⟶ HE(n) induces an inclusion at the E_{2}-terms of the Q_{n}-based Adams spectral sequence:

E_{2}(HY (n)) ↪ E_{2}(HE(n)).

Second, the spectral sequence for E(n) collapses at E_{2}. Thus, third, the same must be true for the Y (n) spectral sequence, completing the proof of the lemma. □

4.6.3. The v_{1}-inverted Ext of the mod 2 Moore spectrum. Let p = 2, let X be the mod 2 Moore spectrum, and let M = A(0) be its homology. In this subsection we discuss Mahowald’s unverified computation in [Mah70] of v_{1}^{−1} Ext_{A}^{∗∗}(F_{2}, M); no proof of this result has been published. Before stating the result, we recall some computations: π_{∗∗}(HA(1)∧M) is a series of “lightning flashes”:

π_{∗∗}(HA(1) ∧ M) ≅ Ext_{A}^{∗∗}(F_{2}, H_{∗}(bo ∧ X))
≅ Ext_{A(1)}^{∗∗}(F_{2}, M)
≅ F_{2}[v_{1}^{4}] ⊗ F_{2}[h_{11}, v_{1}]/(h_{11}^{3}, v_{1}^{2}).

Also, if E′ is the quotient Hopf algebra E′ = A/(ξ_{1}^{2}, ξ_{2}^{2}, ξ_{3}, ξ_{4}, …), then π_{∗∗}(HE′ ∧ M) is a v_{1}-tower:

π_{∗∗}(HE′ ∧ M) ≅ Ext_{A}^{∗∗}(F_{2}, H_{∗}(bu ∧ X))
≅ Ext_{E′}^{∗∗}(F_{2}, M)
≅ F_{2}[v_{1}].

See Figure 4.6.B for a few pictures.

To present Mahowald’s answer, we also need a definition. Let P be the polynomial algebra P = F_{2}[x_{1}, x_{2}, x_{3}, …], which is bigraded by setting |x_{i}| = (2, 2^{i+2} +1). Define a derivation d on P by d(x_{i}) = x_{1}x_{i−1}^{2}. Let H(d) denote the homology with respect to d, and let B(d) denote the image of d.

Conjecture 4.6.11 ([Mah70]). Let p = 2, and let M = F_{2}[ξ_{1}]/(ξ_{1}^{2}) be the homology of the mod 2 Moore spectrum. Let E′ = F_{2}[ξ_{1}, ξ_{2}]/(ξ_{1}^{2}, ξ_{2}^{2}). Then v_{1}^{−1}π_{∗∗}M is a direct sum of copies of v_{1}^{−1}π_{∗∗}(HA(1) ∧ M) and v_{1}^{−1}π_{∗∗}(HE′ ∧ M). More

[figure: Ext_{A(1)}^{st}(F_{2}, M); t − s; s; Ext_{E′}^{st}(F_{2}, M); t − s; s]

Figure 4.6.B. Lightning flashes and v_{1}-towers. Pictures of Ext_{A(1)}^{∗∗}(F_{2}, M) and Ext_{E′}^{∗∗}(F_{2}, M), where M is the homology of the mod 2 Moore spectrum, A(1) is the usual quotient of A, and E′ = A/(ξ_{1}^{2}, ξ_{2}^{2}, ξ_{3}, ξ_{4}, …).

precisely, we have:

v_{1}^{−1}π_{∗∗}M ≅ ⨁_{α∈H(d)} Σ^{|α|}v_{1}^{−1}π_{∗∗}(HA(1) ∧ M)
⊕ ⨁_{β∈B(d)} Σ^{|β|}v_{1}^{−1}π_{∗∗}(HE′ ∧ M).

One can try to compute this using the Q_{1}-based Adams spectral sequence; by a change-of-rings computation, the E_{2}-term for the object M is

E_{2} ≅ F_{2}[v_{1}^{±1}] ⊗ F_{2}[h_{11}, h_{21}, h_{31}, …].

In this case, v_{1}^{4} is a permanent cycle, and the spectral sequence is one of F_{2}[v_{1}^{±4}]- modules.

We view x_{i} as v_{1}h_{i+1,1}. Then Mahowald’s answer would follow from the differentials

• d_{3}h_{i1} = v_{1}^{−2}h_{11}h_{21}h_{i−1,1}^{2} (for i ≥ 3),
• d_{3}v_{1}^{2} = h_{11}^{3},

together with some sort of multiplicativity properties. Unfortunately, we do not know how to produce these differentials, and there is not much apparent multiplicative structure in the spectral sequence. We know the following.

(a) The Q_{1}-based spectral sequence converging to v_{1}^{−1}π_{∗∗}M is a module over the Q_{1}-based spectral sequence for the sphere, which has

E_{2} ≅ F_{2}[v_{1}^{±1}] ⊗ F_{2}[h_{10}, h_{11}, h_{21}, h_{31}, …].

The latter spectral sequence need not converge, and indeed it is not even clear what the abutment is, but it is a spectral sequence of algebras, and the comodule map F_{2} ⟶ M induces a surjection at E_{2}. Hence if one can compute any differentials for the sphere spectral sequence, that has implications for the Moore spectrum.

(b) In the abutment of the spectral sequence for the Moore spectrum, h_{11}^{3} = 0.

(c) Davis and Mahowald [DM88, Theorem 1.3] have computed the v_{1}-inverted Ext of the spectrum Y = H_{∗}(RP^{2} ∧CP^{2}) = F_{2}[x_{1}]/(x_{1}^{4}). They found that

v_{1}^{−1} Ext_{A}^{∗∗}(F_{2}, H_{∗}Y) ≅ F_{2}[v_{1}^{±1}] ⊗ F_{2}[h_{21}, h_{31}, h_{41}, …].

As Eisen points out in [Eis87], this follows from his computation; from our point of view, the Q_{1}-based Adams spectral sequence for Y embeds in the Q_{1}-based Adams spectral sequence for Y (1), and hence collapses. The cofibration M ⟶ Y ⟶ Σ^{0,2}M in Stable(A), corresponding to the Ext class h_{11}, leads to a Bockstein spectral sequence with E_{1}-term the same as the E_{2}-term for the Q_{1}-based spectral sequence converging to v_{1}^{−1}π_{∗∗}M. One might be able to use this to aid in the computation of differentials in that spectral sequence.

We end this section by discussing the odd primary case. When p is odd, Miller [Mil78, Corollary 3.6] calculated v_{1}^{−1} Ext_{A}^{∗∗}(F_{p}, H_{∗}X), where X is the mod p Moore spectrum:

v_{1}^{−1} Ext_{A}^{∗∗}(F_{p}, H_{∗}X) ≅ F_{p}[v_{1}^{±1}] ⊗ Λ[h_{t0} : t ≥ 1] ⊗ F_{p}[b_{t0} : t ≥ 1].

This is the E_{2}-term of the Q_{1}-based Adams spectral sequence. The spectral sequence collapses in this case, because of two things: first, since the mod p Moore spectrum is a ring spectrum, this is a spectral sequence of algebras. Second, for easy degree reasons, the elements h_{t0} and b_{t0} are permanent cycles; a more careful analysis shows that v_{1}^{n} is a permanent cycle for all n—each differential d_{r} takes v_{1}^{n} to a tridegree which is zero. In a spectral sequence of algebras in which each algebra generator is a permanent cycle, the whole spectral sequence consists of permanent cycles.

CHAPTER 5 Quillen stratification and nilpotence

The vanishing line theorem of Section 2.3 is a nilpotence theorem of a sort: if X is a finite spectrum, then any self-map of X with slope smaller than that of the vanishing line for X must be nilpotent (because some power of it will lie above the vanishing line, where “above” is in terms of the Adams spectral sequence grading of Figure 2.3.A). This vanishing line theorem has applications; for example, we used it in Section 2.4 to construct a non-nilpotent self-map of any finite spectrum.

In this chapter we give some related, but stronger, results; we work mostly at the prime 2. We let 𝒬 denote the category of quasi-elementary quotients of A, with morphisms given by quotient maps. We can assemble the individual Hurewicz (restriction) maps S^{0} ⟶ HE into a homotopy limit

S^{0} ⟶ \underleftarrow{lim}_{𝒬}HE.

Since the maximal quasi-elementary quotients are conormal, there is an action of A on the F_{p}-algebra \underleftarrow{lim}_{𝒬} HE_{∗∗}; we use a superscript A to denote the invariants under this action (see Definition 5.1.1). We prove that

π_{∗∗}S^{0} ⟶ (\underleftarrow{lim}_{𝒬}HE_{∗∗})^{A}

is an F -isomorphism—its kernel consists of nilpotent elements, and some pth power of every element in the target is actually in the image. One can view this as an analogue of the Quillen stratification theorem [Qui71, Theorem 6.2], which identifies π_{∗∗}S^{0} up to F -isomorphism in the category Stable(kG^{∗}), for G a finite group and k an algebraically closed field of characteristic p.

Similarly, suppose that we write D for the following quotient of A:

D = A/(ξ_{1}^{p}, ξ_{2}^{p^{2}}, ξ_{3}^{p^{3}}, …, ξ_{n}^{p^{n}}, …).

Then D is conormal, and when p = 2, the Hurewicz map induces an F -isomorphism

π_{∗∗}S^{0} ⟶ (HD_{∗∗})^{A}.

These F -isomorphism theorems first appeared in [Pal99].

Quillen’s theorem about group cohomology with trivial coefficients was generalized to nontrivial coefficients by Avrunin and Scott in [AS82]; their result suggests that a direct analogue of these F -isomorphism theorems with nontrivial coefficients—for example, that [X, X] is F -isomorphic to (\underleftarrow{lim}_{𝒬} [X, HE ∧X]_{∗∗})^{A}—is too much to hope for. In this chapter, we give several weaker results with nontrivial coefficients: if R is a ring spectrum, then an element α ∈ π_{∗∗}R is nilpotent if and only if its image under the Hurewicz map in HD_{∗∗}R is nilpotent, or if its image in HE_{∗∗}R is zero for all quasi-elementary quotients E of A. There are similar results for detecting the nilpotence of self-maps of finite spectra. (These theorems

p = 2  p odd
Figure 5.1.A. Profile functions for D = A/(ξ_{1}^{p}, …, ξ_{n}^{p^{n}}, …).

are improvements on results of the author in [Pal96a].) In the next chapter, we discuss this further: see Theorem 6.2.3 for a statement about lifting A-invariant elements from HD_{∗∗}R to π_{∗∗}R, and Conjecture 6.7.1 for the best generalization of the F -isomorphism theorem we could hope for.

We state our results more precisely in Section 5.1. In Section 5.2, we prove the two theorems—nilpotence and F -isomorphism—involving the quotient Hopf algebra D; we then use these to prove the analogous results for quasi-elementary Hopf algebras in Section 5.3. We end the chapter with Sections 5.4 and 5.5. In the first of these, we indicate why our proofs only work at the prime 2; this leads us to a conjecture about the nilpotence of particular classes in Ext over quotients of A. In the second, we discuss a few possible generalizations of the main results of this chapter.

Except for Sections 5.4 and 5.5, we work at the prime 2 in this chapter.

5.1. Statements of theorems

Let p = 2. In this section we present two F -isomorphism results and two nilpotence results.

Let D be the following quotient Hopf algebra of A:

D = A/(ξ_{1}^{2}, ξ_{2}^{4}, ξ_{3}^{8}, …, ξ_{n}^{2^{n}}, …).

See Figure 5.1.A for its profile function (along with the analogous picture at odd primes). Note that D and A have the same quasi-elementary quotients—i.e., every quasi-elementary quotient map A ↠ E factors as A ↠ D ↠ E. As a result, it turns out that HD plays a similar role in Stable(A) to that of BP in the ordinary stable homotopy category, at least as far as detection of nilpotence goes. We write η_{D}: S^{0} = HA ⟶ HD for the unit map of the ring spectrum HD, and similarly for η_{E}: S^{0} ⟶ HE for E quasi-elementary.

5.1.1. Quillen stratification. In this subsection, we state our results describing π_{∗∗}S^{0} up to F -isomorphism. The results here first appeared in [Pal99].

Definition 5.1.1. (a) Given a Hopf algebra B and a B-comodule M, we define the B-invariants of M to be

M^{B} = Hom_{B}(F_{p}, M) ⊆ M.

If B is connected, this is the same as the primitives PM of M: if ψ: M ⟶ B ⊗ M is the coaction map, then we let PM = {m ∈ M | ψ(m) = 1 ⊗ m}.

(b) Following Quillen [Qui71], if k is a field of characteristic p > 0 and if φ: R ⟶ S is a map of graded commutative k-algebras, we say that φ is an F -isomorphism if it satisfies the following properties.

(i) Every x ∈ ker φ is nilpotent. (Hence \sqrt{ker φ} is the nilradical of R.) (ii) For any element y ∈ S, there is an integer n so that y^{p^{n}} ∈ im φ. In (ii), if one can choose the same n for every y, then we say that φ is a uniform F -isomorphism.

The definition of F -isomorphism is intended to capture the properties of the Frobenius map

F: F_{p}[x_{1}, …, x_{m}] ⟶ F_{p}[x_{1}, …, x_{m}]

and its iterates F^{n}, where F is defined by F (α) = α^{p}. There is another notion, called “N-isomorphism,” in which condition (ii) is replaced by

(ii)′ For any element y ∈ S, there is an integer n so that y^{n} ∈ im φ. When working in characteristic p, F -isomorphism is a more natural concept. It is also stronger than N-isomorphism, and it is what our results produce.

Here is our first result. Theorem 2.1.1 tells us that D is conormal, so by Remark 1.2.13, there is a coaction of A □_{D}F_{2} on HD_{∗∗}.

Theorem 5.1.2 (Quillen stratification, I). The Hurewicz map π_{∗∗}S^{0} ⟶ HD_{∗∗} factors through

φ: π_{∗∗}S^{0} ⟶ HD_{∗∗}^{A □_{D}F_{2}} ,

and φ is an F -isomorphism.

Here is our second result, an analogue of Quillen’s theorem [Qui71, Theorem 6.2], which identifies group cohomology up to F -isomorphism. We have defined in Definition 2.1.10 (see also Proposition 2.1.12) the notion of a quasi-elementary quotient Hopf algebra of A; we let 𝒬 denote the category of quasi-elementary quotients of A, with morphisms given by quotient maps. In Section 5.3 below, we construct a coaction of A on \underleftarrow{lim}_{𝒬} HE_{∗∗}.

Theorem 5.1.3 (Quillen stratification, II). The map π_{∗∗}S^{0} ⟶ \underleftarrow{lim}_{𝒬} HE_{∗∗} factors through

γ: π_{∗∗}S^{0} ⟶ (\underleftarrow{lim}_{𝒬}HE_{∗∗})^{A} ,

and γ is an F -isomorphism.

Both inverse limits here are inverse limits of F_{2}-algebras.

We prove Theorem 5.1.2 in Section 5.2; we show that Theorem 5.1.3 follows from Theorem 5.1.2 in Section 5.3.

Lin proved special cases of Theorems 5.1.2 and 5.1.3 in [Lin77a, Lin77b]; his results applied not to Ext_{A}^{∗∗}(F_{2}, F_{2}) but to Ext_{B}^{∗∗}(F_{2}, F_{2}), where B is any quotient Hopf algebra of A with a finite profile function: for each n, ξ_{n}^{2^{i}} = 0 in B for some i. In this setting, it turns out that some power of every cohomology element is invariant, so one does not have to apply (−)^{A}. There are also weaker forms of our results in [Wil81], [HP93], and [Pal96a]; these papers all focus on the detection of nilpotence (finding a map f that satisfies condition (i) in Definition 5.1.1(b)), rather than on the identification of all of the non-nilpotent elements.

Remark 5.1.4. (a) One can view Theorem 5.1.3 as giving an analogue of the Quillen stratification theorem [Qui71, Theorem 6.2]: given a finite group G and an algebraically closed field k of characteristic p, we let 𝒜 be the category whose objects are the elementary abelian p-subgroups of G, and whose morphisms are generated by inclusions and compositions. Then the natural map

H^{∗}(G; k) ⟶ \underleftarrow{lim}_{E∈𝒜} H^{∗}(E; k)

is an F -isomorphism. The role of the conjugation maps in the category 𝒜 is played, in our results, by the process of taking invariants.

(b) One can also view Theorem 5.1.2 as an analogue of Nishida’s theorem [Nis73]—in the ordinary stable homotopy category, π_{∗}S^{0} is isomorphic to Z, mod nilpotence; however, while Nishida’s theorem does not immediately lead to guesses as to further structure in the ordinary stable homotopy category, our analogue does. For instance, see Section 6.7 for a suggested classification of thick subcategories of finite spectra in Stable(A).

(c) We have an explicit formula for the algebra \underleftarrow{lim} HE_{∗∗} (Proposition 6.5.1), as well as for the coaction of A on \underleftarrow{lim} HE_{∗∗} (Proposition 6.5.6). One can use this to predict the presence of large families of non-nilpotent elements in π_{∗∗}S^{0} = Ext_{A}^{∗∗}(F_{2}, F_{2}). See Section 6.5 for details.

(d) We do not expect the F -isomorphisms in these results to be uniform, although we do not have much evidence either way.

(e) Of course, if A′ is any quotient Hopf algebra of A, then similar results hold: for instance, the map HA′_{∗∗} ⟶ (HD′_{∗∗})^{A′} is an F -isomorphism, where D′ is the apparent quotient of A′. The proofs for Theorems 5.1.3 and 5.1.2 carry over easily.

5.1.2. Nilpotence. We move on to our nilpotence theorems. The first of these is based on the nilpotence theorem of Devinatz, Hopkins, and Smith [DHS88]: the ring spectrum BP detects nilpotence. Note that if E is a ring spectrum and Y is any spectrum, then [Y, E ∧ Y]_{∗∗} is a ring: given f, g ∈ [Y, E ∧ Y]_{∗∗}, define the product fg to be the composite

Y ⟶^{g} E ∧ Y ⟶^{1_{E}∧f} E ∧ E ∧ Y ⟶^{μ∧1_{Y}} E ∧ Y,

where μ is the multiplication map on E. Also, we say that a map of spectra f: F ⟶ X is smash nilpotent if some smash power of f, f^{∧n}: F^{∧n} ⟶ X^{∧n}, is null.

Theorem 5.1.5 (Nilpotence theorem, I). The ring spectrum HD detects nilpotence:

(a) Fix a ring spectrum R and α ∈ π_{∗∗}R. Then α is nilpotent if and only if HD_{∗∗}(α^{n}) = 0 for some n.

(b) Fix a finite spectrum Y and a self-map f: Y ⟶ Y. Then f ∈ [Y, Y]_{∗∗} is nilpotent if and only if HD_{∗∗}f is nilpotent.

(c) Fix a finite spectrum F, an arbitrary spectrum X, and a map f: F ⟶ X. Then f ∈ [F, X]_{∗∗} is smash-nilpotent if 1_{HD} ∧ f ∈ [HD ∧ F, HD ∧ X]_{∗∗} is zero.

The corresponding result for modules is [Pal96a, Theorems 3.1 and 4.2]. In [Pal96a] we prove this in enough generality so that the proof goes through here without difficulty. We also give a (slightly different) proof in Section 5.2.

The second nilpotence theorem is, more or less, an analogue of the K(n) nilpotence theorem in [HS98]. The following appeared for bounded below modules in [Pal96a, Theorems 1.1 and 4.3].

Theorem 5.1.6 (Nilpotence theorem, II). The collection of ring spectra

{HE | E quasi-elementary}

detects nilpotence:

(a) Fix a ring spectrum R and α ∈ π_{∗∗}R. Then α is nilpotent if HE_{∗∗}α = 0 is zero for all quasi-elementary quotients E of A.

(b) Fix a finite spectrum Y and a self-map f: Y ⟶ Y. Then f is nilpotent if and only if HE_{∗∗}f is nilpotent for all quasi-elementary quotients E of A.

(c) Fix a finite spectrum F, an arbitrary spectrum X, and a map f: F ⟶ X. Then f is smash-nilpotent if 1_{E} ∧ f: HE ∧ F ⟶ HE ∧ X is zero for all quasi-elementary quotients E of A.

One might hope to replace part (a) with an “if and only if” statement, but since R need not be finite, and since there are infinitely many quasi-elementary quotients of A, this may be problematic (compare to Theorem 5.1.5(a), in which there is a single detecting ring spectrum). For part (b), on the other hand, the finiteness of Y essentially means that if η_{E} ∧ f is nilpotent for all E, then some iterate of this map is zero for all E. All three parts have an “if and only if” version in ordinary stable homotopy theory (see [HS98, Theorem 3]). The proof of that theorem relies on nice properties of the Morava K-theories, properties that are not shared by the objects HE in our setting; in particular, the K(n)’s are field spectra, and the HE’s are not. Hence we have a slightly weaker theorem than Hopkins and Smith do.

The proof of the analogous result in [Pal96a] does not apply to nonconnective situations, so we give a new proof below.

Remark 5.1.7. (a) Suppose that F and R are ring spectra and fix α ∈ π_{∗∗}R. Then F_{∗∗}α = 0 if and only if η_{F} ∧ α: S^{0} ⟶ F ∧ R is null. So one may replace the hypothesis in Theorems 5.1.5 and 5.1.6 that HD_{∗∗}(α^{n}) = 0 (resp., HE_{∗∗}α = 0) with the condition that η_{D} ∧ α^{n} is null (resp., that η_{E} ∧ α is null).

(b) Note that F_{2} is a quasi-elementary quotient Hopf algebra of A, so HF_{2} is included as one of the detecting spectra in Theorem 5.1.6. (Compare with [HS98], where HF_{p} = K(∞) is included for parts (a) and (c), but not for (b)).

(c) Fix a quotient Hopf algebra A′ of A, and let D′ be the quotient of A′ induced by A ↠ D. One can generalize Theorem 5.1.5 in an obvious sort of way: if Y is a finite spectrum and f: Y ⟶ HA′ ∧ Y is a “self-map,” then

f ∈ [Y, HA′ ∧ Y]_{∗∗} is nilpotent
⇔ η_{D′} ∧ f ∈ [Y, HD′ ∧ Y]_{∗∗} is nilpotent
⇔ η_{E′} ∧ f ∈ [Y, HE′ ∧ Y]_{∗∗} is nilpotent for all E′.

(Here E′ ranges over all quasi-elementary quotients of A′.) There are similar versions of the ring spectrum and smash-nilpotence results. The proofs are straightforward generalizations of the ones below, so we omit them.

(d) The quotient Hopf algebra D is “best possible,” in the sense that if B is a quotient of A which does not map onto D, then there are non-nilpotent elements in π_{∗∗}S^{0} which are in the kernel of π_{∗∗}S^{0} ⟶ HB_{∗∗}. One can see this from Theorem 5.1.2 and Example 6.5.7(a).

(e) Theorem 5.1.5 identifies a single ring spectrum HD which detects nilpotence, but we do not know its coefficient ring completely. See Propositions 5.3.4 and 6.5.1 for partial information. On the other hand, we have computed the coefficient rings of HE for E quasi-elementary in Proposition 2.1.9—they are polynomial rings.

(f) One does not need to use all of the quasi-elementary quotient Hopf algebras of A to detect nilpotence; for instance, one can use only the maximal quasielementary quotients, or only the finite-dimensional ones.

(g) In fact, one can see from the proof of Theorem 5.1.6 that one only needs the following spectra to detect nilpotence:

{h_{m+1,j}^{−1}H(E(m)/(ξ_{m+1}^{2^{j+1}})) | m ≥ 0, 0 ≤ j ≤ m}.

Here E(m) is one of the maximal quasi-elementary quotients of A, defined in Corollary 2.1.8, and E(m)/(ξ_{m+1}^{2^{j+1}}) is a further quotient of that. By Lemmas 1.2.15 and 2.1.2, H(E(m)/(ξ_{m+1}^{2^{j+1}})) has a self-map h_{m+1,j}, and

h_{m+1,j}^{−1}H(E(m)/(ξ_{m+1}^{2^{j+1}})

denotes the mapping telescope of this self-map.

5.2. Nilpotence and F -isomorphism via the Hopf algebra D

In this section we show that the spectrum HD detects a lot of information: we prove that it detects π_{∗∗}S^{0} modulo nilpotent elements (Theorem 5.1.2), and that it detects non-nilpotent elements of π_{∗∗}R for any ring spectrum R (Theorem 5.1.5).

The Hopf algebra D is a conormal quotient of A, by Theorem 2.1.1. It is standard that for conormal quotient Hopf algebras, the restriction (Hurewicz) map h: π_{∗∗}S^{0} ⟶ HD_{∗∗} factors as advertised in Theorem 5.1.2:

φ: π_{∗∗}S^{0} ⟶ HD_{∗∗}^{A □_{D}F_{2}} .

One way to see this is to consider the Hopf algebra extension

A □_{D}F_{2} ⟶ A ⟶ D,

and the associated spectral sequence (as in Section 1.4), which has

E_{2}^{s,t} = Ext_{A □_{D}F_{2}}^{s} (F_{2}, Ext_{D}^{t}(F_{2}, F_{2})) ⇒ Ext_{A}^{s+t}(F_{2}, F_{2}).

The desired factorization is just the edge homomorphism

Ext_{A}^{t}(F_{2}, F_{2}) ⟶ E_{2}^{0,t}.

We have to prove Theorem 5.1.5—the spectrum HD detects nilpotence—and to verify conditions (i) and (ii) of Definition 5.1.1(b)—every element of ker φ is nilpotent, and for every element y ∈ HD_{∗∗}^{A □_{D}F_{2}} , there is an integer m so that y^{2^{m}} ∈ im φ.

Note that condition (i) follows from Theorem 5.1.5(a) with R = S^{0}. Also, the proof of Theorem 5.1.5 and the verification of (ii) are quite similar, so first we lay the groundwork for both.

For each integer n ≥ 1, we let

D(n) = A/(ξ_{1}^{2}, ξ_{2}^{4}, …, ξ_{n}^{2^{n}}).

5.2. NILPOTENCE AND F -ISOMORPHISM VIA THE HOPF ALGEBRA D 113

[figure: n; n+1]

Figure 5.2.A. Profile function for D(n). As in Figure 2.1.B, the diagonal line is an abbreviation for a staircase shape.

We let D(0) = A. See Figure 5.2.A for the profile function. Each D(n) is a conormal quotient Hopf algebra of A, and we have a diagram of Hopf algebra surjections:

A = D(0) ↠ D(1) ↠ D(2) ↠ · · · .

D is the colimit of this diagram.

First we discuss how to lift information from HD_{∗∗} to HD(n)_{∗∗} for some n. We have the following lemma.

Lemma 5.2.1. (a) We have HD_{∗∗} = \underrightarrow{lim} HD(n)_{∗∗}. More generally, if X is any connective spectrum, then HD_{∗∗}X = \underrightarrow{lim} HD(n)_{∗∗}X, and this colimit stabilizes in each bidegree.

(b) We have HD_{∗∗}^{A □_{D}F_{2}} = \underrightarrow{lim} (HD(n)_{∗∗}^{A □_{D(n)}F_{2}} ).

(These colimits are of F_{2}-vector spaces.)

Proof. Since homotopy commutes with direct limits, the first sentence of part (a) is clear. The second follows from an easy connectivity argument, as in the proof of Lemma 2.4.4.

Part (b): The coaction of A on HD(n)_{∗∗} is defined in Remark 1.2.13. Note that each restriction map HD(n)_{∗∗} ⟶ HD(n + 1)_{∗∗} is an A-comodule map (and in fact, a map of comodule algebras). Applying Hom_{A}^{∗}(F_{2}, −) gives the following:

Hom_{A}(F_{2}, HD_{∗∗}) = Hom_{A}(F_{2}, \underrightarrow{lim} HD(n)_{∗∗}) = \underrightarrow{lim} Hom_{A}(F_{2}, HD(n)_{∗∗}),

where the last equality holds because F_{2} is small in the category of A-comodules. Since Hom_{A}(F_{2}, HD(n)_{∗∗}) = HD(n)_{∗∗}^{A} = HD(n)_{∗∗}^{A □_{D(n)}F_{2}} , we have the desired result. □

Now we discuss how to take information about HD(n)_{∗∗} and get information about HD(n − 1)_{∗∗}. We want to know about π_{∗∗}S^{0} = HD(0)_{∗∗}, so we will eventually want to use downward induction on n.

Not only is each D(n) a conormal quotient of A, it is also a conormal quotient of D(n − 1). The Hopf algebra kernel of the quotient map is easy to identify: there is an extension of Hopf algebras

F_{2}[ξ_{n}^{2^{n}}] ⟶ D(n − 1) ⟶ D(n),

where ξ_{n}^{2^{n}} is primitive in the kernel. So given any D(n − 1)-comodule M, there is a change-of-rings spectral sequence (see (1.4.8)) with

E_{2}^{s,t,u}(M) = Ext_{F_{2}[ξ_{n}^{2^{n}}]}^{s,u}(F_{2}, Ext_{D(n)}^{t,∗}(F_{2}, M)) ⇒ Ext_{D(n−1)}^{s+t,u}(F_{2}, M).

By Proposition 1.4.9, in the category Stable(D(n − 1)) of cochain complexes of injective left D(n−1)-comodules, this spectral sequence is the same, up to regrading, as the HD(n)-based Adams spectral sequence. Corollary 1.4.7 extends this from the case when M is a comodule to an arbitrary object X in Stable(D(n − 1)). Because of this, for parts of the proof we will work in the category Stable(D(n − 1)). We also use the grading on the spectral sequence as given here; we do not use the Adams spectral sequence grading from Section 1.4. Throughout, we abuse notation somewhat, writing Ext_{D(n)}^{∗∗}(F_{2}, X) for π_{∗∗}(HD(n) ∧ X).

We need to establish an important property of this spectral sequence, that it has a nice vanishing plane at some E_{r}-term. We write E_{r}^{s,t,u}(X) for the spectral sequence converging to Ext_{D(n−1)}^{∗∗}(F_{2}, X) = HD(n − 1)_{∗∗}X.

Proposition 5.2.2. Fix a connective spectrum X and an integer m ≠ 1. For some numbers r and c, E_{r}^{s,t,u}(X) = 0 when ms + t − u > c.

We prove this using Theorem 1.4.4, which says that vanishing planes in Adams spectral sequences are generic. (And again, this is an Adams spectral sequence, as long as we work in the category Stable(D(n − 1)).)

(As remarked above, the change-of-rings spectral sequence E_{r}^{∗∗∗} is the same as the Adams spectral sequence, up to a regrading. Writing the Adams spectral sequence as ^{‵}^{‵}E_{r}^{∗∗∗}, then with m as in the proposition, the vanishing plane for ^{‵}^{‵}E_{r}^{∗∗∗} has the following form:

‵^{‵}E_{2}^{s,t,u}(W ∧ X) = 0

when ms − t > c, i.e., when

s ≥ \frac{−1}{m − 1}(s + u) + \frac{1}{m − 1}(t + u) + \frac{c}{m − 1}.

In particular, if m ≠ 1, we can apply Theorem 1.4.4.)

We fix a connective spectrum X. “(i_{0}, j_{0})-connectivity” is defined in Definition 1.3.4.

Lemma 5.2.3. For each integer m, there is a finite D(n − 1)-comodule W so that

E_{2}^{s,t,u}(W ∧ X) = 0

when ms+t−u > c, where c depends only on the connectivity of X. More precisely, if X is (i_{0}, j_{0})-connective, then

E_{2}^{s,t,u}(W ∧ X) = 0

when ms + t − u > −j_{0}.

Proof. Choose an integer j ≥ n so that |ξ_{n}^{2^{j}}| = 2^{j}(2^{n} − 1) > m, and let W = W_{j} = F_{2}[ξ_{n}^{2^{n}}]/(ξ_{n}^{2^{j}}), with the apparent D(n − 1) □_{D(n)}F_{2}-comodule structure. Then W_{j} is a trivial D(n)-comodule (by definition), and as coalgebras we have

F_{2}[ξ_{n}^{2^{n}}] ≅ W_{j} ⊗ F_{2}[ξ_{n}^{2^{j}}].

In particular, W_{j} = F_{2}[ξ_{n}^{2^{n}}] □_{F_{2}[ξ_{n}^{2^{j}}]}F_{2}, and for any F_{2}[ξ_{n}^{2^{n}}]-comodule N there is a change-of-rings isomorphism (as in Corollary 1.2.7—see also Example 1.2.4 and Lemma 1.2.5):

Ext_{F_{2}[ξ_{n}^{2^{n}}]}^{∗∗}(F_{2}, W_{j} ⊗ N) ≅ Ext_{F_{2}[ξ_{n}^{2^{j}}]}^{∗∗}(F_{2}, N).

Hence the E_{2}-term of the spectral sequence for W_{j} ∧ X looks like

E_{2}^{s,t,u} ≅ Ext_{F_{2}[ξ_{n}^{2^{n}}]}^{s,u}(F_{2}, Ext_{D(n)}^{t,∗}(F_{2}, W_{j} ∧ X))
≅ Ext_{F_{2}[ξ_{n}^{2^{n}}]}^{s,u}(F_{2}, W_{j} ⊗ Ext_{D(n)}^{t,∗}(F_{2}, X))
≅ Ext_{F_{2}[ξ_{n}^{2^{j}}]}^{s,u}(F_{2}, Ext_{D(n)}^{t,∗}(F_{2}, X)).

The Hopf algebra F_{2}[ξ_{n}^{2^{j}}] is |ξ_{n}^{2^{j}}|-connected, so if L is a comodule which is zero below degree t, then

Ext_{F_{2}[ξ_{n}^{2^{j}}]}^{s,u}(F_{2}, L) = 0

when u < |ξ_{n}^{2^{j}}|s + t. Now note that if X is (i_{0}, j_{0})-connective, then HD(n)_{t,∗}X = Ext_{D(n)}^{t,∗}(F_{2}, X) is zero below degree t + j_{0}. □

Next we need to show that X is in thick(W_{j} ∧ X), in which case genericity of vanishing planes will yield one for X. We start with the following lemma, which describes how to build W_{j} out of F_{2} in a nice way.

Lemma 5.2.4. For j ≥ n, let W_{j} = F_{2}[ξ_{n}^{2^{n}}]/(ξ_{n}^{2^{j}}). Then there is a short exact sequence of D(n − 1) □_{D(n)}F_{2}-comodules

0 ⟶ W_{j} ⟶ W_{j+1} ⟶ s^{|ξ_{n}^{2^{j}} |}W_{j} ⟶ 0.

(As in Section 0.1, s denotes the suspension functor in the category of graded comodules.) The connecting homomorphism in Ext_{D(n−1)}^{∗∗}(F_{2}, −) is multiplication by

h_{nj} = [ξ_{n}^{2^{j}}] ∈ Ext_{D(n−1)}^{∗∗}(F_{2}, F_{2}).

Replacing W_{j} by its injective resolution gives a cofibration sequence

Σ^{1,|ξ_{n}^{2^{j}} |}W_{j} ⟶^{h_{nj}} W_{j} ⟶ W_{j+1} ⟶ Σ^{0,|ξ_{n}^{2^{j}} |}W_{j}.

(We are abusing notation a bit here, by writing W_{j} for both the module and its injective resolution. We will continue this practice for the remainder of this section.)

Proof. This follows from Lemma 1.2.15. □

Lemma 5.2.5. Let Y and Z be spectra. Given a cofiber sequence Y ⟶^{f} Y ⟶ Z, if the map f is nilpotent under composition, then Y ∈ thick(Z).

(As usual, we are omitting suspensions. The self-map f could have nonzero degree here.)

Proof. Write C(f^{i}) for the cofiber of f^{i}. First we show by induction on i that C(f^{i}) is in thick(Z) for each i ≥ 1. The i = 1 case is immediate, because C(f) = Z. Now assume that C(f^{i−1}) ∈ thick(Z), and consider the following diagram, in which each row and column is a cofibration (cf. [HS98, Lemma 2.6]):

Y ⟶^{f^{i−1}} Y ⟶ C(f^{i−1})
∥  ↓^{f}  ↓
Y ⟶^{f^{i}} Y ⟶ C(f^{i})
↓  ↓  ↓
∗ ⟶ Z  Z.

The right-hand column shows that C(f^{i}) is in thick(Z), as desired.

If f^{n} = 0, then C(f^{n}) splits as C(f^{n}) = Y ∨ Y. Since C(f^{n}) ∈ thick(Z), so is Y. □

Suppose that j ≥ n; then a result of Lin [Lin77a], stated as Theorem B.2.1(a) below, tells us that the element h_{nj} ∈ HD(n − 1)_{∗∗} is nilpotent. Hence we have the following:

Lemma 5.2.6. If j ≥ n, then X ∈ thick(W_{j} ∧ X).

Proof. We show by downward induction on i that for j ≥ i ≥ n, W_{i} ∧ X is in thick(W_{j} ∧ X); the lemma is proved when i = n, since W_{n} = S^{0}.

The induction starts (trivially) with i = j. Suppose that i < j, and consider the cofibration sequence in Lemma 5.2.4. Because h_{ni} is nilpotent, Lemma 5.2.5, applied with Y = W_{i} ∧X and Z = W_{i+1} ∧X, shows that W_{i} ∧X ∈ thick(W_{i+1} ∧X), and hence is in thick(W_{j} ∧ X) by induction. □

Proof of Proposition 5.2.2. This follows immediately from Lemma 5.2.3, Lemma 5.2.6, and the genericity of vanishing planes, Theorem 1.4.4. □

5.2.1. Nilpotence: Proof of Theorem 5.1.5. Now we prove Theorem 5.1.5, and hence verify condition (i) of Definition 5.1.1(b). Recall from Remark 5.1.7(a) that if R is a ring, then for α ∈ π_{∗∗}R, HD_{∗∗}(α^{n}) = 0 if and only if η_{D} ∧ α^{n} = 0. Since η_{D}: S^{0} ⟶ HD is the unit of the ring spectrum HD, then in π_{∗∗}HD, η_{D}^{n} = η_{D} for all n; so η_{D} ∧ α^{n} = 0 if and only if η_{D}^{n} ∧ α^{n} = 0.

Proof of Theorem 5.1.5. The basic idea of the proof is, of course, based on that of the nilpotence theorem in [DHS88]. As in that proof, one can reduce to the ring spectrum case—part (a)—in which the ring is connective. So we let R be a connective ring spectrum, we fix α ∈ π_{∗∗}R and assume that η_{D} ∧ α is nilpotent. By raising α to a power, we may assume that η_{D} ∧ α = 0. We want to show that α is nilpotent in π_{∗∗}R = HD(0)_{∗∗}R.

By Lemma 5.2.1, since η_{D} ∧ α = 0, then we must have η_{D(n)} ∧ α = 0 for some n. We want to show that if η_{D(n)} ∧ α = 0, then η_{D(n−1)} ∧ α^{j} = 0 for some j; the result will follow by downward induction on n.

Consider the change-of-rings spectral sequence

E_{2}^{s,t,u}(R) = Ext_{F_{2}[ξ_{n}^{2^{n}}]}^{s,u}(F_{2}, Ext_{D(n)}^{t,∗}(F_{2}, R)) ⇒ Ext_{D(n−1)}^{s+t,u}(F_{2}, R).

Write z for η_{D(n−1)} ∧ α. Since η_{D(n)} ∧ α = 0, then z must be represented by a class z̃ ∈ E_{2}^{s,t,u} with s > 0.

We can find an integer m ≠ −1 so that ms + t − u > 0, and then for any c, we can find a j so that

msj + tj − uj > c.

By Proposition 5.2.2 (with X = R), for some r and c we have E_{r}^{s,t,u} = 0 when ms+t−u > c. As noted above, we can choose j so that z̃^{j} lies above this vanishing plane, so at the E_{r}-term for which we have the vanishing plane, z̃^{j} must be zero. Thus z̃^{j} is zero at E_{∞}, and hence is zero in the abutment modulo terms of higher filtration. The higher filtration pieces are also above the vanishing plane, though, and are therefore zero. So z^{j} = η_{D(n−1)}^{j} ∧ α^{j} = 0 in the abutment, HD(n − 1)_{∗∗}R; since η_{D(n−1)}^{j} = η_{D(n−1)}, we are done. □

5.2.2. F -isomorphism: Proof of Theorem 5.1.2. We need to show that the map

φ: π_{∗∗}S^{0} ⟶ HD_{∗∗}^{A □_{D}F_{2}}

is an F -isomorphism. By Theorem 5.1.5, it satisfies part (i) of Definition 5.1.1(b); i.e., it is a monomorphism mod nilpotents. Now we have to verify condition (ii).

Verification of condition (ii). Fix y ∈ HD_{i,j}^{A □_{D}F_{2}} . We want to show that there is an integer m so that y^{2^{m}} ∈ im φ. By Lemma 5.2.1(b), there is an n so that y lifts to HD(n)_{∗∗}^{A □_{D(n)}F_{2}} . (Alternatively, one can use Lemma 5.2.1(a) to lift y to HD(n)_{∗∗} for some n, and then use Lemma 5.2.7 below to show that some power of that lift is invariant.) Now we show that some power of y lifts to HD(n − 1)_{∗∗}^{A □_{D(n−1)}F_{2}} ; since D(0) = A, then downward induction on n will finish the proof.

Since y is invariant under the A □_{D(n)}F_{2}-coaction, then it is also invariant under the coaction of D(n − 1) □_{D(n)}F_{2} (since the latter is a quotient Hopf algebra of the former). So y represents a class at the E_{2}-term of the change-of-rings spectral sequence

E_{2}^{s,t,u}(F_{2}) = Ext_{F_{2}[ξ_{n}^{2^{n}}]}^{s,u}(F_{2}, Ext_{D(n)}^{t,∗}(F_{2}, F_{2})) ⇒ Ext_{D(n−1)}^{s+t,u}(F_{2}, F_{2}).

Also, by assumption, y lies in the (t, u)-plane, say y ∈ E_{2}^{0,q,v}. Choose m large enough so that m + q − v − 1 is positive. By Proposition 5.2.2 (with X = S^{0}), we know that for some c and r, we have E_{r}^{s,t,u} = 0 when ms + t − u > c (hence the same is true for E_{r′}^{s,t,u}, for all r′ ≥ r).

For each i ≥ 0, Proposition 1.4.11 tells us that the possible differentials on y^{2^{i}} are

d_{j+1}(y^{2^{i}}) ∈ E_{j+1}^{j+1,2^{i}q−j,2^{i}v},

for j ≥ 2^{i}. Choose i so that

2^{i} > max(r − 1, \frac{c − m}{m + q − v − 1}),

and fix j ≥ 2^{i}. Then we have a vanishing plane at the E_{2^{i}+1}-term (and hence at the E_{j+1}-term); we claim that the element d_{j+1}(y^{2^{i}}) lies above the vanishing plane, and so is zero. We just have to verify the inequality specified by the vanishing plane:

m(j + 1) + 2^{i}q − j − 2^{i}v = (m − 1)j + m + 2^{i}(q − v)

≥ (m − 1)2^{i} + m + 2^{i}(q − v)

= 2^{i}(m + q − v − 1) + m

> \frac{c − m}{m + q − v − 1}(m + q − v − 1) + m

= c.

Hence y^{2^{i}} is a permanent cycle. For degree reasons, it cannot be a boundary; hence it gives a nonzero element of E_{∞}, and hence a nonzero element of HD(n − 1)_{∗∗}.

It only remains to show that y^{2^{i}}, or at least some power y^{2^{i+j}}, is invariant under the A-coaction. Let ρ: HD(n − 1)_{∗∗} ⟶ HD(n)_{∗∗} denote the Hurewicz (restriction) map. This map detects nilpotence: if x ∈ ker ρ, then x is nilpotent. (This follows from Remark 5.1.7(c), for instance; alternatively, this is the main inductive step in proving Theorem 5.1.5.) Hence by Lemma 5.2.7 below, y^{2^{i+j}} is invariant for some j.

This completes the verification of condition (ii) of Definition 5.1.1(b), and hence the proof of Theorem 5.1.2. □

We have used the following in the case when R and S are commutative, so there are no concerns about centrality of particular elements. In the next chapter, we use the lemma in its full generality.

Lemma 5.2.7. Suppose that R and S are A-comodule algebras, with an A-linear map ρ: R ⟶ S that detects nilpotence: every x ∈ ker ρ is nilpotent. Given a central element z ∈ R so that ρ(z) ∈ S is invariant under the A-coaction, then z^{2^{n}} is also invariant, for some n.

Proof. Since ρ(z) is invariant, then the coaction on z is of the form

z ⟼ 1 ⊗ z + ∑_{i} a_{i} ⊗ x_{i},

where each x_{i} is in ker ρ, and hence is nilpotent. Since z is central, then so is each x_{i} by Lemma 5.2.8 below. Thus we see that

z^{2^{n}} ⟼ 1 ⊗ z^{2^{n}} + ∑_{i} a_{i}^{2^{n}} ⊗ x_{i}^{2^{n}}.

This is a finite sum (since these are comodules), so for n sufficiently large, z^{2^{n}} is invariant. □

Lemma 5.2.8. Suppose that R is an A-comodule algebra. Given an element z ∈ R with A-coaction

z ⟼ 1 ⊗ z + ∑_{i} a_{i} ⊗ x_{i},

where the elements a_{i} are linearly independent, if z is central, then so is each x_{i}.

Proof. This is an easy verification. We let ψ: R ⟶ A ⊗ R be the coaction map on R. We prove that x_{i} commutes with each element y in R by induction on the number of summands in the coaction on y. If y is invariant—that is, if ψ(y) = 1 ⊗ y—then the commutator [y, z] maps to 1 ⊗ [y, z] + ∑ a_{i} ⊗ [y, x_{i}]. Since z is central, then [y, z] = 0; since the a_{i} are linearly independent, then we conclude that [y, x_{i}] = 0 for each i.

Now suppose that the coaction on y has n terms:

ψ(y) = 1 ⊗ y + ∑_{j=1}^{n−1} b_{j} ⊗ y_{j}.

By coassociativity, each y_{j} has fewer than n terms in its diagonal, and so commutes with each x_{i}. So the formula ψ[y, z] = 0 gives the same identity as above: 1⊗[y, z]+ ∑ a_{i} ⊗ [y, x_{i}] = 0. Again, we conclude that [y, x_{i}] = 0 for each i. □

5.3. Nilpotence and F -isomorphism via quasi-elementary quotients

In this section we use our F -isomorphism and nilpotence theorems for HD— Theorems 5.1.2 and 5.1.5—to prove analogous theorems for the quasi-elementary quotients of A—Theorems 5.1.3 and 5.1.6.

5.3.1. Nilpotence: Proof of Theorem 5.1.6. We start with Theorem 5.1.6, because we use it in the proof of Theorem 5.1.3. We need a few preliminary results. Suppose we have a map f: S^{0} ⟶ X. We define X^{∧∞} to be the homotopy colimit of the following diagram:

S^{0} ⟶^{f} X ⟶^{f∧1} X ∧ X ⟶^{f∧1∧1} X ∧ X ∧ X ⟶ ⋯,

and we let f^{∧∞}: S^{0} ⟶ X^{∧∞} be the obvious map. We recall the following from [HS98, Lemma 2.4].

Lemma 5.3.1. Given a map f: S^{0} ⟶ X and a ring spectrum E with unit map η: S^{0} ⟶ E, the following are equivalent.

(a) E ∧ X^{∧∞} = 0.

(b) η ∧ f^{∧∞}: S^{0} ⟶ E ∧ X^{∧∞} is zero.

(c) η ∧ f^{∧n}: S^{0} ⟶ E ∧ X^{∧n} is zero for n ≫ 0.

(d) 1_{E} ∧ f^{∧n}: E = E ∧ S^{0} ⟶ E ∧ X^{∧n} is zero for n ≫ 0.

For integers q > r ≥ 0, we define the following quotient Hopf algebras of D:

D_{r} = D/(ξ_{1}, …, ξ_{r}),
D_{r,q} = D_{r}/(ξ_{r+2}^{2^{r+1}}, …, ξ_{q}^{2^{r+1}}).

See Figure 5.3.A for their profile functions. Recall from Corollary 2.1.8 that the maximal quasi-elementary quotients of A are called E(r) for r ≥ 0.

Lemma 5.3.2. We have \underrightarrow{lim}_{r} HD_{r} = HF_{2} and \underrightarrow{lim}_{q} HD_{r,q} = HE(r).

These are homotopy colimits, as usual.

Proof. We leave this as an exercise. □

Lemma 5.3.3. Let i, j, q, r, and R be integers.

(a) Suppose that q > r ≥ i ≥ 0 and q − r ≥ j ≥ 0, and consider the Hopf algebra B = D_{r,q}/(ξ_{r+1}^{2^{i+1}}, ξ_{q+1}^{2^{r+1+j}}). By Lemma 2.1.2, there are Hopf algebra extensions

E[ξ_{r+1}^{2^{i}}] ⟶ B ⟶ C_{1},
E[ξ_{q+1}^{2^{r+j}}] ⟶ B ⟶ C_{2},

. . .. . . ⋮  D_{2}  ^{. . .. . . ⋮  . . .. . .. . . ⋮ ⋮ ⋮ D_{2,6}}

Figure 5.3.A. Profile functions for D_{r} and D_{r,q}.

leading to elements h_{r+1,i} and h_{q+1,r+j} in HB_{∗∗}. Then h_{r+1,i}h_{q+1,r+j} is nilpotent in HB_{∗∗}.

(b) Whenever q > r and r ≥ i ≥ 0, there is a Bousfield equivalence

⟨h_{r+1,i}^{−1}H(D_{r}/(ξ_{r+1}^{2^{i+1}}))⟩ = ⟨h_{r+1,i}^{−1}H(D_{r,q}/(ξ_{r+1}^{2^{i+1}}))⟩ .

(c) Whenever q > R, we have a Bousfield class decomposition

⟨HD⟩ = ⟨HD_{R+1}⟩ ∨ ⋁_{r=0}^{R} ⋁_{i=0}^{r} ⟨h_{r+1,i}^{−1}H(D_{r}/(ξ_{r+1}^{2^{i+1}}))⟩

= ⟨HD_{R+1}⟩ ∨ ⋁_{r=0}^{R} ⋁_{i=0}^{r} ⟨h_{r+1,i}^{−1}H(D_{r,q}/(ξ_{r+1}^{2^{i+1}}))⟩ .

Proof. For (a), the nilpotence of h_{r+1,i}h_{q+1,r+j} is due to Lin [Lin77a]; we recall the statement of this result in Theorem B.2.1(b). (The nilpotence of this product is also the content of [Wil81, Theorem 6.4], as well as being essentially equivalent to the classification of quasi-elementary quotients of A in Propositions 2.1.7 and 2.1.12.)

Part (b) follows from part (a) and Corollary 1.5.2, by induction: one can get from D_{r}/(ξ_{r+1}^{2^{i+1}}) to D_{r,q}/(ξ_{r+1}^{2^{i+1}}) by dividing out by one ξ_{t}^{2^{s}} at a time, where s ≥ r+1. In other words, one has a sequence of extensions of the form

E[ξ_{t}^{2^{s}}] ⟶ B ⟶ C,

and hence cofibrations of the form

HB ⟶^{h_{ts}} HB ⟶ HC.

One inverts h_{r+1,i} in each term; then by the nilpotence of h_{r+1,i}h_{ts}, one gets an equivalence of Bousfield classes

⟨h_{r+1,i}^{−1}HB⟩ = ⟨h_{r+1,i}^{−1}HC⟩.

Part (c) is similar to part (b). □

Proof of Theorem 5.1.6. We imitate the proof of [HS98, Theorem 3]. As in that proof, one can reduce to the smash-nilpotence case, and using Spanier-Whitehead duality, one can reduce to the case where F = S^{0}. Suppose we have a map f: S^{0} ⟶ X so that 1_{HE} ∧ f = 0 for all quasi-elementary E. We want to show that 1_{HD} ∧ f^{∧n}: HD ⟶ HD ∧ X^{∧n} is zero for some n; then Theorem 5.1.5(c) will tell us that f is smash-nilpotent. By Lemma 5.3.1, 1_{HD} ∧ f^{∧n} is zero if and only if HD ∧ X^{∧∞} = 0. We use a Bousfield class argument to show this.

By assumption, for all quasi-elementary E, 1_{HE}∧f = 0, and hence HE∧X^{∧∞} = 0. By Lemma 5.3.3(c), we must show that

HD_{R+1} ∧ X^{∧∞} = 0,

for some R, and then that

h_{r+1,i}^{−1}H(D_{r,q}/(ξ_{r+1}^{2^{i+1}})) ∧ X^{∧∞} = 0,

for all r ≤ R, i ≤ r, and some q > R.

First we show that HD_{R+1} ∧ X^{∧∞} = 0. By Lemma 5.3.1, it is equivalent to show that η ∧ f^{∧∞}: S^{0} ⟶ HD_{R+1} ∧ X^{∧∞} is zero. Let R go to infinity; then by Lemma 5.3.2, the map

S^{0} ⟶ \underrightarrow{lim}_{R} HD_{R+1} ∧ X^{∧∞} = HF_{2} ∧ X^{∧∞}

is null. Since homotopy commutes with direct limits, then for some R, the map S^{0} ⟶ HD_{R+1} ∧ X^{∧∞} is null.

One uses the same argument to show that

h_{r+1,i}^{−1}H(D_{r,q}/(ξ_{r+1}^{2^{i+1}})) ∧ X^{∧∞} = 0,

but using the second equality of Lemma 5.3.2, rather than the first. □

5.3.2. F -isomorphism: Proof of Theorem 5.1.3. Now we work on Theorem 5.1.3: the map

π_{∗∗}S^{0} ⟶ (\underleftarrow{lim}_{𝒬}HE_{∗∗})^{A}

is an F -isomorphism. First we construct the coaction of A on \underleftarrow{lim}_{𝒬} HE_{∗∗}. We let \overline{𝒬} denote the full subcategory of 𝒬 consisting of the conormal quasi-elementary quotient Hopf algebras of A. Since the maximal quasi-elementary quotients are conormal, we see that \overline{𝒬} is final in 𝒬; hence we have

\underleftarrow{lim}_{𝒬}HE_{∗∗} = \underleftarrow{lim}_{\overline{𝒬}}HE_{∗∗}.

On the right we have an inverse limit of comodules over A; we give this the induced A-comodule structure. So we have (since taking invariants is an inverse limit)

(\underleftarrow{lim}_{𝒬}HE_{∗∗})^{A} = (\underleftarrow{lim}_{\overline{𝒬}}HE_{∗∗})^{A}

= \underleftarrow{lim}_{\overline{𝒬}}(HE_{∗∗}^{A})

= \underleftarrow{lim}_{\overline{𝒬}}(HE_{∗∗}^{A//E}) .

To finish the proof of Theorem 5.1.3, we show that we can compute HD_{∗∗} up to F -isomorphism in terms of the coefficient rings HE_{∗∗} for E quasi-elementary.

Proposition 5.3.4. The natural map

HD_{∗∗} ⟶ \underleftarrow{lim}_{𝒬}HE_{∗∗},

is an F -isomorphism, as is the induced map

HD_{∗∗}^{A} ⟶ (\underleftarrow{lim}_{𝒬}HE_{∗∗})^{A} .

See Proposition 6.5.1 for an explicit computation of the ring \underleftarrow{lim}_{𝒬} HE_{∗∗}.

Proof of Theorem 5.1.3. This follows immediately from Theorem 5.1.2 and Proposition 5.3.4. □

Proof of Proposition 5.3.4. We need to show that the restriction map

ρ: HD_{∗∗} ⟶ \underleftarrow{lim}_{E∈𝒬} HE_{∗∗}

is an F -isomorphism, so we need to show two things: every element in the kernel of f is nilpotent, and we can lift some 2^{n}th power of any element in the codomain of f.

By the nilpotence theorem 5.1.6 (or more precisely, by the generalization in Remark 5.1.7(c)), we know that an element z ∈ HD_{∗∗} is nilpotent if and only if z^{n} ∈ ker ρ for some n.

The second statement follows from the stronger claim that for every quasielementary quotient E, some 2^{n}th power of any element in HE_{∗∗} lifts to HD_{∗∗}. This in turn follows, almost directly, from a result of Hopkins and Smith (stated as Theorem 0.4.10), combined with the fact that the Hopf algebra D is a direct limit of finite-dimensional Hopf algebras. We explain.

As above, we write D_{r} for the quotient Hopf algebra D_{r} = D/(ξ_{1}, …, ξ_{r}) of D. Then each D_{r} is conormal in D, the kernel K_{r} = D □_{D_{r}}F_{2} is finite-dimensional, and we have D = \underrightarrow{lim}_{r} K_{r}. Note that this colimit stabilizes in any given degree.

Given a quasi-elementary quotient E of D, we define the quotient F_{r} of E to be the pushout of E ⟵ D ⟶ D_{r}, and we let E_{r} be the Hopf algebra kernel of E ⟶ F_{r}. In other words, we have the following diagram of Hopf algebra extensions:

K_{r} ⟶ D ⟶ D_{r}
↓  ↓  ↓
E_{r} ⟶ E ⟶ F_{r}.

(In other words, the profile function for F_{r} is the intersection of the profile functions of D_{r} and E—cf. the Hopf algebra B(n) in Proposition 2.1.5.)

Now, fix an element y ∈ HE_{∗∗}. Since E = \underrightarrow{lim}_{r} E_{r}, then for r sufficiently large, y is in the image of the inflation map Ext_{E_{r}}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{E}^{∗∗}(F_{2}, F_{2}), say ỹ ⟼ y. Also, K_{r} ⟶ E_{r} is a quotient map of finite-dimensional graded connected commutative Hopf algebras; let res_{K_{r},E_{r}} denote the induced map on Ext. Theorem 0.4.10 says that given ỹ ∈ Ext_{E_{r}}^{∗∗}(F_{2}, F_{2}), for some n we have ỹ^{p^{n}} ∈ im(res_{K_{r},E_{r}}). Consider the following commutative diagram:

Ext_{K_{r}}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{D}^{∗∗}(F_{2}, F_{2})
↓  ↓
Ext_{E_{r}}^{∗∗}(F_{2}, F_{2}) ⟶ Ext_{E}^{∗∗}(F_{2}, F_{2}).

Given y ∈ HE_{∗∗} = Ext_{E}^{∗∗}(F_{2}, F_{2}), we can pull y back to ỹ ∈ Ext_{E_{r}}^{∗∗}(F_{2}, F_{2}), and for some n we can lift ỹ^{2^{n}} to Ext_{K_{r}}^{∗∗}(F_{2}, F_{2}). The image of this lift in Ext_{D}^{∗∗}(F_{2}, F_{2}) is a lift of y^{2^{n}}, as desired. This finishes the proof that HD_{∗∗} ⟶ \underleftarrow{lim}_{𝒬} HE_{∗∗} is an F -isomorphism.

Now we want to show that the induced map on the invariants is also an F - isomorphism. As above, we write \overline{𝒬} for the category of conormal quasi-elementary quotients of A. We want to show that the natural map

HD_{∗∗}^{A □_{D}F_{2}} ⟶ \underleftarrow{lim}_{\overline{𝒬}}(HE_{∗∗}^{A □_{E}F_{2}} )

is an F -isomorphism. We treat the A □_{D}F_{2}-coaction on HD_{∗∗} as an A-coaction with trivial D-coaction (and similarly for A □_{E}F_{2} coacting on HE_{∗∗}), so we want to compare HD_{∗∗}^{A} with \underleftarrow{lim} HE_{∗∗}^{A}. We write f and f̃ for the maps

f: HD_{∗∗} ⟶ \underleftarrow{lim}_{\overline{𝒬}}HE_{∗∗},

f̃: HD_{∗∗}^{A} ⟶ \underleftarrow{lim}_{\overline{𝒬}}HE_{∗∗}^{A}.

The maximal quasi-elementary quotients are conormal; hence the conormal quasielementary quotients are final in the inverse system of all quasi-elementary quotients; hence f is an F -isomorphism. So it is clear that if x ∈ HD_{∗∗}^{A} is in the kernel of f̃, then x is nilpotent. Furthermore, given y ∈ \underleftarrow{lim} HE_{∗∗}^{A}, we know that y is in the image of f; hence by Lemma 5.2.7, some 2^{n}th power of y must be in the image of f̃. □

5.4. Further discussion: nilpotence at odd primes

There are two main obstructions to proving the above results at odd primes. The first is clear: we do not have a classification of the quasi-elementary quotients of A; hence we cannot prove the quasi-elementary versions of the theorems. There are two aspects to this: to prove a version of Proposition 5.3.4, one would like to know that the quasi-elementary quotients are in fact quotients of D; and to prove a version of Lemma 5.3.3, one would like to have an odd primary analogue of Theorem B.2.1(b). The second main obstruction is perhaps more important, since it affects the HD versions of the theorems, and more surprising, because the corresponding result at the prime 2 seems rather standard. This is the property that allows one to prove Lemma 5.2.6 and hence Proposition 5.2.2: at the prime 2, the element h_{nj} = [ξ_{n}^{2^{j}}] ∈ HD(n)_{∗∗} is nilpotent, if j ≥ n (see Theorem B.2.1(a)).

Recall from Notation 1.2.14 and Lemma 1.2.15 that a primitive y in a Hopf algebra B gives rise to a class [y] ∈ Ext_{B}^{1,|y|}(k, k). When the characteristic p of the ground field k is odd, an even-dimensional primitive y in B gives rise to a class β𝒫̃^{0} [y] in Ext_{B}^{2,p|y|}(k, k).

Conjecture 5.4.1. Fix an odd prime p. Fix a quotient Hopf algebra B of A, and suppose that ξ_{t}^{p^{s}} is primitive in B. If s ≥ t, then b_{ts} = β𝒫̃^{0} [ξ_{t}^{p^{s}}] is nilpotent in HB_{∗∗}.

As at the prime 2, we define D as follows:

D = A/(ξ_{1}^{p}, ξ_{2}^{p^{2}}, ξ_{3}^{p^{3}}, …).

One can see from the proofs when p = 2 that the odd prime versions of Theorems 5.1.2 and 5.1.5 would follow from this conjecture. (One has to make a slight change in Lemma 5.2.4—as in the proof of Lemma 1.2.15, there are two related short exact sequences of comodules, leading to a cofibration in which the connecting homomorphism is a map of homological degree 2. Other than that, everything goes through as written.)

We discuss the status of Conjecture 5.4.1 in Appendix B.

5.5. Further discussion: miscellany

The nilpotence theorem [DHS88] in stable homotopy theory has far-reaching implications for the global structure of the stable homotopy category—see [Hop87] and [Rav92], for instance. We develop analogues of some of the structural results in the next chapter, but there are still many gaps. We discuss those in Sections 6.7 and 6.9 below.

If Γ is a Hopf algebra, then the quasi-elementary quotients of Γ give the right homology functors to consider for detecting nilpotence, essentially by the definition of “quasi-elementary.” Indeed, if Γ is commutative and finite-dimensional, it is not too hard to prove an analogue of Theorem 5.1.6, either directly (as in [Pal97]) or via Chouinard’s theorem (as in [HPS97, Theorem 9.6.10, Remark 9.6.11]). If, in addition, Γ is connected, then Theorem 0.4.10 allows one to prove an analogue of Theorem 5.1.3—see [Pal97]. Actually, by Theorem 0.4.10, some power of every element in the inverse limit is invariant, so one gets an F -isomorphism

Ext_{Γ}^{∗∗}(k, k) ⟶ \underleftarrow{lim}_{𝒬}Ext_{E}^{∗∗}(k, k),

where the inverse limit is over 𝒬, the category of quasi-elementary quotients of Γ.

Here are some related questions: one already has Quillen stratification for group algebras; can one prove it, as we do for A, with vanishing lines or planes? Can one use our approach to Quillen stratification for the Steenrod algebra to study the cohomology of other Hopf algebras? Given a Hopf algebra Γ, one should study the quotient Hopf algebra which is the pullback of the quasi-elementary quotients of Γ, so one might want to assume that those quotients are somewhat well-behaved. Even more generally, in an arbitrary stable homotopy category one could consider a collection of appropriately chosen ring spectra—versions of the spectra HE—and look at the map from S^{0} to an appropriate inverse limit of them.

Can one refine the description of π_{∗∗}S^{0} as given by Theorem 5.1.2? In particular, can one describe the kernel, determine the nilpotence height of elements in the kernel, or say which powers of classes in the codomain of

φ: π_{∗∗}S^{0} ⟶ HD_{∗∗}^{A □_{D}F_{2}}

lift to the domain?

Since one has a version of Quillen stratification for the Steenrod algebra, one might look for analogues of other group-theoretic results. Chouinard’s theorem [Cho76] is an example: a kG-module M is projective if and only if it is projective upon restriction to kE for every elementary abelian subgroup E of G. One analogue in Stable(A) might be: for any spectrum X, X = 0 if and only if HE ∧ X = 0 for every quasi-elementary quotient E of A. This is false, though: as noted in the proof of Theorem 6.6.1, X = IHD is a counterexample, where I is the Brown-Comenetz duality functor. It is not clear what a better analogue would be, though; other statements, such as “for any spectrum X, X is in loc(HF_{p}) if and only if HE ∧ X is in loc(HF_{p}) for every quasi-elementary quotient E of A,” seem equally dubious.

CHAPTER 6 Periodicity and other applications of the nilpotence theorems

The nilpotence theorem in ordinary stable homotopy theory [DHS88] has a number of important consequences: the periodicity theorem and the thick subcategory theorem of [HS98] are examples. In this chapter we study applications of our nilpotence and Quillen stratification theorems—Theorems 5.1.2, 5.1.3, 5.1.5, and 5.1.6.

One of the main results of this chapter is a version of the periodicity theorem: if X is a finite spectrum, then we produce a number of central non-nilpotent elements of the ring [X, X]_{∗∗} via the “ideal of X,” which is essentially the kernel of

[S^{0}, HD]_{∗∗} ⟶^{−∧X} [X, HD ∧ X]_{∗∗}.

Equivalently, this gives families of central non-nilpotent elements in the homotopy of any finite ring spectrum. This is our analogue of the periodicity theorem of Hopkins and Smith. We state this precisely in Section 6.1, and we prove it in Sections 6.2–6.4.

Theorem 5.1.2 says that π_{∗∗}S^{0} is F -isomorphic to the A-invariants in HD_{∗∗}; in Section 6.5 we discuss some examples of invariant elements in HD_{∗∗}.

In Section 6.6, we show that the objects that detect nilpotence—HD and ⋁_{E} HE—have strictly smaller Bousfield classes than that of the sphere. The role of D, as well as the action of A on D, has led us to a conjectured thick subcategory theorem, which we give in Section 6.7. We end the chapter with two sections of miscellany—one on slope supports of finite spectra, and a short section with a few additional questions and remarks.

As in the previous chapter, we work at the prime 2 unless otherwise stated.

6.1. The periodicity theorem

We start this chapter by giving our version of the periodicity theorem. This is a weak analogue of the Quillen stratification theorem 5.1.2, but with nontrivial coefficients. The following definition was motivated by work in modular group representation theory of Carlson, Avrunin, Scott, and others.

Definition 6.1.1. Given a spectrum X, we define the ideal of X, I(X), to be the radical of the ideal

ker (HD_{∗∗} ⟶^{−∧X} [X, HD ∧ X]_{∗∗}) .

Recall from the start of Subsection 5.1.2 that [X, HD∧X]_{∗∗} is a ring; one can check that the above map is a ring map.

In Section 6.3, we study the ideal I(X). In particular, we show that when X is finite, the ideal I(X) is invariant (Definition 6.3.1) under the A-coaction; hence the A-coaction on HD_{∗∗} induces one on HD_{∗∗}/I(X). The following definition generalizes Definition 2.4.2, the definition of u_{n}-map.

Definition 6.1.2. Given an element y ∈ Ext_{D}^{∗∗}(F_{2}, F_{2}) = HD_{∗∗} and an object X in Stable(A), we say that a map z: X ⟶ X is a y-map if 1_{HD} ∧ z = y^{n} ∧ 1 in [HD ∧ X, HD ∧ X]_{∗∗} for some n.

Theorem 6.1.3 (Periodicity theorem). Let X be a finite spectrum. For every y ∈ HD_{∗∗} which maps to an A-invariant element of HD_{∗∗}/I(X), X has a y-map which is central in the ring [X, X]_{∗∗}.

Remark 6.1.4. (a) There is an equivalent statement involving elements in the homotopy of a finite ring spectrum R, which we give as Theorem 6.2.3 below.

(b) We conjecture that the ring (HD_{∗∗}/I(X))^{A} is F -isomorphic to the center of [X, X]_{∗∗}. See Section 6.7 for this and related issues.

(c) One of the potential flaws of Theorem 6.1.3 is that computing I(X) is difficult, as is finding invariants in HD_{∗∗}/I(X). If one knows which P_{t}^{s}- homology groups of X are nonzero, though, one can find some invariants in HD_{∗∗}/I(X), and hence one can use the theorem to produce some self-maps of X. See Theorem 6.8.3 for details.

6.2. y-maps and their properties

In this section we lay the groundwork for proving Theorem 6.1.3. In particular, we study analogues of v_{n}-maps [Hop87, HS98]. In ordinary stable homotopy theory, v_{n}-maps are defined via the Morava K-theories—these are field spectra, and hence have various convenient properties, such as Künneth isomorphisms. Our analogues are defined via HD, and hence are not quite as easy to work with. Nonetheless, our versions of v_{n}-maps, called “y-maps,” share many of the same properties as v_{n}-maps in ordinary stable homotopy theory; in particular, we show in this section that the property of having a y-map is generic in our setting.

Fix a quotient Hopf algebra B of A which maps onto D. We work in the category Stable(B). Recall from Remark 5.1.7(c) that the nilpotence theorem 5.1.5 holds in this category: HD detects nilpotence. Note that if E is a ring spectrum and y is an element of π_{∗∗}E, then y gives a self map of E, namely the composite

E = S^{0} ∧ E ⟶^{y∧1_{E}} E ∧ E ⟶^{μ} E.

We write y for both the homotopy element and the self-map. We start by expanding Definition 6.1.2 a bit:

Definition 6.2.1. Fix y ∈ Ext_{D}^{∗∗}(F_{2}, F_{2}) = HD_{∗∗}, and let X be an object in Stable(B). A map z: X ⟶ X is a y-map if 1_{HD} ∧ z = y^{n} ∧ 1 ∈ [HD ∧ X, HD ∧ X]_{∗∗} for some n. Similarly, if R is a ring object in Stable(B), then an element α ∈ π_{∗∗}R is a y-element if for some n, we have 1_{HD} ∧ α = y^{n} ∧ 1 as maps HD ⟶ HD ∧ R.

Remark 6.2.2. (a) Given X and z as in the definition, then the condition 1_{HD} ∧ z = y^{n} ∧ 1 is equivalent to the statement that η_{HD} ∧ z = y^{n} ∧ 1 in [X, HD ∧ X]_{∗∗}. This second condition is a natural one to consider, because the change-of-rings isomorphism (Corollary 1.2.7) tells us that [X, HD ∧ X]_{∗∗} is isomorphic to [X, X]_{∗∗}^{D}, the set of D-linear cochain homotopy classes of maps from X to itself. Similarly, given R and α as above, 1_{HD} ∧ α = y^{n} ∧ 1 if and only if η_{HD} ∧ α = y^{n} ∧ 1.

6.2. y-MAPS AND THEIR PROPERTIES 129

(b) Note that the set of y-maps from X to itself is in bijection with the set of y-elements in the ring spectrum X ∧ DX, by Spanier-Whitehead duality. Indeed, there is a ring isomorphism

[HD ∧ X, HD ∧ X]_{∗∗} ≅ [HD, HD ∧ X ∧ DX]_{∗∗}.

So the following is equivalent to Theorem 6.1.3. Theorem 6.2.3 (Periodicity theorem for ring spectra). Let R be a finite ring

spectrum with unit map η: S^{0} ⟶ R. Let I = \sqrt{ker HD_{∗∗}η}—this is an invariant ideal in HD_{∗∗}. For every y ∈ HD_{∗∗} which maps to an A-invariant element in HD_{∗∗}/I, there is a central y-element in π_{∗∗}R.

The main tool for proving Theorems 6.1.3 and 6.2.3 is the following. See Definition 1.3.7 for the definition of “thick subcategory.”

Theorem 6.2.4. Suppose that B is a quotient Hopf algebra of A with A ↠ B ↠ D. Fix y ∈ HD_{∗∗}. The full subcategory 𝒞 consisting of finite objects of Stable(B) having a y-map is a thick subcategory of Stable(B).

In other words, the property of having a y-map is generic.

The proof is a modification of the proof in [HS98, Section 3] that having a v_{n}-map is a generic property. We devote the rest of this section to the details. We start with a variant of the notion of y-map, and a general lemma.

Definition 6.2.5. Fix y = HD_{∗∗}, and let X be an object in Stable(B). A map z: X ⟶ X is a weak y-map if HD_{∗∗}z: HD_{∗∗}X ⟶ HD_{∗∗}X is multiplication by y^{n} for some n. Similarly, if R is a ring object in Stable(B), then an element α ∈ π_{∗∗}R is a weak y-element if for some n, HD_{∗∗}α = y^{n} as maps HD_{∗∗} ⟶ HD_{∗∗}R.

We should justify this terminology.

Lemma 6.2.6. If a self-map z: X ⟶ X of an object X in Stable(B) is a ymap, then it is a weak y-map. If an element α ∈ π_{∗∗}R in the homotopy of a ring spectrum in Stable(B) is a y-element, then it is a weak y-element.

Proof. This is clear. □

Note, though, that since HD_{∗∗} is not likely to satisfy a Künneth isomorphism, then Spanier-Whitehead duality may not yield a bijection between weak y-maps of X and weak y-elements in π_{∗∗}(X ∧ DX). Having such a bijection is important in the manipulations below, which is why we work with y-maps rather than weak y-maps. Compare this to the situation in ordinary stable homotopy theory, where v_{n}-maps are defined in terms of Morava K-theories, which do satisfy a Künneth isomorphism; hence v_{n}-maps are defined in terms of their effect on K(n)_{∗}.

We move on to the proof of Theorem 6.2.4. For the remainder of the section, we consider a fixed element y of HD_{∗∗}. We work at the prime 2.

Lemma 6.2.7. Let R be a ring spectrum in Stable(B), and fix weak y-elements α and β in π_{∗∗}R. If α and β commute, then there exist positive integers i and j so that α^{i} = β^{j}.

Proof. We may assume that HD_{∗∗}(α − β) = 0 by raising α and β to suitable powers; then by the nilpotence theorem 5.1.5, α − β is nilpotent. Since α and β commute, then α^{2^{n}} = β^{2^{n}} for some n. □

Lemma 6.2.8. Let R be a finite ring object in Stable(B), and fix a weak yelement α ∈ π_{∗∗}R. For some i > 0, the element α^{i} is central in π_{∗∗}R.

Proof. Let ℓ(α)_{∗}, r(α)_{∗} ∈ End(π_{∗∗}R) denote left and right multiplication by α, respectively. More precisely, ℓ(α) is the following self-map of R:

R ⟶^{α∧1} R ∧ R ⟶^{μ} R,

and similarly for r(α). Since HD_{∗∗}α is central in HD_{∗∗}R, then ℓ(α) − r(α) maps to zero in End(HD_{∗∗}R); since R is finite, then the nilpotence theorem for selfmaps 5.1.5(b) applies to show that ℓ(α) − r(α) is nilpotent. Since right multiplication by α commutes with left multiplication by α, we conclude that ℓ(α)^{2^{n}} = r(α)^{2^{n}} for some n. Since ℓ(α)^{2^{n}} = ℓ(α^{2^{n}}) and r(α)^{2^{n}} = r(α^{2^{n}}), we conclude that α^{2^{n}} is central in π_{∗∗}R. □

Corollary 6.2.9. Let R be a finite ring spectrum in Stable(B). For any weak y-elements α, β ∈ π_{∗∗}R, there exist positive integers i and j so that α^{i} = β^{j}.

Now we use the adjointness between y-maps and y-elements, so we do not use the weak forms any more.

Corollary 6.2.10. Let X be a finite spectrum in Stable(B), and let f and g be two y-maps of X. Then:

(a) Some power of f is central in [X, X].

(b) f^{i} = g^{j} for some positive integers i and j.

Corollary 6.2.11. Suppose that X_{1} and X_{2} have y-maps y_{1} and y_{2}, respectively. Then there are positive integers i and j so that for every Z and every h : Z ∧ X_{1} ⟶ X_{2}, the following diagram commutes:

Z ∧ X_{1} ⟶^{h} X_{2}
1∧y_{1}^{i}↓  ↓^{y_{2}^{j}}
Z ∧ X_{1} ⟶_{h} X_{2}

Proof. DX_{1} ∧ X_{2} has two y-maps: Dy_{1} ∧ 1 and 1 ∧ y_{2}. Now we apply Corollary 6.2.10 and Spanier-Whitehead duality. □

We are ready to prove the main theorem of this section, that the property of having a y-map is generic.

Proof of Theorem 6.2.4. If a finite spectrum Y has a (central) y-map f and if X is a retract of Y, then the induced self-map of X

X ⟶ Y ⟶^{f} Y ⟶ X

is easily seen to be a y-map.

Suppose that X_{1} ⟶ X_{2} ⟶ X_{3} is a cofibration of finite spectra, and that X_{1} and X_{2} have y-maps y_{1} and y_{2}, respectively. By replacing y_{1} and y_{2} by powers, Corollary 6.2.11 tells us that we may assume that the left square in this diagram commutes:

X_{1}  X_{2}  X_{3} X_{1}  X_{2}  X_{3}. w_{_{y_{1}} u}  w_{_{y_{2}} u}  _{u_{y_{3}}} w  w

Hence there is a map y_{3} : X_{3} ⟶ X_{3} so that the right square commutes. We claim that y_{3}^{2} is a y-map. Since y_{1} and y_{2} are y-maps, then 1_{HD} ∧ y_{1} and 1_{HD} ∧ y_{2} are both multiplication by some power of y, say y^{n}. We compare 1_{HD} ∧ y_{3} and y^{n} ∧ 1 in [HD ∧ X_{3}, HD ∧ X_{3}] via the following commutative diagram:

HD ∧ X_{2} ⟶ HD ∧ X_{3} ⟶ HD ∧ X_{1}
1∧y_{2}−y^{n}∧1↓  ↓^{1∧y_{3}−y^{n}∧1}  ↓^{1∧y_{1}−y^{n}∧1}
HD ∧ X_{2} ⟶ HD ∧ X_{3} ⟶ HD ∧ X_{1}.

(As usual, we are omitting all suspensions from the notation.) The left- and righthand vertical maps are null by assumption, so a simple diagram chase shows that (1 ∧ y_{3} − y^{n} ∧ 1)^{2} is null as well. Since we are working in characteristic 2, and since 1 ∧ y_{3} and y^{n} ∧ 1 commute in [HD ∧ X, HD ∧ X], we find that 1 ∧ y_{3}^{2} = y^{2n} ∧ 1. Hence y_{3}^{2} is a y-map. □

6.3. Properties of ideals

In this section, we establish a key property of the ideal I(X), namely invariance. As usual in this chapter, we work at the prime 2.

Recall from Corollary 1.2.7 that if B is a conormal quotient Hopf algebra of A, then for any objects X and Y of Stable(A), [X, HB ∧ Y]_{∗∗} is the set of B-linear homotopy classes of maps from X to Y, which we write as [X, Y]_{∗∗}^{B}. We define I_{B}(X) to be the radical of the kernel of

[S^{0}, S^{0}]_{∗∗}^{B} ⟶^{−∧X} [X, X]_{∗∗}^{B}.

Note that this definition makes sense for any object X in Stable(B). When X is defined over A rather than B, then we show in Proposition 6.3.2 that I_{B}(X) is invariant in the following sense.

Definition 6.3.1. Suppose that R is a commutative comodule algebra over A (i.e., a commutative algebra and a left A-comodule, compatibly); we write ψ for the comodule structure map. We say that an ideal I of R is invariant under the A-coaction if for all x ∈ I, ψ(x) lies in A ⊗ I.

Proposition 6.3.2. Let B be a conormal quotient Hopf algebra of A.

(a) Let X be a finite object of Stable(B). Then

\sqrt{ker(HB_{∗∗} ⟶ [X, X]_{∗∗}^{B})} = \sqrt{ann_{HB_{∗∗}}(HB_{∗∗}X)}.

(b) Let X be a finite object of Stable(A). Then I_{B}(X) is invariant under the A-coaction on HB_{∗∗}.

The proof of part (a) is based on similar work in [Ben91b, Section 5.7].

Proof. Part (a): Let

I = \sqrt{ker(HB_{∗∗} ⟶ [X, HB ∧ X]_{∗∗})},
J = \sqrt{ann_{HB_{∗∗}}(HB_{∗∗}X)}.

The Yoneda action of HB_{∗∗} on HB_{∗∗}X factors through the composition action, since HB_{∗∗} is commutative. Hence I ≤ J. On the other hand, since X is finite, then it is in thick(S^{0}). So any element of HB_{∗∗} that annihilates [S^{0}, HB ∧ X]_{∗∗} will also annihilate [X, HB ∧ X]_{∗∗}. Hence J ≤ I.

Part (b): Now we show that I(X) is invariant under the A-coaction. By Remark 1.2.13, both HB_{∗∗} and HB_{∗∗}X are A □_{B}F_{p}-comodules, and one can check that the action map

HB_{∗∗} ⊗ HB_{∗∗}X ⟶ HB_{∗∗}X

is a map of A □_{B}F_{p}-comodules. Suppose that y ∈ I(X), and that under the A □_{B}F_{p}- coaction, y maps to ∑_{i} a_{i} ⊗ y_{i}. (We may assume that the a_{i}’s are linearly independent elements of A.) We want to show that each y_{i} is in I(X); i.e., that some power of y_{i} annihilates HB_{∗∗}X.

Fix x ∈ HB_{∗∗}X. If we assume that y^{2^{k}}x = 0, then we claim that y_{i}^{2^{k}}x = 0 for each i. Suppose that under the A □_{B}F_{p}-coaction, we have

(6.3.3)  x ⟼ 1 ⊗ x + ∑_{j=1}^{n} b_{j} ⊗ x_{j}.

We prove, by induction on n, that y_{i}^{2^{k}}x = 0 for all i. When n = 0 (i.e., when x is primitive in HB_{∗∗}X), then we have

HB_{∗∗}X ⟶ (A □_{B}F_{p}) ⊗ HB_{∗∗}X,
0 = y^{2^{k}}x ⟼ (∑_{i} a_{i} ⊗ y_{i})^{2^{k}}(1 ⊗ x) = ∑_{i} a_{i}^{2^{k}} ⊗ y_{i}^{2^{k}}x.

Since the a_{i}’s are linearly independent, we conclude that y_{i}^{2^{k}}x = 0 for all i.

Suppose that y_{i}^{2^{k}} annihilates every element of HB_{∗∗} which has at most n terms in its diagonal, and fix x with diagonal as in (6.3.3). Then by coassociativity, each x_{j} has at most n terms in its diagonal, so we have

0 = y^{2^{k}}x ⟼ (∑_{i} a_{i} ⊗ y_{i})^{2^{k}}(1 ⊗ x + ∑_{j} b_{j} ⊗ x_{j})

= ∑_{i}(a_{i}^{2^{k}} ⊗ y_{i}^{2^{k}}x + ∑_{j} a_{i}^{2^{k}}b_{j} ⊗ y_{i}^{2^{k}}x_{j})

= ∑_{i} a_{i}^{2^{k}} ⊗ y_{i}^{2^{k}}x.

Hence y_{i}^{2^{k}}x = 0 for each i. □

6.4. The proof of the periodicity theorem

In this section we prove Theorem 6.1.3. Fix a finite object X in Stable(A). If y ∈ HD_{∗∗} maps to an invariant element in HD_{∗∗}/I(X), then we want to show that X has a y-map. The basic pattern of the proof is the same as that of Theorem 5.1.2: we inductively work our way from D to A via the Hopf algebras D(n) (defined in Section 5.2).

Since D(n) is a conormal quotient of A for each n, then [X, HD(n) ∧ X]_{∗∗} = HD(n)_{∗∗}(X ∧ DX) is an A-comodule, by Remark 1.2.13. As noted there, the D(n)-coaction is trivial, so there is an induced coaction of A □_{D(n)}F_{2}; we also have a coaction of the quotient D(n − 1) □_{D(n)}F_{2}. The statement we want to prove by downward induction on n is: X has a y-map when viewed as an object in Stable(D(n))—i.e., there is a y-map in [X, HD(n) ∧ X]_{∗∗}—and that y-map is invariant under the A-coaction (and hence under the D(n − 1) □_{D(n)}F_{2}-coaction).

By Lemma 5.2.1, HD_{∗∗} is the colimit of the algebras HD(n)_{∗∗}, so y ∈ HD_{∗∗} lifts to HD(n)_{∗∗} for all sufficiently large n; since X is finite, then for some n, the element y∧1_{X} ∈ [X, HD∧X]_{∗∗} lifts to a y-map ỹ ∈ [X, HD(n)∧X]_{∗∗} = [X, X]_{∗∗}^{D(n)}. To start the induction, we need to check that some power of ỹ is invariant under the A-coaction. Let I_{D(n)}(X) be the radical of the kernel of

HD(n)_{∗∗} ⟶^{−∧X} [X, HD(n) ∧ X]_{∗∗}.

By Proposition 6.3.2, I_{D(n)}(X) is invariant under the A-coaction. We have a commutative diagram of A-comodules:

HD(n)_{∗∗}/I_{D(n)}(X) ⟶ HD_{∗∗}/I(X)
−∧X↓  ↓^{−∧X}
[X, HD(n) ∧ X]_{∗∗} ⟶ [X, HD ∧ X]_{∗∗}.

By choice of n, the element y ∈ HD_{∗∗} lifts to y′ ∈ HD(n)_{∗∗}; since y maps to an invariant in HD_{∗∗}/I(X), then Lemma 5.2.7 says that some 2^{k}th power of y′ maps to an invariant as well. Since the diagram is one of A-comodules, the image of (y′)^{2^{k}} in [X, HD(n) ∧ X]_{∗∗} is invariant under the A-coaction, as desired. This starts the induction.

(To apply Lemma 5.2.7, we need to know that the map HD(n)/I_{D(n)}(X) ⟶ HD_{∗∗}/I(X) detects nilpotence, but this follows from the fact that HD(n)_{∗∗} ⟶ HD_{∗∗} detects nilpotence.)

Now, assume that X has a y-map when viewed as an object in Stable(D(n)). We want to show that X still has a y-map, but when viewed as an object in Stable(D(n − 1)); in other words, we want to show that under the map

[X, HD(n − 1) ∧ X]_{∗∗} ⟶ [X, HD(n) ∧ X]_{∗∗},

some power of the y-map in the target lifts to the source. In Stable(D(n − 1)), for any object Y we have the Adams spectral sequence based on HD(n):

Ext_{D(n−1) □_{D(n)}F_{2}}^{s,u} (F_{2}, HD(n)_{t,∗}(Y)) ⇒ [S^{0}, Y]_{s+t,u}^{D(n−1)}.

(As in Section 5.2, we are writing this Adams spectral sequence using the changeof-rings grading of Proposition 1.4.9.) In particular, if we let W_{j} be defined as in Lemmas 5.2.3 and 5.2.4, then we have

Ext_{D(n−1) □_{D(n)}F_{2}}^{s,u} (F_{2}, HD(n)_{t,∗}(X ∧ W_{j} ∧ DX ∧ DW_{j})) (6.4.1)

\begin{aligned}⇒ & [S^{0}, X ∧ W_{j} ∧ DX ∧ DW_{j}]_{∗∗}^{D(n−1)} \\ & = [X ∧ W_{j}, X ∧ W_{j}]_{∗∗}^{D(n−1)}. \end{aligned}

Lemma 5.2.6 tells us that thick(X) = thick(W_{j} ∧ X) for any j ≥ n. Hence by Theorem 6.2.4, it suffices to show that X ∧ W_{j} has a y-map for some j.

The idea is that since W_{j} approximates the difference between D(n) and D(n− 1), then for j sufficiently large, the endomorphisms of X ∧ W_{j} over D(n − 1) should be more or less the same as the endomorphisms of X over D(n). Since X has a y-map over D(n), then X ∧ W_{j} should have one over D(n − 1).

The details are as follows. Since X is finite, then X ∧ DX is (i_{0}, j_{0})-connective for some i_{0} and j_{0}. The comodule W_{j} is nonzero only between degrees 0 and 2^{j}|ξ_{n}|, so X ∧ DX ∧ DW_{j} is (i_{0}, j_{0} − 2^{j}|ξ_{n}|)-connective. By Lemma 5.2.3, we see that the E_{2}-term for the Adams spectral sequence (6.4.1) has the following vanishing plane:

E_{2}^{s,t,u}(X ∧ W_{j} ∧ DX ∧ DW_{j}) = 0 when 2^{j}|ξ_{n}|s + t − u + j_{0} − 2^{j}|ξ_{n}| > 0.

The inductive hypothesis tells us that we have an A-invariant y-map in [X, HD(n)∧ X]_{∗∗}. We use ỹ to denote this element, as well as its image in [X ∧ W_{j}, HD(n) ∧ X ∧ W_{j}]_{∗∗}^{D(n−1)}. Since the map

[X, HD(n) ∧ X]_{∗∗} ⟶^{−∧W_{j}} [X ∧ W_{j}, HD(n) ∧ X ∧ W_{j}]_{∗∗}

is a map of D(n − 1) □_{D(n)}F_{2}-comodules, then the image of ỹ in the target is also invariant. Hence ỹ represents an element at the E_{2}-term of the spectral sequence. We claim that, when j is large enough, ỹ is a permanent cycle.

Suppose that ỹ ∈ Ext_{D(n)}^{p,q}(X, X); then it gives a class in E_{2}^{0,p,q}. The rth differential on this class would lie in E_{r}^{r,p−r+1,q}. So we only have to check that for some j, this group is above the vanishing plane for all r ≥ 2, and hence zero. We check our inequality:

2^{j}|ξ_{n}|r + (p − r + 1) − q + j_{0} − 2^{j}|ξ_{n}| >^{?} 0.

Whatever p, q, and j_{0} are, we can choose j large enough so that this holds for all r ≥ 2. Hence ỹ is a permanent cycle in the spectral sequence for X ∧ W_{j} for j ≫ 0; it obviously cannot support a differential, so it survives to give a nonzero class at E_{∞}. Since the resulting self-map of X over D(n − 1) restricts to the y-map ỹ over D(n), then it is a y-map over D(n − 1). Now we consider the map

[X, HD(n − 1) ∧ X]_{∗∗} ⟶ [X, HD(n) ∧ X]_{∗∗}.

This is a map of A-comodule algebras which detects nilpotence. By assumption, the element ỹ in the codomain is invariant under the A-coaction, and we know that its lift is a y-map, and hence some power of it is central. Lemma 5.2.7 says that some further power is invariant under the A-coaction, as desired. This completes the inductive step, and with it, the proof of Theorem 6.1.3.

Remark 6.4.2. We have used the nilpotence theorem part of the “Quillen stratification” theorem 5.1.2 (i.e., we have used Theorem 5.1.5) in the proof of Theorems 6.2.4 and 6.1.3. We have not used the other part of Theorem 5.1.2— that some power of any invariant element in HD_{∗∗} lifts to π_{∗∗}S^{0}. This in fact follows from the periodicity theorem, so it gives us an alternate proof of that part of Theorem 5.1.2.

6.5. Computation of some invariants in HD_{∗∗}

Theorem 5.1.3 gives an F -isomorphism

π_{∗∗}S^{0} ⟶ (\underleftarrow{lim}_{𝒬}HE_{∗∗})^{A}.

We compute \underleftarrow{lim} HE_{∗∗} in Proposition 6.5.1 below; in Proposition 6.5.6 we give a formula for the coaction of A on this inverse limit; and then we give a few examples of invariant elements. The computations in this section first appeared in [Pal99].

6.5. COMPUTATION OF SOME INVARIANTS IN HD_{∗∗} 135

At the prime 2, the maximal quasi-elementary quotient Hopf algebras of A are the E(m), m ≥ 0. We recall from Corollary 2.1.8 and Proposition 2.1.9 their definition and the computation of their coefficient rings:

E(m) = A/(ξ_{1}, …, ξ_{m}, ξ_{m+1}^{2^{m+1}}, ξ_{m+2}^{2^{m+1}}, ξ_{m+3}^{2^{m+1}}, …),
HE(m)_{∗∗} = F_{2}[h_{ts} | t ≥ m + 1, 0 ≤ s ≤ m].

The bidegrees of the polynomial generators are given by |h_{ts}| = (1, |ξ_{t}^{2^{s}}|). Since it is easy to see the effects of the maps in 𝒬 on the coefficients, we immediately have the following.

Proposition 6.5.1. There is an algebra isomorphism

\underleftarrow{lim} HE_{∗∗} ⟶^{≅} F_{2}[h_{ts} | 0 ≤ s < t] / (h_{ts}h_{vu} | u ≥ t),

where |h_{ts}| = (1, |ξ_{t}^{2^{s}}|). Hence there is an F -isomorphism

HD_{∗∗} ⟶^{F -iso} F_{2}[h_{ts} | 0 ≤ s < t] / (h_{ts}h_{vu} | u ≥ t).

The second sentence follows from Proposition 5.3.4, which says that there is an F -isomorphism HD_{∗∗} ⟶ \underleftarrow{lim} HE_{∗∗}.

We want to describe the coaction of A on \underleftarrow{lim} HE_{∗∗}. The coaction of A □_{E(m)}F_{2} on HE(m)_{∗∗} is determined by the coaction on the polynomial generators h_{ts}. Furthermore, since the coaction preserves homological degree—HE(m)_{i∗} is an Acomodule for each i—then the coaction on each h_{ts} only involves individual generators h_{ij}, not monomials.

Proposition 6.5.2. Fix m ≥ 0. Let χ: A ⟶ A denote the conjugation map of A, and for n ≥ 1 let ζ_{n} = χ(ξ_{n}). Let ξ_{0} = ζ_{0} = 1. Under the coaction map

HE(m)_{∗∗} ⟶ (A □_{E(m)}F_{2}) ⊗ HE(m)_{∗∗},

we have

h_{ts} ⟼ ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} ζ_{j}^{2^{s}}ξ_{t−i−j}^{2^{i+j+s}} ⊗ h_{i,j+s}.

The ζ_{j} part comes, essentially, from the right coaction of A on itself, while the ξ_{t−i−j} comes from the left coaction. See Lemma 6.5.5 for a few useful facts about χ and the ζ_{n}’s.

Proof. (We assume that t > m + 1 and that s < m; the special cases when t = m + 1 or s = m are even easier to deal with.) First we find all of the terms in the coaction on h_{ts} of the form a ⊗ h_{ij} for a ∈ A primitive, i.e., with a = ξ_{1}^{2^{n}} for some n. For degree reasons, the only possible such terms are ξ_{1}^{2^{s}} ⊗ h_{t−1,s+1} and ξ_{1}^{2^{s+t−1}} ⊗ h_{t−1,s}. By computations as in the proofs of Lemmas A.3 and A.5 of [Pal96b], we see that both of these terms do appear; that is, we can see that

h_{ts} ⟼ 1 ⊗ h_{ts} + ζ_{1}^{2^{s}} ⊗ h_{t−1,s+1} + ξ_{1}^{2^{s+t−1}} ⊗ h_{t−1,s} + other terms,

where the “other terms” are of the form b ⊗ h_{ij}, with b ∈ A non-primitive. (We have also used Lemma 6.5.5(a) to replace ξ_{1}^{2^{s}} with ζ_{1}^{2^{s}}.) So the formula given in the proposition is “correct on the primitives”; once we have verified that the formula is co-associative, we will have finished the proof. This verification is a straight-forward (although slightly messy) computation; it could be left to the diligent reader, but due to lack of space constraints, we include it in Lemma 6.5.3 below. □

Lemma 6.5.3. The formula

h_{ts} ⟼ ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} ζ_{j}^{2^{s}}ξ_{t−i−j}^{2^{i+j+s}} ⊗ h_{i,j+s}.

defines a coassociative coaction of A on HE(m)_{∗∗}.

Proof. We write Δ for the coproduct on A, and we write ψ for the coaction map of A on HE(m)_{∗∗}. We have to verify that (Δ ⊗ 1) ∘ ψ = (1 ⊗ ψ) ∘ ψ, so we compute both of these on h_{ts}. Lemma 6.5.5(b) tells us what Δ(ζ_{n}) is; using that, we compute (Δ ⊗ 1) ∘ ψ(h_{ts}):

(Δ ⊗ 1)(ψ(h_{ts})) = (Δ ⊗ 1) ( ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} ζ_{j}^{2^{s}}ξ_{t−i−j}^{2^{i+j+s}} ⊗ h_{i,j+s})

= ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} Δ(ζ_{j}^{2^{s}})Δ(ξ_{t−i−j}^{2^{i+j+s}}) ⊗ h_{i,j+s}

= ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} (∑_{n=0}^{j} ζ_{n}^{2^{s}} ⊗ ζ_{j−n}^{2^{s+n}}) ( ∑_{q=0}^{t−i−j} ξ_{t−i−j−q}^{2^{i+j+s+q}} ⊗ ξ_{q}^{2^{i+j+s}}) ⊗ h_{i,j+s}

= ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} ∑_{n=0}^{j} ∑_{q=0}^{t−i−j} ζ_{n}^{2^{s}}ξ_{t−i−j−q}^{2^{i+j+s+q}} ⊗ ζ_{j−n}^{2^{s+n}}ξ_{q}^{2^{i+j+s}} ⊗ h_{i,j+s}. (6.5.4)

On the other hand, for (1 ⊗ ψ) ∘ ψ(h_{ts}) we have

(1 ⊗ ψ)(ψ(h_{ts})) = (1 ⊗ ψ)( ∑_{ℓ=0}^{m−s} ∑_{k=m+1}^{t−ℓ} ζ_{ℓ}^{2^{s}}ξ_{t−k−ℓ}^{2^{k+ℓ+s}} ⊗ h_{k,ℓ+s})

= ∑_{ℓ=0}^{m−s} ∑_{k=m+1}^{t−ℓ} ζ_{ℓ}^{2^{s}}ξ_{t−k−ℓ}^{2^{k+ℓ+s}} ⊗ ψ(h_{k,ℓ+s})

= ∑_{ℓ=0}^{m−s} ∑_{k=m+1}^{t−ℓ} ζ_{ℓ}^{2^{s}}ξ_{t−k−ℓ}^{2^{k+ℓ+s}} ⊗ ∑_{b=0}^{m−ℓ−s} ∑_{a=m+1}^{k−b} ζ_{b}^{2^{ℓ+s}}ξ_{k−a−b}^{2^{a+b+ℓ+s}} ⊗ h_{a,b+ℓ+s}

= ∑_{ℓ=0}^{m−s} ∑_{k=m+1}^{t−ℓ} ∑_{b=0}^{m−ℓ−s} ∑_{a=m+1}^{k−b} ζ_{ℓ}^{2^{s}}ξ_{t−k−ℓ}^{2^{k+ℓ+s}} ⊗ ζ_{b}^{2^{ℓ+s}}ξ_{k−a−b}^{2^{a+b+ℓ+s}} ⊗ h_{a,b+ℓ+s}.

Now we replace a and b with the variables i = a and j = b + ℓ, and interchange the order of summation:

= ∑_{j=0}^{m−s} ∑_{i=m+1}^{t−j} ∑_{ℓ=0}^{j} ∑_{k=i+j−ℓ}^{t−ℓ} ζ_{ℓ}^{2^{s}}ξ_{t−k−ℓ}^{2^{k+ℓ+s}} ⊗ ζ_{j−ℓ}^{2^{ℓ+s}}ξ_{k+ℓ−i−j}^{2^{i+j+s}} ⊗ h_{i,j+s}.

Finally, we let n = ℓ and q = k − i − j + ℓ = k − i − j + n, to get the formula in (6.5.4). This finishes the proof. □

We have used the following facts about the conjugation map χ. See [Mil58, Lemma 10] for a formula for ζ_{n} in terms of the ξ_{j}’s.

Lemma 6.5.5. As above, we write ζ_{n} for χ(ξ_{n}).

(a) For any j, ξ_{1}^{j} = ζ_{1}^{j}.

(b) For any n, Δ(ζ_{n}) = ∑_{i=0}^{n} ζ_{i} ⊗ ζ_{n−i}^{2^{i}}, where ζ_{0} = 1.

Proof. Part (a): Since F_{2}[ξ_{1}] is a sub-Hopf algebra of A, then one can compute χ(ξ_{1}^{j}) in that sub-Hopf algebra, and the results will hold in A. We conclude that ξ_{1}^{j} = ζ_{1}^{j} for any j.

Part (b): The conjugation map χ is an anti-homomorphism—see [Swe69, Proposition 4.0.1]; in other words, Δ∘χ = (χ⊗χ)∘T ∘Δ, where T : A⊗A ⟶ A⊗A is the twist map, T (a ⊗ b) = b ⊗ a. So the formula for Δ(ξ_{n}) gives the formula for Δ(ζ_{n}). □

Since the E(m)’s are the maximal quasi-elementary quotients, the formula in Proposition 6.5.2 for the coaction of A on HE(m)_{∗∗} yields the following corollary.

Proposition 6.5.6. Under the coaction map

\underleftarrow{lim}_{𝒬}HE_{∗∗} ⟶ A ⊗ \underleftarrow{lim}_{𝒬}HE_{∗∗},

we have

h_{ts} ⟼ ∑_{j=0}^{⌊ \frac{s+t−1}{2} ⌋} ∑_{i=j+s+1}^{t−j} ζ_{j}^{2^{s}}ξ_{t−i−j}^{2^{i+j+s}} ⊗ h_{i,j+s}.

Milnor’s formula for the conjugation map χ in [Mil58, Lemma 10] says that χ(ξ_{n}) always has a summand of the form ξ_{1}^{2^{n}−1}, so we can pick out all of the terms of the form ξ_{1}^{k} ⊗ h_{i,j} to translate this formula into an action of A^{∗} on \underleftarrow{lim} HE_{∗∗}:

Sq^{n}(h_{ts)} = \begin{cases}h_{t−j,s+j} & if n = 2^{s+j} − 2^{s} and s + j < t − j, \\ h_{t−j−1,s+j} & if n = 2^{s+t−1} + 2^{s+j} − 2^{s} and s + j < t − j − 1, \\ 0 & otherwise. \end{cases}

We give a graphical depiction of the (co)action in Figure 6.5.A, in which we indicate the coaction by the primitives (i.e., the 2^{n}th powers of ξ_{1}).

We end the section with some examples. Example 6.5.7. Let R denote the ring \underleftarrow{lim} HE_{∗∗}.

(a) The element h_{t,t−1} ∈ R in bidegree (1, |ξ_{t}^{2^{t−1}}|) = (1, 2^{t−1}(2^{t} − 1)) is an invariant for all t ≥ 1. Indeed, we know that h_{10} lifts to an element of the same name in π_{1,1}S^{0}; also h_{21}^{4} lifts to an element in π_{4,24}S^{0} (the element known as g or \overline{κ}—see [Zac67]). We do not know which power of h_{t,t−1} lifts for t ≥ 3.

(b) We have some families of invariants. h_{20} ∈ R is not invariant: we have

h_{20} ⟼ 1 ⊗ h_{20} + ξ_{1}^{2} ⊗ h_{10}.

But since h_{10}h_{21} = 0 in R, the monomial h_{20}^{i}h_{21}^{j} is invariant for all i ≥ 0 and j ≥ 1. It turns out that more of these elements lift to π_{∗∗}S^{0} than one might expect from Theorem 5.1.2: the elements in the “Mahowald-Tangora wedge” [MT68] are lifts of the elements h_{20}^{i}h_{21}^{j} for all i ≥ 0 and j ≥ 8. (See [MPT71]; Zachariou [Zac67] first verified this for elements of the form h_{20}^{2i}h_{21}^{2(i+j)} for i, j ≥ 0.) These elements are distributed over a wedge between lines of slope \frac{1}{2} and \frac{1}{5} (in the Adams spectral sequence (t − s, s) grading).

[figure: h_{10}; h_{20}; h_{21}; h_{30}; h_{31}; h_{32}; h_{40}; h_{41}; h_{42}; h_{43}; ⋯; ⋯; ⋯; ⋯; 1; 2; 3; 4; 3; 4; 5; 5; 6; 7; 0; 0; 0; 1; 1; 2]

Figure 6.5.A. Graphical depiction of coaction of A on \underleftarrow{lim} HE_{∗∗}.

An arrow labeled by k represents an action by Sq^{2^{k}} ∈ A^{∗}, or equivalently a “coaction” by ξ_{1}^{2^{k}} ∈ A—in other words, a term of the form ξ_{1}^{2^{k}} ⊗ (target) in the coaction on the source.

(c) Similarly, while h_{30} and h_{31} are not invariant, the monomials

{h_{30}^{i}h_{31}^{j}h_{32}^{k} | i ≥ 0, j ≥ 0, k ≥ 1}

are invariant elements. Hence some powers of them lift to π_{∗∗}S^{0}. We do not know what powers of them lift, but they will be distributed over a wedge between lines of slope \frac{1}{6} and \frac{1}{27}. Continuing in this pattern, we find that for n ≥ 1, we have sets of invariant elements

{h_{n0}^{i_{0}}h_{n1}^{i_{1}} … h_{n,n−1}^{i_{n−1}} | i_{0}, …, i_{n−2} ≥ 0, i_{n−1} ≥ 1}.

The lifts of these elements lie in a wedge between lines of slope \frac{1}{2^{n}−2} and \frac{1}{2^{n−1}(2^{n}−1)−1}. Hence the family of elements in the Mahowald-Tangora wedge is not a unique phenomenon—we have infinitely many such families, and when n ≥ 3 they give more than a lattice of points in π_{∗∗}S^{0}.

(d) Margolis, Priddy, and Tangora [MPT71, p. 46] have found some other non-nilpotent elements, such as x ∈ π_{10,63}S^{0} and B_{21} ∈ π_{10,69}S^{0}. These both come from the invariant

z = h_{40}^{2}h_{21}^{3} + h_{20}^{2}h_{21}^{2}h_{41} + h_{30}^{2}h_{21}h_{31}^{2} + h_{20}^{2}h_{31}^{3}

in R. While we do not know which power of z lifts to π_{∗∗}S^{0}, we do know that B_{21} maps to the product h_{20}^{3}h_{21}^{2}z, and x maps to h_{20}^{5}z. In other words, we have at least found some elements in the ideal

(z) ⊆ (R)^{A}

which lift to π_{∗∗}S^{0}.

(e) Computer calculations have led us to a few other such “sporadic” invariant elements (i.e., invariant elements that do not belong to any family—any family that we know of, anyway):

h_{20}^{8}h_{31}^{4} + h_{30}^{8}h_{21}^{4} + h_{21}^{11}h_{31}

in bidegree (12, 80), an element in bidegree (9, 104) (a sum of 8 monomials in the variables h_{i,0} and h_{i,1}, 2 ≤ i ≤ 5), and an element in bidegree (13, 104) (a sum of 12 monomials in the same variables). We do not know what powers of these elements lift to π_{∗∗}S^{0}, nor are we aware of any elements in the ideal that they generate which are in the image of the restriction map from π_{∗∗}S^{0}.

6.6. Computation of a few Bousfield classes

In [HPS97, Section 5.1] we show that if a spectrum E is Bousfield equivalent to the sphere, then E detects nilpotence. This is true in any stable homotopy category, and it is not very deep. While we do not vouch for the depth of the nilpotence theorems of Chapter 5, we at least point out that they are not examples of this generic nilpotence theorem.

Many of the results in this section hold at all primes; some only hold at the prime 2. Unless otherwise indicated, we work at an arbitrary prime p. Bousfield classes are defined in Section 1.5.

Theorem 6.6.1. We have ⟨S^{0}⟩ > ⟨HD⟩, and when p = 2, ⟨HD⟩ > ⋁_{E}⟨HE⟩, where the wedge is taken over all quasi-elementary quotients E of A.

The proof is quite similar to Ravenel’s computations in [Rav84, Theorem 2.2 and Theorem 3.1] that

⟨S^{0}⟩ > ⟨BP ⟩ > ⋁_{n≥0}⟨K(n)⟩

in the ordinary stable homotopy category. We need a few preliminary results, first. Recall that I is the Brown-Comenetz duality functor, defined in Section 1.5. Here is the main lemma.

Lemma 6.6.2. Suppose that B and C are quotient Hopf algebras of A that fit into a Hopf algebra extension

B □_{C}F_{p} ↪ B ↠ C.

(a) Suppose that dim_{F_{p}} B □_{C}F_{p} = ∞. If B and C are conormal quotient Hopf algebras of A, then [HC, HB]_{∗∗} = 0. Hence HC ∧ IHB = 0; hence

⟨HB⟩ > ⟨HC⟩.

(b) Suppose that dim_{F_{p}} B □_{C}F_{p} < ∞.

(i) If B □_{C}F_{p} contains some τ_{n} or some ξ_{t}^{p^{s}} with s < t, then ⟨HB⟩ >

⟨HC⟩.
(ii) If not, and if p = 2, then ⟨HB⟩ = ⟨HC⟩.

Proof. Part (a): The statement that [HC, HB]_{∗∗} = 0 implies the rest of the lemma, by Proposition 1.5.1, so we only have to verify that. By Shapiro’s lemma (Example 1.2.4), there is an isomorphism

[HC, HB]_{∗∗} ≅ Ext_{B}^{∗∗}(A □_{C}F_{p}, F_{p}).

The proof that this Ext group is zero is somewhat technical, so we relegate it to Lemma 6.6.4 below.

Part (b)(i) follows from Corollary 1.5.2 and induction, using the non-nilpotence of the classes h_{ts} (when s < t), b_{ts} (when s < t), and v_{n}. Part (b)(ii) is similar, but uses the nilpotence of h_{ts} when s ≥ t: see Theorem B.2.1. □

We need to complete an Ext calculation to finish the proof of Lemma 6.6.2. To do this, it will be more convenient to work with modules over the dual A^{∗} of A, rather than with A-comodules. This is because we will be using duality, and while the vector space dual of a module is always a module, this is not in general true for comodules without some sort of finiteness hypotheses.

In particular, for any graded Hopf algebra Γ^{∗} over a field k, there is a duality functor

d: Γ^{∗}-Mod ⟶ Γ^{∗}-Mod,

defined by setting dM = Hom_{k}(M, k). The Γ^{∗}-action on dM is given by γf(−) = f(χ(γ) · −) for γ ∈ Γ^{∗} and f ∈ Hom_{k}(M, k), where χ: Γ^{∗} ⟶ Γ^{∗} is the conjugation map. More generally, there is a Γ^{∗}-action on Hom_{k}(M, N) for any Γ^{∗}-modules M and N—see the end of Section 0.3 for details. As noted after Definition 0.2.2, if Γ^{∗} is either commutative or cocommutative, then χ^{2} = 1. Hence in this case, if M is of finite type (i.e., if M is finite-dimensional in each degree), then ddM ≅ M.

We write Ext_{Γ^{∗}-Mod} for Ext in the category of Γ^{∗}-modules.

Lemma 6.6.3. If M and N are Γ^{∗}-modules, then there is an isomorphism

Ext_{Γ^{∗}-Mod}^{∗∗}(M, N) ≅ Ext_{Γ^{∗}-Mod}^{∗∗}(k, Hom_{k}(M, N)).

Proof. We compute Ext_{Γ^{∗}-Mod}^{∗∗}(M, N) by taking an injective resolution I_{•} of N and then mapping in M: in other words, we are interested in the cohomology of the cochain complex which is Hom_{Γ^{∗}-Mod}(M, I_{n}) in degree n. Of course, Hom_{k} and ⊗_{k} are adjoint, so viewing the M in the first entry as k ⊗ M, we have

Hom_{Γ^{∗}-Mod}(M, I_{n}) ≅ Hom_{Γ^{∗}-Mod}(k, Hom_{k}(M, I_{n})).

Indeed, the cochain complexes with these terms are isomorphic. The same adjointness shows that Hom_{k}(M, I_{n}) is an injective Γ^{∗}-module, and this expression is an exact functor in M; hence the cochain complex Hom_{k}(M, I_{•}) is an injective resolution of Hom_{k}(M, N). □

Here is the missing computation from the proof of Lemma 6.6.2.

Lemma 6.6.4. Suppose that B and C are conormal quotient Hopf algebras of A that fit into a Hopf algebra extension

B □_{C}F_{p} ↪ B ↠ C.

If dim_{F_{p}} B □_{C}F_{p} = ∞, then Ext_{B}^{∗∗}(A □_{C}F_{p}, F_{p}) = 0.

Proof. Lemma 0.3.3 tells us that there is an inclusion functor J: A-Comod ⟶ A^{∗}-Mod. Writing Ext_{B^{∗}-Mod} for Ext in the category of B^{∗}-modules, we want to show that

Ext_{B^{∗}-Mod}^{∗∗}(J(A □_{C}F_{p}), F_{p}) = 0.

One can see that J(A □_{C}F_{p}) = d(A^{∗}//C^{∗}), where A^{∗}//C^{∗} = A^{∗} ⊗_{C^{∗}} F_{p}. Given an extension of cocommutative Hopf algebras over F_{p}

C^{∗} ↪ B^{∗} ↠ B^{∗}//C^{∗},

and given a B^{∗}-module M, there is a spectral sequence, the change-of-rings spectral sequence, with E_{2}-term

E_{2}^{∗∗∗} ≅ Ext_{B^{∗}//C^{∗}-Mod}^{∗∗}(F_{p}, Ext_{C^{∗}-Mod}^{∗∗}(F_{p}, M)),

converging (strongly) to Ext_{B^{∗}-Mod}^{∗∗}(F_{p}, M). This is the same as the spectral sequence described in Section 1.4 but in the module setting; this module version is more standard and has been described elsewhere, such as [Sin73, II, §5] and [Ben91b, Section 3.5].

We are interested in Ext_{B^{∗}-Mod}^{∗∗}(d(A^{∗}//C^{∗}), F_{p}); since A^{∗}//C^{∗} is of finite type, then this Ext group is isomorphic to Ext_{B^{∗}-Mod}^{∗∗}(F_{p}, A^{∗}//C^{∗}). The change-of-rings spectral sequence for this has E_{2}-term equal to

Ext_{B^{∗}//C^{∗}-Mod}^{∗∗}(F_{p}, Ext_{C^{∗}-Mod}^{∗∗}(F_{p}, A^{∗}//C^{∗})).

Since C is a conormal quotient of A, then C^{∗} is a normal sub-Hopf algebra of A^{∗}; hence A^{∗}//C^{∗} is a trivial C^{∗}-module, and the E_{2}-term is isomorphic to

Ext_{B^{∗}//C^{∗}-Mod}^{∗∗}(F_{p}, A^{∗}//C^{∗} ⊗ Ext_{C^{∗}-Mod}^{∗∗}(F_{p}, F_{p})).

Now, B^{∗}//C^{∗} is a sub-Hopf algebra of A^{∗}//C^{∗}; hence A^{∗}//C^{∗} is a free module over B^{∗}//C^{∗}, by a theorem of Milnor and Moore [MM65]. The dual of Lemma 0.3.2 (or equivalently, [Mar83, Proposition 12.4]) tells us that A^{∗}//C^{∗} ⊗ Ext_{C^{∗}-Mod}^{∗∗}(F_{p}, F_{p}) is also free over B^{∗}//C^{∗}; therefore, the E_{2}-term is isomorphic to

Hom_{B^{∗}//C^{∗}-Mod}(F_{p}, free B^{∗}//C^{∗}-module).

So it suffices to show that

Hom_{B^{∗}//C^{∗}-Mod}(F_{p}, B^{∗}//C^{∗}) = 0.

Dualizing the module B^{∗}//C^{∗} and switching back to the world of comodules, we need to show that

Hom_{B □_{C}F_{p}} (B □_{C}F_{p}, F_{p}) = 0.

This is where the infinite-dimensionality of B □_{C}F_{p} comes into play. We put the details in Lemma 6.6.5. □

Lemma 6.6.5. Suppose B and C are conormal quotient Hopf algebras of A, with B ↠ C. If dim_{F_{p}} B □_{C}F_{p} = ∞, then

Hom_{B □_{C}F_{p}} (B □_{C}F_{p}, F_{p}) = 0.

Proof. First, for any B □_{C}F_{p}-comodule M and any comodule map f: M ⟶ F_{p}, the kernel of f contains all elements of M which appear in the coproducts of other elements. We claim that when M = B □_{C}F_{p}, the kernel must be all of M, so the map f must be zero.

Fix a nonzero element x of B □_{C}F_{p}; hence x is a polynomial in the ξ_{t}’s (and τ_{n}’s, if p is odd). If we can find an element ξ_{t}^{p^{s}} ∈ B □_{C}F_{p} which is not a factor of any term of x, then x appears in the coproduct of ξ_{t}^{p^{s}}x; thus our goal is to find such a ξ_{t}^{p^{s}}. Since dim_{F_{p}} B □_{C}F_{p} = ∞, the classification of (conormal) quotients of A in Theorem 2.1.1 tells us that there are two cases: either for all t ≫ 0, there is an s so that ξ_{t}^{2^{s}} ∈ B □_{C}F_{p}, or there is a t so that for all s ≫ 0, ξ_{t}^{p^{s}} ∈ B □_{C}F_{p}. In either case, we can find an element ξ_{t}^{p^{s}} so that |ξ_{t}^{p^{s}}| > |x|, and this does the job. □

Here are some consequences of Lemma 6.6.2(a).

Corollary 6.6.6. [HD, S^{0}]_{∗∗} = 0. Hence HD ∧ IS^{0} = 0; hence ⟨S^{0}⟩ > ⟨HD⟩.

Proof. Apply part (a) of the lemma with B = A and C = D. □

Corollary 6.6.7. Let p = 2 and let E be a quasi-elementary quotient of A. Then HE ∧ IHD = 0; hence ⟨HD⟩ > ⟨HE⟩.

Proof. When E is conormal, Lemma 6.6.2(a) says that [HE, HD]_{∗∗} = 0, so the corollary follows in that case. Since the maximal quasi-elementary quotient Hopf algebras of A are conormal quotients of both A and D, then if E is an arbitrary quasi-elementary quotient, there is a maximal quotient E′ with E′ ↠ E. This induces HE′ ⟶ HE, making HE a module spectrum over HE′. Hence Proposition 1.5.1 tells us that ⟨HE⟩ ≤ ⟨HE′⟩. □

Corollary 6.6.8. [A, HD]_{∗∗} = 0. Hence if X is a finite object in loc(A), then X = 0.

Proof. For the first statement, we apply the lemma with B = D and C = F_{p}. This then implies that [Y, HD]_{∗∗} = 0 for all Y in loc(A). If X is finite with Spanier-Whitehead dual DX, then [X, HD]_{∗∗} = HD_{∗∗}(DX). So it suffices to show that if X is finite and nontrivial, then HD_{∗∗}X ≠ 0. Well, HD_{∗∗}X = 0 if and only if X is contractible when viewed as a cochain complex of comodules over D, in which case the homology of X must be zero. On the other hand, by the Hurewicz Theorem 1.3.5, if X is finite and nontrivial, then it has nonzero homology. □

Proof of Theorem 6.6.1. It is clear (by Proposition 1.5.1(c), for instance) that

⟨S^{0}⟩ ≥ ⟨HD⟩ ≥ ⋁_{E quasi- ; elem.}⟨HE⟩

(where the second inequality is only known to be valid when p = 2). By Corollary 6.6.6, the first inequality is strict. When p = 2, Corollary 6.6.7 says that the second is, also. □

Remark 6.6.9. Let p = 2 and let X = ⋁ HE. One can in fact show a stronger result—that X has no complement in HD. Suppose otherwise: suppose that there were an object G so that ⟨HD⟩ = ⟨X⟩ ∨ ⟨G⟩ and 0 = X ∧ G. Smashing the former equality with IHD gives

⟨HD ∧ IHD⟩ = ⟨X ∧ IHD⟩ ∨ ⟨G ∧ IHD⟩,

But HE ∧ IHD = 0 for all quasi-elementary E by Corollary 6.6.7 and Proposition 1.5.1(b), so we have

⟨HD ∧ IHD⟩ = ⟨G ∧ IHD⟩.

Also, ⟨IHD⟩ ≤ ⟨HF_{p}⟩ ≤ ⟨X⟩ by Proposition 1.5.1(a), so we have

⟨HD ∧ IHD⟩ = ⟨G ∧ IHD⟩ ≤ ⟨G ∧ X⟩ = ⟨0⟩.

But HD ∧ IHD ≠ 0.

6.7. Ideals and thick subcategories

In this section, we examine a possible relationship between the ideals I(X) and a classification of thick subcategories of finite objects in Stable(A). Since this section consists primarily of conjectures, we work for the most part at an arbitrary prime p (although we have more evidence for the conjectures when p = 2). There are two subsections: the first one, which is the longer of the two, describes the thick subcategory conjecture and some related results and conjectures. The second discusses rank varieties, and is even more conjectural.

6.7.1. The thick subcategory conjecture. The Quillen stratification theorem 5.1.2 and the periodicity theorem 6.1.3 provide support for several conjectures about the “global structure” of the category Stable(A). For instance, we have the following suggested analogue of the result of Hopkins and Smith [HS98, Theorem 11], in which they identify the center of [X, X]_{∗} up to F -isomorphism, for any finite p-local spectrum X. Given a ring R, we let Z(R) denote the center of R.

Conjecture 6.7.1. For any finite spectrum X, there is an F -isomorphism

Z[X, X]_{∗∗} ⟶ (HD_{∗∗}/I(X))^{A} .

We should point out there is not even an obvious map between these two rings, but there is a diagram

[X, X]_{∗∗}
↓
(HD_{∗∗}/I(X))^{A} ⟶^{f} [X, HD ∧ X]_{∗∗}.

When p = 2, the periodicity theorem implies that large enough pth powers of elements in the image of the horizontal map f lift to the center of [X, X]_{∗∗}.

Theorem 6.1.3 also suggests a conjectured classification of thick subcategories of finite spectra in Stable(A); we spend most of this subsection discussing this conjecture and related ideas. We start with a finite-generation result for the ideal of a finite spectrum. Let R denote the ring

R = \underleftarrow{lim} HE_{∗∗} = F_{2}[h_{ts} | 0 ≤ s < t] / (h_{ts}h_{vu} | u ≥ t),

where |h_{ts}| = (1, |ξ_{t}^{2^{s}}|). Proposition 6.5.1 says that there is an F -isomorphism f: HD_{∗∗} ⟶ R. This gives a bijection between the radical ideals of the two rings HD_{∗∗} and R; given a radical ideal I ≤ HD_{∗∗}, we write f_{∗}I for the corresponding radical ideal in R.

Proposition 6.7.2. Let p = 2. If X is a finite object of Stable(D), then f_{∗}I(X) is a finitely generated ideal of R.

Since we only use this result as motivation for some conjectures, we sketch the proof. (To fill in the details would require a careful discussion of the Atiyah-Hirzebruch spectral sequence in the category Stable(A).)

Sketch of proof. To see that f_{∗}I(X) is finitely generated, note that since X is finite, then “most” of D acts trivially on X: as in Subsection 5.3.1, we let D_{n} denote the following quotient Hopf algebra of D:

D_{n} = D/(ξ_{1}, ξ_{2}, …, ξ_{n}) = F_{2}[ξ_{n+1}, ξ_{n+2}, …]/(ξ_{n+1}^{2^{n+1}}, ξ_{n+2}^{2^{n+2}}, …).

Then D_{n} should “act trivially on X” for n large enough. More precisely, there is an Atiyah-Hirzebruch spectral sequence with

E_{2} = (HF_{p})_{∗∗}X ⊗ (HD_{n})_{∗∗} ⇒ (HD_{n})_{∗∗}X.

Since X is finite, then its homology (HF_{p})_{∗∗}X is bounded. So if n is large enough, then for degree reasons, the elements in the image of the edge homomorphism

(HF_{p})_{∗∗}X ⟶ E_{2} = (HF_{p})_{∗∗}X ⊗ (HD_{n})_{00}

are all permanent cycles. This is a spectral sequence of modules over (HD_{n})_{∗∗}, so everything must be a permanent cycle, and we find that

(HD_{n})_{∗∗}X ≅ (HF_{p})_{∗∗}X ⊗ (HD_{n})_{∗∗}.

Consider the ring map HD_{∗∗} ⟶ (HD_{n})_{∗∗}. The annihilator in HD_{∗∗} of HD_{∗∗}X is contained in the annihilator in HD_{∗∗} of (HD_{n})_{∗∗}X. Since

ann_{(HD_{n})_{∗∗}}((HD_{n})_{∗∗}X) = (0),

then I(X) is contained in the radical of the kernel of HD_{∗∗} ⟶ (HD_{n})_{∗∗}. We can calculate this kernel, up to radical, by imitating the proof of Proposition 5.3.4; we find that we have an F -isomorphism

(HD_{n})_{∗∗} ⟶ F_{2}[h_{ts} | s < t, n < t] / (h_{ts}h_{vu} | u ≥ t),

and hence the radical of the kernel of HD_{∗∗} ⟶ (HD_{n})_{∗∗} maps to the ideal

K = (h_{ts} | s < t ≤ n)

in the Noetherian ring

F_{2}[h_{ts} | s < t ≤ n]/(h_{ts}h_{vu} | u ≥ t) ⊆ R.
□

We say that a radical ideal I of HD_{∗∗} is essentially finitely generated if the ideal f_{∗}I ≤ \underleftarrow{lim} HE_{∗∗} is finitely generated.

Now we state our main conjecture of this section.

Conjecture 6.7.3. The thick subcategories of finite spectra in Stable(A) are in one-to-one correspondence with the essentially finitely generated radical ideals of HD_{∗∗} which are invariant under the coaction of A □_{D}F_{p}.

The conjectured bijection should send an invariant ideal I to the full subcategory 𝒟(I) with objects

{X finite | I(X) ⊇ I}.

This is clearly a thick subcategory. The other arrow in the bijection should send a thick subcategory 𝒟 of the finite objects in Stable(A) to the ideal I(𝒟), defined by

I(𝒟) = ⋂_{X∈ob 𝒟} I(X).

This is an essentially finitely generated radical invariant ideal by Propositions 6.3.2 and 6.7.2.

Example 6.7.4. Here is a little evidence for Conjecture 6.7.3.

(a) By Example 6.5.7(a), we have maps

h_{10} : S^{1,1} ⟶ S^{0},
h_{21}^{4} : S^{4,24} ⟶ S^{0}.

We let S^{0}/h_{10} and S^{0}/h_{21}^{4} denote the cofibers of these. One can easily compute the ideals of these cofibers:

f_{∗}I(S^{0}/h_{10}) = \sqrt{(h_{10})},
f_{∗}I(S^{0}/h_{21}^{4}) = \sqrt{(h_{21})}.

Since h_{10}h_{21} = 0 in \underleftarrow{lim} HE_{∗∗}, then Conjecture 6.7.3 would tell us that

thick(S^{0}/h_{10}, S^{0}/h_{21}^{4}) = thick(S^{0}).

Let 𝒞 = thick(S^{0}/h_{10}, S^{0}/h_{21}^{4}). Clearly, 𝒞 ⊆ thick(S^{0}). One can also show that S^{0} is contained in 𝒞, using the octahedral axiom: the cofiber of the map h_{10}h_{21}^{4} : S^{5,25} ⟶ S^{0} fits into a cofibration with S^{0}/h_{10} and S^{0}/h_{21}^{4}, and hence is in 𝒞. But since h_{10}h_{21}^{4} is zero in π_{∗∗}S^{0}, then this cofiber is just S^{0} ∨ S^{4,25}; hence S^{0} is in 𝒞.

(b) By Example 6.5.7(b), we have non-nilpotent self-maps of the sphere spectrum called h_{20}^{i}h_{21}^{j}, for certain exponents i and j. If i and j are both positive, then f_{∗}I(S^{0}/(h_{20}^{i}h_{21}^{j})) = \sqrt{(h_{20}h_{21})}, so Conjecture 6.7.3 would imply that thick(S^{0}/(h_{20}^{i}h_{21}^{j})) is independent of i and j. Arguing as in part (a), one can see that this is true.

(c) Lastly, we point out that thick(S^{0}) ≠ thick(S^{0}/(h_{20}^{i}h_{21}^{j})). For instance, if we let d_{0} = h_{20}^{2}h_{21}^{2}, then S^{0}/d_{0} ∧ d_{0}^{−1}S^{0} = 0, and hence X ∧ d_{0}^{−1}S^{0} = 0 for every X in thick(S^{0}/d_{0}). Note that for degree reasons, d_{0} induces the zero map on P_{t}^{s}-homology for every s and t; hence no P_{t}^{s}-homology theory can distinguish between the spectra S^{0}/d_{0} and S^{0}∨S^{0}. Therefore P_{t}^{s}-homology groups are not a fine enough invariant to detect thick subcategories. See Section 6.8 for related ideas.

Here is a sketch of part of the proof of Conjecture 6.7.3. Conjecture 6.7.5. Given any invariant essentially finitely generated radical

ideal I of HD_{∗∗}, there is a finite spectrum X so that I(X) = I.

Idea of proof. Since the ideal f_{∗}I ≤ \underleftarrow{lim} HE_{∗∗} is finitely generated, we can choose generators for it. After raising them to powers, if necessary, we pull those generators back to elements y_{1}, y_{2}, …, y_{n} ∈ I, with y_{k} ∈ HD_{i_{k},j_{k}}. We order these so that i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}; then (y_{1}, …, y_{k}) is invariant for each k ≤ n. For each k ≥ 0, we want to define spectra X_{k} inductively so that I(X_{k}) = (y_{1}, …, y_{k}). We start by letting X_{0} = S^{0}; then I(X_{0}) = \sqrt{(0)}. Given X_{k−1} with I(X_{k−1}) = (y_{1}, …, y_{k−1}), Theorem 6.1.3 tells us that then X_{k−1} has a y_{k}-map; we let X_{k} be the cofiber of this map. Clearly I(X_{k}) ⊇ (y_{1}, …, y_{k}); if we could prove equality, we would be done. □

This would show that the composite I ↦ 𝒟(I) ↦ I(𝒟(I) ) is the identity. For the composite 𝒟 ↦ I(𝒟) ↦ 𝒟(I(𝒟) ), one still needs to show that every thick subcategory 𝒟 is of the form 𝒟 = 𝒟(I) for some invariant ideal I.

In the category Stable(kG), where G is a finite p-group and k is an algebraically closed field of characteristic p, there are two interesting bijections: one is between thick subcategories of finite spectra and subsets of Proj π_{∗}S^{0} which are closed under specialization, where Proj π_{∗}S^{0} means the set of nonmaximal homogeneous prime ideals in the graded ring π_{∗}S^{0}. The other bijection is between Bousfield classes and all subsets of Proj π_{∗}S^{0}. (See [BCR97] and [HPb], as well as Section 6.9, for details.)

By analogy, one might conjecture that there is a bijection between the set of Bousfield classes in Stable(A) and the set of all radical ideals of HD_{∗∗}, but that seems a bit much to expect without further evidence. Indeed, the presence of Bousfield classes ⟨X⟩ with ⟨X⟩ < ⟨HF_{p}⟩, such as ⟨X⟩ = ⟨IS^{0}⟩, may be evidence against such a conjecture. Such Bousfield classes make Stable(A) look more similar to the ordinary stable homotopy category than to Stable(kG), despite the Quillen stratification theorem 5.1.3.

We end this subsection with the following result. This is related to Conjecture 6.7.5, and is also a generalization of Theorem 2.5.3, which describes the construction of the spectra F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}).

Theorem 6.7.6. Let p = 2, and fix an essentially finitely generated invariant radical ideal I ≤ HD_{∗∗}. Write I = (y_{1}, …, y_{m}), and order the generators so that (y_{1}, …, y_{i}) is invariant for each i ≤ m. For any integers k_{1}, …, k_{m}, there are integers j_{1}, …, j_{m} with k_{i} ≤ j_{i} for each i, so that there is a finite spectrum F = F (y_{1}^{j_{1}}, …, y_{m}^{j_{m}}) satisfying the following.

(a) When I = \sqrt{(0)}, then F = S^{0}.

(b) I is contained in the ideal of F.

(c) Hence if y ∈ HD_{∗∗}/I is invariant, then F has a y-map, u.

(d) F is self-dual, as are its y-maps.

(e) For any finite spectrum W with I ≤ I(W), then W is in thick(F).

(f) Hence the Bousfield class of F is independent of the choice of exponents j_{i}.

(g) Fix y and u as in (c). For any finite spectrum W and any y-map v on W, there are integers i and j so that u^{i} ∧ 1_{W} = 1_{F} ∧ v^{j} as self-maps of F ∧ W.

(h) Suppose that ℓ_{1}, …, ℓ_{m} are integers so that F (y_{1}^{ℓ_{1}}, …, y_{m}^{ℓ_{m}}) exists. If ℓ_{i} ≫ j_{i} for each i, then there is a map F (y_{1}^{ℓ_{1}}, …, y_{m}^{ℓ_{m}}) ⟶ F (y_{1}^{j_{1}}, …, y_{m}^{j_{m}}) commuting with projection to the top cell.

See Section 6.9 for an application of this theorem to the construction of certain finite localization functors.

6.7.2. Rank varieties. In this subsection we discuss another body of ideas, based on work of Nakano and the author [NP98] (and this was based, in turn, on work of Friedlander and Parshall [FP86, FP87], among others). One might want to work over the algebraically closed field \overline{F}_{p} instead of F_{p} in this subsection.

Let p = 2. We let W be the vector space

W = Span_{F_{2}}(P_{t}^{s} | s < t).

We view W as being an inhomogeneous sub-vector space of the Steenrod algebra A^{∗}, and we let V_{D}(F_{2}) be the following subset of W:

V_{D}(F_{2}) = {y ∈ W | y^{2} = 0}.

We do not require that the elements y be homogeneous. One can show (as in the proof of [NP98, Theorem 1.7]) that V_{D}(F_{2}) consists precisely of linear combinations of commuting P_{t}^{s}’s—i.e., it is a union of affine spaces, one such space for each maximal elementary quotient of A. Therefore it is “equal” to the prime ideal spectrum of \underleftarrow{lim}_{𝒬} HE_{∗∗}, and hence is homeomorphic to the prime ideal spectrum of HD_{∗∗}. At odd primes, we define

W_{ev} = Span_{F_{p}}(P_{t}^{s} | s < t),
W_{odd} = Span_{F_{p}}(Q_{n} | n ≥ 0),

and then let V_{D}(F_{p}) be the set of all elements y = (y_{1}, y_{2}) in W_{ev} ⊕ W_{odd} with y_{1}^{p} = 0 and y_{2}^{2} = 0. We do not have as nice a description of V_{D}(F_{p}) at odd primes; because of this, and for other technical reasons, it might be best to restrict the following discussion to the case p = 2.

Given an element y ∈ V_{D}(F_{p}), we can construct the y-homology spectrum H(y), just as we did the P_{t}^{s}-homology spectrum in Definition 2.2.1. Since y need not be homogeneous, H(y) has only one natural grading, the homological grading. Given a spectrum X, we define its rank variety, V_{D}(X), to be the following subset of V_{D}(F_{p}):

V_{D}(X) = {y ∈ V_{D}(F_{p}) | H(y)_{∗}X ≠ 0}.

We say that a subset V of V_{D}(F_{p}) is realizable if V = V_{D}(X) for some finite spectrum X.

Conjecture 6.7.7. Let X be a finite spectrum. The rank variety V_{D}(X) determines the thick subcategory generated by X. In other words, there is a bijection between the invariant essentially finitely generated radical ideals of HD_{∗∗} and the realizable subsets of V_{D}(F_{p}).

This bijection should come about by the following: if X is a finite spectrum, then there should be a homeomorphism (actually, an “inseparable isogeny”—see [Ben91b, p. 172]) between the prime ideal spectrum of HD_{∗∗}/I(X) and V_{D}(X).

6.8. Further discussion: slope supports

In this section, we discuss slope supports. This material was originally introduced in [Pal96b] as a P_{t}^{s}-homology approach to the periodicity theorem and the thick subcategory conjecture. Since those topics now seem more closely related to I(X), the ideal of X, slope supports appear to be more peripheral. (Because of this, and because the results in this section have appeared elsewhere, we only gives sketches of proofs.) On the other hand, since P_{t}^{s}-homology groups do determine vanishing lines and other useful information, studying slope supports may be worthwhile.

The two main results in this section are Proposition 6.8.2, which gives a classification of the possible supports of finite spectra; and Theorem 6.8.3, which uses the periodicity theorem 6.1.3 to produce a number of non-nilpotent self-maps of a finite spectrum.

Definition 6.8.1. Let p be a prime, and let Slopes denote the set of slopes of A (Notation 2.2.8). Given a spectrum X, we define its slope support to be the set

supp(X) = {n | Z(n)_{∗∗}X ≠ 0} ⊆ Slopes .

We let Slopes′ be the following set:

Slopes′ = \begin{cases}\{(t, s) | t > s ≥ 0\}, & p = 2, \\ \{(t, s) | t > s ≥ 0\} ∪ \{n | n ≥ 0\}, & p odd. \end{cases}

Then there is bijection between Slopes and Slopes′, with the slope ^{\frac{p|ξ_{t}^{p^{s}} |}{2}} ∈ Slopes corresponding to (t, s) ∈ Slopes′, and when p is odd, |τ_{n}| ∈ Slopes corresponding to n ∈ Slopes′. We say that a subset T of Slopes′ is admissible if T satisfies the following conditions:

(a) When p = 2:

(t, s) ∈ T ⇒ (t + 1, s) ∈ T,
(t, s) ∈ T ⇒ (t + 1, s − 1) ∈ T,
card(Slopes′ \ T) < ∞.

(b) When p is odd: the above conditions, as well as:

n ∈ T ⇒ n + 1 ∈ T.

We call such sets “admissible” because they are the possible slope supports of finite spectra, at least at the prime 2.

Proposition 6.8.2. [Pal96b, Prop. 3.10 and Thm. A.1] For any prime p, if T ⊆ Slopes′ is admissible, then T is the slope support of some finite spectrum. If p = 2, then the converse holds: if X is any finite spectrum, then supp X is admissible.

We conjecture that the converse is also true when p is odd. We can at least prove the following: for X finite, if (Q_{n})_{∗∗}X = 0, then (Q_{n−1})_{∗∗}X = 0 (see [Pal96b, A.8]). We also note that there are no restrictions on the slope support of an arbitrary spectrum: the connective P_{t}^{s}-homology spectrum p_{t}^{s} has slope support equal to {(s, t)}, so any subset of Slopes′ may be realized as the support of a wedge of p_{t}^{s}’s (and similarly at odd primes, using the p_{t}^{s}’s and the q_{n}’s).

Sketch of proof. We include a sketch of the proof that every admissible T is the slope support of some finite spectrum, since the corresponding result in [Pal96b] assumed that p = 2. We focus on the case when p is odd; the reader can imitate this proof or refer to [Pal96b] for the p = 2 case.

Every admissible T ⊆ Slopes′ can be written as

T = T (t_{1}, s_{1}) ∩ ⋯ ∩ T (t_{n}, s_{n}) ∩ T (m)

for some numbers t_{i}, s_{i}, m, where T (t, s) is the largest admissible set not containing (t, s)—i.e., it is the complement of

{(t, s), (t − 1, s), (t − 1, s + 1), (t − 2, s), (t − 2, s + 1), (t − 2, s + 2), … }.

T (m) is the complement of {0, …, m}. See Figure 6.8.A. If we can find sufficiently nice finite spectra X(t, s) and X(m) so that supp(X(t, s)) = T (t, s) and supp(X(m)) = T (m), then we can realize any slope support T by the spectrum

X(t_{1}, s_{1}) ∧ ⋯ ∧ X(t_{n}, s_{n}) ∧ X(m).

One might worry that although Z(d)_{∗∗}X(t_{i}, s_{i}) ≠ 0 and Z(d)_{∗∗}X(t_{j}, s_{j}) ≠ 0 for some i, j, and d, one might have Z(d)_{∗∗}X(t_{i}, s_{i})∧X(t_{j}, s_{j}) = 0. This does not happen for injective resolutions of finite comodules—see [NP98, Corollary 1.5]—and it certainly does not happen for the examples we construct in the next paragraph. This is what we mean by “sufficiently nice.”

We recall from [Mit85] (see also Section 4.3) that the quotient Hopf algebra

P (n) = A/(ξ_{1}^{p^{n+1}}, ξ_{2}^{p^{n}}, …, ξ_{n+1}^{p}, ξ_{n+2}, ξ_{n+3}, …; τ_{0}, τ_{1}, τ_{2}, …)

[figure: s; t; T (t, s)]  m T (m)

Figure 6.8.A. T (t, s) and T (m) as subsets of Slopes′.

has the structure of an A-comodule, extending the P (n)-comodule structure. The same also holds for

V (m) = Λ[τ_{0}, …, τ_{m}].

We will use P (n) and V (m) to refer to the Hopf algebras, the comodules, or their injective resolutions, depending on the context. According to Theorem 2.3.6(c), H(P (n), P_{t}^{s}) is zero if and only if ξ_{t}^{p^{s}} ≠ 0 in P (n); the corresponding result holds for V (m). Hence as subsets of Slopes′, we have

supp P (n) = T (n + 1, 0),
supp V (m) = T (m).

Recall from Section 4.5 that Φ denotes the “pth power” functor on the category of A-comodules: if either p = 2 or M is evenly graded, then we define ΦM by

(ΦM)_{n} = \begin{cases}M_{r}, & if n = pr, \\ 0, & if n is not divisible by p, \end{cases}

with the comodule structure induced by the Frobenius on A. One can readily see that if s < t, then

supp Φ^{s}P (t − 1) = T (t, s).

Hence an injective resolution of the comodule

Φ^{s_{1}}P (t_{1} − 1) ⊗ ⋯ ⊗ Φ^{s_{n}}P (t_{n} − 1) ⊗ V (m)

has slope support contained in

T = T (t_{1}, s_{1}) ∩ ⋯ ∩ T (t_{n}, s_{n}) ∩ T (m).

By a Poincaré series argument, one can check that this is exactly the slope support. See [Pal96b, A.1] for the proof of the converse when p = 2. □

Recall from Notations 2.2.8 and 2.4.1 that for each slope n, there is a nonnilpotent element u_{n} ∈ Z(n)_{∗∗}. There is a ring map HD_{∗∗} ⟶ Z(n)_{∗∗}, and Proposition 6.5.1 implies that some power of u_{n} lifts to HD_{∗∗}. Hence we can consider the existence of u_{n}-maps, à la Definition 6.1.2: a u_{n}-map is a map z: X ⟶ X so that z maps to u_{n}^{j} ∧ 1_{X} in [Z(n), Z(n) ∧ X]_{∗∗}. Arguing as in Remark 6.2.2(a), one can see that this is the same as the definition of u_{n}-map given in Definition 2.4.2.

This first appeared for modules as [Pal96b, Theorem 4.1]. The proof uses the A-comodule \frac{1}{2}A(n), described in Section 4.3.

Theorem 6.8.3. Let p = 2. Let X be a finite spectrum and let T ⊆ Slopes be the slope support of X. If n ∈ T and T \ {n} is admissible (viewed as a subset of Slopes′), then there is a non-nilpotent u_{n}-map in [X, X]_{∗∗}.

Sketch of proof. Theorem 6.2.4 tells us that the property of having a u_{n}- map is generic, so it suffices to find a finite spectrum Y with two properties: Y has a u_{n}-map, and the thick subcategory generated by Y contains all X with supp(X) = T as in the statement.

Let P_{t}^{s} = Z(n). We claim that Y = Φ^{s} \frac{1}{2}A(t − 1) has the required properties. First of all, in the terminology used in the proof of Proposition 6.8.2, the slope support of Y is supp(Y) = T (t, s) \ {(t, s)}. Proposition 4.5.6 describes a map Σ^{1} \frac{1}{2}A(t − 1) ⟶ \frac{1}{2}A(t − 1), and one can check that this is an h_{t,0}-map; applying Φ^{s} turns it into an h_{ts}-map.

Theorem 4.5.1 says that the thick subcategory generated by \frac{1}{2}A(t − 1) contains all finite spectra Z with Z(d)_{∗∗}Z = 0 for all d < |ξ_{t}|; the proof uses vanishing lines to conclude this. In our setting, we replace the vanishing line argument with an application of the nilpotence theorem 5.1.5, to conclude that thick(Φ^{s} \frac{1}{2}A(t − 1)) contains all finite spectra with support as described. □

For example, when X = S^{0}, this gives the maps described in Example 6.5.7(a). This result gives incomplete information—it does not give all of the non-nilpotent self-maps of X—but since Z(n)-homology groups are easier to compute than the ring (HD_{∗∗}/I(X))^{A}, it is perhaps more accessible than the full periodicity theorem.

6.9. Further discussion: miscellany

We have already mentioned several conjectures related to global structure of the category Stable(A) when p = 2—see Section 6.7. We note that some of these issues have been resolved by Hovey and the author for finite-dimensional quotient Hopf algebras of A in [HPb] and [HPa]. In [HPb], we classify the thick subcategories and Bousfield classes in the category Stable(B) for any finite-dimensional quotient B of A ⊗_{F_{2}} \overline{F}_{2}, where \overline{F}_{2} is the algebraic closure of F_{2}. In [HPa], we show how to translate a classification of thick subcategories over an algebraically closed field to a similar result over a finite field. Combining the main results of the two papers, we get this result: if B is a finite-dimensional quotient of A, then there is a bijection between thick subcategories of finite objects in Stable(B) and subsets of Proj π_{∗∗}S^{0} which are closed under specialization. Here Proj means the set of nonmaximal bihomogeneous prime ideals, and a set T of primes is closed under specialization if p ∈ T and p ⊆ q implies q ∈ T.

As far as getting results like Theorem 6.1.3 at odd primes, one has to prove analogues of Theorems 5.1.2 and 5.1.5 first; see Section 5.4 for a discussion of those issues.

We presented some preliminary computations of A-invariants in HD_{∗∗} in Section 6.5; it would be nice to have further results. Extensive computer calculations could be useful, and obviously it would be nice to find new families of invariants. Along similar lines, we could use more information about invariant ideals of HD_{∗∗}— basic properties, examples, and of course a classification would be helpful.

It is also natural to wonder if one can prove a version of the periodicity theorem 6.1.3 which uses quasi-elementary quotients of A instead of D.

Finally, we point out that one can use the periodicity theorem 6.1.3 to generalize the “killing construction” given in Theorem 3.1.6. Theorem 3.1.6 was proved using the finite spectra F (u_{1}^{j_{1}}, …, u_{n}^{j_{n}}), which were constructed in Theorem 2.5.3 so as to have specific vanishing properties with respect to P_{t}^{s}-homology groups. We generalize this by using Theorem 6.7.6 and the finite spectra constructed there. Let I be a finitely generated invariant ideal of HD_{∗∗}, and let 𝒞 be the thick subcategory of finite spectra X with I(X) ⊇ I. Then we have a finite localization functor L_{I}^{f} and a cofibration

C_{I}^{f}X ⟶^{f} X ⟶^{g} L_{I}^{f}X.

If the ideal I is generated by classes u_{d_{i}}, then Z(d)_{∗∗}f is an isomorphism for d ∉ {d_{i}}, and Z(d)_{∗∗}g is an isomorphism if d ∈ {d_{i}}. The analogue of Proposition 3.1.3 holds. Our proof of Theorem 3.1.6(b), on the other hand, does not work in this situation; it may be that these finite type and connectivity results only hold when I is as in that theorem, i.e., when I is the radical ideal generated by classes {u_{d} | d ≤ n}, for some n.

APPENDIX A An underlying model category

Let Γ be a graded commutative Hopf algebra over a field k. In this section we (briefly) describe a model category whose associated homotopy category is equivalent to Stable(Γ). The main results here (Theorems A.1.3 and A.1.4) are due to Hovey; see [Hov99] for the details.

Recall that Quillen [Qui67] defined the notion of a closed model category; this is a category 𝒞, like the category of topological spaces, in which one has a notion of a well-behaved homotopy relation between maps. Briefly, a closed model structure is determined by specifying three classes of maps—weak equivalences, fibrations, and cofibrations—satisfying certain properties. See [Qui67], as well as [DS95] and [Hov99], for details. Given this structure, one can define a new category Ho 𝒞, the associated homotopy category, in which one has inverted the weak equivalences. Rather than work with the homotopy category, it is often more convenient to work with an equivalent category, in which the objects are certain well-behaved objects of 𝒞 (the objects which are both “fibrant” and “cofibrant”), and the morphisms are homotopy classes of maps.

These days, by the way, one often says “model category” rather than “closed model category.”

Model categories are useful because, while one can do many constructions working entirely in a homotopy category, for certain more delicate operations one needs to work at the “point-set” level—i.e., in the model category. For example, while Adams’ definition in [Ada74] of the homotopy category of spectra is extremely useful, there are constructions one cannot do unless one has a model category underlying it. Hence today one has various definitions of model categories of spectra, each with its own advantages and disadvantages.

In our work here, we have not needed a model category underlying Stable(Γ). Nonetheless, it seems like a good idea, pedagogically and for future applications, to set one up.

We assume that Γ is a graded commutative Hopf algebra over a field k. We let Ch(Γ) denote the category whose objects are cochain complexes of Γ-comodules (not necessarily injective ones), and whose morphisms are cochain maps. We put a model category structure on Ch(Γ); to do this, we need to specify the weak equivalences, the fibrations, and the cofibrations in Ch(Γ).

Notation A.1.1. Given cochain complexes X and Y, let [X, Y] denote the set of cochain homotopy classes of degree zero maps from X to Y. Given a Γ-comodule M, let S^{i}M denote the cochain complex which is M in degree i, and zero elsewhere. Let L(k) denote an injective resolution of the trivial module k, and let 𝒮 denote the set of simple comodules of Γ.

Given an integer i, M ∈ 𝒮, and X any object of Ch(Γ), we define π_{i}(X; M) (the “ith homotopy group of X with coefficients in M”) by

π_{i}(X; M) = [S^{i}M, L(k) ⊗ X].

We say that a map f: X ⟶ Y in Ch(Γ) is a weak equivalence if and only π_{i}(f; M) is an isomorphism for all integers i and all simple comodules M. We say that a map f: X ⟶ Y is a fibration if and only if each component f_{n}: X_{n} ⟶ Y_{n} of f is an epimorphism with injective kernel. We say that a map f: X ⟶ Y is a cofibration if and only if each component f_{n}: X_{n} ⟶ Y_{n} is a monomorphism.

Remark A.1.2. One can show (see [Hov99]) that a map f is a weak equivalence if and only if 1_{L(k)} ⊗ f is a cochain homotopy equivalence. Also, note that π_{i}(X; k) = [S^{i}k, L(k) ⊗ X] is the ith cohomology group of the cochain complex of primitives of L(k) ⊗ X; this is a useful alternate description of π_{∗}(−; k).

Theorem A.1.3. [Hov99] With weak equivalences, fibrations, and cofibrations defined as above, Ch(Γ) is a model category. Its associated homotopy category is equivalent to Stable(Γ).

Note that with this model structure, every object of Ch(Γ) is cofibrant.

One can in fact give Ch(Γ) the structure of a cofibrantly generated model category, as follows; we refer to [Hov99] for the proofs. Given a Γ-comodule M, we let D^{n}M denote the (contractible) cochain complex which is M in degrees n and n + 1, zero elsewhere, with differential given by the identity map:

⋯ ⟶ 0 ⟶ 0 ⟶ M ⟶^{1_{M}} M ⟶ 0 ⟶ 0 ⟶ · · · .

Both S^{n} (defined in Notation A.1.1) and D^{n} are functors from Ch(Γ) to itself. We let J denote the following set of maps in Ch(Γ):

J = {D^{n}f | n ∈ Z, f: M ↪ N an inclusion of finite-dimensional comodules}.

We define I by

I = J ∪ {S^{n+1}M ↪ D^{n}M | n ∈ Z, M simple},

where the map S^{n+1}M ↪ D^{n}M is the obvious inclusion.

Here is the second main theorem of this section.

Theorem A.1.4. [Hov99] With I and J defined as above, Ch(Γ) has the structure of a cofibrantly generated model category, in which I is the set of generating cofibrations and J is the set of generating trivial cofibrations.

APPENDIX B Steenrod operations and nilpotence in Ext_{Γ}^{∗∗}(k, k)

Let A be the dual of the mod p Steenrod algebra. In this appendix we recall a few results about Steenrod operations in the cohomology of any commutative Hopf algebra Γ over F_{p}, and we focus in particular on the case when Γ is a quotient Hopf algebra of A. Then we discuss the nilpotence of certain classes in Ext_{Γ}^{∗∗}(F_{p}, F_{p}), for use in proving the nilpotence theorems of Chapter 5.

B.1. Steenrod operations in Hopf algebra cohomology

In this short section we recall a few facts about Steenrod operations in Hopf algebra cohomology. May’s paper [May70, Section 11] is our basic reference; see also [Sin73], [Wil81], [Saw82], [BMMS86], and [Rav86] for related results.

Suppose that Γ is a commutative Hopf algebra over the field F_{p}. Then there are Steenrod operations acting on Ext_{Γ}^{∗∗}(F_{p}, F_{p})—as usual, Ext here is taken in the category of Γ-comodules. The operations are of this form:

• If p = 2, then for each n ≥ 0 there is an operation

S̃q^{n}: Ext_{Γ}^{s,t}(F_{2}, F_{2}) ⟶ Ext_{Γ}^{s+n,2t}(F_{2}, F_{2}).

• If p is odd, then for each n ≥ 0 there are operations

𝒫̃^{n} : Ext_{Γ}^{s,2t}(F_{p}, F_{p}) ⟶ Ext_{Γ}^{s+qn,2pt}(F_{p}, F_{p}),
β𝒫̃^{n} : Ext_{Γ}^{s,2t}(F_{p}, F_{p}) ⟶ Ext_{Γ}^{s+qn+1,2pt}(F_{p}, F_{p}).

Here, q = 2p − 2. Note that β𝒫̃^{n} is a single operation, not the composite of two operations.

These satisfy the usual properties of Steenrod operations: the Cartan formula, the Adem relations, and some instability conditions; for example, if x ∈ Ext_{Γ}^{s,t}(F_{p}, F_{p}) (with s and t even if p is odd), then

\begin{matrix}S̃q^{s}(x) = x^{2}, & p = 2, \\ 𝒫̃^{s/2} (x) = x^{p}, & p odd. \end{matrix}

Note that at odd primes, the operations are zero on classes in Ext_{Γ}^{s,t}(F_{p}, F_{p}) when t is odd. This is an artifact of the grading conventions on the operations. To remedy this, one can either use a different grading convention (see [May70] and [BMMS86, IV.2] for two different conventions), or one can define operations indexed by half-integers, as in [Rav86, Theorem A1.5.2].

156 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext_{Γ}^{∗∗}(k, k)

Example B.1.1. (a) Fix a commutative Hopf algebra Γ over F_{p}. Recall from Lemma 0.4.8 that Ext_{Γ}^{1}(F_{p}, F_{p}) is isomorphic to the vector space of primitives in Γ. If x is primitive, then we write [x] for the corresponding element of Ext. We have the following Steenrod operations: when p is odd, then 𝒫̃^{0} [x] = [x^{p}]. At the prime 2, S̃q^{0}[x] = [x^{2}]. One can compute 𝒫̃^{0} and S̃q^{0} on general Ext classes by a similar formula—these operations are induced by the following map on the cobar complex:

[x_{1}|x_{2}| ⋯ |x_{n}] ↦ [x_{1}^{p}|x_{2}^{p}| ⋯ |x_{n}^{p}].

See [May70, Proposition 11.10] for a proof.

(b) Recall that E[x] and D[x] are defined in Notation 0.5.1. If Γ = E[x], then Ext_{Γ}^{∗∗}(F_{p}, F_{p}) ≅ F_{p}[h], where h = [x]. If p = 2, then S̃q^{1}(h) = h^{2}, and S̃q^{n}(h) = 0 when n ≠ 1. If p is odd (in which case |x| must be odd), then all operations vanish on h (because of our grading conventions).

(c) If p is odd and Γ = D[x], then Ext_{Γ}^{∗∗}(F_{p}, F_{p}) ≅ Λ[h] ⊗ F_{p}[b], where h = [x] and b is the p-fold Massey product of h with itself. In this case, β𝒫̃^{0} (h) = b, and all other operations on h are zero; also, 𝒫̃^{1} (b) = b^{p}, and all other operations on b are zero.

B.2. Nilpotence in Ext over quotients of A: p = 2

In this section we recall a result of Lin’s which we use in the proofs of the main results of Chapter 5. Lin’s result first appeared as [Lin77a, Theorem 3.1 and Corollary 3.2]. Wilkerson [Wil81, Theorem 6.4] has also proved a related result.

Let A be the dual of the mod 2 Steenrod algebra. Recall from Notation 1.2.14 and Remark 2.1.3 that if ξ_{t}^{2^{s}} is primitive in a quotient Hopf algebra B of A, then we let h_{ts} = [ξ_{t}^{2^{s}}] denote the corresponding element of HB_{1,∗}.

Theorem B.2.1. Suppose that

B = A/(ξ_{1}^{2^{n_{1}}} , ξ_{2}^{2^{n_{2}}} , …)

is a quotient Hopf algebra of A so that for some integer m, the exponents n_{i} are finite when i = 1, 2, …, m − 1. Fix μ so that ξ_{m}^{2^{μ}} is primitive in B.

(a) If μ ≥ m, then h_{m,μ} is nilpotent in HB_{∗∗}.

(b) Fix an integer ℓ ≤ m, and suppose that for some λ, ξ_{ℓ}^{2^{λ}} is primitive in B. If ℓ ≤ μ, then h_{ℓ,λ}h_{m,μ} is nilpotent in HB_{∗∗}.

Note that if μ ≥ max(n_{1}, …, n_{m−1}), then ξ_{m}^{2^{μ}} will be primitive. Note also that part (a) is a corollary of part (b)—just set ℓ = m and λ = μ.

One example of this theorem is that the class h_{11} = [ξ_{1}^{2}] ∈ Ext_{A}^{1,2}(F_{2}, F_{2}) is nilpotent. The proof is easy, and demonstrates the techniques used by Lin to prove the general result: the (reduced) diagonal ξ_{2} ↦ ξ_{1}^{2} ⊗ ξ_{1} in A gives the relation h_{11}h_{10} = 0 in Ext_{A}^{∗∗}(F_{2}, F_{2}). Applying the Steenrod operation S̃q^{1} gives

h_{12}h_{10}^{2} + h_{11}^{3} = 0,

so multiplying through by h_{11} and using h_{11}h_{10} = 0 yields h_{11}^{4} = 0. Note that repeatedly applying S̃q^{0} to this relation gives h_{1n}^{4} = 0 for all n ≥ 1.

For our purposes, part (a) is one of the key ingredients in the proof that restriction to the quotient Hopf algebra D detects nilpotence. Part (b) is used in the classification of quasi-elementary quotients of A, essentially, and is also used in showing that restricting to the quasi-elementary quotients detects nilpotence.

B.3. NILPOTENCE IN EXT OVER QUOTIENTS OF A: p ODD 157

B.3. Nilpotence in Ext over quotients of A: p odd

In this section we discuss the odd-primary analogue of Theorem B.2.1.

Fix an odd prime p, and let A be the dual of the mod p Steenrod algebra. Recall from Remark 2.1.3 that if ξ_{t}^{p^{s}} is primitive in a quotient Hopf algebra B of A, then h_{ts} = [ξ_{t}^{p^{s}}] is the corresponding element of HB_{1,∗} = Ext_{B}^{1,∗}(F_{p}, F_{p}), and b_{ts} ∈ HB_{2,∗} is defined to be β𝒫̃^{0} (h_{ts}). Alternatively, b_{ts} is equal to the p-fold Massey product of h_{ts} with itself. If τ_{n} is primitive in B, we let v_{n} = [τ_{n}] be the corresponding element of HB_{1,∗}.

For convenience, we restate Conjecture 5.4.1. This would be the odd-primary analogue of Theorem B.2.1(a).

Conjecture B.3.1. Fix integers s and t. Suppose that

B = A/(ξ_{1}^{p^{n_{1}}} , ξ_{2}^{p^{n_{2}}} , …; τ_{0}^{e_{0}}, τ_{1}^{e_{1}}, …)

is a quotient Hopf algebra of A in which ξ_{t}^{p^{s}} is nonzero and primitive. If s ≥ t, then b_{ts} is nilpotent in HB_{∗∗}.

One can also consider a partial analogue of Theorem B.2.1(b): under conditions on s, t, and n, the product b_{ts}v_{n} is nilpotent.

Proposition B.3.2. Fix integers s, t, and n. Suppose that B is a quotient Hopf algebra of A in which ξ_{i}^{p^{s}} = 0 for i < t, and τ_{j} = 0 for j < n; hence ξ_{t}^{p^{s}} and τ_{n} are primitive in B. If n ≤ s, then b_{ts}v_{n} is nilpotent in Ext_{B}^{∗∗}(F_{p}, F_{p}).

Proof. The coproduct

τ_{n+t} ↦ ∑_{i=0}^{n+t} ξ_{n+t−i}^{p^{i}} ⊗ τ_{i} + τ_{n+t} ⊗ 1,

together with the conditions on B, gives the following relation in the cobar complex for B:

∑_{i=n}^{n+t−1} h_{n+t−i,i}v_{i} = 0.

Hence h_{tn}v_{n} = − ∑_{i=n+1}^{n+t−1} h_{n+t−i,i}v_{i}. Applying Steenrod operations gives the following (here one needs to use half-integer indexed operations 𝒫̃^{\frac{n}{2}} , or one needs to use operations 𝒫^{k} which are indexed differently):

b_{tn}v_{n}^{p} = ± ∑_{i=n+1}^{n+t−1} b_{n+t−i,i}v_{i}^{p},

b_{t,n+1}v_{n}^{p^{2}} = ± ∑_{i=n+1}^{n+t−1} b_{n+t−i,i+1}v_{i}^{p^{2}},

⋮

b_{ts}v_{n}^{p^{s+n−1}} = ± ∑_{i=n+1}^{n+t−1} b_{n+t−i,i+s−n}v_{i}^{p^{s+n−1}}.

Since ξ_{i}^{p^{s}} = 0 for i < t, each term on the right hand side of this equation is zero. □

Note that from Miller’s computation in [Mil78, Corollary 3.6] of the v_{1}-inverted Ext of the Moore spectrum (see also Subsection 4.6.3), one can see that b_{10}v_{1} need not be nilpotent; hence this result may be best possible.

Another analogue of Theorem B.2.1(b) would be that under conditions on s, t, u, and v, the product b_{ts}b_{vu} is nilpotent. Conjecture B.3.1 should be a special case of such a result, and since we do not know how to prove this conjecture, we will not be able to prove such an analogue of Theorem B.2.1(b).

For the remainder of this section, we provide evidence for and partial results related to Conjecture B.3.1.

We point out that if s < t, then F_{p}[ξ_{t}]/(ξ_{t}^{p^{s+1}}) is a quotient Hopf algebra of B; the cohomology of this quotient is

Λ[h_{t0}, …, h_{ts}] ⊗ F_{p}[b_{t0}, …, b_{ts}].

The element b_{ts} is non-nilpotent when restricted to this quotient, and hence nonnilpotent in Ext_{B}. When s ≥ t, though, surprisingly little seems to be known about the nilpotence (or lack thereof) of b_{ts}. For example, while it is easy to verify that h_{11}^{4} = 0 in Ext_{A}^{∗∗}(F_{2}, F_{2}) at the prime 2 (see Section B.2), we have not been able to locate or prove a similar result for b_{11} at an arbitrary odd prime. Working at the prime 3, Nakamura [Nak75] proved that b_{11}^{2} = h_{11}z for some z. Since h_{11} is in odd total degree, then h_{11}^{2} = 0; hence b_{11}^{4} = 0.

Note that if an element ξ_{t}^{p^{s}} is primitive in a Hopf algebra B, then so is ξ_{t}^{p^{r}} for all r ≥ s.

Lemma B.3.3. Fix a Hopf algebra B, and fix integers s ≥ t ≥ 1. Suppose that ξ_{t}^{p^{s}} is primitive in B.

(a) If b_{ts} is nilpotent, then so is b_{t,s+1}.

(b) Conversely, assume that ξ_{1}^{p^{s−t}} = ⋯ = ξ_{t−1}^{p^{s−t}} = 0 in B, and that s ≥ t. If

b_{t,s+1} is nilpotent, then so is b_{ts}.

As an application of (a), Nakamura’s calculation implies that b_{1,n}^{4} = 0 for all n ≥ 1, when p = 3. One might conjecture (based on very little evidence) that at any odd prime p, b_{1,n}^{q} = 0 for all n ≥ 1, where q = 2(p − 1). As an application of (b), b_{11} is nilpotent in Ext_{B} if B is a finite-dimensional quotient of A, because in that case, ξ_{1}^{p^{k}} = 0 in B for some k, hence b_{1k} = 0 in Ext_{B}.

Proof. Part (a) follows from the relation 𝒫̃^{0} (b_{ts}^{i}) = b_{t,s+1}^{i}.

For part (b), we have the following coproduct in B:

ξ_{2t} ↦ ∑_{i=0}^{2t} ξ_{2t−i}^{p^{i}} ⊗ ξ_{i}.

Since ξ_{i}^{p^{s−t}} = 0 when i < t, this simplifies to

ξ_{2t}^{p^{s−t}} ↦ ξ_{2t}^{p^{s−t}} ⊗ 1 + ξ_{t}^{p^{s}} ⊗ ξ_{t}^{p^{s−t}} + 1 ⊗ ξ_{2t}^{p^{s−t}}.

Also since ξ_{i}^{p^{s−t}} = 0 when i < t, then ξ_{t}^{p^{r}} is primitive in B for every r ≥ s − t. So the above coproduct in B translates to the relation h_{t,s−t}h_{ts} = 0 in Ext_{B}. We apply Steenrod operations to this relation: applying 𝒫̃^{p^{t}} ⋯ 𝒫̃^{p} (β𝒫̃^{1} )(β𝒫̃^{0} ) gives

b_{ts}^{p^{t}+1} = ±b_{t,s−t}^{p^{t}}b_{t,s+t}

If b_{t,s+1} is nilpotent, then so is b_{t,s+t}, by part (a). Hence so is b_{ts}. □

We have the following conjecture, as a special case of Conjecture B.3.1. To some extent, this would generalize Nakamura’s result at the prime 3; however, he determines a smaller nilpotence height (4) than this would (the number N = p\frac{p^{p−1}−1}{(p−1)^{2}} = 6 produced in Subsection B.3.1 below).

Conjecture B.3.4. Fix an odd prime p.

(a) The element b_{11} ∈ Ext_{A}^{2,2p^{2}(p−1)}(F_{p}, F_{p}) is nilpotent.

(b) Fix t ≥ 1 and let j = \frac{p+1}{2}. Then b_{tt} is nilpotent in Ext_{B}^{∗∗}(F_{p}, F_{p}), where

B = F_{p}[ξ_{t}, ξ_{2t}, …, ξ_{jt}, ξ_{jt+1}, ξ_{jt+2}, ξ_{jt+3}, …] ⊗ Λ[τ_{0}, τ_{1}, …].

We have a proof for this conjecture when p = 3, so it generalizes Nakamura’s result.

Proposition B.3.5. Conjecture B.3.4 holds when p = 3.

The proof (which contains a gap when p > 3), is a bit lengthy, and so we relegate it to a subsection. We also include a few other technical results in that subsection.

B.3.1. Sketch of proof of Conjecture B.3.4, and other results.

Sketch of proof. The proof involves some Massey product manipulations. May’s paper [May69] is the standard reference for Massey products; many of the key results are reproduced in [Rav86, Section A.1.4].

As remarked after Lemma 1.2.15, the element b_{ts} is the p-fold Massey product of h_{ts} = [ξ_{t}^{p^{s}}] with itself.

Part (a): In A, we have the coproduct

Δ: ξ_{2} ⟼ ξ_{2} ⊗ 1 + ξ_{1}^{p} ⊗ ξ_{1} + 1 ⊗ ξ_{2}.

This produces the relation h_{10}h_{11} = 0 in Ext_{A}. Applying the Steenrod operation β𝒫̃^{0} gives the relation

h_{11}b_{11} − b_{0}h_{12} = 0.

Then for any k ≥ 1, we apply 𝒫̃^{p^{k−1}} ⋯ 𝒫̃^{p} 𝒫̃^{1} to get

h_{1,k+1}b_{11}^{p^{k}} = b_{10}^{p^{k}}h_{1,k+2}.

Using induction gives the following formula, valid for all k ≥ 2:

(B.3.6)  h_{11}b_{11}^{1+p+p^{2}+···+p^{k−2}} = h_{1k}b_{10}^{1+p+p^{2}+···+p^{k−2}}.

So we let

N = 1 + 1 + (1 + p) + (1 + p + p^{2}) + ⋯ + (1 + p + p^{2} + ⋯ + p^{p−2})

= p\frac{p^{p−1} − 1}{(p − 1)^{2}}.

and we look at b_{11}^{N}. Recall that b_{11} is the p-fold Massey product ⟨\underbrace{h_{11}, …, h_{11}}_{p}⟩.

This Massey product has no indeterminacy.

Lemma B.3.7. The element b_{11}^{N} is contained in the p-fold Massey product

⟨h_{11}, h_{11}b_{11}, h_{11}b_{11}^{1+p}, …, h_{11}b_{11}^{1+p+···+p^{p−2}}⟩.

Proof. By the definition of Massey products, if y = ⟨a_{1}, …, a_{n}⟩ (with no indeterminacy), then for any elements x_{1}, …, x_{n}, we have

yx_{1} ⋯ x_{n} = ⟨a_{1}, …, a_{n}⟩x_{1} ⋯ x_{n}
∈ ⟨a_{1}x_{1}, …, a_{n}x_{n}⟩.

So we apply this to b_{11}^{N} = ⟨h_{11}, …, h_{11}⟩b_{11}^{N−1}. □

By our computations with Steenrod operations (i.e., equation (B.3.6)), we have

\begin{matrix}⟨h_{11}, h_{11}b_{11}, h_{11}b_{11}^{1+p}, …, h_{11}b_{11}^{1+p+···+p^{p−2}}⟩ \\ = ⟨h_{11}, h_{12}b_{10}, h_{13}b_{10}^{1+p}, …, h_{1p}b_{10}^{1+p+···+p^{p−2}}⟩. \end{matrix}

Lemma B.3.8. The Massey product

⟨h_{11}, h_{12}b_{10}, h_{13}b_{10}^{1+p}, …, h_{1p}b_{10}^{1+p+···+p^{p−2}}⟩

contains ⟨h_{11}, h_{12}, h_{13}, …, h_{1p}⟩b_{10}^{N−1} = 0.

We let ζ_{n} = χ(ξ_{n}), where χ: A ⟶ A is the canonical anti-automorphism. Hence ζ_{1} = −ξ_{1}, and

Δ(ζ_{n}) = ∑_{i=0}^{n} ζ_{i} ⊗ ζ_{n−i}^{p^{i}}.

Proof. We have to prove two things: that the Massey product contains ⟨h_{11}, h_{12}, h_{13}, …, h_{1p}⟩b_{10}^{N−1}, and that ⟨h_{11}, h_{12}, h_{13}, …, h_{1p}⟩ = 0. To prove the first of these, one follows the proof of Lemma B.3.7. To prove the second, we show that for all n ≥ 2, the n-fold Massey product ⟨h_{10}, h_{11}, h_{12}, …, h_{1,n−1}⟩ = 0; this goes by induction on n. Once we know this, then applying the Steenrod operation (𝒫̃^{0} )^{i} (to either the Massey product or to the proof) gives ⟨h_{1,i}, h_{1,i+1}, h_{1,i+2}, …, h_{1,i+n−1}⟩ = 0 for any i.

Indeed, we show that for each n, the cobar element d[ζ_{n}] is equal to

⟨h_{10}, h_{11}, h_{12}, …, h_{1,n−1}⟩;

hence this Massey product is cohomologous to zero. We also show that this Massey product has no indeterminacy. As a consequence, d[ζ_{n}^{p^{i}}] kills ⟨h_{1,i}, …, h_{1,i+n−1}⟩.

When n = 2, the coproduct

ζ_{2} ↦ ζ_{1} ⊗ ζ_{1}^{p} = ξ_{1} ⊗ ξ_{1}^{p}

gives have the relation h_{10}h_{11} = 0. This starts the induction. When n = 3, we have

ζ_{3} ⟼ ζ_{2} ⊗ ζ_{1}^{p^{2}} + ζ_{1} ⊗ ζ_{2}^{p}.

Since d[ζ_{2}] = h_{10}h_{11}, then d[ζ_{2}^{p}] = h_{11}h_{12}, so we have

⟨h_{10}, h_{11}, h_{12}⟩ = [ζ_{2}]h_{12} + h_{10}[ζ_{2}^{p}] = d[ζ_{3}] = 0.

Since h_{1n} ∈ Ext_{A}^{1,p^{n}q}(F_{p}, F_{p}), then the indeterminacy of this Massey product is of the form

h_{10}x + yh_{12},

where x ∈ Ext_{A}^{1,q(p+p^{2})}(F_{p}, F_{p}) and y ∈ Ext_{A}^{1,q(1+p)}(F_{p}, F_{p}). Since we know that Ext^{1,∗} is in one-to-one correspondence with the primitives of A, then Ext^{1,i} is nonzero only when i is of the form p^{j}q for some j. Hence both x and y must be zero, and there is no indeterminacy.

Now, we assume that ⟨h_{10}, h_{11}, …, h_{1,i−1}⟩ = 0 via ζ_{i}, for all i < n. Then ζ_{i}^{p^{j}} kills ⟨h_{1,j}, h_{1,j+1}, …, h_{1,j+i−1}⟩ for all i < n and all j, and so

⟨h_{10}, h_{11}, …, h_{1,n−1}⟩ = ∑_{i=1}^{n−1}[ζ_{i}|ζ_{n−i}^{p^{i}}].

Taking the coproduct on ζ_{n} kills this Massey product; there is no indeterminacy for the same reason as when n = 3. □

So the Massey product in Lemma B.3.7 contains both b_{11}^{N} and 0; hence b_{11} is an element of the indeterminacy. Therefore we need some information about that indeterminacy. The indeterminacy of a p-fold Massey product is contained in the union of certain (p − 1)-fold matric Massey products (see [May69, Proposition 2.3]); if every entry in the p-fold Massey product has odd total degree, the same is true of each entry in each matrix in the shorter Massey products.

Here is a general conjecture about “short” Massey products at an odd characteristic. This is the gap in our proof.

Conjecture B.3.9. Fix an odd prime p, and fix n < p. Consider an n-fold matric Massey product ⟨V_{1}, …, V_{n}⟩, in which each entry of each matrix V_{i} has odd total degree. Then any element of this matric Massey product is nilpotent.

(As in [Rav86, Section A.1.4], whenever we discuss matric Massey products, we assume that the matrices involved have entries with compatible degrees, so that their products have homogeneous degrees, etc. See [May69, Notation 1.1] for details.)

The conjecture is trivially true when n < 3, by graded commutativity; in particular, it is true when p = 3. Indeed, when p = 3, we see that every element of the indeterminacy is nilpotent of height p. One can also show that the conjecture is true when taking the Massey product of one-dimensional classes in the cohomology of a space [Dwy]; otherwise, we do not have much evidence for it. Meanwhile, it has the following consequence.

Conjecture B.3.10. If n ≤ p and if a_{i} has odd total degree for i = 1, …, n, then any element x in the indeterminacy of ⟨a_{1}, …, a_{n}⟩ is nilpotent.

This would imply that b_{11}^{N} is nilpotent, as desired.

Part (b) of the conjecture would be proved similarly. One starts with the relation h_{t0}h_{tt} = 0 in Ext_{B} and applies Steenrod operations to get the following replacement for (B.3.6):

h_{tt}b_{tt}^{p^{t−1}+p^{2t−1}+···+p^{kt−1}} = ±h_{t,(k+1)t}b_{t0}^{p^{t−1}+p^{2t−1}+···+p^{kt−1}}.

If we set

M = 1 + p^{t−1} + (p^{t−1} + p^{2t−1}) + ⋯ + (p^{t−1} + p^{2t−1} + ⋯ + p^{(p−1)t−1}),

then we get

b_{tt}^{M} = ⟨h_{tt}, …, h_{tt}⟩b_{tt}^{M−1}
∈ ⟨h_{tt}, h_{tt}b_{tt}^{p^{t−1}}, h_{tt}b_{tt}^{p^{t−1}+p^{2t−1}}, …, h_{tt}b_{tt}^{p^{t−1}+···+p^{(p−1)t−1}}⟩
= ⟨h_{tt}, h_{t,2t}b_{t0}^{p^{t−1}}, h_{t,3t}b_{t0}^{p^{t−1}+p^{2t−1}}, …, h_{t,pt}b_{t0}^{p^{t−1}+···+p^{(p−1)t−1}}⟩
∋ ⟨h_{tt}, h_{t,2t}, h_{t,3t}, …, h_{t,pt}⟩b_{t0}^{M−1}
= 0.

So Conjecture B.3.10 implies that b_{tt}^{M} is nilpotent. □

As remarked above, the gap in the proof—Conjecture B.3.9—is not a gap when p = 3, so Conjecture B.3.4 holds at the prime 3. Arguing similarly, we have the following.

Proposition B.3.11. Fix p = 3. The element b_{22} ∈ Ext_{B}^{2,432}(F_{3}, F_{3}) is nilpotent, where B = A/(ξ_{1}).

Proof. The coproduct ξ_{4} ↦ ξ_{2}^{p^{2}} ⊗ ξ_{2} gives the relation h_{20}h_{22} = 0 in Ext_{B}. Hence h_{2,i}h_{2,i+2} = 0 for all i.

As in the “proof” of Conjecture B.3.4, we find that b_{22}^{16} is contained in the Massey product ⟨h_{22}, h_{24}b_{20}^{3}, h_{26}b_{20}^{3+9}⟩; this Massey product also contains the element ⟨h_{22}, h_{24}, h_{26}⟩b_{20}^{15}. We claim that the three-fold Massey product ⟨h_{22}, h_{24}, h_{26}⟩ is nilpotent.

This Massey product is defined because the product h_{2,i}h_{2,i+2} is killed by ξ_{4}^{3^{i}}. So we consider the diagonal on ξ_{6}:

ξ_{6} ↦ ξ_{4}^{3^{2}} + ξ_{3}^{3^{3}} ⊗ ξ_{3} + ξ_{2}^{3^{4}} ⊗ ξ_{4}
= ⟨h_{20}, h_{22}, h_{24}⟩ + h_{30}h_{33}.

We have used the fact that ξ_{3}^{3^{i}} is primitive in A/(ξ_{1}), and hence gives rise to a 1-dimensional Ext class h_{1,i}. Hence we have

h_{30}h_{33} = −⟨h_{20}, h_{22}, h_{24}⟩.

By graded commutativity, (h_{30}h_{33})^{2} = 0; hence the same is true of the Massey product.

We find that b_{22}^{16} = ⟨h_{22}, h_{24}, h_{26}⟩b_{10}^{15} + x for some class x in the indeterminacy of a three-fold Massey product. Both x and ⟨h_{22}, h_{24}, h_{26}⟩ are nilpotent, and hence so is b_{22}. □

Corollary B.3.12. Fix p = 3. The element b_{22} ∈ Ext_{C}^{2,432}(F_{3}, F_{3}) is nilpotent, where C = A/(ξ_{1}^{3}).

Proof. If we know that b_{22} is nilpotent in Ext_{A/(ξ_{1})} and we want to know about its nilpotency in Ext_{A/(ξ_{1}^{3})}, then by Lemma 1.2.15 it suffices to determine whether b_{10}b_{22} is nilpotent in Ext_{A/(ξ_{1}^{3})}: b_{22} is nilpotent over C if and only if b_{10}b_{22} is nilpotent over C.

This is easy, though: the coproduct of ξ_{3} reduces in C to

ξ_{3} ↦ ξ_{2}^{3} ⊗ ξ_{1},

giving the relation h_{10}h_{21} = 0 in Ext_{C}^{∗∗}(F_{3}, F_{3}). Applying β𝒫̃^{0} gives

h_{11}b_{21} − b_{10}h_{22} = 0,

and so b_{10}h_{22} = 0 (h_{11} = 0 over C, since ξ_{1}^{3} = 0 in C). Applying β𝒫̃^{3} then gives

b_{10}^{3}b_{22} = 0.

□

This implies that when p = 2, b_{22} is nilpotent in Ext_{B}^{∗∗}(F_{3}, F_{3}) for any quotient Hopf algebra B in which ξ_{2}^{9} is primitive.

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Index

\frac{1}{2} A(n), 85–99

A, 36 act nilpotently, see nilpotent action Adams spectral sequence, see spectral se-

quence, Adams Adams tower, 25 admissible, 147–150 algebra, 2 A(n), 38, 50, 54, 58, 71, 74, 85–99

A-comodule structure of, 85–99 a_{n}, 80 A(n)_{k}, 91–95 antipode, 2, 3 augmentation, 2

bicommutative, 3 bounded below, 1 Bousfield class, 29, 61, 120, 139 Bousfield equivalence, 29, 120, 139 BP, 108 β𝒫̃^{0} , 21, 123 β𝒫̃^{n} , 155 Brown category, 15 Brown-Comenetz duality, see dual, Brown-

Comenetz b_{ts}, 38, 55, 57, 58, 92, 124, 157

nilpotence of, 123, 157–163

category derived, 15 model, 15, 153–154

cofibrantly generated, 154 stable homotopy, 14

connective, 22 cellular tower, 22

strong, 22 change-of-rings isomorphism, 18 Chouinard’s theorem, 31, 124 chromatic convergence, 75–76 chromatic tower, 75 CL, 24, 49, 58, 68 closed model category, see category, model coaction, 4, 20, 109, 121, 135, 137

diagonal, 4 coalgebra, 2

quotient, 16, 18, 19, 59 coaugmentation, 2 cobar complex, 8, see complex, cobar cocommutative, 3 cofiber sequence, 14, 16, 21, 30 cofibration, 154, see also cofiber sequence cofree, 6, 7, 72, 87 cohomology functor, see functor, cohomol-

ogy coideal

coaugmentation, 20 colimit

homotopy, 15 commutative, 3 comodule, 4

simple, 14, 22, 31, 153 comodule-like, see CL complete, 26, 80 complex

cobar, 8 comultiplication, 2 conjugation, 2, 3, 135–137, 140 connective, 23, 25

weakly, 23 conormal, see Hopf algebra, quotient, conor-

mal coproduct, 15 cotensor product, 16 counit, 2

D, 108, 112 degree, 1 derived category, see category, derived detect nilpotence, 110–112, 139 diagonal, 2 D(n), 112, 132 D_{r}, 119 D_{r,q}, 119 dual

Brown-Comenetz, 30, 125, 139 graded, 3, 5 module, 6, 140 Spanier-Whitehead, 48, 60, 129

DX, see dual, Spanier-Whitehead D[x], 9

classification of comodules over, 10 Ext over, 11

D[ξ_{t}^{p^{s}}], 37, 55 E-complete, see complete elementary, see Hopf algebra, elementary e_{n}, 97–99 E[τ_{n}], 37, 55 E[x], 9

classification of comodules over, 10 Ext over, 11

Ext, 8, 15 Ext_{A}, 49 extension, see Hopf algebra extension; group

extension

F -isomorphism, 109, 110, 122, 128, 134, 135,

143 uniform, 109

fibration, 154 field spectrum, see spectrum, field finite spectrum, see spectrum, finite finite type, 1, 27, 30, 140 finitely generated

essentially, 144–146 flat, 25, 26, 80, 101 F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}), 60–63, 66–71, 91 F (U_{n}), see F (u_{d_{1}}^{j_{1}}, …, u_{d_{m}}^{j_{m}}) functor

cohomology, 16 exact, 72 homology, 16 pth power, 94, 149

Γ □_{B}k, 20 Γ//C, 20 generated, 14 generic, 24, 26, 114, 128, 129 grading, 1 group, 3 group algebra, 3, 31 group extension, 31 grouplike, 4

H, 16–22, 50

Tate version of, 71–75 Ĥ, see H, Tate version of HB-complete, see complete HD, 108, 110, 112, 122, 128, 135, 146, 151 \underline{Hom}, 72 homology functor, see functor, homology homotopy group, 16 hook length, 97 Hopf algebra, 2, 153

connected, 3, 23 elementary, 39, 41, 53, 57 quasi-elementary, 41, 108, 109, 111, 121,

139 quotient, 16, 19, 36, 51, 108, 139, 140

conormal, 20, 27, 36, 109, 112, 121, 139 maximal elementary, 40 maximal quasi-elementary, 119, 134, 137 sub-

normal, 20 Hopf algebra extension, 20, 30, 37, 51, 52,

55, 56, 112, 139, 140 horizontal vanishing line, see vanishing line,

horizontal h_{ts}, 38, 51, 57, 58, 92, 115, 135, 137, 156,

157 nilpotence of, 51, 116, 120, 123, 156

Hurewicz map, 23, 58, 109, 112 Hurewicz theorem, 23 hypercohomology, 16, 29

ideal, 127, 131–132 augmentation, 20 invariant, 127, 131, 144, 146, 151

idempotent, 96–99 induction, 17, 71 injective, 6–8, 17–20, 44, 59, 72 intercept, 50 invariants, 108, 134 isomorphism

change-of-rings, 18 IX, see dual, Brown-Comenetz I(X), see ideal

kG, 31 K(n), see Morava K-theories Künneth isomorphism, 8, 20

L_{𝒞}^{f} , see localization, finite L_{n}^{f}, see localization, finite Lie algebra, 3

restricted, 4 limit

homotopy, 15 L_{n}^{f}, 73 L_{n}^{f}A(m), 74 loc(A), 70, 142 loc(Γ), 24 loc(Y), 24 localization

Bousfield, 66, 71

finite, 65, 66, 72, 75

smashing, 66, 71 localizing subcategory, see subcategory, lo-

calizing locally finite, 6

Margolis’ killing construction, 65, 68 Massey product, 21, 156, 157, 159–163 M⟨n + 1, ∞⟩, 70 M_{n}^{f}, 75 module, 4 monogenic, 14, 22, 29 Moore spectrum, 104–106 Morava K-theories, 57, 128 morphism, 14, 72

grading, 14

stable, 72 multiplication, 2

nilpotence theorem, 110, 111 nilpotent action, 51, 56 Nishida’s theorem, 110 normal, see Hopf algebra, sub-, normal

𝒫̃, 155 P -algebra, 38 periodicity theorem, 128, 129 Φ, 94, 149 π_{∗∗}, 16, 49 PM, 108 P (n), 86–88, 90–99 P (n)_{k}, 91–95 Postnikov tower, 23 primitive, 4, 8, 21, 59, 108, 123, 154, 156–

158 profile function, 37 projective, 6 P_{t}^{s}, 42, 53, 57, 146 p_{t}^{s}, 42, 148 P_{t}^{s}-homology, 42, 43, 128

module, 42, 45 P_{t}^{s}-homology spectrum, 42, 45, 147

𝒬, 109, 121, 134 \overline{𝒬} , 121 Q_{n}, 42, 43, 57, 80–85, 147 q_{n}, 43, 80–81, 88–90 q_{n}-complete, see complete Q_{n}-homology, 43 Q_{n}-homology spectrum, 43 (Q_{n})_{∗∗}(Q_{n}), 88, 101–102 (q_{n})_{∗∗}(q_{n}), 99 quasi-elementary, see Hopf algebra, quasi-

elementary Quillen stratification, 109, 124

rank variety, see variety, rank relative vanishing line, see vanishing line,

relative resolution

injective, 8, 10, 153 restriction, 4, 17, 71 restriction map, 19, 58, 112 ring spectrum, see spectrum, ring

self-map, 57, 110, 111

central, 59, 128 Shapiro’s lemma, 18 shearing isomorphism, 18 Σ_{n}, see symmetric group Σ^{i,j}, see suspension slope, 46, 49, 57, 60, 66, 73, 75, 147–150 slope support, 147–150 Slopes, 147–150 Slopes′, 147–150 small, 24 smash product, 15 Spanier-Whitehead dual, see dual, Spanier-

Whitehead spectral sequence

Adams, 25–29, 80–85, 114 construction, 25 convergence, 26

Atiyah-Hirzebruch, 57, 144

change-of-rings, see spectral sequence, ex-

tension extension, 28, 112, 113 Lyndon-Hochschild-Serre, 28

spectrum, 14 field, 19, 45, 80, 81, 128 finite, 24, 63, 65, 67, 110, 111, 128, 142,

143, 146–148 ring, 16, 19, 25, 45, 110, 111, 128, 129

sphere, 15 Sq^{n}, 137 S̃q^{n}, 155, 156 stable homotopy category, see category, sta-

ble homotopy Stable(A), 36 Stable(Γ), 14, 153 StComod(B), 72 Steenrod algebra, 4, 15, 36 Steenrod operation, 21, 29, 155–156 subcategory

localizing, 24, 49, 66

thick, 24, 26, 61, 65, 72, 90–95, 116, 129,

143–146 suspension, 1, 14, 72 internal, 1, 10

symmetric group, 96–99

tame, 6

telescope, 30, 63, 66 telescope conjecture, 71 tensor product, 4 thick subcategory, see subcategory, thick thick(Y), 24 τ_{n}, 36 trivial comodule, 22 twist map, 2 type n, 57, 63, 67

u_{n}, 57, 59, 149

u_{n}-map, 57, 60–62, 66

unit, 2

unit map, 16

universal enveloping algebra, 3

v_{0}^{−1}π_{∗∗}, 101

vanishing line, 49–60, 70 horizontal, 49, 53, 57 relative, 58

vanishing plane, 26–27, 83, 88–90, 114 variety

rank, 146–147 V (n), 86–88 v_{n}, 38, 55, 57, 59, 80–85, 92, 157 v_{n}-map, 57, 81, 128

weak equivalence, 154 wedge, 15 W_{j}, 114, 115

⟨X⟩, see Bousfield class [x], 21, 156 |x|, see degree X-acyclic, 29 ξ_{n}, 36

[y], see [x] y-element, see y-map y-map, 128–131, 146

weak, 129 Y (n), 102–104 y_{n}, 46, 59, 73 Young diagram, 97–99 Young tableau, 97–99

Z_{n}, 96–99 Z(n), 46, 49, 57, 60–62, 66, 68, 75 z(n), 46, 57, 59 ζ_{n}, 135