H^{∗}V Approximations

— The structure of unstable algebras with applications to spaces

Jesper Grodal

Author address: Matematisk Institut, Universitetsparken 5, DK–2100 København Ø E-mail address: jg@math.ku.dk 1991 Mathematics Subject Classification. 55S45,(55S10,55M30,55P60).

1. PREFACE i

1. Preface

This thesis is written for the Cand.Scient. degree in Mathematics at The University of Copenhagen in the field of Algebraic Topology.

The goal of the thesis is to explore how the general structure of the category of unstable algebras over the Steenrod algebra can be used to determine the large scale structure of cohomology rings of spaces.

The results stated in this thesis are original unless specified otherwise. Section I-2, Section I-3 and the entire Chapter III together constitute the paper [25] (submitted 11/11-96 for publication in Proceedings of The Fields Institute conference on unstable homotopy theory 1996), modulo a few adjustments made after the submission deadline.

1.1. The creation of this thesis—a historical account. The first results of this thesis dates back to May 96 while I was staying at The Fields Institute, Toronto. At this point of time my advisor J. Møller and his colleague C. McGibbon were interested in determining different properties of n-connected covers of finite complexes, seen in the light of the amazing Neisendorfer’s theorem, which states that a finite complex, X, up to p-completion can be naturally recovered from its n-connected cover (cf. [45] and Remark III-4.7). Amongst other things they were interested in determining the Lusternik-Schnirelmann (LS) category of these n-connected covers. They suspected that this should always be infinite, under suitable connectivity assumptions on the space, and they were able to show this under various further assumptions on the space X. They however suspected this to be true in full generality, the reason being that they suspected that every space with only finitely many nontrivial homotopy groups (i.e. a finite Postnikov system) should have elements of infinite height in its mod p cohomology, by analogy to the situation for Eilenberg-Mac Lane spaces. By looking at the Serre spectral sequence of the natural fibration relating the space and its n-connected cover, this would imply the existence of elements of infinite height in the mod p cohomology of the n-connected cover, and thus in a very strong way guarantee infinite LS category.

They asked a number of homotopy theorists whether they knew the answer to this conjecture. J. Møller also asked me the question. I started out by playing around with Steenrod squares and two stage Postnikov systems, and was able to verify their conjecture for some (very) special two stage Postnikov systems, using seemingly coincidental Adem relations and a fair amount of work. However, I suspected that the problem was attackable via the theory of unstable modules which I had just learned in the course of R. Kane at The Fields Institute. After some attempts at the problem I succeeded in putting together a proof of the conjecture using the theory of unstable modules (Theorem III-1.6).

This was just before going to the AMS summer school on group cohomology held in June 96 in Seattle. Here I talked to a number of people about the result and its possible generalizations. I suspected that a strong generalization of the result should hold, namely that the mod p cohomology of these spaces should almost always have infinitely many algebraically independent elements of infinite height. Searching for a way to prove this new conjecture I found an article of Lannes and Schwartz [38], where they actually prove the first conjecture under the assumptions that p = 2 and that the space assumed to be simply connected. Their proof much resembles my original proof.

Upon returning to The University Copenhagen in September 1996 I found after some searching a proof of the generalized conjecture, actually being able to show that, up to F_{p}-equivalence, the only finite Postnikov systems with only finitely many algebraically independent elements in their mod p cohomology are classifying spaces of p-toral groups (Theorem III-3.13). This class of spaces is of course radically smaller than the class of all finite Postnikov systems, and is furthermore a class of spaces whose cohomology is quite well understood. The method of the proof had as an important byproduct an easily applicable formula for determining the maximal number of algebraically independent elements in an arbitrary unstable algebra (Theorem III-3.10).

It took most October and November to get the results written down in their final form. Also I was working on further generalizations, everywhere replacing finite Postnikov systems with the larger class of spaces polyGEMs, which can be seen as the ‘dual spaces’ of spaces with finite LS category, and which play a special role in unstable localization theory. This is still an open problem. I state these conjectures towards the end of the thesis, together with some partial results.

It was only some time in the fall that I decided to make the above results the contents of my Cand. Scient. thesis, not writing it on p-compact groups, which had been my original plan.

I have chosen not to alter the parts of the thesis originating from the paper. Instead I include two additional sections which I hope puts the results into a context.

The first section, Section I-1 is an introductory section which describes the body of mathematics which surrounds this thesis, introducing the concepts needed along the way.

In the course of proving the results I will need some results from a paper of Henn, Lannes and Schwartz [27] , which can not be considered standard knowledge. The second additional section, Section II-1, therefore gives a complete proof of their main theorem and along the way proves the facts which I will need later on. In Section III-3, taken from the paper, we will reintroduce some notation and state some facts already proven in this additional section. However, in order not make the later parts dependent on this section, I have chosen to retain it this way. The reader who has read Section II-1 should just skip the first part of Section III-3 and go right to Theorem III-3.10.

To help guide for instance fellow students through the thesis we will in the next Section I-2 describe which prerequisites are assumed on the part of the reader, with references on where to obtain further background material.

1.2. The contents. Section I-1 contains an informal introduction to a theorem of Quillen and a theorem of Serre which the results of this thesis can be seen to evolve around.

Section I-2 is a shorter and more technical introduction to the contents of this thesis. It also quotes some general theorems which we will need.

Section I-3 states the notation used.

Section II-1 is devoted to giving a proof of a theorem of Henn, Lannes and Schwartz, which can be seen as a generalization of Quillen’s theorem.

Section III-1 and Section III-3 are mainly concerned with giving proofs of Theorem 1.6, Theorem 3.10 and Theorem 2.2.

Section III-2 is used to give corollaries of the results to n-connected covers.

Section III-4 discusses generalizations of the above results on finite Postnikov systems to the more general class of spaces polyGEMs. We give some partial results and state two conjectures.

Section III-5 is an appendix which states and proves a small fact about nilpotent actions which has been used several times in the thesis.

1.3. Acknowledgment. I would like to direct a cordial thank to whole topology group at The University of Copenhagen. Thanks to Kasper Andersen for being a good

1. PREFACE iii

friend and collaborator through my entire education, from linear algebra through the nth stable stem and beyond. Thanks to Jesper Møller for being a good friend and advisor, for arousing my interest in Algebraic Topology, and for suggesting me to go to The Fields Institute for a year.

I am grateful to The Fields Institute for giving me the opportunity to participate in the ‘Emphasis year in Homotopy Theory 95/96’, and especially to the organizers and participants for making the stay so enjoyable and profitable. A special thank to K. Andersen, W. Chachólski, P. Lambrechts, J. Scherer and B. Schuster for so many hours of interesting discussions. Also thanks to R. Kane for introducing me to the beautiful theory of unstable modules which this thesis is based on.

Financial support from Det Internationale Kontor, Det Naturvidenskablige Fakultet, Knud Højgaards fond, and Julie Damms fond has made the year at The Fields Institute and subsequent trips to Seattle and Barcelona possible. The support is greatly appreciated.

I would like to thank W. Chachólski for helpful suggestions and references. Likewise thanks to my advisor J. Møller for his support and for suggesting improvements in the presentation. Thanks to H. Minh Le directing my attention to the conjecture of E. Dror Farjoun. Finally thanks to C. McGibbon and J. Møller whose questions related to Theorem III-1.6 got me interested in the cohomology of finite Postnikov systems in the first place.

Copenhagen, January 1997

.

Jesper Grodal

2. Prerequisites

As a rule of thumb this thesis is written to the working algebraic topologist, or to a fairly advanced graduate student.

As a service, especially minded at fellow students, we will briefly try to account for approximately what material we will use where.

Section I-1 stands out from the rest of the thesis since we here deliberately have tried to keep things as nontechnical as possible, sacrificing a bit of precision on that account. Some objects are not rigorously defined in this section. Most will be defined in Section I-3 on notation, where to the reader is referred. Since this section is supposed to be of mainly motivational nature, we will also mention some things in this section which will not be used later on.

For the thesis in general, the reader is assumed to be familiar with basic elements of homotopy theory as presented in e.g. Bredon [8], Spanier [53] or Whitehead [58]. The main reference is Spanier, although the reader unfamiliar with homotopy theory might be better off by starting with Bredon. The recent book of Selick [50] is also a good introduction to many of the more advanced aspects of the theory.

We will not give specific references to theorems or definitions stated in basic textbooks on topology, but in most cases objects or theorems considered will be stated by name, and the diligent reader with a weaker background in topology should have no problems in looking them up.

Since this thesis takes its point of departure in the theory of unstable modules as described in Schwartz [49], this book stands out as the number one reference. Actually all the results we will need from this book will be stated in the introduction to the results, but a familiarity with this book can definitely help to increase the readers understanding of what goes on ‘behind the scene’. We mention for instance that even though the Steenrod algebra plays a very central role in this thesis, we will actually not do one single calculation involving explicit manipulations with Steenrod squares.

We will be dealing with localizations and F_{p}-equivalence of spaces throughout but mostly on a non-technical level. The main reference for this theory is Bousfield and Kan [7].

In Section II-1 and Section II-3 we will use a bit of basic category theory. The main reference for this theory is Mac Lane [32]. We will not give references as we go along since all notion used is standard and can be found in e.g. Mac Lane.

Likewise in Section II-1 and Section II-3 we will assume that the reader is familiar with basic properties of profinite sets. What we will use is very little, and everything can be found in a concise form in Weibel [57].

In Section III-2 we will talk about the Lusternik-Schnirelmann (LS) category. Whitehead [58] covers this adequately. Since the index in that book next to nonexistent, we will give a concrete reference: The treatment starts on page 457.

The motivation for the contents of Section II-4 comes from the novel developments in unstable localization theory. A good reference for this evolving theory is the recent book of Dror Farjoun [20].

Many of the things carried out in this thesis has roots in the theory of group cohomology. A good reference of this theory is Evens [19]. Its presence is most strongly being felt in Section I-1, and is not needed for understanding the proofs later on. In fact it is a central point in the thesis that our modern treatment of e.g. Quillen’s theorem surpasses the use of group cohomology entirely.

We will at times also use some standard facts from commutative algebra. Good references for these are Matsumura [40] or Atiyah & MacDonald [2]. Also, Benson [4] has a concise chapter which includes much of what one needs to know for our purposes— moreover he does (almost) everything for graded rings as well. However the references to

2. PREREQUISITES v

commutative algebra mainly appear in Section I-1, and are not central for understanding the rest of the thesis. We will also in Section I-1 motivate things with a bit of Galois theory, but here we will state what we need.

vi

Contents

1. Preface i 2. Prerequisites iv

Chapter 1. Introduction 1 1. Introduction to the theorems of Quillen and Serre 1 2. Introduction 9 3. Notation 12

Chapter 2. The generalized Quillen’s theorem of Henn, Lannes and Schwartz 15 1. The generalized Quillen’s theorem 15

Chapter 3. The large scale structure of the cohomology rings of certain spaces 23 1. Growth Properties 23 2. Applications to n-connected covers 26 3. The transcendence degree of H^{∗}(E) 26 4. Generalizations to PolyGEMs 32 5. Appendix: Nilpotent actions 34

Bibliography 37

viii CONTENTS

CHAPTER 1 Introduction

1. Introduction to the theorems of Quillen and Serre

The results of this thesis have origins in two ‘classical’ theorems in homotopy theory. Namely Quillen’s theorem from 1971 about the structure of the mod p cohomology ring of a finite group [46], and Serre’s theorem from 1953 about the duality between homotopy and homology [51]. These two theorems have been vastly generalized during the 1980s and 1990s due to the novel insight into a certain category in which the category of cohomology rings of spaces are embedded. We will in this section give an informal introduction to these two theorems and their ramifications, aimed to motivate and perspectivate the results of this thesis. Some readers may want to skip directly to the next section, where we introduce the results, which can be read independently of this one.

Throughout this thesis we will by cohomology mean mod p cohomology, where p is a fixed but arbitrary prime, and this will be suppressed from the notation. Our reason for working with mod p cohomology is the availability of the Steenrod algebra of cohomology operations, which will be our ever ambient tool.

1.1. Quillen’s theorem. We start by describing Quillen’s theorem and its development. The main references for this part are [46, 59, 60, 1, 47, 31, 27, 25]. Furthermore the book of Evens [19] contains a good exposition of the theorems in [46], and the book of Kane [30] contains an excellent description of much of the work in [59, 60, 1].

Recall that one defines the mod p group cohomology of a finite group G as H^{∗}(G) = Ext_{F_{p}G}^{∗}(F_{p}, F_{p}). One of the most fundamental results in group cohomology is the Evens-Venkov theorem [19, 56], which states that H^{∗}(G) is a noetherian graded commutative ring. This allows for the use of tools from commutative ring theory to be applied to group cohomology.

Remember that a space with only one nontrivial homotopy group G in dimension n is called an Eilenberg-Mac Lane space and is denoted K(G, n) (it is uniquely determined up to homotopy equivalence). It is well known that the group cohomology of a finite group G is equal to the topological cohomology of the Eilenberg-Mac Lane space K(G, 1) (cf. e.g. [10]). Thus the theory group cohomology can also be studied from a topological point of view.

Recall that an elementary abelian p-group V is a finite group of the form V ≃ (Z/p)^{v}. The number v is called the rank of V, rk_{p} V.

Example 1.2. The cohomology ring of an elementary abelian group is given by:

H^{∗}(V) = \{ \begin{matrix}F_{p}[x_{1}, …, x_{v}], |x_{i}| = 1 & for p = 2 \\ F_{p}[x_{1}, …, x_{v}] ⊗ ∧(y_{1}, …, y_{v}), |x_{i}| = 2, |y_{i}| = 1 & for p odd \end{matrix}

where ∧(y_{1}, …, y_{v}) denotes an exterior algebra on v generators.

Quillen proved that the cohomology of an arbitrary finite group G can be approximated by the cohomology of elementary abelian p-groups. Let A(G) denote the ‘Quillen category’ of G, i.e. the category having as objects the elementary abelian p-subgroups of G, and as morphisms the homomorphisms of elementary abelian p-subgroups, which are induced by conjugation by an element in G. We have a natural contravariant functor from A(G) to the category of cohomology rings, which to each elementary abelian group V associates H^{∗}(V). Quillen proved the following theorem:

Theorem 1.3. [46] The canonical map

φ : H^{∗}(G) → lim_{V ∈A(G)} H^{∗}(V)

is an F-isomorphism, i.e. it has a kernel consisting of nilpotent elements, and every element x ∈ lim_{V ∈A(G)} H^{∗}(V) satisfies that x^{p^{n}} ∈ φ(H^{∗}(G)) for some n.

Quillen’s theorem thus gives a description of the cohomology of G in terms of an inverse limit of cohomology rings which are well understood, patched together according to information relating to purely group theoretic properties of G.

Quillen’s proof uses methods from commutative algebra and algebraic geometry combined with a heavy use of equivariant cohomology, and the theorem is actually formulated more generally in terms of a compact Lie group G acting on a compact space X. This more general formulation is essential for the proof even if one is only interested in the case where G is a finite group. Quillen’s proof is, in his own words, inspired by the proof of the Atiyah-Segal completion theorem^{1} using equivariant cohomology given in [3].

That Quillen’s method of proof is still timely is seen by the very recent preprint of Greenlees and Strickland [24], where they use the equivariant methods of Quillen’s proof to obtain various generalizations of Quillen’s theorem, where ordinary cohomology is replaced by a periodic generalized cohomology theory satisfying certain conditions.

Other (still ‘classical’) proofs of Quillen’s theorem have used a detection theorem for elementary abelian groups of Serre [52] combined with techniques from commutative algebra to analyze the relationship between prime ideals in H^{∗}(G) and elementary abelian subgroups of G (cf. e.g. [19]). This gives more algebraic proofs avoiding equivariant cohomology.

Remember that the Krull dimension of a ring is defined as the maximal number of proper inclusions in a chain of prime ideals in the ring. For a graded, connected noetherian algebra it is equal to the maximal number of algebraically independent elements in the ring, the transcendence degree [4]. We have that F-isomorphic F_{p}-algebras have the same Krull dimension, since if A → B is an F-isomorphism then the induced map on prime spectra spec(B) → spec(A) is a bijection, by an easy application of the going-up theorem from commutative algebra.

An important consequence of Quillen’s theorem is that it settles a earlier conjecture of Atiyah and Swan showing that the Krull dimension of the mod p cohomology ring of G, dim(H^{∗}(G)), is equal to the maximal rank of an elementary abelian p-group in G, rk_{p} G. We’ll sometimes also refer to this fact as Quillen’s theorem. Since this corollary is especially important we’ll take time to derive it from the main theorem.

The inequality dim(H^{∗}(G)) ≥ rk_{p} G follows without the use of Quillen’s theorem. Note that for all elementary abelian p-subgroups V of G,

H^{∗}(V) = H^{∗}(V; F_{p}) = H^{∗}(G; Hom_{F_{p}V} (F_{p}G, F_{p}))

by the Eckmann-Shapiro lemma (cf. e.g. [19]). Thus H^{∗}(V) is finite over H^{∗}(G) by the Evens-Venkov theorem, since Hom_{F_{p}V} (F_{p}G, F_{p}) is finite over F_{p}. This means that dim(H^{∗}(G)) ≥ dim(H^{∗}(V)) = rk_{p} V for all elementary abelian p-subgroups V of G (where we again have used the going-up theorem). Equality now follows since by Quillen’s theorem

dim(H^{∗}(G)) = dim( lim_{V ∈A(G)} H^{∗}(V)) ≤ max_{V ∈A(G)} dim(H^{∗}(V)) = rk_{p}(G).

1The Atiyah-Segal completion theorem states in its basic form that K^{0}(BG) ≃ R̂(G) where K denotes complex K-theory, BG is the classifying space of G, R(G) is the complex representation ring of G, and the hat denotes completion at the augmentation ideal.

Quillen’s theorem can easily be reduced to a statement about p-groups, since we know from Cartan-Eilenberg [13] that:

H^{∗}(G) →^{≃} lim_{P ∈Syl_{p} G} H^{∗}(P),

where Syl_{p} G denotes the category with objects p-Sylow subgroups in G and morphisms being conjugation by elements in G. Thus Quillen’s theorem is proved once we know it for p-groups. Note however that in practice the Cartan-Eilenberg theorem is no great computational tool, since the cohomology of a groups p-Sylow subgroups is often are more complicated than that of the group itself.

1.4. The advent of unstable modules. Quillen’s theorem shows that H^{∗}(V) plays a very special role in the cohomology of groups. One of the great discoveries of the 1980s was that this is a special case of a much more general phenomenon.

It has been known since Steenrod in 1946 constructed the Steenrod operations [54] that the mod p cohomology of a space (or group) is not only a graded F_{p}-algebra. It is also endowed with a structure as a module over the Steenrod algebra of cohomology operations. This module action satisfies certain relations, so called ‘instability conditions’, and is compatible with the algebra structure. Algebras with such a structure are called unstable algebras^{2}. During the 1980s it was realized that many facts about the cohomology of spaces was just manifestations of the fact that they were unstable algebras. Actually many facts are just reflections of even less, namely just that the mod p cohomology of spaces are (graded) modules over the Steenrod algebra satisfying these instability conditions. Modules over the Steenrod algebra satisfying the instability conditions are called unstable modules. The category of unstable modules has the technical advantage that is an abelian category. Furthermore for an unstable algebra much of the ring structure is actually encoded in the unstable module structure, so that when viewing them as modules, one does not throw away too much information—this is for instance exemplified by the Lannes linearization principle which will be stated in the introduction to the results.

1.5. The Adams-Wilkerson theorem. One of the first purely algebraic results about unstable algebras is the Adams-Wilkerson embedding theorem from 1978 [1], completing a program set up by Wilkerson in [59, 60]. The Adams-Wilkerson embedding theorem states that any unstable algebra, K, which is furthermore an integral domain of finite transcendence degree, d(K), admits an embedding in H^{∗}(V), where V is an elementary abelian group of rank d(K). Adams and Wilkerson used techniques from field and Galois theory together with concrete calculations with the Steenrod algebra to show that each unstable algebra K satisfying the assumptions above has an ‘algebraic closure’ in the category of unstable algebras. They then identified that algebraic closure as H^{∗}(V)/\sqrt{0}, where \sqrt{0} denotes the nil radical, which led them to the wanted embedding.

They defined the concepts of ‘separability’ and ‘normality’ in this new category, each time making appropriate modifications in order to take the grading and Steenrod operations into account, and checking that everything worked smoothly.

Adams and Wilkerson were not primarily concerned with Quillen’s theorem, but instead interested in solving the Steenrod problem of determining which polynomial algebras arise as the cohomology of spaces (i.e. which gradings can occur). For ‘large primes’ they completely settled this problem showing that in this case the polynomial algebra would be the fixed point algebra of a suitable finite group, G, acting on H^{∗}(V)/\sqrt{0}.

By the Chevalley-Sheppard-Todd theorem (cf. e.g. [30, 4]) we have that for p |̸|G|, (H^{∗}(V)/\sqrt{0})^{G} is polynomial iff G is an F_{p} reflection group. Since p |̸|G| this is equivalent

2Some people (more correctly) refer to these as unstable algebras over the Steenrod algebra, but since we’ll not discuss unstable algebras over anything else, we’ll stick to the shorter form.

4 1. INTRODUCTION

to G being a p-adic reflection group [14, 15], and p-adic reflection groups have been classified by Clark and Ewing [14].

This completely settles the Steenrod problem for ‘large primes’, since Clark and Ewing already had pointed out that all these spaces were realizable by the Borel construction X = K(Ẑ_{p}^{n}, 2) ×_{G} EG, (p ≠ 2), where EG is a contractible free G-space and G acts on K(Ẑ_{p}^{n}, 2) via the representation of G as a p-adic reflection group G ↪ Aut(Ẑ_{p}^{n}). This realizes (H^{∗}(V)/\sqrt{0})^{G} as a cohomology ring since H^{∗}(X) = H^{∗}(K(Ẑ_{p}^{n}, 2))^{G}, as is easily seen, e.g. by looking at the Serre spectral sequence for the fibration K(Ẑ_{p}^{n}, 2) → X → BG.

To see the relation to Quillen’s theorem, recall the following theorem from Galois theory (cf. e.g. [33]).

Proposition 1.6. Let L ⊂ M be a normal extension of fields^{3}, in the sense that M is the splitting field of a family of (not necessarily separable) polynomials in L[X], and let G denote the Galois group of this extension.

Then we get a tower of extensions L ⊂ M^{G} ⊂ M, where M^{G} ⊂ M will be separable (every irreducible polynomium in M^{G} does not have any multiple roots in M) and L ⊂ M^{G} will be purely inseparable (for every α ∈ M^{G}, Irr(α, L) = X^{p^{n}} − a, for some n ∈ N_{0}, a ∈ L).

The results of Adams and Wilkerson in [1] (later clarified by Wilkerson (cf. [30])) show that if one, in addition to the assumptions on K in the Adams-Wilkerson embedding theorem, furthermore assume that K is integrally closed in its field of fractions, this theorem carries over to the category of unstable algebras.

Especially, let ι : K ↪ H^{∗}(V) be the Adams-Wilkerson embedding, and let G be the group of automorphisms of H^{∗}(V) fixing K, then for all x ∈ H^{∗}(V)^{G}, x^{p^{n}} ∈ K for some n. The Adams-Wilkerson theorem can thus be seen as a first step towards a ‘Quillen’s theorem’ for unstable algebras.

The added ‘integrally closed in its field of fractions’ assumption seems natural for the approach of Adams-Wilkerson, since one needs some way of getting from the fields of fractions back to the algebras themselves (it is also an assumption which e.g. appears in the going-down and transitivity part of Cohen-Seidenberg’s theorem from commutative algebra). Note however that F_{p}[x^{n}|n ≥ 2] ⊂ H^{∗}(CP^{∞}), gives us an example of an unstable algebra which is an integral domain but not integrally closed in its field of fractions.

1.7. Quillen’s theorem revisited. In 1982 Rector [47] was able to show the following generalization of Quillen’s theorem:

Theorem 1.8. [47] Let K be a noetherian unstable algebra, and assume that K is of finite transcendence degree. Then there exists a (finite) generalized Quillen category A(K) and an F-isomorphism of unstable algebras:

K → lim_{A(K)} H^{∗}(V).

The category A(K) of Rector becomes the usual Quillen category when taking K = H^{∗}(G), G a finite group. Rector’s proof uses the Adams-Wilkerson theorem together with methods inspired by classical proofs of Quillen’s theorem. Similar results but with slightly different proofs were obtained more or less independently by Lam [31].

In 1983 Carlsson (V = Z/2) and Miller (V = Z/p, p arbitrary) proved the following theorem about the role of H^{∗}(V) in the category of unstable modules, which clarified earlier results and has served as a cornerstone for the later developments.

3Beware that our definition of normality differs from for instance the one taught at The University of Copenhagen (cf. [29]).

Theorem 1.9. [11, 44, 39] H^{∗}(V) is an injective object in the category of unstable modules.

The theorem as stated above, i.e. for all elementary abelian p-groups V, was proved by Lannes and Zarati [39]. The above theorem combined with the ‘linearization principle’ of Lannes implies that H^{∗}(V) is also injective in the category of unstable algebras. The reader might want to play around with Steenrod operations on H^{∗}(Z/p) for a short while to convince himself that they really do act in a ‘divisible’ way (i.e. according to the principle ‘if the Adem relations allow an element to be reached by a Steenrod operation, it is reached’).

During the 1980s the general theory of unstable modules has greatly expanded by the work of a number of people amongst them Broto, Carlsson, Dwyer, Henn, Lannes, Miller, Schwartz, Wilkerson and Zarati.

Using theory of unstable modules over the Steenrod algebra, Henn, Lannes and Schwartz in 1990 proved a further generalization of Quillen’s theorem:

Theorem 1.10. [27] Let K be an unstable algebra, and assume that K is of finite transcendence degree less than or equal to d. Then there exists a category s_{d}(K) and an F-isomorphism of unstable algebras:

K → lim_{s_{d}(K)} H^{∗}(V).

The category s_{d}(K) might now in some (extreme) cases, turn out to be infinite, in which case the above statement has to be interpreted correctly, taking profinite topologies into account. Their short and conceptual proof does not use the Adams-Wilkerson embedding theorem but instead uses three general facts about unstable modules (injectivity of H^{∗}(V), Lannes linearization principle and the characterization of nilpotent unstable modules). Furthermore their theorem has the Adams-Wilkerson theorem as an immediate consequence. Their category s_{d}(K) is different from the Quillen category, although related, the idea being that one allows for all compatible maps to and between elementary abelian groups. Especially the maps between the elementary abelian groups need not necessarily be monomorphisms as in the Quillen or Rector category.

The theorem of Henn, Lannes and Schwartz has the advantage over the theorem of Rector, that it does not require the unstable algebra to be noetherian. In algebraic topology, contrary to algebraic geometry or commutative ring theory, the rings at hand are often not noetherian, or at least not a priori known to be so (of course the main example here is the cohomology ring of the classifying space of a finite loop space—although the cohomology in this case by now is known to be noetherian by the seminal work of Dwyer and Wilkerson [17], the proof is of course nontrivial).

One immediate problem with the above theorem is that it requires a priori knowledge of the transcendence degree of K in order to determine the category s_{d}(K). We will in this thesis show an easily applicable formula for determining the transcendence degree of K.

Theorem 1.11. Let K be an unstable algebra of transcendence degree d(K) and assume that Hom_{K}(K, H^{∗}(V)) is finite for all V. If d(K) is finite then

log_{p} | Hom_{K}(K, H^{∗}(V))| ∼ d(K)v as v → ∞.

If d(K) is infinite then log_{p} | Hom_{K}(K, H^{∗}(V))| grows faster that linearly in v.

Here K denotes the category of unstable algebras and v = rk_{p} V. The numbers log_{p} | Hom_{K}(K, H^{∗}(V))| can be thought of as ‘unstable Betti numbers’ of the algebra. The formula can thus be seen as an analog of the classical formula relating the growth of

6 1. INTRODUCTION

the Betti numbers (the coefficients in the Poincaré series) and the transcendence degree of a noetherian connected graded F-algebra.

d(K) = min{k ∈ N_{0}| there exists a C such that dim_{F} K^{n} ≤ Cn^{k−1} for all n} .

The new formula is in general easily applicable and has for instance the classical Quillen’s theorem about the Krull dimension of the cohomology ring of a finite group as a consequence. To see this just note that

Hom_{K}(H^{∗}(P), H^{∗}(V)) = [BV, BP] = Rep(V, P),

when P is a finite p-group, using a theorem of Lannes for the first equality. Since it is easily seen that log_{p} | Rep(V, P)| ∼ rk_{p} P v for v → ∞, we are done.

1.12. Serre’s theorem and beyond. The above development in the algebraic theory of unstable modules has interesting applications for the cohomology of spaces. The main link for translating the algebra into topology is given by a theorem of Lannes [34, 35], generalizing earlier work of Miller [43, 44], stating that H^{∗}(map(BV, X)) = T_{V} (H^{∗}(X)) under mild restrictions on X. Here T_{V} is the so-called Lannes T-functor, the left adjoint of H^{∗}(V) ⊗ − in K. The theorem says that the cohomology of the mapping space map(BV, X) can be determined algebraically from H^{∗}(X). Looking at dimension zero the theorem says that [BV, X] = Hom_{K}(H^{∗}(X), H^{∗}(V)) (the reader might want to note the contrast to CP^{∞}, where we for instance have nontrivial ‘phantom’ maps CP^{∞} → S^{3} [23], which of course induce the trivial map on cohomology).

These developments in the theory of unstable modules and its connections to mapping spaces has allowed various generalizations the classical Serre’s theorem from 1953.

Theorem 1.13. [51] A 1-connected finite Postnikov system, E, with finitely generated homotopy groups and nontrivial cohomology, has cohomology in infinitely many dimensions.

We remind the reader that a Postnikov system is a space with only finitely many nontrivial homotopy groups.

Serre used an induction on the number of homotopy groups in the Postnikov system together with his newly invented spectral sequence and his calculation of the cohomology of Eilenberg-Mac Lane spaces to get estimates on the asymptotic size of Betti numbers of H^{∗}(E), showing that these grew rapidly with the dimension.

Serre’s theorem is called a duality theorem since it states a certain duality between homotopy and homology. It says that a space with finitely generated homotopy groups cannot have finite homotopy and homology at the same time. Serre’s theorem is often stated in the following dual way, equivalent to the one above: A 1-connected complex X with finitely generated homotopy groups, H̄^{∗}(X) ≠ 0, and H^{∗}(X) finite dimensional has homotopy groups containing p-torsion or of infinite order in infinitely many dimensions. This was actually the form stated by Serre in his original paper, where he also conjectured that the part ‘...or of infinite order’ could be left out (Serre actually only worked at p = 2 because he hadn’t worked out the cohomology of Eilenberg-Mac Lane spaces at odd primes yet, but once this was done (by Cartan [12]) the generalization was immediate [55]).

Serre’s theorem has been generalized in several ways by a number of people (Dwyer-Wilkerson [16], Lannes-Schwartz [36, 38], McGibbon-Neisendorfer [42], and the author [25]) all utilizing the theory of unstable modules. One of the main advantages of this approach is that it, by relating certain properties of the cohomology to questions about mapping spaces, gives a grip on how these properties behave with respect to fibrations— i.e. it turns traditional spectral sequence questions into long exact sequence questions. This allows one to get information about the ring structure and Steenrod algebra action on H^{∗}(E).

1. INTRODUCTION TO THE THEOREMS OF QUILLEN AND SERRE 7

McGibbon and Neisendorfer [42] verified in 1983 Serre’s conjecture that ‘...or of infinite order’ could be removed, and removed furthermore the finitely generated hypothesis on the homotopy, using the celebrated Miller’s theorem saying that map_{∗}(BZ/p, X) is (weakly) contractible when H^{∗}(X) is finite dimensional [44]. In 1985 Lannes and Schwartz [36] further weakened the assumption on X by only assuming H^{∗}(X) to be of finite type and locally finite over the Steenrod algebra^{4}. In 1988 Dwyer and Wilkerson [16] further relaxed the assumptions by replacing the locally finiteness assumption on H^{∗}(X) with a locally finiteness assumption on the indecomposables QH^{∗}(X), though then having to assume X 2-connected in order to avoid problems with CP^{∞}.

Likewise in 1988 Lannes and Schwartz [38] derived another generalization of Serre’s theorem going in a slightly different direction. They proved that if p = 2 and if X is a 1connected space with H^{∗}(X) of finite type, such that H^{∗}(X) contains no element of infinite height, then X has homotopy groups which are not uniquely 2-divisible in infinitely many dimensions (note how this becomes the old ‘containing 2-torsion or of infinite order’, when the homotopy groups are assumed to be finitely generated). They furthermore point out that SU, the infinite special unitary group serves as a counterexample if the assumption ‘not uniquely 2-divisible’ is replaced by ‘containing 2-torsion’. To see this note that SU has no torsion in its homotopy by Bott periodicity, but has a cohomology ring given by an exterior algebra on infinitely many generators.

Our results generalize the results in [38]. In the papers above ‘Serre’s theorem’ is stated in its ‘dual form’. We will however return to using the other version. One reason for this is that we feel that it for our purpose is more natural, the other being that it fits better with Quillen’s theorem.

Both Quillen’s and Serre’s theorem are theorems about the cohomology ring of finite Postnikov systems. To get a common ground for these two theorems our main object of study will be the mod p cohomology ring of a connected, nilpotent, finite Postnikov system E with finite π_{1}(E) and H^{∗}(E) of finite type. Remember that a space is called nilpotent if π_{1} is a nilpotent group and π_{1} acts nilpotently on the higher homotopy groups, that is there exists a finite filtration of each of the higher homotopy groups, such that π_{1} acts trivially on the filtration quotients. Especially, K(G, 1) is a nilpotent spaces when G is a finite p-group, and likewise a 1-connected space is nilpotent. Thus a space E as above is general enough to include the spaces of both theorems.

Quillen’s theorem especially tells us that the cohomology ring of a finite group always is of positive, finite transcendence degree. It is natural to ask how these facts generalize to finite Postnikov systems. Our knowledge of the cohomology of Eilenberg-Mac Lane spaces tells us that at least here the cohomology is always of positive transcendence degree. It also reveals that the transcendence degree can not always be finite since this holds for very few Eilenberg-Mac Lane spaces (cf. e.g. [51, 18, 12]).

By using the formula for the transcendence degree given in terms of the ‘unstable Betti numbers’ of Theorem 1.11 together with a determination of those numbers by an induction on the number of nontrivial homotopy groups in the Postnikov system, we are able to derive the following vast generalization of Serre’s theorem, also generalizing the results in [38]:

Theorem 1.14. Let E be a connected nilpotent finite Postnikov system with finite π_{1}(E). Assume that H^{∗}(E) is of finite type and that H̄^{∗}(E) ≠ 0. Then H^{∗}(E) has positive transcendence degree. It has finite transcendence degree iff it is F_{p}-equivalent to the classifying space of a p-toral group.

4Locally finite over the Steenrod algebra means that Ax is finite dimensional for all x ∈ H^{∗}(X), where A denotes the Steenrod algebra

8 1. INTRODUCTION

Here we recall that two spaces are F_{p}-equivalent iff they are equivalent in the quotient category of the category of spaces, obtained by formally inverting all maps inducing isomorphism on mod p (co)homology. Especially two F_{p}-equivalent spaces have the same mod p cohomology. Furthermore we recall that a p-toral group is a group G which fits into an extension of the form 1 → T^{n} → G → P → 1, where T^{n} is an n dimensional torus, and P is a finite p-group.

The result shows that just imposing a very mild finiteness restriction on the cohomology of a finite Postnikov system, requiring it to be of finite transcendence degree, puts us right back in the realm of classifying spaces of groups.

In the case where H^{∗}(E) is of finite transcendence degree we can still use Theorem 1.11 to get a concrete formula for its transcendence degree in terms of the homotopy groups of E together with ‘gluing data’, which has Quillen’s theorem about the Krull dimension of the cohomology ring of a finite group as a special case.

2. INTRODUCTION 9

2. Introduction

In 1953 Serre showed his celebrated result that a 1-connected finite Postnikov system E with finitely generated homotopy always has homology in infinitely many dimensions, using his newly invented spectral sequence [51]. His methods, however, although revealing the asymptotic size of Betti numbers (the coefficients in the Poincaré series) of H^{∗}(E; F_{p}), did not in general give information about the ring or A-module structure of H^{∗}(E; F_{p}) (here A denotes the Steenrod algebra). Serre’s theorem has since then been generalized in several ways by a number of people (Dwyer-Wilkerson [16], Lannes-Schwartz [36, 38], McGibbon-Neisendorfer [42]) all utilizing the theory of unstable modules over the Steenrod algebra as developed by Lannes, Schwartz and others. One of the main advantages of this approach is that, by relating certain properties of the cohomology to questions about mapping spaces, it gives a grip on how these properties behave with respect to fibrations—i.e. it turns traditional spectral sequence questions into long exact sequence questions. This paper is a contribution along these lines.

We offer the following two main theorems:

Theorem 2.1. Let E be a connected nilpotent finite Postnikov system with finite π_{1}(E). Assume that H^{∗}(E; F_{p}) is of finite type and that H̄^{∗}(E; F_{p}) ≠ 0. Then H̄^{∗}(E; F_{p}) contains an element of infinite height.

Theorem 2.2. Let E be a connected nilpotent finite Postnikov system with finite π_{1}(E). Assume that H^{∗}(E; F_{p}) is of finite type. Then H^{∗}(E; F_{p}) has finite transcendence degree iff E is F_{p}-equivalent to a space E′ fitting into a principal fibration sequence of the form

CP^{∞} × ⋯ × CP^{∞} → E′ → K(P, 1) ,

where P is a finite p-group.

Here we define the transcendence degree, d(K), of a graded algebra K as the maximal number of homogeneous algebraically independent elements in K. When K is connected noetherian, d(K) is equal to the Krull dimension of K. Note that Theorem 2.1 can be reformulated as saying that the transcendence degree of H^{∗}(E; F_{p}) is always positive. Theorem 2.1 was previously known in the case p = 2 for E 1-connected by work of Lannes and Schwartz [38], but was actually rediscovered independently by the author and formed the starting point for this work.

From now on H^{∗}(X) denotes the mod p cohomology of X for some fixed but arbitrary prime p. In the rest of this introduction we use some standard notation concerning unstable modules over the Steenrod algebra. In the next section we briefly introduce these concepts, for a general reference see e.g. Schwartz [49].

The key topological result needed in proving the results about H^{∗}(E) is the following asymptotic growth formula in rk_{p} V, where V is an elementary abelian group:

log_{p} |[BV, K(π_{n}(E), n)]| ∼^{<} log_{p} |[BV, E]| ∼^{<} C log_{p} |[BV, K(π_{n}(E), n)]| ,

where n is the dimension of the highest homotopy group which is not uniquely p-divisible.

The above growth formula relates to the algebra structure of H^{∗}(E) by the following theorem of Lannes, generalizing earlier work of Miller [43]:

Theorem 2.3. [34, 35] Let X be a connected space. Suppose that X is nilpotent with finite π_{1}(X) and H^{∗}(X) of finite type. Then the natural map f ↦ f^{∗}, [BV, X] → Hom_{K}(H^{∗}(X), H^{∗}(V)) is a bijection.

Here K denotes the category of unstable algebras over the Steenrod algebra.

We have the following theorem of Lannes and Schwartz:

10 1. INTRODUCTION

Theorem 2.4. [48, 49] Let M be an unstable module over the Steenrod algebra. Then the following two conditions are equivalent:

1. M is nilpotent.

2. Hom_{U}(M, H^{∗}(V)) = 0 for all elementary abelian p-groups V.

Here U denotes the category of unstable modules over the Steenrod algebra.

The Lannes linearization principle [35]

Hom_{U}(K, H^{∗}(V))′ ≃ F_{p}^{Hom_{K}(K,H^{∗}(V))} ,

where′ denotes the (continuous) vector space dual, now immediately leads to an analogous theorem concerning unstable algebras.

Theorem 2.5. [37] For an unstable algebra K with 1 ≠ 0 the following conditions are equivalent:

1. Hom_{K}(K, H^{∗}(V)) = 0 for all elementary abelian p-groups V.

2. K is F -equivalent to the trivial unstable algebra F_{p}.

3. K̄ is nilpotent as an unstable module over the Steenrod algebra.

4. K̄ is a nil ideal in K, i.e. it consists of nilpotent elements.

The equivalence of 2., 3., and 4. are immediate consequences of the definitions. Theorem 2.1 now follows from the growth formula and the preceding general theorems.

It is worth noting that in the preceding theorems one has to consider elementary abelian groups V of an arbitrary size. Restriction to e.g. V = Z/p would not be enough, which is seen by for example setting K = H^{∗}(K(Z, 3)). This makes the property ‘nilpotent’ a bit less well behaved than for example the property ‘locally finite’.

To prove Theorem 2.2 we examine work of Henn, Lannes and Schwartz [27] on the structure of unstable algebras and look at it in terms of growth properties. This leads to a characterization of the transcendence degree of an unstable algebra K in terms of the growth of log_{p} | Hom_{K}(K, H^{∗}(V))| in v = rk_{p} V, under mild restrictions on K. More precisely we prove:

Theorem 2.6. Let K be an unstable algebra of transcendence degree d(K) and assume that Hom_{K}(K, H^{∗}(V)) is finite for all V. If d(K) is finite then log_{p} | Hom_{K}(K, H^{∗}(V))| ∼ d(K)v. If d(K) is infinite then log_{p} | Hom_{K}(K, H^{∗}(V))| grows faster than linearly in v.

Theorem 2.6 is a powerful tool for calculating the transcendence degree of unstable algebras. To demonstrate this we give a two line proof of Quillen’s theorem in the pivotal finite p-group case, by proving that the Krull dimension of H^{∗}(P) is equal to rk_{p} P for every finite p-group P.

The numbers log_{p} | Hom_{K}(K, H^{∗}(F_{p}^{⊕n}))| (which are not necessarily integers) can in some sense be viewed as an unstable algebra alternative to the traditional Betti numbers of a graded algebra, β_{n}(K) = dim_{F_{p}} K^{n}. One should think of the log_{p} as a replacement of dim_{F_{p}} nessesary since Hom_{K}(K, H^{∗}(F_{p}^{⊕n})) in general is not an F_{p}-vector space. Theorem 2.6 shows that an analog of the well known formula relating the growth of the Betti numbers and the transcendence degree (Krull dimension) of a noetherian connected graded algebra holds for these new numbers, now with much weaker restrictions on the unstable algebra.

Applying Theorem 2.6 to the obtained growth formula for log_{p} |[BV, E]|, where E is a finite Postnikov system, and doing some work now leads to Theorem 2.2.

We also include a section where we see that the above results imply that the nconnected cover of a finite complex always has infinite LS category, generalizing earlier partial results of McGibbon and Møller.

2. INTRODUCTION 11

Finally we discuss and conjecture generalizations of the above results to polyGEMs, correcting a small mistake put forth in [20].

In the proofs of the theorems we several times need a small but useful fact about nilpotent actions. Since we have been unable to find this fact stated in the literature, and believe that it ought to be better known, we give a proof in a short appendix.

Acknowledgment: I am grateful to The Fields Institute for giving me the opportunity to participate in the ‘Emphasis year in Homotopy Theory 95/96’, and especially to the organizers and participants for making the stay so enjoyable and profitable. A special thanks to K. Andersen, W. Chachólski, P. Lambrechts, J. Scherer and B. Schuster for so many hours of interesting discussions. Also thanks to R. Kane for introducing me to the beautiful theory of Lannes and Schwartz which this paper is based on.

I would like to thank W. Chachólski for helpful suggestions and references. Likewise thanks to my advisor J. Møller for his support and for suggesting improvements in the presentation. Thanks to H. Minh Le for directing my attention to the conjecture of E. Dror Farjoun. Finally thanks to C. McGibbon and J. Møller whose questions related to Theorem 2.1 got me interested in the cohomology of finite Postnikov systems in the first place.

12 1. INTRODUCTION

3. Notation

By a space we, for simplicity, mean an object in the pointed homotopy category Ho_{∗} of CW-complexes. The bracket [−, −] means a free homotopy class of maps (Theorem 2.3 is the main reason why this is convenient.) We will also need to refer to pointed homotopy classes of pointed maps which we denote by [−, −]_{pt}.

We define the p-rank, rk_{p}, of a group G as the maximal rank of an elementary abelian subgroup V contained in G. At all times we employ the convention that v = rk_{p} V, for the elementary abelian group V in question.

When talking about the asymptotic behavior of some sequence of numbers, we mean by the symbol ∼^{<} that for all ϵ > 0 there exists an N such that (left-hand side) ≤ (1+ϵ)(righthand side) for all n ≥ N.

In our notation involving unstable modules over the Steenrod algebra, we follow the standard notation used in e.g. Schwartz [49]. We quickly review the basic definitions:

Definition 3.1. An unstable module M is a graded module over the Steenrod algebra A satisfying the following instability conditions:

• If p = 2 then Sq^{i} x = 0 for i > |x|.

• If p > 2 then β^{e}P^{i}x = 0 for e + 2i > |x|.

Let U denote the category whose objects are unstable modules and whose morphisms are (degree 0) A-module maps. Note that this is an abelian category.

Definition 3.2. An unstable algebra K is an unstable module equipped with two maps η : F_{p} → K and μ : K ⊗ K → K making K into a commutative unital F_{p}-algebra such that

• μ is A-linear (i.e. the Cartan formula holds.)

• Sq^{|x|}x = x^{2} for any x ∈ K if p = 2; P^{|x|/2}x = x^{p} for any x ∈ K of even degree if p > 2.

Let K denote the category whose objects are unstable algebras, and whose morphisms are those (degree 0) F_{p}-algebra maps which are A-linear.

Furthermore, throughout this paper, for convenience we make the standing assumption that all unstable algebras has 1 ≠ 0, i.e. we exclude the unstable algebra 0.

Definition 3.3. We say that an unstable module is nilpotent if the following holds:

• If p = 2 then for every x ∈ M there exists an integer N such that Sq^{2^{N−1}|x|} Sq^{2^{N−2}|x|} ⋯ Sq^{|x|}x = 0.

• If p > 2 then for every x ∈ M where |x| is even there exists an integer N such that P^{p^{N−1}|x|/2}P^{p^{N−2}|x|/2} ⋯ P^{|x|/2}x = 0.

Let N il denote the full subcategory of U whose objects are nilpotent modules.

Note that this definition extends the usual definition if I is an ideal in an unstable algebra, in the sense that we have that I lies in N il iff it is nil as an ideal, that is if all its elements are nilpotent. Beware however, for a non-noetherian algebra a nil ideal I need not be nilpotent (i.e. there exists n such that I^{n} = 0), which makes the terminology slightly ambiguous. Note also that an unstable algebra with 1 ≠ 0 of course can not be nilpotent.

We need a last definition:

Definition 3.4. A morphism of unstable algebras φ : K → K′ is said to be an F - monomorphism if the kernel of φ in the (abelian) category of unstable modules is nilpotent as an unstable module. We define F-epimorphism and F-isomorphism analogously.

3. NOTATION 13

This definition coincides with Quillen’s original definition of an F-isomorphism, and is also the same as what is sometimes referred to as a (purely) inseparable isogeny.

Finally note that everything in this paper only depends on the space E up to F_{p}- equivalence (E here, as everywhere, connected nilpotent with finite π_{1}.) As noted by Miller [44] map_{∗}(BV, E) ≃ map_{∗}(BV, Ê_{p}), and of course H^{∗}(E; F_{p}) ≃ H^{∗}(Ê_{p}; F_{p}) – Likewise we have that [BV, E] ≃ [BV, Ê_{p}] [7, 5]. Thus we could reformulate everything by writing: “Let E be F_{p}-equivalent to...” – For the sake of clarity we refrain from doing so and instead leave reformulations like that to the reader.

14 1. INTRODUCTION

CHAPTER 2 The generalized Quillen’s theorem of Henn, Lannes and Schwartz

1. The generalized Quillen’s theorem

We will in this section prove a structural theorem for the category of unstable algebras due to Henn, Lannes and Schwartz [27], which can be seen as a vast generalization of Quillen’s theorem. We will furthermore show how this gives the Adams-Wilkerson embedding theorem as a corollary.

We include this section mainly to prove those results from [27] that we will be needing later on and thus to make this thesis as self contained as possible. Secondly we hope to demonstrate to the reader how this new approach of Henn, Lannes and Schwartz to Quillen’s theorem gives a short ‘tautological’ proof of a vast generalization of Quillen’s theorem, granted some fact about the category of unstable algebras. Indeed all that goes into the proof apart from linear algebra and basic category theory (and a small calculation of an invariant ring) is the following three deep but fundamental theorems from the theory of unstable modules: Injectivity of H^{∗}(V) in U, Lannes linearization principle and the characterization of nilpotent unstable modules.

Our proof is basically a reproduction of the proof given in [27], with a few exceptions. We have substantially rearranged the order of which the facts are presented in order to get a more linear presentation, and to single out the facts which we’ll need later. Also we have included a proposition which precisely states some equivalent definitions of the transcendence degree of a type of functors we’ll be considering. At least part of this proposition is used implicitly in [27] in their proof of the main theorem, but only fully justified in the later section about the notion of the kernel of an element in an End V -set.

Definition 1.1. For any unstable algebra K define its transcendence degree d(K) as the maximal number of algebraically independent homogeneous elements in K.

Remark 1.2. If K is a connected graded noetherian algebra, the maximal number of algebraically independent elements, d(K), will be finite. In this case the number d(K) will coincide with the Krull dimension of K, and we can furthermore choose d(K) algebraically independent elements such that K will be finite over the algebra spanned by those elements. If K is an integral domain, d(K) will be the same as the classical transcendence degree of the field of fractions of K. A nice and graded proof of these standard facts can be found in [4].

In the following we will need to refer to elementary abelian groups of different rank. Therefore we will sometimes equip the elementary abelian group V with a subscript, which will then indicate the rank of V.

Definition 1.3. Let E denote the category of elementary abelian p-groups, i.e. of finite dimensional F_{p} vector spaces. Let PS denote the category of profinite sets and let G be the category of functors E → (PS)^{op}. Note that we can view G^{op} as the category of contravariant functors E → PS.

16 2. THE GENERALIZED QUILLEN’S THEOREM

One should realize that objects in G contain a rich structure. This stems from the fact that not only do we to each vector space associate a profinite set, but we do this in a natural way, which, loosely speaking, ties the profinite sets together.

By inverting all F-isomorphisms in K we obtain a quotient category of K which will be denoted by K/N il.

Let g : K → G be given by g(K)(V) = Hom_{K}(K, H^{∗}(V)) for all V. Here we will equip Hom_{K}(K, H^{∗}(V)) with the profinite topology induced by writing

Hom_{K}(K, H^{∗}(V)) = Hom_{K}(colim_{α} K_{α}, H^{∗}(V)) = lim_{α} Hom_{K}(K_{α}, H^{∗}(V)) ,

where α runs over the finitely generated A-subalgebras K_{α} of K.

By Theorem 2.4 and the Lannes linearization principle we have that a map between unstable algebras K → K′ is an F-monomorphism (resp. F-epi) iff g(K′)(V) → g(K)(V) is a surjective (resp. injective) map of sets for all V. This show that g induces a a faithful functor K/N il → G (likewise denoted g).

Define a : G → K: a(G) = Hom_{G}(H^{∗}(−), G(−)) = Hom_{G^{op}}(G(−), H^{∗}(−)), where the unstable algebra structure comes from that on H^{∗}(V). It is easy to see that a is right adjoint to g:

Hom_{G}(g(K), G) = Hom_{G}(Hom_{K}(K, H^{∗}(−)), G(−))
= Hom_{G^{op}}(G(−), Hom_{K}(K, H^{∗}(−)))
= Hom_{K}(K, Hom_{G^{op}}(G(−), H^{∗}(−)))
= Hom_{K}(K, a(G))

Definition 1.4. Let PS − End V_{d} denote the category whose objects are profinite sets equipped with a continuous right action of the monoid End V_{d} and whose morphisms are maps of profinite sets respecting the End V_{d}-action.

To each G ∈ G^{op} and each vector space V_{d} we can associate a profinite right End V_{d}-set G(V_{d}). Namely define the right action of End V_{d} on G(V_{d}) by to each (s, φ) ∈ G(V_{d}) × End V_{d} associating sφ = G(φ)s. This gives us a ‘restriction functor’

e_{d} : G^{op} → PS − End V_{d} .

Likewise we can define an ‘induction functor’

i_{d} : PS − End V_{d} → G^{op} : S ↦ (W ↦ S ×_{End V_{d}} Hom(W, V_{d})) ,

where S ×_{End V_{d}} Hom(W, V_{d}) denotes the coequalizer in the category of profinite sets (coinciding with the coequalizer in the category of topological spaces) of the action of End V_{d} on S and Hom(W, V_{d}) respectively.

Now note that the canonical map G(V_{d})×Hom(W, V_{d}) → G(W) induces a well defined map on the coequalizer G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W). Therefore we can to each map S → G(V_{d}) of profinite End V_{d}-sets and each W associate a map

S ×_{End V_{d}} Hom(W, V_{d}) → G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W) .

Likewise a morphism in G^{op}, i_{d}(S) → G of course induces a map S → G(V_{d}) of profinite End V_{d}-sets. Since these operations are inverses of each other we get that Hom_{G^{op}}(i_{d}(S), G) = Hom_{PS−End V_{d}}(S, e_{d}(G)), naturally in G ∈ G^{op}, S ∈ PS − End V_{d}, so i_{d} is left adjoint of e_{d}. It is immediate that 1_{PS−End V_{d}} →^{≃} e_{d} ∘ i_{d}, given by the unit of the adjunction, so we get an embedding of PS − End V_{d} as a subcategory of G^{op}. Define s_{d} = e_{d}^{op} ∘ g, where e_{d}^{op} : G → (PS − End V_{d})^{op} is the opposite functor of e_{d}.

We can define a functor b_{d} : (PS − End V_{d})^{op} → K by setting

b_{d}(S) = Hom_{PS−End V_{d}}(S, H^{∗}(V_{d})),

1. THE GENERALIZED QUILLEN’S THEOREM 17

where the unstable algebra structure comes from H^{∗}(V_{d}), and H^{∗}(V_{d}) is equipped with the canonical End V_{d}-action. It follows directly from the definitions that s_{d} is left adjoint to b_{d} since

Hom_{(PS−End V_{d})^{op}}(s_{d}(K), S) = Hom_{PS−End V_{d}}(S, Hom_{K}(K, H^{∗}(V_{d})))
= Hom_{K}(K, Hom_{PS−End V_{d}}(S, H^{∗}(V_{d})))
= Hom_{K}(K, b_{d}(S))

Like g, s_{d} induces a functor on K/N il which will also be denoted s_{d}. Furthermore, we can when convenient view b_{d} and a as functors to K/N il.

Let us write the introduced functors in a (non-commutative!) diagram:

\xymatrix{K \ar[r]^{g} \ar[dr]^{s_{d}} & G \ar[l]^{a} \ar[d]^{e_{d}^{op}} \\  & (PS − End V_{d})^{op} \ar[ul]^{b_{d}} \ar[u]^{i_{d}^{op}}}

Lemma 1.5. [27] The morphism (i_{d}∘e_{d})(G) → G given by the counit of the adjunction is a monomorphism in G^{op} for all G ∈ G^{op}, i.e. we have that (i_{d} ∘ e_{d})(G)(V) → G(V) is an injective map of profinite sets for all V.

Proof. We want to prove that G(V_{d}) ×_{End V_{d}} Hom(V, V_{d}) → G(V) is injective for all V. Suppose therefore that φ, φ′ ∈ Hom(V, V_{d}), s, s′ ∈ G(V_{d}) and G(φ)s = G(φ′)s′.

Note that we can choose commuting projections π, π′ ∈ End V_{d}, such that ker π = ker φ and ker π′ = ker φ′ (to see this, write V = V_{0} ⊕ V_{1} ⊕ V_{2} ⊕ V_{3} such that V_{0} = ker φ ∩ ker φ′, V_{0} ⊕ V_{1} = ker φ and V_{0} ⊕ V_{2} = ker φ′, and let π be the projection onto V_{2} ⊕ V_{3} (in this decomposition of V!) and let π′ be the projection onto V_{1} ⊕ V_{3}). Write π = ψφ, π′ = ψ′φ′, for some ψ, ψ′ ∈ Hom(V_{d}, V).

We now calculate in G(V_{d}) ×_{End V_{d}} Hom(V, V_{d}):

(s, φ) = (s, φπ) = (s, φψφ) =
(G(φψ)s, φ) = (G(φ′ψ)s′, φ) = (s′, φ′ψφ) =
(s′, φ′π′π) = (s′, φ′ππ′) = (s′, φ′ψφψ′φ′) =
(G(φ′ψφψ′)s′, φ′) = (G(φψφψ′)s, φ′) = (G(φψ′)s, φ′) =
(G(φ′ψ′)s′, φ′) = (s′, φ′ψ′φ′) = (s′, φ′)

We thus get that G(V_{d}) ×_{End V_{d}} Hom(V, V_{d}) → G(V) is injective as wanted.

This gives us a filtration of G which we will later see turns out to coincide with the filtration of K/N il by transcendence degree.

Note that we have a natural equivalence of functors b_{d} ≃ a ∘ i_{d}^{op} since

b_{d}(S) = Hom_{PS−End V_{d}}(S, H^{∗}(V_{d}))
= Hom_{G^{op}}(i_{d}^{op}(S), H^{∗}(−))
= a(i_{d}^{op}(S))

where the second equality is because PS−End V_{d} is a full subcategory in G^{op} by Lemma 1.5.

Definition 1.6. Let G ∈ G^{op}. Define the transcendence degree of G as

d(G) = min{d | ((i_{d} ∘ e_{d})G)(W) → G(W) is bijective for all W } ,

where we take d(G) = ∞ if none such d exists. For an element G ∈ G we define the transcendence degree by viewing it as lying in G^{op}.

18 2. THE GENERALIZED QUILLEN’S THEOREM

We shall need some alternative ways of expressing the transcendence degree of an object in G^{op}. These can be found implicit in [27]. Since they are important in their own right we find it useful to state them explicitly. First a useful definition:

Definition 1.7. Let S be an End V_{d}-set, and let s ∈ S. Define the rank of s ∈ S as

rk s = min{rk φ| where s = tφ for some t ∈ S, φ ∈ End V_{d}} .

We say that s is regular if rk s = d, i.e. if s = tφ implies that φ is a regular (invertible) matrix.

Proposition 1.8. We have the following formula for the transcendence of G ∈ G^{op}:

d(G) := min{d|((i_{d} ∘ e_{d})G)(W) → G(W) is bijective for all W }
= max{rk s|s ∈ G(V_{d}) for some d}
= max{d|G(V_{d}) contains a regular element} .

Proof. We start by proving that

max{d|G(V_{d}) contains a regular elt.} = max{rk s|s ∈ G(V_{d}) for some d}. (1.1)

First note that ‘≤’ is obvious. To prove ‘≥’ let s ∈ G(W) and suppose that rk s = d. We can thus choose π ∈ End W, t ∈ G(W) such that s = tπ and rk π = d. By changing t we can assume that π is a projection. Let ρ : W → πW and i : πW → W be the canonical projection and inclusion associated to π. Now set s′ = G(i)s ∈ G(πW) and note that G(ρ)s′ = G(ρ)G(i)s = G(ρ)G(i)G(π)t = G(πiρ)t = s. We claim that s′ is regular. Suppose we have φ ∈ End(πW), u ∈ G(πW) such that s′ = uφ. Then we have

s = G(ρ)s′ = G(ρ)G(φ)u = G(ρ)G(φ)G(i)G(ρ)u = G(iφρ)G(ρ)u = (G(ρ)u)(iφρ)

so φ has to be regular, since rk s = d. This shows the wanted inequality.

We now prove that the number given by (1.1) actually coincides with the transcendence degree. We will first see that it is less than or equal to the transcendence degree. Suppose therefore that we have d such that G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W) is bijective for all W, and let s ∈ G(W) be arbitrary. Choose t ∈ G(V_{d}) and φ ∈ Hom(W, V_{d}) such that G(φ)t = s. Since the rank of φ : W → V_{d} must be less that or equal to d, we can choose a projection π ∈ End W of rank less that or equal to d such that φπ = φ. But this gives us that s = G(φ)t = G(π)G(φ)t = (G(φ)t)π, which shows that rk s ≤ d, as wanted.

We finish the proof by showing d(G) ≤ max{d|G(V_{d}) contains a regular element}. Let d be the maximal number such that G(V_{d}) contains a regular element (if there exists regular elements in infinitely many dimension we are done). We want to prove that G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W) is surjective for all W. Let s ∈ G(W) be arbitrary and set n = rk s. Note that n ≤ d by (1.1). Choose an element π ∈ End W of rank n and t ∈ G(W) such that s = tπ. We can by changing the choice of t assume that π is a projection. Letting ρ : W → πW and i : πW → W denote the canonical projection and inclusion we get that s = G(π)t = G(πiρ)t = G(ρ)(G(πi)t). Therefore s is in the image of G(V_{n})×_{End V_{n}}Hom(W, V_{n}) which implies that s is in the image of G(V_{d})×_{End V_{d}}Hom(W, V_{d}), since n ≤ d. This completes the proof, since G(V_{d})×_{End V_{d}}Hom(W, V_{d}) → G(W) is injective by Lemma 1.5.

Example 1.9. Let X_{n} be the End V_{n}-set consisting of two elements x_{1} and x_{0}, with the End V_{n}-action given as follows:

x_{0}α = x_{0} for all α ∈ End V_{n}

x_{1}α = \{ \begin{matrix}x_{1} & for α ∈ Aut V_{n} \\ x_{0} & for α ∉ Aut V_{n} . \end{matrix}

1. THE GENERALIZED QUILLEN’S THEOREM 19

We have that x_{1} has rank n and x_{0} has rank 0. The End V_{n}-set X_{n} is thus the smallest End V_{n}-set containing a regular element. It has the universal property, that any End V_{n}-set containing a regular element surjects onto X_{n} as an End V_{n}-set.

We will determine the unstable algebra b_{n}(X_{n}) ⊂ H^{∗}(V) ⊕ H^{∗}(V). Assume first that p = 2. Note that it is enough to restrict attention to homogeneous elements, since all maps preserve grading. In degree zero it is obvious that all elements have to be of the form α(1, 1) ∈ H^{∗}(V) ⊕ H^{∗}(V).

For an arbitrary homogeneous element of positive degree, (f, g), we have to have (f, g) = (f, 0) since x_{0}0 = x_{0}. Let {e_{1}, …, e_{n}} be the canonical basis for V_{n} = F_{2}^{n}, and let ρ_{i} denote the projection onto the subspace spanned by {e_{1}, …, ê_{i}, …, e_{n}}. Write H^{∗}(V) = F_{2}[y_{1}, …, y_{n}] where y_{i} is identified with e_{i}^{∗} under the identification H^{∗}(V) = F_{2}[V], where ∗ denotes the vector space dual. Since x_{1}ρ_{i} = x_{0} we conclude y_{i}|f for all i. Thus ω|f, where ω = y_{1} ⋯ y_{n}, so f = ωh. Since x_{1}g = x_{1} for all g ∈ Aut V_{n}, we conclude that f, and thus h, is invariant under the action of Aut V_{n}. Elements of this form (ωh, 0) on the other hand satisfies the conditions for lying in the inverse limit. We have thus shown that one can naturally identify

b_{n}(X_{n}) = F_{2} ⊕ ωF_{2}[y_{1}, …, y_{n}]^{Aut V_{n}} ⊂ H^{∗}(V)

(where the direct sum naturally is as modules only).

The algebra F_{2}[y_{1}, …, y_{n}]^{Aut V_{n}} is of course the well known Dickson algebra, D_{n} = F_{2}[d_{0}, …, d_{n−1}], |d_{i}| = p^{n} − p^{i} (cf. e.g. [61]). Especially b_{n}(X_{n}) is of Krull dimension (transcendence degree) d (the reader might want to note that it is really not nessesary to appeal to knowledge of the Dickson algebra in order to determine the transcendence degree of b_{n}(X_{n})—any invariant ring F_{2}[y_{1}, …, y_{n}]^{G}, G a finite group, has transcendence degree n since F_{2}[y_{1}, …, y_{n}]^{G} is noetherian and F_{2}[y_{1}, …, y_{n}] is finite over F_{2}[y_{1}, …, y_{n}]^{G} by the Hilbert-Noether theorem (cf. [4]), so d(F_{2}[y_{1}, …, y_{n}]^{G}) = d(F_{2}[y_{1}, …, y_{n}]) = n).

For p odd we will only do the analysis modulo nilpotent elements, since this is all we are really interested in anyway. In this case we have H^{∗}(V) = F_{p}[y_{1}, …, y_{n}] ⊗ ∧(z_{1}, …, z_{n}), |y_{i}| = 2, |z_{i}| = 1. Thus H^{∗}(V)/\sqrt{0} = F_{p}[y_{1}, …, y_{n}] and so the analysis proceeds as before revealing

b_{n}(X_{n})/\sqrt{0} = F_{p} ⊕ ωF_{p}[y_{1}, …, y_{n}]^{Aut V_{n}} ⊂ H^{∗}(V),

but now with |y_{i}| = 2.

Remark 1.10. A profinite End V -set is said to be free if it is isomorphic to a profinite End V -set of the form T ×End V, where T is a profinite set, and End V acts on the second factor of T × End V.

Every profinite End V -set has a canonical free representation, given as the coequalizer in PS −End V (here coinciding with the coequalizer in PS) of the following double arrow:

S × End V × End V → S × End V
(s, α, β) ↦ (sα, β)
(s, α, β) ↦ (s, αβ)

Proposition 1.11. We have a natural equivalence of functors i_{d}^{op} →^{≃} g ∘ b_{d}.

Proof. Since g and a are adjoints we have a natural unit of the adjunction id_{G} → g∘a, and thus a natural transformation i_{d}^{op} → g ∘ a ∘ i_{d}^{op} ≃ g ∘ b_{d}. We claim that this map is an equivalence of functors.

By definition b_{d} turns colimits into limits. By the Lannes linearization principle and the injectivity of H^{∗}(V) in U, g turns finite limits into colimits. Thus g ∘ b_{d} especially preserves coequalizers. Since it is also clear that i_{d}^{op} preserves coequalizers, and since every End V_{d}-set admits a free resolution, it is thus enough to check that i_{d}^{op}(S) → (g ∘ b_{d})(S) is an isomorphism for every free End V_{d}-set S.

20 2. THE GENERALIZED QUILLEN’S THEOREM

Suppose that S = T × End V_{d}. Writing out the definitions we get that the map we need to check is an isomorphism is the canonical map:

T × Hom(V, V_{d}) → Hom_{K}(Hom_{PS}(T, H^{∗}(V_{d})), H^{∗}(V))  (1.2)

Since g turns finite limits into colimits and since Hom(V, V_{d}) = Hom_{K}(H^{∗}(V_{d}), H^{∗}(V)) the isomorphism is immediate in the case where T is finite.

We now want see that this extends to the profinite case, since both the right hand side and the left hand side of (1.2) preserves topological limits in T.

To see this let T be a profinite and represent it in the canonical way, i.e. as a limit of its finite topological quotient spaces T = lim_{α} T_{α}.

The only thing that needs checking is that the following map is an isomorphism in K.

φ : colim_{α} Hom_{PS}(T_{α}, H^{∗}(V_{d})) → Hom_{PS}(lim_{α} T_{α}, H^{∗}(V_{d}))

The map is obviously a monomorphism. To see that it is an epimorphism, let f ∈ Hom_{PS}(lim T_{α}, H^{∗}(V_{d})). Since T is compact by the definition of the profinite topology, we have that f(T) is compact, and hence finite. Thus we can define the finite quotient set (T/ ∼) where we set t_{1} ∼ t_{2} if f(t_{1}) = f(t_{2}). Thus f induces a map f̃: (T/ ∼) → H^{∗}(V_{d}). By identifying f̃ with its image in the colimit we can thus write φ(f̃) = f, which shows the wanted isomorphism, and concludes the proof of the proposition.

Proposition 1.12. We have that d(K) = d(g(K)).

Proof. Note that since g is a faithful functor K/N il → G we have that

d(g(K)) = min{d|(i_{d} ∘ e_{d})(g(K)) → g(K) is an isomorphism in G^{op}}
= min{d|(g ∘ b_{d} ∘ s_{d})(K) → g(K) is an isomorphism in G^{op}}
= min{d|K → (b_{d} ∘ s_{d})(K) is an F-isomorphism}

where we have used Proposition 1.11 in the rewriting.

This immediately show that d(K) ≤ d(g(K)), since it is obvious that d(b_{d}(S)) ≤ d for all S ∈ PS − End V_{d}.

Conversely, suppose that g(K)(V_{n}) contains a regular element. Then we can by Example 1.9 choose an epimorphism i PS − End V_{n}, g(K)(V_{n}) → X_{n}. But this gives us a monomorphism b_{n}(X_{n}) → (b_{n} ∘ e_{n}^{op} ∘ g)(K) = (b_{n} ∘ s_{n})(K) in K. Thus d((b_{n} ∘ s_{n})(K)) ≥ n. But since g((b_{n} ∘ s_{n})(K)) = (i_{d} ∘ e_{d})(g(K)) → g(K) is a monomorphism in G^{op} by Lemma 1.5, K → (b_{n} ∘ s_{n})(K) is an F-epimorphism, so we conclude that d(K) ≥ d((b_{n} ∘ s_{n})(K)) ≥ n. By Proposition 1.8 his shows that d(K) ≥ d(g(K)) as wanted.

Definition 1.13. Let K_{d} denote the full subcategory of K having as objects all unstable algebras of transcendence degree less that or equal to d.

We are now ready to prove the main theorem of this section:

Theorem 1.14. [27] The functor

s_{d} : K_{d}/N il → (PS − End V_{d})^{op}, K ↦ Hom_{K}(K, H^{∗}(V_{d}))

induces an equivalence of categories K_{d}/N il → (PS − End V_{d})^{op}. The inverse equivalence is induced by the functor

b_{d} : (PS − End V_{d})^{op} → K_{d}/N il, S ↦ Hom_{PS−End V_{d}}(S, H^{∗}(V_{d})).

More precisely: The natural map S → (s_{d} ∘ b_{d})(S) in PS − End V_{d} is an isomorphism for all S ∈ PS − End V_{d}. The natural map K → (b_{d} ∘ s_{d})(K) is an F-epimorphism for all unstable algebras K and an F-isomorphism iff the transcendence degree of K is less that or equal to d.

1. THE GENERALIZED QUILLEN’S THEOREM 21

Proof. That the natural map S → (s_{d} ∘ b_{d})(S), is an equivalence follows from Proposition 1.11, since

s_{d} ∘ b_{d} = e_{d}^{op} ∘ g ∘ b_{d} ≃ e_{d}^{op} ∘ i_{d}^{op} ≃ id_{(PS−End V_{d})^{op}} .

To prove the statements about the map η : K → (b_{d} ∘ s_{d})(K) it is enough to prove the corresponding statements for the map g ∘ η : g(K) → (g ∘ b_{d} ∘ s_{d})(K) = (i_{d}^{op} ∘ e_{d}^{op})(g(K)) in G. By Lemma 1.5 and the definition of the transcendence degree of an object in G, g ∘ η is always an epimorphism, and is an isomorphism iff d(g(K)) ≤ d. Since d(g(K)) = d(K) by Proposition 1.12, we are done.

Remark 1.15. One should note how the above theorem can be seen as a generalization of Quillen’s theorem about the structure of the cohomology ring of a finite group. Indeed we can, at least when s_{d}(K) is finite, by changing the notation write (b_{d} ∘ s_{d})(K) = lim_{s_{d}(K)} H^{∗}V_{d}. Therefore part of the statement in the theorem is that we for any unstable algebra K of transcendence degree less that or equal to d with s_{d}(K) finite have an Fisomorphism

K → lim_{s_{d}(K)} H^{∗}(V_{d})

Note however that the theorem, as stated here, assumes a priori knowledge of the transcendence degree of K in order to determine s_{d}(K). We shall later derive a theorem which gives an effective way of determining the transcendence degree of an unstable algebra.

1.16. The Adams-Wilkerson embedding theorem. Using the previous theorem we can now easily (following [27]) show the Adams-Wilkerson embedding theorem.

Theorem 1.17. [1] Let K be an unstable algebra which is an integral domain of finite transcendence degree less than or equal to d. Then there exists an embedding of unstable algebras K → H^{∗}(V).

Proof. Since K is an integral domain it does not contain any nilpotent elements, so the map K → (b_{d} ∘ s_{d})(K) is injective by Theorem 1.14. By definition (b_{d} ∘ s_{d})(K) is a subalgebra of H^{∗}(V)^{g(K)(V_{d})}. If g(K)(V_{d}) is finite, K has to embed into a single factor since it is an integral domain, and we are done.

If g(K)(V_{d}) was infinite this might no longer be true (think of the p-adics), so we proceed by a clever bootstrap argument, to show that this cannot happen since K is an integral domain of finite transcendence degree. Note that if an unstable algebra L embeds into H^{∗}(V_{d}) then |g(L)(V_{d})| ≤ | End V_{d}|, since we in this case have a surjection

End V_{d} = Hom_{K}(H^{∗}(V_{d}), H^{∗}(V_{d})) → Hom_{K}(L, H^{∗}(V_{d}))

by the injectivity of H^{∗}(V_{d}) in K.

Now express K as a direct limit of its sub-unstable algebras K_{α}, which are finitely generated as A-algebras (A still being the Steenrod algebra). We thus have g(K)(V_{d}) expressed as an inverse limit over g(K_{α})(V_{d}). Since K_{α} is finitely generated we have that each g(K_{α})(V_{d}) is finite, and hence each K_{α}, being an integral domain of transcendence degree less than or equal to d, embeds into H^{∗}(V_{d}). Thus |g(K_{α})(V_{d})| ≤ | End V_{d}| by the first part of the proof. We now have that g(K)(V_{d}) is expressed as an inverse limit of an upwards filtering system of sets and projections, each set of cardinality less that equal to End V_{d}. From this we conclude that |g(K)(V_{d})| ≤ | End V_{d}|. Especially g(K)(V_{d}) is finite, which concludes the proof of the theorem.

22 2. THE GENERALIZED QUILLEN’S THEOREM

CHAPTER 3 The large scale structure of the cohomology rings of certain spaces

1. Growth Properties

In this section we establish the growth formula for log_{p} |[BV, E]| mentioned in the introduction. To do this we first need to prove a couple of lemmas. The first lemma in its present form is due to Lannes and Schwartz [38].

Lemma 1.1. Let G be an abelian group, and assume that t = rk_{p}(_{p}G) and s = rk_{p}(G⊗ Z/p) are finite. Furthermore let V be an elementary abelian p-group of rank v. Then the Poincaré series of H^{∗}(V; G), P_{G}(x) = ∑_{k>0} dim_{F_{p}}(H^{k}(V; G) ⊗ Z/p)x^{k} is given by

P_{G}(x) = \frac{t + sx}{1 + x}((1 − x)^{−v} − 1).

Especially

rk_{p}(H^{1}(V; G)) ∼ tv

for v → ∞ and

rk_{p}(H^{k}(V; G)) ∼ \{ \begin{matrix}\frac{tv^{k}}{k!} & if t > 0 \\ \frac{sv^{k−1}}{(k−1)!} & if t = 0 \end{matrix}

for v → ∞, k ≥ 2.

Proof. Note that induction on the rank v of V, using the cohomology of H^{∗}(Z/p; Z) together with the Künneth formula, gives us that pH^{k}(V; Z) = 0 for all k > 0 and all V.

The universal coefficient theorem now gives us the following exact sequence

0 → H^{k}(V; Z) ⊗ G → H^{k}(V; G) → Tor(H^{k+1}(V; Z), G) → 0,

i.e. for k ≥ 1

0 → H^{k}(V; Z) ⊗ (G ⊗ Z/p) → H^{k}(V; G) → H^{k+1}(V; Z) ⊗ (_{p}G) → 0.
This shows that xP_{G}(x) = (t + sx)P_{Z}(x). Setting G = Z/p we obtain

P_{Z}(x) = \frac{x}{1 + x}P_{Z/p}(x) = \frac{x}{1 + x}((1 − x)^{−v} − 1)

and hence

P_{G}(x) = \frac{t + sx}{1 + x}((1 − x)^{−v} − 1).

From this formula the growth formulas are immediate.

Lemma 1.2. Let X be an arbitrary space and let E be a connected finite Postnikov system. Then

|[X, E]_{pt}| ≤ ∏_{i>0} |H^{i}(X, π_{i}(E))|

when E is simple.

If more generally E is nilpotent, then there exists, for each i, a filtration 0 = F_{i,0} ⊲ ⋯ ⊲ F_{i,t_{i}} = π_{i}(E) such that π_{1}(E) acts trivially on F_{i,j}/F_{i,j−1} and

|[X, E]_{pt}| ≤ ∏_{i>0} ∏_{j=1}^{t_{i}} |H^{i}(X; F_{i,j}/F_{i,j−1})| .

Proof. The proof is by induction on the number of nontrivial homotopy groups. Assume that the top homotopy group sits in dimension n. Consider first the case where E is simple. We have a fibration sequence, which is principal since E is simple:

K(π_{n}(E), n) → E → P_{n−1}E,

where P_{k}E denotes the kth Postnikov stage of E. Since the fibration is principal we have an action E×K(π_{n}(E), n) → E, and thus an induced action ∗ : [X, E]_{pt}×[X, K(π_{n}(E), n)]_{pt} → [X, E]_{pt}. This action has the property that in the exact sequence

[X, K(π_{n}(E), n)]_{pt} →^{ι} [X, E]_{pt} →^{κ} [X, P_{n−1}E]_{pt}

we have that κ(f) = κ(g) iff there exists h ∈ [X, K(π_{n}(E), n)] such that f ∗ h = g. This implies that |[X, E]_{pt}| ≤ |[X, P_{n−1}E]_{pt}||H^{n}(X, π_{n}(E))|, so by induction we get that |[X, E]_{pt}| ≤ ∏_{i>0} |H^{i}(X, π_{i}(E))| as wanted.

Now assume that E is only nilpotent. The fibration K(π_{n}(E), n) → E → P_{n−1}E might no longer be principal, but it does have a principal refinement corresponding to a filtration 0 = F_{i,0} ⊲ ⋯ ⊲ F_{i,t_{i}} = π_{i}(E) of π_{i}(E) (cf. [28].) Using induction as before finishes the proof in this case too.

We are now ready to prove the key growth theorem:

Theorem 1.3. Let E be a connected nilpotent finite Postnikov system with finite π_{1}(E). Assume that H^{∗}(E) is of finite type and that H̄^{∗}(E) ≠ 0. Let n denote dimension of the highest homotopy group π_{n}(E) which is not uniquely p-divisible and set k = n if π_{n}(E) has p-torsion, k = n − 1 if not. Then

cv^{k} ∼^{<} log_{p} |[BV, E]| ∼^{<} Cv^{k} ,

for v → ∞, where c, C are positive constants.

Remark 1.4. The condition that H^{∗}(X) is of finite type is equivalent to the condition that H_{∗}(X; F_{p}) is of finite type. By the universal coefficient theorem this is again equivalent to requiring _{p}H_{n}(X; Z) and H_{n}(X; Z) ⊗ Z/p to be finite for all n. If X is connected nilpotent with finite π_{1}, this is equivalent to asking that _{p}π_{n}(X) and π_{n}(X) ⊗ Z/p are finite for all n ≥ 2, e.g. by the Whitehead theorem modulo Serre classes. Note also that for a connected nilpotent space, X, the condition H̄^{∗}(X) ≠ 0 is equivalent to π_{∗}(X) not being uniquely p-divisible (by the above or [7].) Also note that the k which appears in the theorem also could be defined as the dimension of the highest non-trivial mod p homotopy group minus 1.

Proof of Theorem 1.3. By replacing E by the F_{p}-equivalent space P_{n}E, we can assume that n is the dimension of the top non-trivial homotopy group. Since E is connected and π_{1}(E) is finite we might as well show the theorem for [BV, E]_{pt}, which is what we do. We have a principal fibration sequence ΩP_{n−1}E → K(π_{n}(E), n) → E so we get an exact sequence of pointed sets

[BV, ΩP_{n−1}E]_{pt} → [BV, K(π_{n}(E), n)]_{pt} → [BV, E]_{pt}.

We can also write this as

[BV, Ω_{0}P_{n−1}E]_{pt} → H^{n}(V; π_{n}(E)) → [BV, E]_{pt} ,  (1.1)

1. GROWTH PROPERTIES 25

where [BV, Ω_{0}P_{n−1}E]_{pt} acts on H^{n}(V; π_{n}(E)) as described in the previous proof and Ω_{0}P_{n−1}E denotes the zero component of ΩP_{n−1}E.

Now by Lemma 1.2 and 1.1 we get:

log_{p} |[BV, Ω_{0}P_{n−1}E]_{pt}| ≤ ∑_{i=1}^{n−2} log_{p} |H^{i}(V; π_{i+1}(E))|

= ∑_{i=1}^{n−2} rk_{p} H^{i}(V; π_{i+1}(E))
∼^{<} Kv^{n−2}

for v → ∞, since Ω_{0}P_{n−1}E is an H-space and hence simple. Since log_{p} |H^{n}(V; π_{n}(E))| ∼ cv^{k}, where k is as defined in the theorem, (1.1) shows us:

log_{p} |H^{n}(V; π_{n}(E))| ∼ log_{p} |H^{n}(V; π_{n}(E))| − log_{p} |[BV, Ω_{0}P_{n−1}E]|_{pt}
∼^{<} log_{p} |[BV, E]|_{pt} .

We want to get the other inequality by appealing to Lemma 1.2. In the simple case it is immediate that

log_{p} |[BV, E]_{pt}| ≤ ∑_{i=1}^{n} log_{p} |H^{i}(V; π_{i}(E))| ∼^{<} Cv^{k} .

If π_{n}(E) does contain p-torsion we can take C = c, whereas C in the case where π_{n}(E) does not contain p-torsion is given in terms of π_{n}(E) and π_{n−1}(E).

In the non-simple case we need to worry a little bit about actions. We now have that

log_{p} |[BV, E]_{pt}| ∼^{<} ∑_{i=1}^{n} ∑_{j=1}^{t_{i}} log_{p} |H^{i}(V; F_{i,j}/F_{i,j−1})| .

It is still clear that log_{p} |[BV, E]_{pt}| ≤ Cv^{n} for some C. This takes care of the case where π_{n}(E) does contain p-torsion. In the case where π_{n}(E) does not contain p-torsion however we want the better estimate Cv^{n−1}. This is not obvious since the filtration quotients could have p-torsion, even though π_{n}(E) did not. To resolve this problem we have to appeal to Proposition 5.1.

By replacing E by an F_{p}-equivalent space, we can assume that π_{1}(E) is a p-group, since every finite nilpotent group is a product of p-groups. By Proposition 5.1 π_{1}(E) acts trivially on π_{n}(E), so we can take the filtration of π_{n}(E) to be the trivial filtration, and thus there is no p-torsion introduced in dimension n. This shows that, in this case, we can conclude that log_{p} |[BV, E]_{pt}| ∼^{<} Cv^{n−1}.

Remark 1.5. It follows from the proof of the theorem above that one can actually get concrete estimates for c and C. This does have some interest, especially in the case k = 1 where c and C actually turn out to give lower and upper bounds on the transcendence degree of H^{∗}(E).

The first main theorem now follows from the preceding growth theorem.

Theorem 1.6. Let E be a connected nilpotent finite Postnikov system with finite π_{1}(E). Assume that H^{∗}(E) is of finite type and that H̄^{∗}(E) ≠ 0. Then H̄^{∗}(E) contains an element of infinite height.

Proof. Theorem 1.3 shows the asymptotic growth of log_{p} |[BV, E]|—in particular it shows that [BV, E] is nontrivial for some V. But by Theorem 2.3 and 2.5 this now implies that H̄^{∗}(E) ∉ N il, so H̄^{∗}(E) has to contain an element of infinite height.

26 3. THE COHOMOLOGY OF CERTAIN SPACES

2. Applications to n-connected covers

We now turn to investigating the consequences for n-connected covers of finite complexes:

Corollary 2.1. Let X be a finite complex, 1-connected with finite π_{2}(X). Assume furthermore that H̄^{∗}(P_{n}X) ≠ 0. Then H̄^{∗}(X⟨n⟩) contains an element of infinite height.

Remark 2.2. The assumption H̄^{∗}(P_{n}X) ≠ 0 could be formulated in a number of alternative ways. Since X is 1-connected the assumption is equivalent to the natural map H^{∗}(X) → H^{∗}(X⟨n⟩) not being an isomorphism by a Serre spectral sequence argument. Also, since X is of finite type, H̄^{∗}(P_{n}X) ≠ 0 iff H̄^{∗}(P_{n}X; Z) is not entirely q-torsion for primes q ≠ p, which, by for example the Whitehead theorem modulo Serre classes, is equivalent to π_{∗}(P_{n}X)_{(p)} ≠ 0.

Proof of Corollary 2.1. We have an exact sequence

[BV, ΩX] → [BV, ΩP_{n}X] → [BV, X⟨n⟩] → [BV, X] .

By Miller’s theorem [BV, ΩX] = [BV, X] = 0, so [BV, ΩP_{n}X] = [BV, X⟨n⟩]. Since H̄^{∗}(P_{n}X) ≠ 0 we have that H̄^{∗}(ΩP_{n}X) ≠ 0. Now observe that ΩP_{n}X satisfies the assumptions of Theorem 1.3, so in particular we get that [BV, X⟨n⟩] = [BV, ΩP_{n}X] ≠ 0 for some V. By Theorem 2.3 and 2.5 this is equivalent to H̄^{∗}(X⟨n⟩) containing an element of infinite height.

Remark 2.3. The relatively strong assumption on the connectivity of X cannot be weakened, as is shown by the Hopf fibration S^{1} → S^{3} → S^{2}.

Corollary 2.4. Let X be a finite complex, 1-connected with finite π_{2}(X). Assume that X ≠ X⟨n⟩. Then there exists a prime q such that H̄^{∗}(X⟨n⟩; F_{q}) contains an element of infinite height.

Proof. If X ≠ X⟨n⟩, we have to have that H̄^{∗}(P_{n}X; F_{q}) ≠ 0 for some prime q. The statement is now obvious from Corollary 2.1.

Corollary 2.5. Let X be a finite complex, 1-connected with finite π_{2}(X). Assume that X ≠ X⟨n⟩. Then X⟨n⟩ has infinite LS category.

Proof. By Corollary 2.4 there exists a prime q such that H̄^{∗}(X⟨n⟩; F_{q}) has infinite cup-length, so especially X⟨n⟩ has infinite LS category.

These corollaries generalize earlier partial results of Møller and McGibbon [41]. As they point out, it can be interesting to note the radical difference between these results and the results obtained in the rational case. Here the rational LS category of X⟨n⟩ is always less than or equal to the rational LS category of X by the mapping theorem of Félix and Halperin [21]. But, on the other hand, in the rational setting we have no Serre’s theorem either—indeed the rational cohomology of K(Z, 3) does not contain an element of infinite height.

3. The transcendence degree of H^{∗}(E)

In this section we give a complete classification of the Postnikov systems whose cohomology is of finite transcendence degree. We start by using work of Henn, Lannes and Schwartz [27] to relate the growth of log_{p} | Hom_{K}(K, H^{∗}(V))| in v to the transcendence degree of K, under mild restrictions on K. Combining these results with our growth formula for [BV, E] = Hom_{K}(H^{∗}(E), H^{∗}(V)) now leads to a classification theorem for finite Postnikov systems of finite transcendence degree.

In order to state and prove our results we need to review some work of Henn, Lannes and Schwartz. For the ease of the reader we follow their notation; we refer the reader to [27] for more details.

3. THE TRANSCENDENCE DEGREE OF H^{∗}(E) 27

Definition 3.1. For any unstable algebra K define its transcendence degree d(K) as the maximal number of algebraically independent homogeneous elements in K.

Remark 3.2. If K is a connected graded noetherian algebra, the maximal number of algebraically independent elements, d(K), is finite. In this case the number d(K) coincides with the Krull dimension of K, and we can furthermore choose d(K) algebraically independent elements such that K will be finite over the algebra spanned by those elements. If K is an integral domain, d(K) is equal to the classical transcendence degree of the field of fractions of K. A nice and graded proof of these standard facts can be found in [4].

In the following we need to refer to elementary abelian groups of different rank. Therefore we sometimes equip the elementary abelian group V with a subscript which in those cases indicate the rank of V.

Definition 3.3. Let E denote the category of elementary abelian p-groups, i.e. of finite dimensional F_{p} vector spaces. Let PS denote the category of profinite sets and let G be the category of functors E → (PS)^{op}. Note that we can view G^{op} as the category of contravariant functors E → PS.

One should realize that objects in G contain a rich structure. This stems from the fact that not only do we associate to each vector space a profinite set, but we do this in a natural way, which, loosely speaking, ties the profinite sets together.

By inverting all F-isomorphisms in K we obtain a quotient category of K which will be denoted by K/N il. Let g : K → G be given by g(K)(V) = Hom_{K}(K, H^{∗}(V)) for all V. Here we equip Hom_{K}(K, H^{∗}(V)) with the profinite topology induced by writing

Hom_{K}(K, H^{∗}(V)) = Hom_{K}(colim_{α} K_{α}, H^{∗}(V)) = lim_{α} Hom_{K}(K_{α}, H^{∗}(V)) ,

where α runs over the finitely generated A-subalgebras K_{α} of K.

By Theorem 2.4 and the Lannes linearization principle we have that a morphism between unstable algebras K → K′ is an F-monomorphism (resp. F-epi) iff g(K′)(V) → g(K)(V) is a surjective (resp. injective) map of sets for all V. This shows that g induces a a faithful functor K/N il → G (likewise denoted g.) In [27] Henn, Lannes and Schwartz actually identify the image of g—we shall however not need this.

Definition 3.4. Let PS − End V_{d} denote the category whose objects are profinite sets equipped with a continuous right action of the monoid End V_{d} and whose morphisms are maps of profinite sets respecting the End V_{d}-action.

To each G ∈ G^{op} and each vector space V_{d} we can associate a profinite right End V_{d}-set G(V_{d}). Namely define the right action of End V_{d} on G(V_{d}) by to each (s, φ) ∈ G(V_{d}) × End V_{d} associating sφ = G(φ)s. This gives us a ‘restriction functor’

e_{d} : G^{op} → PS − End V_{d} .

Likewise we can define an ‘induction functor’

i_{d} : PS − End V_{d} → G^{op} : S ↦ (W ↦ S ×_{End V_{d}} Hom(W, V_{d})) ,

where S ×_{End V_{d}} Hom(W, V_{d}) denotes the coequalizer in the category of profinite sets of the action of End V_{d} on S and Hom(W, V_{d}) respectively.

Now note that the canonical map G(V_{d})×Hom(W, V_{d}) → G(W) induces a well defined map on the coequalizer G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W). Therefore, to each map S → G(V_{d}) of profinite End V_{d}-sets and each W, we can associate a map

S ×_{End V_{d}} Hom(W, V_{d}) → G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W) .

Likewise a morphism in G^{op}, i_{d}(S) → G of course induces a map S → G(V_{d}) of profinite End V_{d}-sets. Since these operations are inverses of each other we get that Hom_{G^{op}}(i_{d}(S), G) =

28 3. THE COHOMOLOGY OF CERTAIN SPACES

Hom_{PS−End V_{d}}(S, e_{d}(G)), naturally in G ∈ G^{op}, S ∈ PS − End V_{d}, so i_{d} is left adjoint of e_{d}. It is immediate that we have an equivalence 1_{PS−End V_{d}} →^{≃} e_{d} ∘ i_{d}, given by the unit of the adjunction, so we get an embedding of PS − End V_{d} as a subcategory of G^{op}. Define s_{d} = e_{d}^{op} ∘ g, where e_{d}^{op} : G → (PS − End V_{d})^{op} is the opposite functor of e_{d}.

In [27] Henn, Lannes and Schwartz prove the following key results about the structure of G^{op}.

Proposition 3.5. [27] The morphism (i_{d} ∘ e_{d})(G) → G given by the counit of the adjunction is a monomorphism in G^{op} for all G ∈ G^{op}, i.e. we have that (i_{d} ∘ e_{d})(G)(V) → G(V) is an injective map of profinite sets for all V.

This gives us a filtration of G which turns out to coincide with the filtration of K/N il by transcendence degree.

Proposition 3.6. [27] Let G ∈ G^{op}. Define the transcendence degree of G as

d(G) = min{d | ((i_{d} ∘ e_{d})G)(W) → G(W) is bijective for all W } ,

where we take d(G) = ∞ if none such d exists. For an element G ∈ G we define the transcendence degree by viewing it as lying in G^{op}. With these definitions we have that d(K) = d(g(K)).

We shall need some alternative ways of expressing the transcendence degree of an object in G^{op}. These can be found implicit in [27]. Since they are important in their own right we find it useful to state them explicitly—we include proofs for the convenience of the reader. First a useful definition:

Definition 3.7. Let S be an End V_{d}-set, and let s ∈ S. Define the rank of s ∈ S as

rk s = min{rk φ| where s = tφ for some t ∈ S, φ ∈ End V_{d}} .

We say that s is regular if rk s = d, i.e. if s = tφ implies that φ is a regular (invertible) matrix.

Proposition 3.8. We have the following formula for the transcendence degree of G ∈ G^{op}:

d(G) := min{d|((i_{d} ∘ e_{d})G)(W) → G(W) is bijective for all W }
= max{rk s|s ∈ G(V_{d}) for some d}
= max{d|G(V_{d}) contains a regular element} .

Proof. We start by proving that

max{d|G(V_{d}) contains a reg. elt.} = max{rk s|s ∈ G(V_{d}) for some d}.  (3.1)

First note that ‘≤’ is obvious. To prove ‘≥’ let s ∈ G(W) and suppose that rk s = d. We can thus choose π ∈ End W, t ∈ G(W) such that s = tπ and rk π = d. By changing t we can assume that π is a projection. Let ρ : W → πW and i : πW → W be the canonical projection and inclusion associated to π. Now set s′ = G(i)s ∈ G(πW) and note that G(ρ)s′ = G(ρ)G(i)s = G(ρ)G(i)G(π)t = G(πiρ)t = s. We claim that s′ is regular. Suppose we have φ ∈ End(πW), u ∈ G(πW) such that s′ = uφ. Then we have

s = G(ρ)s′ = G(ρ)G(φ)u = G(ρ)G(φ)G(i)G(ρ)u = G(iφρ)G(ρ)u = (G(ρ)u)(iφρ)

so φ has to be regular, since rk s = d. This shows the wanted inequality.

We now prove that the number given by (3.1) actually coincides with the transcendence degree. We first see that it is less than or equal to the transcendence degree. Suppose therefore that we have d such that G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W) is bijective for all W, and let s ∈ G(W) be arbitrary. Choose t ∈ G(V_{d}) and φ ∈ Hom(W, V_{d}) such that G(φ)t = s. Since the rank of φ : W → V_{d} must be less than or equal to d, we can choose

3. THE TRANSCENDENCE DEGREE OF H^{∗}(E) 29

a projection π ∈ End W of rank less than or equal to d such that φπ = φ. But this gives us that s = G(φ)t = G(π)G(φ)t = (G(φ)t)π, which shows that rk s ≤ d, as wanted.

We finish the proof by showing d(G) ≤ max{d|G(V_{d}) contains a regular element}. Let d be the maximal number such that G(V_{d}) contains a regular element (if there exist regular elements in infinitely many dimension we are done.) We want to prove that G(V_{d}) ×_{End V_{d}} Hom(W, V_{d}) → G(W) is surjective for all W. Let s ∈ G(W) be arbitrary and set n = rk s. Note that n ≤ d by (3.1.) Choose an element π ∈ End W of rank n and t ∈ G(W) such that s = tπ. By changing the choice of t we can assume that π is a projection. Letting ρ : W → πW and i : πW → W denote the canonical projection and inclusion we get that s = G(π)t = G(πiρ)t = G(ρ)(G(πi)t). Therefore s is in the image of G(V_{n})×_{End V_{n}}Hom(W, V_{n}) which implies that s is in the image of G(V_{d})×_{End V_{d}}Hom(W, V_{d}), since n ≤ d. This completes the proof, since G(V_{d})×_{End V_{d}}Hom(W, V_{d}) → G(W) is injective by Proposition 3.5.

Example 3.9. Let X_{n} be the End V_{n}-set consisting of two elements x_{1} and x_{0}, with the End V_{n}-action given as follows:

x_{0}α = x_{0} for all α ∈ End V_{n}

x_{1}α = \{ \begin{matrix}x_{1} & for α ∈ Aut V_{n} \\ x_{0} & for α ∉ Aut V_{n} . \end{matrix}

We have that x_{1} has rank n and x_{0} has rank 0. The End V_{n}-set X_{n} is thus the smallest End V_{n}-set containing a regular element. It has the universal property that any End V_{n}-set containing a regular element surjects onto X_{n} as an End V_{n}-set.

Just knowing the growth properties of log_{p} |g(K)(V)| in v is actually enough to determine the transcendence degree of K, under mild restrictions on K.

Theorem 3.10. Let K be an unstable algebra and assume that Hom_{K}(K, H^{∗}(V)) is finite for all V. If d(K) is finite then log_{p} |g(K)(V)| ∼ d(K)v. If d(K) is infinite then log_{p} |g(K)(V)| grows faster than linearly in v.

Remark 3.11. The assumption that Hom_{K}(K, H^{∗}(V)) is finite for all V is a technical assumption which (by Theorem 1.3) is satisfied in all the applications we have in mind. Also it is clear that if for instance K is finitely generated as an A-algebra, then Hom_{K}(K, H^{∗}(V)) is likewise finite. Furthermore if K is N il-closed (cf. [9, 27]) and of finite type then Hom_{K}(K, H^{∗}(V)) is finite [27]. Note however that H^{∗}(∏_{i=1}^{∞} K(Z/p, i)) serves as an example of an unstable algebra of finite type where Hom_{K}(K, H^{∗}(V)) is not finite.

Proof of Theorem 3.10. We want to establish the general growth formulas by establishing them for the ‘largest’ and the smallest finite End V_{n}-set containing a regular element. Let X_{n} be the End V_{n}-set defined in Example 3.9. For the elements in i_{n}(X_{n})(V) = X_{n} ×_{End V_{n}} Hom(V, V_{n}) we have the following relations:

(x_{0}, φ) ∼ (x_{0}, 0) for all φ ∈ Hom(V, V_{n})
(x_{1}, φ) ∼ (x_{0}, 0) if rk φ < n
(x_{1}, φ) ∼ (x_{1}, ψ) if rk φ = rk ψ = n and ker φ = ker ψ .

This shows that |i_{n}(X_{n})(V)| = #( n dimensional subspaces in V ) + 1 For v > n, the number of n dimensional subspaces of V is as follows:

#( n dimensional subspaces in V ) = \frac{(p^{v} − 1) ⋯ (p^{v} − p^{n−1})}{(p^{n} − 1) ⋯ (p^{n} − p^{n−1})}
∼ Cp^{nv}

for v → ∞.

30 3. THE COHOMOLOGY OF CERTAIN SPACES

Let T be a finite set and consider the End V_{n}-set T × End V_{n}. We have that i_{n}(T × End V_{n})(V) = T × Hom(V, V_{n}) so

|i_{n}(T × End V_{n})(V)| = |T |p^{nv} .

From the above we conclude that

log_{p} |i_{n}(X_{n})(V)| ∼ log_{p} |i_{n}(T × End V_{n})(V)| ∼ nv .

Now let K be an unstable algebra and assume that s_{n}(K) contains a regular element. Since s_{n}(K) is finite we can find a surjection of End V_{n}-sets T × End V_{n} → s_{n}(K) for some finite set T. Also, since s_{n}(K) contains a regular element, we can find a surjection s_{n}(K) → X_{n} of End V_{n}-sets. Now this means that, for all V, we have surjections of sets

i_{n}(T × End V_{n})(V) → i_{n}(s_{n}(K))(V) → i_{n}(X_{n})(V) .

This shows that log_{p} |i_{n}^{op}(s_{n}(K))(V)| behaves asymptotically as nv.

If K has transcendence degree d < ∞, then g(K) also has transcendence degree d so log_{p} |g(K)(V)| = log_{p} |(i_{d}^{op} ∘ e_{d}^{op} ∘ g)(K)(V)| = log_{p} |i_{d}^{op}(s_{d}(K))(V)| which grows asymptotically as dv, since s_{d}(K) contains a regular element by Proposition 3.8. If K has transcendence degree d = ∞ then by Proposition 3.5

log_{p} |g(K)(V)| ≥ log_{p} |(i_{n}^{op} ∘ e_{n}^{op} ∘ g)(K)(V)| = log_{p} |i_{n}^{op}(s_{n}(K))(V)| .

Since s_{n}(K) contains a regular element for infinitely many n by Proposition 3.8, we get that log_{p} |g(K)(V)| ∼^{>} nv for all n as wanted.

The characteristic numbers log_{p} |g(K)(F_{p}^{⊕n})| can, in some sense, be viewed as an unstable algebra alternative to the classical Betti numbers of a graded algebra. In the classical case we have a formula relating the growth of the Betti numbers and the transcendence degree of a noetherian connected graded F-algebra given by

d(K) = min{k ∈ N_{0}| there exists a C such that dim_{F} K^{n} ≤ Cn^{k−1} for all n} .

Theorem 3.10 establishes a different but analogous formula for these new numbers, which has among its advantages that it holds with much weaker restrictions on the unstable algebra. Moreover, when applied to the cohomology of spaces, it is easy to see how these numbers behave with respect to fibrations of the underlying spaces.

Remark 3.12. In [27] Henn, Lannes and Schwartz use Proposition 3.5 and 3.6 to derive a far reaching generalization of Quillen’s theorem about the structure of the cohomology ring of a finite group [46]. They prove that for any unstable algebra K of transcendence degree less than or equal to d there is an F-isomorphism

K → lim_{s_{d}(K)} H^{∗}(V_{d}) .

Here we view s_{d}(K) as a category by taking as objects the elements in s_{d}(K), and as morphisms the maps induced on s_{d}(K) from endomorphisms of V_{d}. (The theorem is strictly speaking only true as stated here when s_{d}(K) is finite—when s_{d}(K) is infinite one has only to consider the elements in lim_{s_{d}(K)} H^{∗}(V_{d}) which are continuous as functions s_{d}(K) → H^{∗}(V_{d}).)

The theorem however has the inherent weakness that it requires a priori knowledge of the transcendence degree of K. Theorem 3.10 can remedy this defect, since it gives a very easily applicable way of calculating the transcendence of an unstable algebra K.

To illustrate the power of this approach we will rederive Quillen’s theorem in the pivotal finite p-group case, by showing that the Krull dimension of H^{∗}(P) is equal to rk_{p} P, the maximal rank of an elementary abelian group in the p-group P. It was shown by Hurewicz that [BV, BP] = Rep(V, P) = Hom(V, P)/(conj. by p ∈ P). It is furthermore an easy exercise to see that log_{p} | Rep(V, P)| grows asymptotically as rk_{p} P v. Theorem 3.10

3. THE TRANSCENDENCE DEGREE OF H^{∗}(E) 31

now implies that d(H^{∗}(P)) = rk_{p} P, and since we know that H^{∗}(P) is noetherian by the Evens-Venkov theorem (cf. [19]) we are done. Tracing back what elements go into this proof one sees that one of the main ingredients is the use of Proposition 3.8, which in some sense can be seen as being the replacement of Serre’s theorem about cohomological detection of elementary abelian groups.

Theorem 3.13. Let E be a connected nilpotent finite Postnikov system with finite π_{1}(E). Assume that H^{∗}(E) is of finite type. Then H^{∗}(E) has finite transcendence degree iff E is F_{p}-equivalent to a space E′ fitting into a principal fibration sequence of the form:

CP^{∞} × ⋯ × CP^{∞} → E′ → K(P, 1) ,

where P is a finite p-group.

Proof. By Theorem 1.3 and 3.10 all spaces E′ as in Theorem 3.13 have cohomology of finite transcendence degree. To prove the converse, assume that H^{∗}(E) has finite transcendence degree. We see from Theorem 1.3 that Hom_{K}(H^{∗}(E), H^{∗}(V)) is finite for all V. Theorem 1.3 and 3.10 now imply that E has to be F_{p}-equivalent to P_{2}E, and furthermore that π_{2}(E) cannot contain p-torsion. Since π_{1}(P_{2}E) is finite nilpotent, and thus a product of p-groups, we can by passing to a covering replace P_{2}E with an F_{p}- equivalent space whose fundamental group is a finite p-group. These observations show that, without restriction, we can assume that E has nontrivial homotopy groups only in dimension 1 and 2, π_{2}(E) being without p-torsion and π_{1}(E) being a finite p-group. But now Proposition 5.1 tells us that the action of π_{1}(E) on π_{2}(E) has to be trivial. This means that the Postnikov fibration K(π_{2}(E), 2) → E → P_{1}E is principal (cf. [28].) We thus have a fibration sequence

E → P_{1}E →^{k} K(π_{2}(E), 3) .

Upon passing to F_{p}-completion we obtain a fibration sequence (cf. [7])

Ê_{p} → P_{1}E →^{k} K(Ext(Z/p^{∞}, π_{2}(E)), 3) .

Now Ext((Z/p^{∞}, π_{2}(E)) is a torsion free Ext −p complete abelian group, and thus classified by the F_{p}-dimension of Ext(Z/p^{∞}, π_{2}(E)) ⊗ Z/p (cf. [7, p. 181], [26]), which is finite since H^{∗}(E) is of finite type. In other words Ext(Z/p^{∞}, π_{2}(E)) ≃ Ẑ_{p} × ⋯ × Ẑ_{p}. Note that H^{3}(π_{1}(E); Z) = H^{3}(π_{1}(E); Ẑ_{p}) by the universal coefficient theorem. We can hence let k′ : P_{1}E → K(Z × ⋯ × Z, 3) be the map corresponding to k under the equivalence [P_{1}E, K(Ext(Z/p^{∞}, π_{2}(E)), 3)] ≃ [P_{1}E, K(Z×⋯×Z, 3)]. Letting E′ denote the homotopy fiber of k′ we obtain a diagram

[figure: E′; f; P_{1}E; k′; K(Z × ⋯ × Z, 3); F_{p}≃; Ê_{p}; P_{1}E; k; K(Ext(Z/p^{∞}, π_{2}(E)), 3)]

where f is any lifting which makes the diagram commute. From this diagram we see that f : E′ → Ê_{p} has to be an F_{p}-equivalence as well. By construction E′ is F_{p}-equivalent to E and fits into a principal fibration sequence of the form stated in the theorem.

Remark 3.14. Remember that a p-toral group G is a group which arises as a group extension 1 → T^{n} → G → P → 1, where T^{n} is an n-dimensional torus and P is a finite p-group. The spaces E′ which appear in Theorem 3.13 are just those classifying spaces of p-toral groups which arise as central extensions. They are classified by n, P and their one extension class (= Postnikov invariant) k ∈ [BP, K(Z×⋯×Z, 3)] = H^{3}(P; Z×⋯×Z) ≃ H^{2}(P; T^{n}), where in the cohomology P acts trivially on the coefficients.

32 3. THE COHOMOLOGY OF CERTAIN SPACES

Remark 3.15. By Venkov’s theorem [56], classifying spaces of p-toral groups have noetherian cohomology. Theorem 3.13 thus in particular shows that for the cohomology of a finite Postnikov system, being of finite transcendence degree is equivalent to being noetherian. This is indeed a very striking and unusual property which of course does not hold for spaces in general. The cohomology of the loop space of a (1-connected, say) finite complex for instance is always non-noetherian and has transcendence degree 0. Just knowing this intriguing property of finite Postnikov systems would actually be enough to rederive the above theorem using the original Betti numbers estimates of Serre—knowing this equivalence would secure that the wild growth of the Betti numbers could only be caused by an infinite number of polynomial generators in the ring.

We end this section by showing a proposition which precisely determines the transcendence degree of the cohomology the spaces E′ of Theorem 3.13, and whose proof illustrates a calculation using Theorem 3.10. The proposition can also be obtained by using Quillen’s theorem for compact Lie groups.

Proposition 3.16. Let E be a space which fits into a fibration sequence of the form

\underbrace{CP^{∞} × ⋯ × CP^{∞}}_{n} → E → K(P, 1) →^{k} K(Z × ⋯ × Z, 3) ,

where P is a finite p-group. Then

d(H^{∗}(E)) = n + max{rk_{p} V |V ↪^{ι} P, ι^{∗}(k) = 0 ∈ H^{3}(V; Z × ⋯ × Z)} .

Especially n + rk_{p} P ≥ d(H^{∗}(E)) ≥ n + 1, when P is non-trivial.

Proof. Consider the sequence

[BV, E] → [BV, K(P, 1)] →^{k_{∗}} [BV, K(Z × ⋯ × Z, 3)] .

Let ι ∈ [BV_{d}, BP] = Rep(V_{d}, P) be an injection and assume that ι ∈ ker(k_{∗}). We have that ι naturally gives rise to ∼ Cp^{dv} elements in ker(k_{∗}) as v → ∞ coming from maps arising by precomposing ι with projections onto d dimensional subspaces of V. Since Rep(V_{rk_{p} P} , P) is finite we obtain that

log_{p} | ker(k_{∗})| ∼ max{rk_{p} V |V ↪^{ι} P, ι^{∗}(k) = 0}v .

But the exact sequence now shows that

log_{p} |[BV, E]| ∼ nv + max{rk_{p} V |V ↪^{ι} P, ι^{∗}(k) = 0}v ,

so d(H^{∗}(E)) = n + max{rk_{p} V |V ↪^{ι} P, ι^{∗}(k) = 0} by Theorem 3.10. To get the last part of the Proposition, note that H^{3}(Z/p; Z) = 0.

Example 3.17. Consider the space E = Fib(K(Z/p × Z/p, 1) →^{k} K(Z, 3)), where k is some nonzero element in H^{3}(Z/p × Z/p; Z) ≃ Z/p. From the above Proposition it follows that d(H^{∗}(E)) = 2.

Remark 3.18. Note that the formula for the transcendence degree of H^{∗}(E) involves the behavior of a certain class in the integral cohomology of P when restricting to elementary abelian subgroups.

4. Generalizations to PolyGEMs

Recall that a GEM is a (possibly infinite) product of K(G, n)’s, where G is an abelian group. A space is a polyGEM if it belongs to the smallest full subcategory ( polyGEMs) of spaces containing all GEMs and which is closed under taking extensions by fibrations, i.e. which satisfies that if F → E → B is an arbitrary fibration then B, F ∈ polyGEMs ⇒ E ∈ polyGEMs. We say that a space is an oriented polyGEM if it belongs to the smallest

4. GENERALIZATIONS TO POLYGEMS 33

full subcategory polyGEMs^{ori} of spaces containing all GEMs and which is closed under taking extensions by principal fibrations, i.e. which satisfies that if F → E → B is a principal fibration then F, B ∈ polyGEMs^{ori} ⇒ E ∈ polyGEMs^{ori}. Note that a nilpotent finite Postnikov system is an oriented polyGEM by [28].

In [20] Dror Farjoun conjectures that if X is a non-trivial p-complete polyGEM then [BZ/p, X] is always non-trivial. As stated the conjecture is false—Ŝ_{p}^{1} and K(Ẑ_{p}, 3) serve as counterexamples. The problem with Ŝ_{p}^{1} is easy—the fundamental group is not finite. The problem with K(Ẑ_{p}, 3) is deeper, it demonstrates that it in general is not enough to look at maps from just a single BZ/p (it is however interesting to note that for any nilpotent, connected space X with finite π_{1} and with H^{∗}(X) assumed to be noetherian, we have that [BV, X] ≠ 0 ⇒ [BZ/p, X] ≠ 0 (cf. [27, Prop II 7.2])). Since the idea that polyGEMs should behave like finite Postnikov systems still seems very plausible, we dare to state the following more modest conjecture:

Conjecture 4.1. Let E be a connected nilpotent polyGEM with finite π_{1}(E). Assume that H^{∗}(E) is of finite type and that H̄^{∗}(E) ≠ 0. Then H̄^{∗}(E) contains an element of infinite height.

Note that imposing the restriction on the fundamental group, we are also led to the simplification that [BV, X] = [BV, X̂_{p}], so the p-completion doesn’t matter.

We likewise believe that Theorem 3.13 should generalize:

Conjecture 4.2. Let E be a connected nilpotent polyGEM with finite π_{1}(E). Assume that H^{∗}(E) is of finite type. Then the transcendence degree of H^{∗}(E) is finite iff E is F_{p}- equivalent to a space E′ fitting into a principal fibration sequence of the form:

CP^{∞} × ⋯ × CP^{∞} → E′ → K(P, 1) ,

where P is a finite p-group.

One piece of evidence for the first conjecture is a theorem of Félix, Halperin, Lemaire and Thomas [22] which (in the language of Dror Farjoun) states that a 1-connected oriented polyGEM E with H^{∗}(E) of finite type and H̄^{∗}(E) ≠ 0 has infinite LS category.

We will give another piece of evidence for the conjecture, whose proof makes use of the Neisendorfer localization functor L = L_{HZ/p}P_{BZ/p} (cf. [45]). Here L_{HZ/p} denotes localization with respect to mod p homology (cf. [5]) and P_{BZ/p} denotes BZ/p-nullification (cf. [20]). First we need a definition.

Definition 4.3. Let polyGEMs^{ft} denote the smallest full subcategory of spaces which is closed under taking extensions by fibrations and which contains all connected GEMs with finite π_{1}.

Remark 4.4. The class polyGEMs^{ft} contains all connected finite Postnikov systems with finite solvable π_{1}, and is probably very close to being equal to all connected polyGEMs with finite π_{1}.

Remember that an unstable module M is called locally finite if for all x ∈ M we have that Ax is finite dimensional.

Proposition 4.5. Let E be a nilpotent polyGEM which belongs to polyGEMs^{ft}. Assume that H^{∗}(E) is of finite type and that H̄^{∗}(E) ≠ 0. Then H^{∗}(E) is not locally finite.

Before giving the proof we need a lemma:

Lemma 4.6. We have that LE = ∗ for all E ∈ polyGEMs^{ft}, where L denotes the Neisendorfer localization functor.

34 3. THE COHOMOLOGY OF CERTAIN SPACES

Proof of Lemma 4.6. Our main technical tool is a theorem of Dror Farjoun which states that if L_{f} is any localization functor with respect to some map f, and if F → E → B is any fibration sequence, then L_{f}F = ∗ implies that L_{f}E →^{≃} L_{f}B (cf. [20]). This also applies to the Neisendorfer localization functor, since we can view L as localization with respect to just one (large) map. We first note that the class of spaces which is acyclic with respect to Neisendorfer localization, i.e. the spaces E which satisfies LE = ∗, is closed under taking extensions by fibrations. It is therefore enough to prove the claim for connected GEMs with finite π_{1}. Furthermore note that it is enough to prove the claim for 1-connected GEMs, since we can apply the theorem of Dror Farjoun to the fibration sequence E⟨1⟩ → E → K(π_{1}(E), 1), where LK(π_{1}(E), 1) is easily seen to be zero.

Now let E be a 1-connected GEM and write this E = ΩẼ where Ẽ is 2-connected. By a theorem of Dror Farjoun [20, Prop 7.B.5] P_{BZ/p} and P_{M(Z/p,1)} coincide on GEMs (where M(Z/p, 1) denotes the mod p Moore space). This gives us

P_{BZ/p}E = P_{M(Z/p,1)}E = ΩP_{M(Z/p,2)}Ẽ ,

where the last equality is by another theorem of Dror Farjoun [20, Prop 3.A.1]. Bousfield [6] has shown that

π_{i}(P_{M(Z/p,2)}Ẽ) = π_{i}(Ẽ) ⊗ Z[\frac{1}{p}] ,

where we use that Ẽ is a 2-connected GEM. But this shows that P_{BZ/p}E is a connected nilpotent space with uniquely divisible homotopy groups, and hence LE = L_{HZ/p}P_{BZ/p}E = ∗.

Proof of Proposition 4.5. By Lemma 4.6 we have that LE = ∗, so especially map_{∗}(BZ/p, E) ≠ ∗, since map_{∗}(BZ/p, E) = ∗ would imply that LE = Ê_{p} ≠ ∗. But saying that map_{∗}(BZ/p, E) ≠ ∗ is equivalent to saying that H^{∗}(E) is not locally finite (cf. [44, 37]).

Remark 4.7. Neisendorfer’s theorem [45], stating that LX⟨n⟩ = X̂_{p} for a 1-connected finite complex X with finite π_{2}, follows immediately from Lemma 4.6, by applying L to the fibration sequence ΩP_{n}X → X⟨n⟩ → X and using Miller’s theorem to conclude that LX = X̂_{p}.

Remark 4.8. There is no direct relation between the LS category and the property locally finite. For instance ΣK(Z/2, 2) is an example of a space with non-locally finite cohomology but with LS category 1, whereas ∏_{n∈N} S^{n} is an example of a space of infinite LS category whose cohomology is locally finite.

5. Appendix: Nilpotent actions

In this short appendix we prove a proposition about nilpotent actions, which we have used several times in the paper. The author thanks W. Chachólski for pointing out a proof somewhat simpler than the author’s original proof.

Proposition 5.1. Let M be an abelian group which does not contain any p-torsion and let P be a p-group. Assume that P acts nilpotently on M. Then P acts trivially on M.

Proof. Suppose that P acts nilpotently on M and let 0 = F_{0} ⊂ ⋯ ⊂ F_{n} = M be a filtration on M such that P acts trivially on the filtration quotients. We want to do an induction on the length of the filtration, the initial step being trivial. Let g ∈ P and x ∈ F_{2} be arbitrary. We can write gx = x + x_{1} where x_{1} ∈ F_{1}. By iterating this we get that g^{n}x = x + nx_{1}. Setting n = |g| gives us |g|x_{1} = 0. But since M does not contain p-torsion we have that multiplication by |g| is injective on M so x_{1} = 0. Since x and g

5. APPENDIX: NILPOTENT ACTIONS 35

were arbitrary we conclude that P acts trivially on F_{2}. Induction on the length of the filtration now finishes the proof.

36 3. THE COHOMOLOGY OF CERTAIN SPACES

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