Stable homotopy over the Steenrod algebra
John H. Palmieri
Author address:
Department of Mathematics, University of Notre Dame, Notre
Dame, IN 46556
Email address: palmieri@member.ams.org
1991 Mathematics Subject Classification. 55S10, 55U15, 18G35, 55U35, 55T15,
55P42, 55Q10, 55Q45, 18G15, 16W30, 18E30, 20J99.
Research partially supported by National Science Foundation grant DMS9407459.
Abstract.We apply the tools of stable homotopy theory to the study of mod
ules over the Steenrod algebra A*; in particular, we study the (triangula*
*ted)
category Stable(A) of unbounded cochain complexes of injective comodules
over A, the dual of A*, in which the morphisms are cochain homotopy class*
*es
of maps. This category satisfies the axioms of a stable homotopy category*
* (as
given in [HPS97]); so we can use Brown representability, Bousfield locali*
*za
tion, BrownComenetz duality, and other homotopytheoretic tools to study
Ext**A(Fp; Fp), which plays the role of the stable homotopy groups of sph*
*eres.
We also have nilpotence theorems, periodicity theorems, a convergent chro
matic tower, and a number of other results.
Contents
List of Figures v
Preface vii
Chapter 1. Stable homotopy over a Hopf algebra 1
1.1. Recollections 2
1.1.1. Hopf algebras 2
1.1.2. Comodules 4
1.1.3. Homological algebra 5
1.2. The category Stable() 6
1.3. The functor H 8
1.3.1. Remarks on Hopf algebra extensions 10
1.4. Some classical homotopy theory 13
1.5. The Adams spectral sequence 15
1.6. Bousfield classes and BrownComenetz duality 18
1.7. Further discussion 19
Chapter 2. Basic properties of the Steenrod algebra 23
2.1. Quotient Hopf algebras of A 23
2.1.1. Quasielementary quotients of A 28
2.2. Psthomology 29
2.3. Vanishing lines for homotopy groups 34
2.3.1. Proof of Theorems 2.3.1 and 2.3.2 for p = 2 35
2.3.2. Changes necessary when p is odd 40
2.4. Selfmaps via vanishing lines 42
2.5. Further discussion 44
Chapter 3. Quillen stratification and nilpotence 47
3.1. Statements of theorems 48
3.1.1. Quillen stratification 48
3.1.2. Nilpotence 50
3.2. Nilpotence and Fisomorphism via the Hopf algebra D 51
3.2.1. Nilpotence: Proof of Theorem 3.1.5 54
3.2.2. Fisomorphism: Proof of Theorem 3.1.2 55
3.3. Nilpotence and Fisomorphism via quasielementary quotients 56
3.3.1. Nilpotence: Proof of Theorem 3.1.6 56
3.3.2. Fisomorphism: Proof of Theorem 3.1.3 58
3.4. Further discussion: nilpotence at odd primes 60
3.5. Further discussion: miscellany 61
iii
iv CONTENTS
Chapter 4. Periodicity and other applications of the nilpotence theorems 63
4.1. The periodicity theorem 63
4.2. Properties of ymaps 65
4.3. The proof of the periodicity theorem 67
4.4. Computation of some invariants in HD** 69
4.5. Computation of a few Bousfield classes 73
4.6. Ideals and thick subcategories 76
4.6.1. Ideals 76
4.6.2. A thick subcategory conjecture 78
4.7. Construction of spectra of specified type 80
4.8. Further discussion: slope supports 84
4.9. Further discussion: miscellany 86
Chapter 5. Chromatic structure 87
5.1. Margolis' killing construction 87
5.2. A Tate version of the functor H 94
5.3. Chromatic convergence 97
5.4. Further discussion 99
Appendix A. Two technical results 101
A.1. An underlying model category 101
A.2. Vanishing planes in Adams spectral sequences 102
A.2.1. Vanishing lines in ordinary stable homotopy 107
Appendix B. Steenrod operations and nilpotence in Ext**(k; k) 109
B.1. Steenrod operations in Hopf algebra cohomology 109
B.2. Nilpotence in HB**= Ext**B(F2; F2) 110
B.3. Nilpotence in HB**= Ext**B(Fp; Fp) when p is odd 111
B.3.1. Sketch of proof of Conjecture B.3.4, and other results 113
Bibliography 117
Index 121
List of Figures
2.1.A Graphical representation of a quotient Hopf algebra of A. 25
2.1.B Profile function for A(n). 26
2.1.C Profile functions for maximal elementary quotients of A, p = 2. 28
2.3.A Vanishing line at the prime 2. 34
3.1.A Profile function for D. 48
3.2.B Profile function for D(n). 52
3.3.C Profile functions for Dr and Dr;q. 57
4.4.A Graphical depiction of coaction of A on limHE** 72
4.8.A T(t; s) and T(m) as subsets of Slopes0. 85
5.1.A Vanishing curve for ssij(CfnS0). 93
5.2.A The coefficients of HA(1)and bHA(1). 96
v
vi LIST OF FIGURES
Preface
The object of study for this book is the mod p Steenrod algebra A and its co*
*ho
mology ExtA. Various people (including the author) have approached this subject
by taking results in stable homotopy theory and then trying to prove analogous
results for Amodules. This has proven to be successful, but the analogies were*
* just
that_there was no formal setting in which to do anything more precise than to
make analogies.
In [HPS97 ], Hovey, Strickland, and the author developed "axiomatic stable
homotopy theory." In particular, we gave axioms for a stable homotopy category;*
* in
any such category, one has available many of the tools of classical and modern *
*stable
homotopy theory_tools like Brown representability and Bousfield localization. It
turns out that a category Stable(A) (defined in the next paragraph) related to *
*the
category of Amodules is such a category; as one might expect, the trivial modu*
*le
Fp plays the role of the sphere spectrum S0, and Ext**A( ; ) plays the role of
homotopy classes of maps. Since so many of the tools of stable homotopy theory
are focused on the study of the homotopy groups of S0 (and of other spectra),
one should expect the corresponding tools in Stable(A) to help in the study of
Ext**A(Fp; Fp) (and related groups). In this book we apply some of these tools
(nilpotence theorems, periodicity theorems, chromatic towers, etc.) to the stud*
*y of
Ext over the Steenrod algebra. It is our hope that this book will serve two pur*
*poses:
first, to provide a reference source for a number of results about the cohomolo*
*gy of
the Steenrod algebra, and second, to provide an example of an indepth use of t*
*he
language and tools of axiomatic stable homotopy theory in an algebraic setting.
First we describe the category in which we work. We fix a prime p, let A* be
the mod p Steenrod algebra, and let A = Hom Fp(A*; Fp) be the (graded) dual of
the Steenrod algebra. We let Stable(A) be the category whose objects are cochain
complexes of injective left Acomodules, and whose morphisms are cochain homo
topy classes of maps. This is a stable homotopy category (of a particularly nice
sort_it is a monogenic Brown category_see [HPS97 , 9.5]). We prove a number
of results in Stable(A); some of these are analogues of results in the ordinary*
* stable
homotopy category, and some are not. Some of these are new, and some already
known, at least in the setting of A*modules; the old results often need new pr*
*oofs
to apply in the more general setting we discuss here.
Note. This work arose from the study of the abelian category of (left) A*
modules; to apply stable homotopy theoretic techniques, though, it is most con
venient to work in a triangulated category. One's first guess for an appropria*
*te
category might have objects which are chain complexes of projective A*modules;
it turns out that this category has some technical difficulties (see Remark 1.2*
*.1).
It is much more convenient to work with Acomodules instead of A*modules, and
fortunately, one does not lose much by doing this. Most A*modules of interest *
*can
vii
viii PREFACE
be viewed as Acomodules; the main effects of using comodules are things of the
following sort: various arrows go the "wrong" way, Ext**A(k; k) is covariant in*
* A,
and one studies A by means of its quotient Hopf algebras (because those are dual
to the subHopf algebras of A*).
Each chapter is divided into a number of sections; at the beginning of each
chapter, we give a brief description of its contents, section by section. In t*
*his
introduction, we give a brief overview of each chapter. We note that each chapt*
*er
has at least one "Further discussion" section, in which we discuss issues auxil*
*iary
to the general discussion.
In Chapter 1, we set up notation and discuss results that hold in the catego*
*ry
Stable() for any graded commutative Hopf algebra over a field k, e.g., the dual
of a group algebra, the dual of an enveloping algebra, or the dual of the Steen*
*rod
algebra. Aside from setting up notation for use throughout this book, the main
topics of this chapter include: construction of cellular and Postnikov towers, *
*an
examination of the Adams spectral sequence associated to particular homology
theories on Stable(), and some remarks on Bousfield classes and BrownComenetz
duality.
Note. While some of Chapter 1 may be wellknown, we recommend that the
reader look over Section 1.2 and the first part of Section 1.3 (at least the de*
*finition
of the functor H) before reading later parts of the book. These sections introd*
*uce
notation that gets used throughout the book.
In Chapter 2 we specialize to the case in which p is a prime, k = Fpis the f*
*ield
with p elements, and A is the dual of the mod p Steenrod algebra. Recall from
[Mil58] that as algebras, we have
(
A ~= F2[1; 2; 3; : :]:; if p = 2,
Fp[1; 2; 3; : :]: E[o0; o1; o2;i:f:]:;p is odd.
The coproduct on A is determined by
Xn i
: n 7! pni i;
i=0
Xn i
: on 7! pni oi+ on 1:
i=0
In this chapter, we discuss two tools with which to study Stable(A): quotient H*
*opf
algebras of A and Psthomology. We use these tools to prove two theorems: the
first is a vanishing line theorem (given conditions on the Psthomology groups *
*of X,
then Exts;tA(Fp; X) = 0 when s > mtc, for some numbers m and c). The second is
a "selfmap" theorem: given a finite Acomodule M, we construct a nonnilpotent
element of Ext**A(M; M) satisfying certain properties.
Let p = 2. In Chapter 3 we develop analogues in the category Stable(A) of the
nilpotence theorem of Devinatz, Hopkins, and Smith, as well as the stratificati*
*on
theorem of Quillen. In fact we give two nilpotence theorems: in one we describe
a single ring object (like BP) that detects nilpotence; more precisely, there i*
*s a
quotient Hopf algebra D of A so that, if M is a finitedimensional Acomodule,
an element z 2 Ext**A(M; M) is nilpotent under Yoneda composition if and only
if its restriction to Ext**D(M; M) is nilpotent. The second nilpotence theorem*
* is
similar, but uses a family of ring objects (somewhat like the Morava Ktheories)
to detect nilpotence. These are versions in Stable(A) of the nilpotence theorems
PREFACE ix
of [DHS88 ] and [HSb ]. We strengthen these results when studying Ext**A(F2; F*
*2),
by "identifying" the image of Ext**A(F2; F2) !Ext**D(F2; F2) (and similarly fo*
*r the
other nilpotence theorem). One can view this as an analogue of Quillen's theorem
[Qui71 , 6.2], in which he identifies the cohomology of a compact Lie group up *
*to
Fisomorphism.
Again, let p = 2. In Chapter 4, we discuss applications of the theorems from
the previous chapter. In ordinary stable homotopy theory, the nilpotence theore*
*ms
lead to the periodicity theorem and the thick subcategory theorem (see [Hop87 ]*
*);
in our setting, things are a bit harder, so we get a weak version of a periodic*
*ity
theorem, and only a conjecture as to a classification of the thick subcategorie*
*s of
finite objects in Stable(A). More precisely, if M is a finitedimensional Acom*
*odule,
then we produce a number of central nonnilpotent elements in Ext**A(M; M) by
using the "variety of M": the kernel of Ext**D(F2; F2) !Ext**D(M; M).
One of our analogues of Quillen's theorem says that the elements of the kern*
*el
of Ext**A(F2; F2) ! Ext**D(F2; F2) are nilpotent, and it identifies the image.*
* This
identification is not explicit, so we discuss a small list of examples. We also*
* imi
tate [Rav84 ] to show that the objects that detect nilpotence have strictly sma*
*ller
Bousfield class than the sphere.
Let p be any prime. In Chapter 5 we consider Steenrod algebra analogues
of chromatic theory and the functors Ln and Lfn. The latter turns out be more
tractable; in fact, it is a generalization (from the module setting) of Margoli*
*s'
killing construction [Mar83 , Chapter 21]. We show that Ln 6= Lfnif n > 1, at
least at the prime 2. We compute Lfnon some particular ring spectra, and show
that, at least for these rings, it turns "group cohomology" into "Tate cohomolo*
*gy."
We use this result to show that the chromatic tower constructed using the funct*
*ors
Lfnconverges for any finite object. (This is an extension of a theorem of Margo*
*lis
[Mar83 , Theorem 22.1].)
We also have several appendices: In Appendix A.1, we describe a model cat
egory whose associated homotopy category is Stable(A); the results in this sect*
*ion
are due to Hovey. In Appendix A.2 we prove a theorem due to Hopkins and Smith
[HSa ], that the property of having a vanishing line with given slope, at some *
*term
of the Adams spectral sequence and with some intercept, is generic. (We prove t*
*his
in the context of Adams spectral sequences in Stable(A), which are trigraded; h*
*ence
we actually discuss vanishing planes.) In Appendix B, we discuss the nilpotence*
* of
certain classes in Ext**A(Fp; Fp) when A is the Steenrod algebra; we use these *
*results
in Chapter 3 to prove our nilpotence theorems.
In this book we have a mix of results: some are extensions of older results
to the cochain complex setting, and some are new. For each older result, if the
proof in the literature extends easily to our setting, then we do not include a
proof; otherwise, we at least give a sketch. It appears that when one uses the
language of stable homotopy theory, one tends to change arguments with spectral
sequences into simpler arguments with cofibration sequences (see Lemma 1.3.10,
for example), so even though the setting is potentially more complicated, some *
*of
the proofs simplify. In such cases, we often give in to temptation and include *
*the
new proof in its entirety (as, for example, with the vanishing line theorem 2.3*
*.1).
Obviously, we include full proofs of all of the new results, and we give comple*
*te
references for all of the old results.
x PREFACE
Acknowledgments: I have had a number of entertaining and illuminating dis
cussions with Mark Hovey, Mike Hopkins, and Haynes Miller on this material.
CHAPTER 1
Stable homotopy over a Hopf algebra
In this chapter we discuss stable homotopy theory over any graded commutative
Hopf algebra over a field k; a major focus of study is Ext**(k; k), where Ext
denotes comodule Ext_derived functors of Hom in the category of left Acomodule*
*s.
This material applies when is the dual of a group algebra, the dual of an enve*
*loping
algebra, or the dual of the Steenrod algebra; in these cases, Ext**(k; k) is the
ordinary cohomology of * with coefficients in k.
Our goal in this chapter is to establish some notation, make some basic def
initions, and prove some general facts about the category Stable() of cochain
complexes of injective comodules.
In more detail: we start in Section 1.1 with brief recollections about Hopf *
*al
gebras, comodules, and homological algebra. In Section 1.2 we define our setting
for the rest of the book, the category Stable(). We also set up the some im
portant notation; for instance, we explain the grading conventions on morphisms
and (co)homology functors in Stable()_if X is an injective resolution of a left
comodule M, then the (s; t)homotopy group sss;tX is equal to Exts;t(k; M). In
Section 1.3 we construct some particular ring objects in our category, one obje*
*ct
HB for every quotient Hopf algebra B of . To be precise, HB is an injective res
olution of 2Bk; this is a ring spectrum in Stable(). So for instance, if we we*
*re
working with = (kG)*, we would have one such object HB for every subgroup
B of G, and the object HB would have homotopy groups ss*HB = H*(B; k). In
Subsection 1.3.1 we establish some notation for Hopf algebra extensions, and we
prove one or two useful results about extensions with small kernel. For example,
given an extension of Hopf algebras of the form
E[x] !B !C;
the associated extension spectral sequence has only one possible differential; *
*if B is
a quotient Hopf algebra of , then in the category Stable(), this manifests itse*
*lf as
a cofibration sequence HB !HB !HC. In Section 1.4 we set up cellular towers
and Postnikov towers in Stable(), and we prove a Hurewicz theorem and a few
useful lemmas. In Section 1.5 we discuss the Adams spectral sequence based on t*
*he
homology theory associated to the ring spectrum HB, for B a "conormal" quotient
of . This turns out to be the same, up to a regrading, as the LyndonHochschild
Serre spectral sequence associated to the Hopf algebra extension
2Bk ! !B:
In Section 1.6 we define Bousfield classes and BrownComenetz duality, and we
recall some results of Ravenel's relating the two.
In Section 1.7 we apply some of this work to the study of stable homotopy ov*
*er
a group algebra. We point, for example, out that a corollary of work of Benson,
1
2 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
Carlson, and Rickard is a classification of the Bousfield lattice in Stable(kG**
*), for
G a pgroup and k a field of characteristic p.
1.1.Recollections
In this section we review material on Hopf algebras, comodules, and homologi*
*cal
algebra. This material is standard; many readers are probably familiar with it.
1.1.1. Hopf algebras. We start with the definition of a Hopf algebra. This
is standard; one reference is [MM65 ].
DefinitionL1.1.1.Fix a field k. A Hopf algebra over k is a graded kvector
space = i2Zitogether with the following structure maps:
oa unit map j :k !,
oa multiplication map : k !,
oa counit map ": !k,
oa comultiplication map : ! k ,
oand a conjugation map O: !.
The maps j and give the structure of an associative unital kalgebra_i.e., the
following two diagrams commute (all tensor products are over k):
1!
? ?
1 ?y ?y
! ;
and
k[

 [ =
1j  [
 []
u
u _____:w

 aeaeo
j1  =
 ae
ae
k
Dually, the maps " and make into a coassociative coalgebra_the dual diagrams
to those above (i.e., diagrams as above, but with all the arrows reversed) comm*
*ute.
We give an algebra structure via the composite
1T1! ! ;
where T : ! is the "twist" map: T(a b) = (1)abb a. (Dually,
we can give the structure of a coalgebra.) We insist that the maps and "
be algebra maps; equivalently, we insist that and j be coalgebra maps. Lastly,
1.1. RECOLLECTIONS 3
the conjugation map O makes the following diagram commute:
1O
_______w _____w
 
 
"  
 
 
u j u
k ___________________:w
Also, the same diagram, except with O 1 replacing 1 O, also commutes.
Convention 1.1.2.We work throughout with graded vector spaces; every map
between graded spaces is a graded map, and every element from such a vector spa*
*ce
is assumed to be homogeneous, unless otherwise indicated. Given v 2 V , we write
v for the degree of v. Also, all unmarked tensor products are over the ground*
* field.
Since the structure maps in the previous definition are assumedLto be graded
maps, then the image of j lies in 0, and the kernel of " contains i6=0i.
Definition 1.1.3.Let be a Hopf algebra over a field k. We say that is
commutative if it is commutative as an algebra_i.e., the following diagram com
mutes:
[
 [
T  [[]

u
_____:w
(As in Definition 1.1.1, T is the twist map.) is cocommutative if the dual dia*
*gram
commutes; isLbicommutative if it is both commutative and cocommutative. We
say that = ii is connected if i = 0 when i < 0, and j :k ! 0 is an
isomorphism.
Note that if each homogeneous piece iis finitedimensional, then the graded
dual * of has the structure of a Hopf algebra. Then is commutative if and
only if * is cocommutative, and so forth.
Example 1.1.4. (a)The homology of a topological group G with coeffi
cients in a field k is a cocommutative Hopf algebra; it is connected if and
only if G is connected.
(b)For any group G, the group algebra kG is a cocommutative Hopf algebra.
It is commutative if and only if G is abelian. It is graded trivially: eve*
*ry
element is homogeneous of degree zero. If G is finite, then the vector spa*
*ce
dual of kG is a commutative Hopf algebra.
(c)For any Lie algebra L, its universal enveloping algebra U(L) is a cocommu
tative Hopf algebra. It is commutative if and only if L is abelian. As with
kG, it is graded trivially (unless L is graded itself).
(d)Similarly, if k has characteristic p, for any restricted Lie algebra L, it*
*s re
stricted universal enveloping algebra V (L) is a cocommutative Hopf algebr*
*a.
(e)Fix a prime p. The mod p Steenrod algebra A* is a cocommutative Hopf
algebra; its dual A* is a commutative Hopf algebra. Both A* and A* are
connected. Starting in the next chapter, we will focus almost exclusively *
*on
this example.
4 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
Definition 1.1.5.An element fl of a Hopf algebra is primitive if (fl) =
fl 1 + 1 fl. We write P for the vector space of all primitives of . We say th*
*at
fl is grouplike if (fl) = fl fl.
1.1.2. Comodules. We move on to a brief discussion of modules and comod
ules. Again, [MM65 ] is one of the standard references; [Boa ] is also quite u*
*seful.
Definition 1.1.6.Let be a coalgebra over a field k. A kspace M is a (left)
comodule if there is a structure map :M ! M, called the coaction map,
making the following diagram commute:
M ! M:
? ?
?y ?y1
M 1! M
(In other words, defines a "coassociative" coaction.) A (left) module over a *
*k
algebra is defined dually, of course. We say that M = iMi is bounded below if
Mi= 0 for all sufficiently small i.
Throughout, we use left comodules and left modules; from here on, we will
omit the word "left."
Given two comodules M and N over a Hopf algebra , then M N is
naturally a comodule, via the structure map
M N MN! M N 1T1! M N 11! M N:
This is called the "diagonal" coaction. We can also put the "left" coaction on
M N, in which the structure map is
M N M1! M N:
We rarely use this comodule structure; when we do, we denote the comodule by
L
M N. If we want to explicitly distinguish the diagonal coaction from the left
coaction, we write M N for the tensor product with the diagonal coaction.
We will use the following lemma once or twice. It is fairly standard; see [B*
*oa ,
5.7], for example.
Lemma 1.1.7.Let M be a comodule. Then M with the diagonal coaction
L
is naturally isomorphic, as a comodule, to M with the left coaction. In
particular, M is a direct sum of copies of .
Proof. One can check that the following two composites are mutually inverse
comodule maps:
L 1 N 1O1
N ! N ! N ! N;
L
N 1N! N 1! N:
___
Lemma 1.1.8.Let be a Hopf algebra over k, and assume that each i is
finitedimensional. Let * denote the (graded) dual of . Then every comodule
1.1. RECOLLECTIONS 5
has a natural *module structure. If is finitedimensional, then the categories
Comod and *Mod are equivalent.
Proof. Let M be a comodule. Then we make it a *module via the struc
ture map
* M 1!* M ev1!k M = M:
If is finitedimensional, then one chooses dual bases (fli) and (gi) for and *
**.
Given a *module N with structure map ': * M ! M, we make N into a
comodule via the map
N ! N;
X
n 7! flj nj;
j
where we sum over all j so that n is "hit" by gj: we have '(gjnj) = n. Since is
finitedimensional, this is a finite sum, and hence an element of the tensor_pr*
*oduct.
We leave the rest of the proof to the reader. __
1.1.3. Homological algebra. Now we discuss a little homological "coalge
bra." [Boa ] is a good reference for this material, as well. One can also dua*
*lize
discussions of homological algebra for modules, as found in any number of places
(such as [CE56 , Wei94, Ben91a]).
Since we are working with comodules rather than modules, we work with the
notions of cofree and injective comodules, which are dual to the notions of fre*
*e and
projective, respectively.
Definition 1.1.9.Let be a kcoalgebra. A comodule M is injective if the
functor Hom (; M) is exact. A comodule M is projective if Hom (M; ) is exact.
The forgetful functor U: Comod !kMod has a right adjoint, C:
Hom k(UM; N) ~=Hom (M; CN):
CN is called the cofree comodule on N.
See [Boa ] for the following two results.
Lemma 1.1.10.A comodule is injective if and only if it is a summand of a
cofree comodule.
Lemma 1.1.11. is cofree, and hence injective, as a comodule.
In other words, every comodule map M ! is adjoint to a vector space map
M !k, and hence is determined by the preimage of 1 2 .
Injective comodules are much more important to us than projective comodules,
because of the following.
Example 1.1.12. (a)On one hand, if is finitedimensional, then injective
comodules are the same as projective comodules.
(b)On the other hand, if = A* is the dual of the mod p Steenrod algebra, then
it seems likely that there are no nonzero projective comodules. Essentiall*
*y,
an element in a projective comodule over A* should need to have infinitely
many elements in its diagonal, so it would have to be viewed as a completed
comodule, not a comodule proper.
6 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
By Lemma 1.1.7, M is injective for any comodule M; indeed, M is
isomorphic to C(UM) . The map M ! M which is adjoint to the identity on
UM gives us the start of an injective resolution of M.
Definition 1.1.13.Given a comodule M, a sequence of injective comodules
Io = (I0 !I1 !I2 !: :):is an injective resolution of M if the sequence
0 !M !I0 !I1 !I2 !: : :
is exact.
As is wellknown, injective resolutions are unique up to chain homotopy equi*
*va
lence. We are interested in injective comodules because they are useful in comp*
*uting
various derived functors, particularly Ext.
Definition 1.1.14.Let be a Hopf algebra over k, and let M and N be
comodules. Then Exts(M; N) is the sth derived functor of Hom (M; N). To
compute it, one takes an injective resolution Io of N, and defines Exts(M; N) to
be the sth cohomology group of the cochain complex Hom (M; Io).
Throughout this book, when we say Ext, we mean Ext in this setting: the
category of comodules over a Hopf algebra.
Lemma 1.1.15.Let be a connected Hopf algebra over a field k. Then the
vector space Ext1(k; k) is isomorphic to P, the space of primitives of .
Proof. This is dual to the statement that in the category of modules over an
algebra, Ext1is in bijection with the generators of the algebra. One proves thi*
*s by
constructing the first few terms of a minimal resolution of k as a comodule._We
leave the details to the reader. __
We point out that for any comodule, there is a canonical injective resolutio*
*n,
known as the cobar complex. We will do a few computations with it in Appendix B*
*.3;
we refer the reader to [Ada56 , HMS74 ] for details.
1.2.The category Stable()
In this section, we define the category in which we work, and we introduce
notation which we will use for the rest of this book.
Let be a graded commutative Hopf algebra over a field k; we work in the
category Stable() whose objects are cochain complexes of injective graded left 
comodules, and whose morphisms are cochain homotopy classes of graded maps.
We call objects of this category spectra, and we write the morphisms from X to Y
as [X; Y ]. We will omit the word "graded" from this point on; all comodules and
maps are understood to be graded.
The category Stable() is a stable homotopy category in the sense of [HPS97 ];
hence one can perform many standard stable homotopy theoretic constructions
in Stable(). For instance, rather than having exact sequences, one has "exact
triangles," also known as "cofibrations" or "cofiber sequences." We freely use *
*other
language and results from [HPS97 ], often without explicit citation. Stable() is
generated by the injective resolutions of the simple comodules_that is, if X is
an object so that [S; X]** = 0 whenever S is an injective resolution of a simple
comodule, then X is a contractible cochain complex. If the trivial comodule k is
the only simple (say, if is connected, or if is the dual of the mod p group a*
*lgebra
1.2. THE CATEGORY Stable() 7
of a pgroup), then we say that Stable() is monogenic, at least in the graded s*
*ense.
In this case, the stable homotopy constructions are even more familiar.
If Stable() is monogenic and the homotopy of the sphere object (defined belo*
*w)
is countable, then Stable() is a Brown category, so that homology functors are
representable. This is the case when = A, the dual of the Steenrod algebra.
(Stable() can be a Brown category even if it is not monogenic; since we are foc*
*using
on the Steenrod algebra in this book, though, we often assume monogenicity.)
Remark 1.2.1. (a)Mahowald and Sadofsky studied the category Stable()
in their paper [MS95 ], with = A.
(b)If is finitedimensional, then one could just as well work with the cat
egory of cochain complexes of injective *modules, because in the finite
dimensional case, the categories of comodules and *modules are equiv
alent, with injective comodules corresponding to injective modules. When
* is not finitedimensional, in particular when * = A* is the Steenrod al
gebra, there are technical problems with the category of cochain complexes
of injective A*modules. For example, there are no maps from F2to A*, so
the "homotopy" of the (injective) module A* would be zero; therefore in the
module setting, we would not have the implication ss*X = 0 ) X = 0.
(c)We also note that, regardless of the dimension of , the category Stable() *
*is
rather different from the derived category of comodules, because homology
isomorphisms are not necessarily invertible in Stable(). For instance, if
= F2[x]=(x2) with x primitive, then the (periodic) cochain complex
: ::! ! ! ! !: :;:
x 7! 1; x 7! 1;
has no homology, and hence is zero in the derived category. On the other
hand, it is noncontractible in Stable(); if we write Ext**(F2; F2) = F2[v*
*],
then this complex is a ring spectrum with homotopy groups (as defined
below) equal to F2[v; v1].
The category Stable() has arbitrary coproducts; we use the symbol _ to denote
the coproduct.
For objects X and Y of Stable(), we write X ^ Y for X k Y , and we call
this the smash product of X and Y . This operation is commutative, associative,
and unital: if S is an injective resolution of the trivial comodule k, then S i*
*s the
unit of the smash product. We call S the sphere spectrum. We grade morphisms
as follows: first of all, we let [X; Y ]0;0= [X; Y ]. We write
S = (I0 !I1 !I2 !: :):;
and we let Si;jbe the cochain complex which is sjIn+iin homological degree n
(here s denotes the "internal" suspension functor s: Comod !Comod). For
integers i and j we define the (i; j)suspension functor i;jby
i;j:Stable() !Stable();
X 7! Si;j^ X:
L
Then we let [X; Y ]i;j= [i;jX; Y ]0;0, and we write [X; Y ]**for i;j[X; Y ]i;*
*j.
Remark 1.2.2.The grading is the usual Ext grading: if X and Y are injective
resolutions of comodules M and N, respectively, then [X; Y ]i;j= Exti;j(M; N).
8 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
Hence if is concentrated in degree 0 (e.g., if = (kG)*), then one may as well
work with comodules concentrated in degree 0, in which case [; ]ij= 0 if
j 6= 0, and [; ]i0= Exti( ; ). We will follow the (somewhat odd) tradition *
*in
homotopy theory of drawing pictures of Ext (and hence of [; ]) using the Adams
spectral sequence grading: Exts;tis drawn with s on the vertical axis and t  s*
* on
the horizontal axis.
Unfortunately, because of the form of long exact sequences in Ext, cofiber
sequences look like this:
: ::!1;0Z !X !Y ! Z !1;0X !: :::
So one needs to take a little care when translating proofs from ordinary homo
topy theory to this setting. Given a spectrum X, we define the homology functor
associated to X, Xij, by
Xij:Stable() !Ab;
Y 7! [S; X ^ Y ]i;j;
and we define the cohomology functor associated to X, Xij, by
Xij:Stable()op! Ab;
Y 7! [Y; X]i;j:
When X = S, we have a special notation for Xij: we define the (i; j)homotopy
group of Y to be ssijY = SijY = [S; Y ]ij. Note that if X ! Y ! Z is a cofiber
sequence, then we have
: ::!ssi1;jZ !ssi;jX !ssi;jY ! ssi;jZ !ssi+1;jX !: : :
(andLsimliarly for other homology functors). As with morphisms, we write ss**()
for i;jssi;j() (and similarly for other homology and cohomologyLfunctors). A*
*lso,
given a spectrum X, we write Xi;jfor ssi;jX = Xi;jS, and X**for i;jXi;j.
We say that an object R in Stable() is a ring spectrum if there is a multipl*
*ica
tion map : R ^R !R and a unit map S !R, making the appropriate diagrams
commute.
We often abuse notation and let S0 = S0;0= S. One of our main goals is to
get as much information as possible about ss**S0 = Ext**(k; k).
1.3. The functor H
We assume that is a graded commutative Hopf algebra over a field k, and
we work in the category Stable(). The quotient coalgebras and Hopf algebras of
carry useful information; in this section we construct a spectrum HB in Stable()
for each quotient coalgebra B of , and we study the properties of the functor H.
Recall that if B is a quotient coalgebra of and if M is a Bcomodule, then
the cotensor product 2BM is defined to be the equalizer of the two maps
M 1M! B M;
M 1M! B M:
Here is the right Bcomodule structure map on , and M is the left Bcomodule
structure map on M. (The tensor products are over k.)
1.3. THE FUNCTOR H 9
Definition 1.3.1.We define a covariant functor H from quotient coalgebras
of to spectra by defining HB to be an injective resolution of 2Bk.
For example, Hk = , so that Hk**(X) is the homology of the cochain com
plex X; also, H = S0. H provides a useful source of (co)homology functors on
Stable(). The general philosophy is that if one has a quotient B of , rather th*
*an
studying B by working in Stable(B), one studies B by looking at HB in the cate
gory Stable(). This is borne out by Corollary 1.3.5, as well as the other resul*
*ts in
this section.
Fix a quotient coalgebra B of . Given an comodule M, we let M#B denote
its restriction to B; this is the Bcomodule with structure map M ! M !
B M. Recall from [MM65 ] and [Rad77 ] that if B happens to be a quotient
Hopf algebra of , then #B is injective as a right (and a left) Bcomodule.
Proposition 1.3.2.For any quotient coalgebra B of over which #B is in
jective as a right Bcomodule, we have HBij~=ExtijB(k; k). Furthermore, we have
the following.
(a)If B is a quotient Hopf algebra of , then HB is a commutative ring spec
trum.
(b)If B is a quotient coalgebra of B over which #B is injective as a right
comodule, then HB has many of the properties of a ring spectrum:
(i)HB**is a kalgebra, and for any spectrum Y , HB**Y is a right module
over HB**.
(ii)There is a "unit map" S0 ! HB which induces an algebra map
ss**S0 !HB**.
(iii)More generally, if B i C are quotient coalgebras of over which
is injective, then the induced map HB**! HC**is an algebra map.
Proof. That HB**~=Ext**B(k; k) follows from Lemma 1.3.4 below.
Part (a) is clear_since is commutative as an algebra, then so is B, and the
commutative product on B induces one on HB. The quotient map i B induces
the unit map S0 = H ! HB, and hence a Hurewicz map ss**(X) ! HB**(X).
This Hurewicz map is the same as the restriction map res;B: Ext**(k; k) !
Ext**B(k; k).
For part (b), the unit map is induced from the quotient i B, as in (a).
The induced map in (iii) is just the restriction map. The rest of (b) follows_f*
*rom
Lemma 1.3.4 and Corollary 1.3.5. __
Example 1.3.3.Note that if B is a quotient coalgebra of , then HB need
not be a ring spectrum. For example, suppose that k has characteristic p, and l*
*et
= k[x] with x primitive. Then B = k[xp] is a quotient coalgebra of , and HB
is an injective resolution of M = k[x]=(xp). A multiplication on HB would induce
one on M (by taking homology), and it is easy to see that M is not a comodule
algebra: the comodule structure on M is given by x 7! 1 x + x 1 2 M, so
if this coaction were multiplicative, we would have
0 = xp 7! xp 1 6= 0:
Lemma 1.3.4.Suppose that B is a quotient coalgebra of so that #B is in
jective as a right Bcomodule.
(a)Given a comodule M and a Bcomodule N, we have an isomorphism
Ext**B(M#B ; N) ~=Ext**(M; 2BN ):
10 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
(b)If N is a comodule, then we have an isomorphism of comodules
2BN ~=( 2Bk) N:
(c)Hence if M and N are both comodules, then we have
Ext**B(M#B ; N#B) ~=Ext**(M; ( 2Bk) N ):
Proof. Part (a) is (the coalgebra version of) Shapiro's lemma. See [Wei94 ,
Lemma 6.3.2], [Ben91a , Corollary 2.8.4], or any other good reference on homolo*
*g
ical algebra, for a statement in terms of modules and algebras. We leave it to *
*the
reader to dualize to our version; see [Rav86 , A1.3.13] for a related result.
Part (b), the "shearing isomorphism," seems to be wellknown, but it may be
difficult to locate a proof in the published literature (see [HSb ] and [Boa ],*
* for
instance). Let Ntrdenote N with the trivial coaction of , so that Ntris the
L
tensor product N with the left coaction (as compared with N = N, the
tensor product with the diagonal coaction). The desired isomorphism is induced *
*by
one between Ntrand N, because 2BN is a subcomodule of the former,
and ( 2Bk) N a subcomodule of the latter. Hence Lemma 1.1.7 completes the
proof of (b). __
Part (c) follows from parts (a) and (b). __
Given a quotient coalgebra B of and given objects X and Y of Stable(),
we let [X; Y ]B denote the set of cochain homotopy classes of maps from X to Y ,
viewed as cochain complexes of left Bcomodules.
Corollary 1.3.5.Suppose that B is a quotient coalgebra of so that #B
is injective as a right Bcomodule. Given objects X and Y of Stable(), we have
[X; HB ^ Y ]**~=[X; Y ]B**. In particular, HB**is an algebra and HB**Y is a rig*
*ht
module over it.
Corollary 1.3.6.The spectrum Hk is a field spectrum: it is a ring spectrum,
and any Hkmodule spectrum is a wedge of suspensions of Hk. In particular, for
any X, Hk ^ X is a wedge of suspensions of Hk.
Proof. By Proposition 1.3.2(b), Hk = is a ring spectrum. By Lemma 1.1.7,
for any comodule M, M is a direct sum of copies of ; one can check (by
[Boa , 5.7], for instance) that given a comodule map M ! N, the induced map
M ! N sends each summand of either isomorphically to a summand,
or to zero. So if X is any cochain complex of comodules, then Hk ^ X = X
splits into a direct sum of cochain complexes of the forms
0 ! =! !0
and
0 ! !0:
___
1.3.1. Remarks on Hopf algebra extensions. Many results about group
cohomology (and indeed about Hopf algebra cohomology in general) are proved
using the spectral sequence associated to an extension. If one has a group exte*
*nsion
in which the quotient is cyclic of prime order, the associated spectral sequenc*
*e is
particularly tractable, and hence quite useful. In this subsection we remind t*
*he
reader of standard notation related to Hopf algebra extensions, and then we foc*
*us
1.3. THE FUNCTOR H 11
on extensions with small kernel. In Section 1.5 we discuss the spectral sequence
associated to a general Hopf algebra extension.
Definition 1.3.7. (a)Suppose that C ,! is an inclusion of augmented
algebras over a field k, and let IC be the augmentation ideal of C_the ker*
*nel
of C !k. We say that C is normal in if the left ideal of generated by
C (i.e., . IC) is equal to the right ideal generated by C (i.e., IC . ). *
*If C
is normal in , then we let ==C = C k = k C . In this case,
C ! !==C
is an extension of augmented kalgebras.
(b)Dually, suppose that !B is a surjective map of coaugmented coalgebras
over k, and let JB denote the coaugmentation coideal of B_the cokernel of
k !B. We say that B is a conormal quotient of if 2Bk = k 2B. If B
is a conormal quotient of , then
2Bk ! !B
is an extension of coaugmented kcoalgebras.
As far as this definition goes, for us will usually be a Hopf algebra with
commutative multiplication, so that every subalgebra of will be normal. We will
be more interested in quotients of ; if !B is a surjective map of commutative
Hopf algebras over k, then 2Bk is the algebra kernel. So B is conormal if the
algebra kernel is also a subcoalgebra of .
Remark 1.3.8.If is a commutative Hopf algebra and B is a quotient coalge
bra of over which #B is injective as a right comodule, then Lemma 1.3.4 tells *
*us
that Ext**B(k; k) ~=Ext**(k; 2Bk). If, in addition, B is conormal, then there *
*is a
right coaction of on 2Bk, which induces a left coaction of on Ext**B(k; k). *
*The
coaction of B is clearly trivial, so we get a left coaction of 2Bk on Ext**B(k*
*; k);
indeed, we get a left coaction of 2Bk on Ext**B(k; M) for any left Bcomodule *
*M.
For the remainder of this section, we assume that the ground field k has cha*
*r
acteristic p > 0.
Notation 1.3.9.Given a homogeneous element x in a graded vector space, let
x denote its degree. Let E[x] denote the Hopf algebra k[x]=(x2) with x primit*
*ive,
where x is odd if p is odd. Let D[x] = k[x]=(xp) with x primitive, where x
is even if p is odd. Recall from Lemma 1.1.15 that if B is a Hopf algebra, then
Ext1;*B(k; k) = HB1;*is isomorphic to the vector space of primitives of B. If y*
* 2 B
is primitive, we let [y] denote the associated element of HB1;*. Recall further*
* that
if p is odd, then there is a Steenrod operation
fifP0: Exts;tB(k; k) !Exts+1;ptB(k; k):
(See [May70 , Wil81], as well as Appendix B.1.)
Here is our analysis of Hopf algebra extensions with small kernel.
Lemma 1.3.10.Fix a graded commutative Hopf algebra over a field k of char
acteristic p > 0.
(a)Suppose that there is a Hopf algebra extension of the form
E[x] !B !C;
12 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
where B is a quotient Hopf algebra of . Then x is primitive in B, so
there is a nonzero element h = [x] in Ext1;xB(k; k) = HB1;x. We also
let h: 1;xHB !HB denote the corresponding selfmap of HB_i.e., the
composite
1;xHB = S1;x^ HB h^1!HB ^ HB !HB;
where is the multiplication map. Then there is a cofiber sequence
1;xHB h!HB !HC !0;xHB:
(b)Suppose that p is odd and there is a Hopf algebra extension of the form
D[x] !B !C;
where B is a quotient Hopf algebra of . Then x is primitive in B; we let
b = fifP0[x] in Ext2;pxB(k; k) = HB2;px. We also let b: 2;pxHB !HB
denote the corresponding selfmap of HB. Then there is a cofiber sequence
2;pxHB b!HB !gHC !1;pxHB;
where gHCis defined by a cofibration
1;xHC !gHC !HC !0;xHC:
The element b = fifP0[x] in (b) is also the pfold Massey product of [x] with
itself, as mentioned in [May70 , 11.11].
Proof. Part (a): It is clear that x is primitive, so we only need to discuss*
* the
putative cofibration. There is a short exact sequence of E[x]comodules
0 !k !E[x] !xk !0:
We apply the (exact) functor 2B to this, noting that E[x] = B 2Ck:
0 ! 2Bk ! 2Ck !x 2Bk !0:
Taking injective resolutions then gives the desired cofibration.
Part (b): Note that there are two short exact sequences of D[x]comodules:
0 !k !D[x] !xk[x]=(xp1) !0;
0 !k[x]=(xp1) !D[x] !(p1)xk !0:
We apply the functor 2B to these and take injective resolutions; writing gHB
for the injective resolution of 2B(k[x]=(xp1)), we have the following cofiber*
* se
quences:
1;xgHBj!HB !HC !0;xgHB;
0
1;(p1)xHB j!gHB! HC !0;(p1)xHB:
It is standard (e.g., [Ben91b , pp. 1378]) that the map b is the composite j O*
* j0,
and the 3 x 3 lemma (or the octahedral axiom_see [HPS97 , A.1.1A.1.2]) allows *
* __
us to identify the cofiber of j O j0in terms of the cofibers of j and j0. *
* __
1.4. SOME CLASSICAL HOMOTOPY THEORY 13
1.4.Some classical homotopy theory
We assume that is a graded commutative Hopf algebra over a field k, and
we work in the category Stable(). For this section, we assume that Stable() is
monogenic (i.e., the trivial comodule k and its suspensions are the only simple
comodules). We also assume that is nonnegatively graded: n = 0 if n < 0.
Because ssijS0 = Exti;j(k; k), then ss**S0 is concentrated in the first quad*
*rant.
More precisely, ssijS0 = 0 if j < 0 and (unless i = j = 0) if i 0; furthermore,
ss00S0 = k. Since ss**S0 is "connected" in this sense, we can construct cellul*
*ar
towers and hence Postnikov towers, as in the usual stable homotopy category (and
indeed in any connective stable homotopy category_see [HPS97 , Section 7]). We
also have a Hurewicz theorem. Since we are working in a bigraded setting rather
than the singly graded setting of [HPS97 , Section 7], we state the relevant re*
*sults;
we leave the proofs as an exercise for the reader. (See also the appropriate pa*
*rts of
[Mar83 ].)
Definition 1.4.1.Given a spectrum X, we say that
: ::!Xn1 !Xn !Xn+1 !: : :
is a cellular tower for X if
(i)wcolimXn = X,
(ii)The fiber of Xn !Xn+1 is a coproduct of spheres Si;jwith i + j = n,
(iii)limHk**Xn = 0.
Because of the connectivity properties of ssijS0, we have the following theo*
*rem,
guaranteeing existence of cellular and Postnikov towers.
Theorem 1.4.2. (a)Any spectrum X has a cellular tower.
(b)For any spectrum X and integer n, there is a cofiber sequence
X[n; 1] !X !X[1; n  1];
so that
(i)ssijX[n; 1] = 0 if i + j < n,
(ii)ssijX[1; n  1] = 0 if i + j n,
(c)If for some integer n we have spectra X and Y with X = X[n; 1] and
Y = Y [1; n  1], then [X; Y ]**= 0.
(d)Hence for any spectrum X, we can construct a Postnikov tower:
: ::! X[1; r]! X[1; r  1]! X[1; r  2]!: : :
x? x x
? ?? ??
X[r] X[r  1] X[r  2]
(Here X[r] is a coproduct of factors of the form i;jHk, where i + j = r.)
The sequential colimit of X[1; r] is 0, and the sequential limit of X[1;*
* r]
is X.
(e)Dually, we can construct a diagram
: ::! X[r; 1]! X[r  1; 1]!X[r  2; 1]!: : :
?? ? ?
y ?y ?y
X[r] X[r  1] X[r  2]
14 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
The sequential colimit of the "connective covers" X[r; 1] is X, and the
sequential limit is 0.
Parts (d) and (e) are easy in our context: if k is the only simple comodule,*
* then
every injective is a direct sum of copies of = Hk. So to construct X[r; 1], for
example, one just truncates the cochain complex X at bidegrees (s; t) with s+t *
*< r.
Note that if is a connected Hopf algebra (Definition 1.1.3), then ssijS0 = 0
if j < i; since the Steenrod algebra is connected, we use this pattern for our
definition of connectivity. At the other extreme, if is concentrated in degree*
* zero,
then ssstS0 = 0 if t 6= 0; this leads to a weaker notion of connectivity.
Definition 1.4.3.Given a spectrum X, if there exist numbers i0 and j0 so
that ssijX = 0 when i < i0 or j  i < j0, then we say that X is (i0; j0)connec*
*tive.
We say that X is connective if X is (i0; j0)connective for some unspecified i0*
* and
j0. If for some i0 and j0, we have ssijX = 0 when i < i0 or j < j0, we say that*
* X
is weakly (i0; j0)connective.
Here is the second main theorem of this section, a bigraded version of the
Hurewicz theorem.
Theorem 1.4.4.If X is (i0; j0)connective, then the Hurewicz map ssijX !
HkijX is an isomorphism when i < i0 or j  i j0. Similarly, if X is weakly
(i0; j0)connective, then ssijX !HkijX is an isomorphism when i < i0 or j j0.
Remark 1.4.5.We point out that one can use the bigrading to generalize these
results somewhat; for example, one can discuss cellular towers with "slope u_v"_
towers as above, but with the fiber of Xn !Xn+1 equal to a coproduct of spheres
Si;jwith vi + uj = n. Using the form of connectivity of the sphere spectrum, one
can construct cellular towers of any given nonpositive slope. Similar remarks h*
*old
for Postnikov towers.
Similarly, one can define "connectivity with slope u_v" for nonpositive u_v_*
*i.e.,
ss**X = 0 below a line of slope u_v_and then show that the Hurewicz map is an
isomorphism at and below the line of connectivity.
Furthermore, if happens to be connected, then ssijS0 = 0 when j < i and
when i < 0. In this case, one can construct cellular towers and prove Hurewicz
theorems for any slope m 0 or m > 1.
One could also work with connectivity determined by arbitrary nonincreasing
curves rather than lines of nonpositive slope and get the same sorts of results*
*. We
have no need to work with anything approaching this level of generality.
If is connected and X is an injective resolution of a bounded below comod
ule M, then X is (0; 0)connective, but it also satisfies a stronger property. *
*This
property occurs several times in this work, so we make it into a definition.
Definition 1.4.6.We say that a spectrum X is comodulelike, or CL, if X
satisfies the following conditions:
(i)There exists an integer i0 such that ssi*X = 0 if i < i0,
(ii)There exists an integer j0 such that ssijX = 0 if j  i < j0,
(iii)There exists an integer i1 so that (Hk)i*X = 0 for i > i1.
(When X is an injective resolution of M, we may take i0 = 0 = i1, and j0 to
be the degree of the bottom class of M.)
1.5. THE ADAMS SPECTRAL SEQUENCE 15
Note that X is a CLspectrum if and only if X has a cellular tower built of
spheres Si;jwith i0 i i1 and j  i j0.
We will need the following lemmas later. First we need to recall a few defin*
*itions
from [HPS97 , 1.4.3, 2.1.1].
Definition 1.4.7. (a)A full subcategory D of Stable() is localizing if it is
"closed under cofibrations and coproducts": if X !Y ! Z is a cofibration
and two of X, Y , andLZ are in D, then so is the third; if {Xff} is a set *
*of
objects in D, then Xffis in D. Given an object Y , we let loc(Y ) denote
the localizing subcategory generated by Y , i.e., the intersection of all *
*of the
localizing subcategories containing Y .
(b)Similarly, a full subcategory D of Stable() is thick if it is closed under
cofibrations and retracts (if Y is in D and there are maps X !Y ! X so
that the composite is an isomorphism, then X is in D); and thick(Y ) denot*
*es
the thick subcategory generated by Y .
(c)A property P of spectra is generic if the full subcategory of spectra sati*
*sfying
P is thick.
(d)An object X of Stable() is finite if and only if it is small (in the categ*
*orical
sense), if and only if it is in thick(S0).
If is connected, then an object X is finite if and only if X is connective *
*and
has dimkHk**X < 1. So if is connected and B is a quotient Hopf algebra of ,
then an object X of Stable() is finite if and only if its restriction X#B is fi*
*nite in
Stable(B).
Lemma 1.4.8.Suppose that X is a spectrum and that there is a line of nonpos
itive slope above which the homotopy of X is zero (i.e., for some u; v with u_v*
* 0,
there is an n so that if vi + ij n, then ssijX = 0). Then X is in the localizi*
*ng
subcategory generated by = Hk.
Proof. The Postnikov tower of slope u_vfor such an X displays X as being a *
* __
colimit of objects of loc(); hence X is itself an object of loc(). *
*__
Lemma 1.4.9.If D is a localizing subcategory of Stable() which contains a
nonzero finite spectrum, then loc() D.
Proof. It suffices to show that 2 obD if D is as given. Let Y be a nonzero
finite object of D; then Y ^ is nonzero (by the Hurewicz theorem 1.4.4_remember
that = Hk) and is contained in obD. On the other hand, Corollary 1.3.6 tells us
that Y ^ is a direct sum of suspensions of , so by the Eilenberg swindle_[HPS97*
* ,
1.4.9], 2 obD. __
We will see in Corollary 4.5.7 that as long as is infinitedimensional, the
containment loc() D is strict (i.e., loc() contains no nonzero finite spectrum*
*).
1.5.The Adams spectral sequence
As in the rest of this chapter, we assume that is a graded commutative
Hopf algebra over a field k. We discuss the generalized Adams spectral sequence
associated to the homology theory HB**in this section, for B a conormal quotient
of (see Definitions 1.3.1 and 1.3.7). In particular, we note that it is the sa*
*me as
the spectral sequence associated to a Hopf algebra extensions, and we derive a *
*few
consequences.
16 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
For the general approach to the Adams spectral sequence, see [Ada74 ]. We
also recall a construction of the spectral sequence in Appendix A.2.
Theorem 1.5.1.Suppose that B is a conormal quotient Hopf algebra of and
fix a spectrum X. Then the Adams spectral sequence based on the homology theory
HB**has E2term
Es;t;u2= Exts;t;uHB**HB(HB**; HB**X);
with differentials
dr: Es;t;ur!Es+r;t+r1;ur+1r;
and abuts to sss+u;t+uX.
Given a comodule M, the changeofrings spectral sequence associated to the
extension
2Bk ! !B
has E2term
8Ep;q;v2= Extp;v q;*
(1.5.2) 2Bk (k; ExtB(k; M))
and converges to
Extp+q;v(k; M):
(The action of 2Bk on Ext**B(k; M) was discussed in Remark 1.3.8.) The differ
entials are indexed as follows:
dr: 8Ep;q;vr!8Ep+r;qr+1;vr:
(See [Sin73, II, x5], for example; alternatively, one can dualize the construct*
*ion
of the LyndonHochschildSerre spectral sequence for the computation of group
cohomology. See [Ben91b , 3.5] for a construction of this as the spectral seque*
*nce
associated to a double complex, for instance.)
Proposition 1.5.3.Suppose that B is a conormal quotient of , and suppose
that X is an injective resolution of a comodule M. Then the HBbased Adams
spectral sequence abutting to ss**X is isomorphic, up to a regrading, to the ch*
*ange
ofrings spectral sequence associated to the extension
2Bk ! !B;
abutting to Ext**(k; M). The regrading is as follows: for all r 2, the Es;t;ur*
*term
of the Adams spectral sequence is isomorphic to the 8Es;u;t+urterm of the chan*
*ge
ofrings spectral sequence.
An example of the regrading is the isomorphism of E2terms
Extp;v2Bk(k; Extq;*B(k;)M)~=Extp;vq;qHB**HB(HB**; HB**X):
Proof. This is an exercise in homological algebra. The key observation is th*
*at
since B is conormal, then 2Bk is a trivial Bcomodule. Hence
HB**HB = Ext**B(k; 2Bk) ~=( 2Bk) Ext**B(k; k) = 2Bk HB**:
If X is an injective resolution of a comodule M, then combining X with an Adams
tower for X yields a double complex; using the above isomorphism, one can easily
show that the resulting spectral sequence is the changeofrings spectral seque*
*nce.
1.5. THE ADAMS SPECTRAL SEQUENCE 17
The shift in gradings comes from a more precise statement of the above iso
morphism:
M `;j
Ext`;mB(k; 2Bk) ~= ( 2Bk)i ExtB(k; k):
i+j=m
___
Hence the HBbased Adams spectral sequence has some nice properties. For
example, we have a convergence result: if X and M are as in the proposition, th*
*en
the Adams spectral sequence converges to ss**X = Ext**(k; M). (We also show
in Proposition A.2.5 that the spectral sequence converges to ss**X whenever is
a connected Hopf algebra and X is connective_i.e., every connective spectrum is
"HBcomplete".)
Also, if X = S0 (or more generally if X is an injective resolution of a comm*
*uta
tive comodule algebra), then one has Steenrod operations acting on this spectr*
*al
sequence, as described in [Sin73] (see also [Saw82 ]). We need the following re*
*sult
in Chapter 3.
Proposition 1.5.4.Suppose that is a Hopf algebra over the field Fp, and
suppose that B is a conormal quotient of . Consider the HBbased Adams spectral
sequence converging to ss**S0. Givennyn2 E0;t;u2, with t and u even if p is odd*
*, then
for each n, ypn survives to E0;ppt;pnu+1.
(Note that the result is the same whether one is using the Adams grading
(Theorem 1.5.1) or the changeofrings grading (equation (1.5.2)): the elements*
* in
8E0;q;v2in the changeofrings grading correspond to elements in E0;vq;q2= E0;*
*t;u2
in the Adams grading; those with q and v even correspond to those with t and u
even.)
Proof. This follows from properties of Steenrod operations on this spectral
sequence, as discussed in [Sin73] and [Saw82 ]. Suppose we have a spectral se
quence Es;trwhich is a spectral sequence of algebras over the Steenrod algebra.*
* Fix
z 2 Es;tr, and fix an integer k. If p = 2, then [Sin73, 1.4] tells us to which *
*term
of the spectral sequence Sqkz survives (the result depends on r, s, t, and k). *
*For
instance, if z 2 E0;tr, then Sqt(z) = z2 survives to E0;2t2r1. [Saw82_,_2.5] *
*is the
corresponding result at odd primes. __
By the way, Singer's results [Sin73] are stated in the case of an extension *
*of
commutative Hopf algebras
B ! !C
where C is also cocommutative (actually, he works in the dual situation). This
cocommutativity condition is not, in fact, necessary, as forthcoming work of Si*
*nger
shows [Sin].
We also need the theorem of Hopkins and Smith [HSa ] that the presence of
a vanishing plane in the Adams spectral sequence is a generic property. Suppose
that E is a spectrum satisfying the following conditions (cf. [Ada74 , III.15] *
*and
[Rav86 , 2.2.5]):
(a)E is a commutative associative ring spectrum.
(b)E**E is flat over E**.
18 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
(c)E is (0; 0)connective (Definition 1.4.3), and the unit map j :S0 !E in
duces an isomorphism on ss0;0.
(d)HkijE is a finitedimensional kvector space for each i and j.
Theorem 1.5.5 (HopkinsSmith).Suppose that is a connected Hopf algebra.
Consider the Adams spectral sequence Es;t;urbased on a ring spectrum E satisfyi*
*ng
the above conditions. Fix numbers m 0 and n. The full subcategory of Stable()
consisting of all spectra X that satisfy the property "there exist numbers r an*
*d b so
that Es;t;ur(X) = 0 when s m(s + u) + n(t + u) + b" is thick.
Since we are not aware of any published proof of this, we give a proof in
Appendix A.2.
1.6.Bousfield classes and BrownComenetz duality
Again, we suppose that is a graded commutative Hopf algebra over a field
k. We assume that Stable() is monogenic. In this section we collect some useful
results about Bousfield classes, BrownComenetz duality, and their interaction.
We remind the reader that the Bousfield class of an object X in an arbit*
*rary
stable homotopy category is the collection of Xacyclic objects_all objects Z w*
*ith
X ^ Z = 0. We order Bousfield classes by reverse inclusion, so means
that X ^ Z = 0 ) Y ^ Z = 0. X and Y are Bousfield equivalent if = . We
define the operation _ on Bousfield classes by _ = ; one can s*
*how
that this is the leastupper bound of and . A theorem of Ohkawa [Ohk89 ]
says that there is a set of Bousfield classes; hence there is also a greatestl*
*ower
bound operation (take the leastupper bound of all the classes less than or equ*
*al to
both and ). For certain wellbehaved classes of spectra, the greatestl*
*ower
bound is given by ^, but that is not true in general.
These concepts were introduced (in the ordinary stable homotopy category)
by Bousfield in [Bou79a ] and [Bou79b ]; Ravenel proved a number of fundamen
tal results about them in [Rav84 ]. See [HPS97 , Section 3.6] for a discussion *
*of
Bousfield classes in a general stable homotopy category.
For any spectrum X, we define its BrownComenetz dual IX to be the spectrum
that represents the cohomology functor
Y 7! Hom**k(ss**(X ^ Y ); k):
Hence ss**IX = Hom **k(ss**X; k). (In a general stable homotopy category, one
should take Hom over R = ss0(S0) into the injective hull of R, essentially. See
[BC76 ] for the analogue in the ordinary stable homotopy category.)
We say that the homotopy of a spectrum X is of finite type if ssijX is finit*
*e
dimensional over k for each bidegree (i; j).
Proposition 1.6.1.[Rav84 ] Let X and Y be spectra.
(a)Given a map f :X ! X with cofiber X=f and telescope f1X, we have
_ = and ^ = 0.
(b)If X is weakly connective (Definition 1.4.3), then IX is in the localizing
subcategory generated by Hk = .
(c)Suppose that X has finite type homotopy. If [Y; X]**= 0, then ss**(Y ^IX) =
0.
(d)If X is a ring spectrum and Y is an Xmodule spectrum, then we have
= .
1.7. FURTHER DISCUSSION 19
(e)If X is a ring spectrum, then IX is an Xmodule spectrum; hence,
.
(f)Suppose that X is a noncontractible ring spectrum with finite type homoto*
*py,
and Y is an Xmodule spectrum. If [Y; X]**= 0, then < .
See [Rav84 ] for the proofs. (See also Lemma 1.4.8 for (b).)
Combining part (a) of this with Lemma 1.3.10 gives us a corollary.
Corollary 1.6.2. (a)Suppose that there is a Hopf algebra extension of the
form
E[x] !B !C;
giving a cofiber sequence
1;xHB h!HB !HC !0;xHB:
Then = _ , and HC ^ h1HB = 0.
(b)Suppose that there is a Hopf algebra extension of the form
D[x] !B !C;
with x > 0, giving cofiber sequences
2;pxHB b!HB !gHC !1;pxHB;
1;0HC !1;xHC !gHC !HC:
Then = _ , and HC ^ b1HB = 0.
Proof. Part (a) is immediate; part (b) follows once we show that =
. By the second cofibration above, it suffices to show that f1HC is con_
tractible. For degree reasons, though, ss**f1HC = 0. __
1.7.Further discussion
In this section, we note that we can apply the technology of this chapter to
group algebras, and we give one or two examples.
Let k be a field of characteristic p > 0, and let G be a finite group. Then *
*kG is
a finitedimensional cocommutative Hopf algebra, so we can use stable homotopy
theory to study Stable(kG*). By Remark 1.2.1, we could just as well work with t*
*he
category of cochain complexes of injective kGmodules; in either case, morphisms
in the category relate to group cohomology, as mentioned in Remark 1.2.2. Since
kG is concentrated in degree 0, then morphisms are only singly graded.
As a result, most of the results in this chapter apply to Stable(kG*). If G
is pgroup, then the trivial module k is the only simple; so all of the results*
* of
the chapter apply to Stable(kG*) for G a pgroup. To illustrate, Lemma 1.3.10
translates to the following, when p = 2.
Corollary 1.7.1.Let k be a field of characteristic 2, and let G be a finite
group. Any group extension
1 !B !G !Z=2 !1
gives rise to a cofibration in Stable(kG*):
1HG z!HG !HB;
where z 2 H1(G; k) is the inflation of the polynomial generator in H1(Z=2; k).
20 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
As mentioned in Subsection 1.3.1, when one has a group extension like
1 !B !G !Z=2 !1;
one can often replace arguments involving the associated spectral sequence with
simpler arguments involving the cofibration in the corollary. It is an amusing *
*exer
cise to prove Chouinard's theorem [Cho76 ] this way, for instance.
We should also discuss the Bousfield lattice in Stable(kG*). In any stable
homotopy category, the partially ordered set of Bousfield classes, together with
the meet and join operations, contains important structural information about t*
*he
category. It turns out that when G is a finite pgroup, there is a complete des*
*cription
of the Bousfield lattice, essentially due to Benson, Carlson, and Rickard [BCR9*
*6 ,
BCR ].
Given a pgroup G and an algebraically closed field k of characteristic p, t*
*hey
define VG(k) to be the maximal ideal spectrum of the (graded) commutative Noe
therian ring H*(G; k). Then for each closed homogeneous irreducible subvariety
V of VG(k), they define a kGmodule (V ) satisfying various properties, as out
lined below. (Throughout, they work in the category StMod(kG), so that (V )
is only welldefined up to projective summands. One could just as easily work in
Stable(kG*), in which case (V ) is a cochain complex, welldefined up to chain
homotopy equivalence. Naturally, we choose the latter course.)
Theorem 1.7.2 ([BCR96 ]).The objects (V ) satisfy the following:
(a)There is a Bousfield class decomposition of :
_
= <(V )>
V VG(k)
(where the wedge is taken over all closed homogeneous irreducible V ).
(b)If V 6= W, then (V ) ^ (W) = 0.
(c)For any V VG(k) and objects X and Y , if (V ) ^ X ^ Y = 0, then either
(V ) ^ X = 0 or (V ) ^ Y = 0.
By [HPS97 , 5.2.3], this implies the following.
Corollary 1.7.3 ([BCR ]).When G is a pgroup, the thick subcategories of
finitely generated kGmodules are in onetoone correspondence with collections*
* of
closed homogeneous subvarieties of VG(k) which are closed under specialization.
Their result also has the following corollary.
Corollary 1.7.4.Each <(V )> is a minimal nonzero Bousfield class. Hence
the Bousfield lattice is a Boolean algebra on the classes <(V )>. Indeed, for e*
*very
X,
_
= <(V )>:
V :(V )^X6=0
W
Proof. To show that <(V )> is minimal, we let E = W6=V(W). Then
= <(V )> _ , and (V ) ^ E = 0_in other words, (V ) is complemented. If
X is any object with < <(V )>, then there is a spectrum Z with X ^ Z = 0
but (V ) ^ Z 6= 0. Hence (V ) ^ X ^ Z = 0; by Theorem 1.7.2(c), this implies
that (V ) ^ X = 0. Since = <(V )> _ , then we have = . On
the other hand, we have (V ) ^ E = 0 and <(V )> > , so X ^ E = 0. Hence
= = <0>, so X = 0.
1.7. FURTHER DISCUSSION 21
Hence for any spectrum Y , if (V ) ^ Y 6= 0, then <(V ) ^ Y > = <(V )>.
Combined with Theorem 1.7.2(a), this gives the desired description of_the Bousf*
*ield
lattice. __
These results describe much of the global structure of the category Stable(k*
*G*)
(and also of StMod(kG)), but they leave open the question of classifying all lo*
*cal
izing subcategories (Definition 1.4.7). We conjecture that they are in onetoo*
*ne
correspondence with arbitrary collections of closed homogeneous subvarieties of
VG(k). This sort of general structure is discussed in [HPS97 , Chapter 6].
One can construct the objects (V ) in Stable() for any whose cohomology
ring is Noetherian_see [HPS97 , 6.0.8, 6.1.4] (this includes all finitedimensi*
*onal
commutative Hopf algebras, by work of Friedlander and Suslin [FS97 ]). The ana
logues of parts (a) and (b) of Theorem 1.7.2 hold, but it is not clear whether *
*part
(c) does. It is natural to conjecture that a similar description of the Bousfie*
*ld lat
tice is valid, as long as the Hopf algebra is suitably wellbehaved. For insta*
*nce,
if every quasielementary quotient of is elementary (these terms are defined in
Section 2.1), then one might expect this. In such a situation, one could try to
imitate the work in [BCR96 ]; if the quasielementaries do not coincide with t*
*he
elementaries, one still might be able to imitate Benson et al., if one had a go*
*od
enough understanding of the quasielementary quotients of .
(In particular, if is a finitedimensional quotient of the dual of the mod
2 Steenrod algebra, its quasielementary quotients are all elementary; if is a
finitedimensional quotient of the dual of the odd primary Steenrod algebra, the
quasielementary quotients are more complicated. In both cases, though, we would
conjecture that the analogues of 1.7.21.7.4 hold.)
22 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA
CHAPTER 2
Basic properties of the Steenrod algebra
The results and constructions in Chapter 1 hold as long as is a graded comm*
*u
tative Hopf algebra over a field (occasionally with the assumption that Stable()
is monogenic or that is connected). Now we start to make use of particular
properties of the Steenrod algebra.
We define the dual A of the Steenrod algebra, and we give a classification of
the quotient Hopf algebras of A in Section 2.1. We also define several important
families of quotient Hopf algebras of A: the A(n)'s, the elementary quotients, *
*and
the quasielementary quotients. We classify the latter two families, at least a*
*t the
prime 2, and for each (quasi)elementary E, we compute the homotopy groups of
the ring spectrum HE. In Section 2.2 we introduce Psthomology, a wellknown
tool for studying Ext over the Steenrod algebra. In our setting, Psthomology is
a homology theory, and hence is represented by an object Pstin Stable(A); we
compute ss**Pst, and we perform a few other computations.
Starting in Section 2.3, we begin to get to the main results. We discuss a
vanishing line theorem in Section 2.3: given conditions on the Psthomology gro*
*ups
of an object X, then ssijX = 0 when mi > j +c for some numbers m and c. (This is
an extension to the cochain complex setting of theorems of AndersonDavis [AD73*
* ]
and MillerWilkerson [MW81 ].) In Section 2.4 we use the vanishing line theorem
to construct "selfmaps of finite objects" in Stable(A). For example, if M is *
*an
Acomodule and dimFpM is finite, and if X is an injective resolution of M, then
we construct a cochain map nX !X which is nonnilpotent under composition.
We also establish that these selfmaps have certain nice properties. (This is *
*an
extension to the cochain complex setting of a result of the author [Pal92].)
In Section 2.5 we mention a few topological applications of the vanishing li*
*ne
and selfmap results, and one or two other issues.
2.1.Quotient Hopf algebras of A
In this section we define the dual A of the mod p Steenrod algebra, we give a
classification of quotient Hopf algebras of A, and we discuss two important fam*
*ilies
of these quotient Hopf algebras: namely, the A(n)'s and the elementary quotient*
*s.
Margolis' book [Mar83 ] is a good reference for all of these topics. In a subse*
*ction,
we also discuss the quasielementary quotients of A.
Fix a prime number p and let A be the dual of the mod p Steenrod algebra.
Recall from [Mil58] Milnor's description of A: as an algebra, we have
(
A = F2[1; 2; 3; : :]:; if p = 2,
Fp[1; 2; 3; : :]: E[o0; o1; o2;i:f:]:;p;is odd
23
24 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
where n = 2n  1 if p = 2, n = 2(pn  1) and on = 2pn  1 if p odd. The
coalgebra map : A !A A is determined by
Xn i
: n 7! pni i;
i=0
Xn i
: on 7! pni oi+ on 1:
i=0
(In both of these formulas, we take 0 to be 1.)
One can check that A is a connected commutative Hopf algebra over the field
Fp (or see [Mil58]); as a result, the work in the previous chapter applies to t*
*he
study of Stable(A).
The quotient Hopf algebras of A have been classified by AndersonDavis (p = *
*2)
and AdamsMargolis (p odd). We state the classification theorem here; see [Mar8*
*3 ]
or the original papers for the proof. (Recall that we defined conormal quotient*
* Hopf
algebras in Definition 1.3.7.)
Theorem 2.1.1.[AD73 , AM74 ]
(a)Every quotient Hopf algebra B of A is of the form
( 2n1 2n2
B = A=(1pn;12pn;2: :):;e e p = 2;
A=(1 ; 2 ; : :;:o00; o11;p:o:):;dd;
for some exponents n1; n2; . .2.{0; 1; 2; : :}:[ {1} and e0; e1; . .2.{1; *
*2}.
These exponents satisfy the following conditions:
(i)For all primes p: for each i and r with 0 < i < r, either nr nri i
or nr ni.
(ii)For p odd: if er = 1, then for each i and j with 1 i r and
i + j = r, either ni< j or ej= 1.
(b)Conversely, any set of exponents {ni} and {ei} satisfying conditions (i)(*
*ii)
above determines a quotient Hopf algebra of A.
(c)Let B be a quotient Hopf algebra of A as in (a). Then B is a conormal
quotient Hopf algebra of A if and only if, for p = 2,
(i)n1 n2 n3 : :,:
and for p odd,
(i)n1 n2 n3 : :,:
(ii)e0 e1 e2 : :,:
(iii)ek = 1 ) nk = 0.
For part (a), if some ni= 1, that means that one does not divide out by any
power of i. Similarly, if some ei= 2, then one does not mod out by oi.
Part (a) says that there is a (monomorphic) function from the set of quotient
Hopf algebras of A to the set of sequences either of the form (n1; n2; : :):, o*
*r of the
form (n1; n2; : :;:e0; e1; : :):; part (b) gives the image of this function. Fo*
*r p = 2,
given a Hopf algebra B, one can view the sequence of exponents n1; n2; : :a:s a
function
{1; 2; ::}:!{0; 1; 2; : :}:[ {1};
i7! ni:
We refer to this as the profile function of B. There is, of course, a similar f*
*unction
when p is odd. We will occasionally give graphical representations of quotient *
*Hopf
2.1. QUOTIENT HOPF ALGEBRAS OF A 25
 
.  2r ..: : : 
.. n__. 
r1 2r1 ..
.  .: ::n::: p2.
.. 42 43.. 1
22__2_ p___ ..: : :
1 12_3 1 ____.
  : : :
0 _________________________1 __________________________1
1 2 3: ::::n:: : : : :o:non+1: : :
p = 2 __________p_odd__________
Figure 2.1.A. Graphical representation of a quotient Hopf alge
bra of A. For p = 2, this is a barrchart; the nth column is height
r  1 if one is dividing out by 2n. For p odd, this is a similar bar
chart, together with an extra row at the bottom; in this row, one
marks which on's are nonzero in the quotient.
algebras via their profile functions, as in [Mar83 , p. 2345]. See Figure 2.1.*
*A, for
example.
Here is a simple, but useful, result. See Notation 1.3.9 for the definition *
*of D[x]
and E[x].
Lemma 2.1.2. (a)Suppose that B is a quotient Hopf algebra of A, and that
for somess and t, we have
 pt6=s0+in1B,
 pts = 0 in B,
 pj= 0 in B for all j < t.
Then there is a Hopf algebra extension
s
D[pt] !B !C:
(b)Fix p odd. Suppose that B is a quotient Hopf algebra of A, and that for
some n we have
 on 6= 0 in B,
 oj= 0 in B for all j < n,
 j= 0 in B for all j n.
Then there is a Hopf algebra extension
E[on] !B !C:
s Proof. For part (a), one only has to check that given the conditions on B, t*
*hen
ptis primitive in B (and similarly for part (b)). This check is straightforward*
*. ___
Remark 2.1.3. (a)Hence the results of Lemma 1.3.10 apply. The usual
notation is:
s 1;ps
hts= [pt] 2 HB1;pst= ExtB t(Fp; Fp);
ps
bts= fifP0(hts) 2 HB2;ppst= Ext2;ptB(Fp; Fp);
vn = [on] 2 HB1;on= Ext1;onB(Fp; Fp):
(b)Also, note that if B is a finitedimensional quotient of A, then one can a*
*lways
find an integer n or a pair (s; t) so that the hypotheses of Lemma 2.1.2 h*
*old.
26 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
n+1 n 
 
n@ n1@
 @ @
 @  @
 
 @  @
 @  @
 @  @
_________________________nn+1@_________________________n@
o0 : : :on
p = 2 __________p_odd__________
Figure 2.1.B. Profile function for A(n). The actual profile func
tions have a staircase shape, which we abbreviate as lines with
slope 1.
At the prime 2, for instance, one can take s and t to be as follows:
t = min{n  n 6= 0 inB};
i
s = max{i  2t6= 0 inB}:
This provides an inductive procedure for studying finite quotients of A.
We need to use several different families of quotient Hopf algebras of A; he*
*re
is the first family. These quotients are quite wellknown; see [Mar83 , p. 235]*
*, for
example.
Example 2.1.4.We define A(n) as follows; see also Figure 2.1.B:
( n+1 n
A=(21 ; 22; : :;:2n+1; n+2; n+3; : :):; p = 2;
A(n) = pn pn1
A=(1 ; 2 ; : :;:pn; n+1; n+2; : :;:on+1; on+2;p:o:):;dd:
Then A(n) is a quotient Hopf algebra of A, and the map A !A(n) is an isomor
phism below degree
2n+11n= 2n+1; p = 2;
p1 = 2(p  1)pn;p odd:
One important property of the A(n)'s is that the dual A* of A is the union
of the duals of the A(n)'s, and this gives A* the structure of a "Palgebra" (s*
*ee
[Mar83 , Chapter 13] for the precise definition). In our setting, this translat*
*es into
the following (cf. [Mar83 , Proposition 13.4]).
Proposition 2.1.5.Suppose that Y is a spectrum so that for each i, ssijY = 0
when j 0. Then Y is the sequential limit of
: ::!HA(3)^ Y ! HA(2)^ Y ! HA(1)^ Y ! HA(0)^ Y:
Hence for any spectrum X, there is a Milnor exact sequence
0 !lim1[X; HA(n)^ Y ]i1;j![X; Y ]i;j!lim[X; HA(n)^ Y ]i;j!0:
Proof. We write the cochain complex Y as : ::!Yj !Yj+1 !: :.:We
may assume that for each j, the injective comodule Yj is bounded below. Because
A i A(n) is an isomorphism in a range of dimensions increasing with n, the inve*
*rse
system of comodules
: ::!(A 2A(n)Fp) Yj! (A 2A(n1)Fp) Yj! (A 2A(n2)Fp) Yj! : : :
2.1. QUOTIENT HOPF ALGEBRAS OF A 27
stabilizes in any given degree, and the inverse limit is Yj. We let (A 2A(n)Fp)*
* Y
denote the cochain complex which is (A 2A(n)Fp) Yj in degree j; then the inver*
*se
system of cochain complexes stabilizes in each bidegree, and the inverse limit *
*is Y .
This finishes the proof. (Note that (A 2A(n)Fp) Y is isomorphic to_HA(n)^ Y in
Stable(A).) __
We consider one other interesting family of quotient Hopf algebras. See [Wil*
*81 ]
for some results related to these sorts of Hopf algebras.
Definition 2.1.6.We say that a connected commutative Hopf algebra B over
a field k of characteristic p is elementary if it is isomorphicn(as a Hopf alge*
*bra) to
a tensor product of Hopf algebras of the forms k[x]=(xp ) with x primitive (and*
* x
even, if p is odd), and E[y] with y primitive (and y odd, if p is odd). (In o*
*ther
words, B is bicommutative and its dual B* has zp = 0 for all z in the augmentat*
*ion
ideal.)
See [Mar83 ], [Lin] and [Wil81 ] for the following.
Proposition 2.1.7. (a)Suppose that p = 2. A quotient Hopf algebra B of
A is elementary if and only if it has the form
n1 2n2
B = A=(21 ; 2 ; : :):;
where for some r, we have
(i)if i < r, then ni= 0,
(ii)if i r, then ni r.
Conversely, any quotient algebra B given by exponents nisatisfying (i)(ii)
is an elementary quotient Hopf algebra of A.
(b)Suppose that p is odd. A quotient Hopf algebra B of A is elementary if and
only if it has the form
B = A=(1; 2; : :;:oe00; oe11; : :):~=E[o1e00; o1e11; : :]:
for any exponents e0, e1, : :,:or
n1 pn2 e e
B = A=(p1 ; 2 ; : :;:o00; o11; : :):;
where for some r, we have
(i)if i < r, then ni= 0,
(ii)if i r, then ni r,
(iii)if i < r, then ei= 1.
Conversely, any quotient algebra B given by these exponents niand eiis an
elementary quotient Hopf algebra of A.
We also describe the maximal elementary quotient Hopf algebras of A; see
Figure 2.1.C. Every elementary quotient Hopf algebra of A is a quotient of one *
*of
these.
Corollary 2.1.8. (a)Suppose that p = 2. The maximal elementary quo
tient Hopf algebras of A are
m+1 2m+1 2m+1
E(m) = A=(1; : :;:m ; 2m+1; m+2 ; m+3 ; : :):; m 0:
28 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA

 .: : :
 ..
 ________________
4 :: :
 .. E(4)
3 :.:.:
2 .. 
 :.:.: 
1_______________________.
 E(0) 
0_________________________
1 2 3 4 5 6
p = 2
Figure 2.1.C. Profile functions for maximal elementary quotients
of A at the prime 2. For reference,swe have included a staircase
above which are the elements 2twith s t.
(b)Suppose that p is odd. The maximal elementary quotient Hopf algebras of A
are
E(1)0= A=(1; 2; : :):~=E[o0; o1; : :]:;
m+1 pm+1 pm+1
E(m)0= A=(1; : :;:m ; pm+1; m+2 ; m+3 ; : :;:o0; : :;:om ); m 0:
Note that the quotient Hopf algebras E(m) and E(m)0are conormal for all m.
We use the elementary quotient Hopf algebras of A to prove the vanishing line
theorem of Section 2.3; we need to know their coefficient rings. The following *
*is
standard.
Proposition 2.1.9. (a)Let p = 2, and assume that E is an elementary
quotient Hopf algebra of A. Then
s
HE**= F2[hts 2t6= 0 inE]:
Here, hts = (1; 2st).
(b)Let p be odd, and assume that E is an elementary quotient Hopf algebra of
A. Then
s ps
HE**= E[hts pt6= 0 inE] Fp[bts t 6= 0 inE]
Fp[vn  on 6= 0 inE]:
s ps
Here, hts = (1; pt), bts = (2; pt ), and vn = (1; on).
s ps
Indeed, if E is an elementary quotient of A with pt6= 0, then t issprimitive
in E. Following Notation 1.3.9 and Remark 2.1.3(a), we have hts= [pt]; similarl*
*y,
if on is nonzero in E, then it is primitive and we have vn = [tn].
2.1.1. Quasielementary quotients of A. We also need to consider one
other family of quotient Hopf algebras, the "quasielementary" Hopf algebras. It
turns out that when p = 2, these coincide with elementary Hopf algebras; when p
is odd, there are quasielementary Hopf algebras which are not elementary, but *
*we
do not have a complete classification.
The quasielementary quotients of A arise in the nilpotence theorem 3.1.6 of
Section 3.1, and indeed in many of the results of Chapters 3 and 4; the lack of*
* a
classification at odd primes is one obstacle to proving those theorems when p i*
*s odd.
So when p is 2, we have already studied these; when p is odd, we have no immedi*
*ate
2.2. PstHOMOLOGY 29
use for them. Hence the contents of this subsection may be safely ignored, exce*
*pt
for the term "quasielementary."
Definition 2.1.10.Recall from Notation 1.3.9 that we have a Steenrod oper
ation fifP0 acting on Ext, when p is odd. We say that a connected commutative
Hopf algebra B over a field k is quasielementary if no product of the form
Q
Qw2Sw; Q p = 2;
( u2Soddu)( v2SevenfifP0v);p odd;
is nilpotent, for any finite sets S Ext1;*B(k; k)  {0}, Sodd Ext1;oddB(k; k) *
* {0},
and Seven Ext1;evenB(k; k)  {0}.
Every elementary Hopf algebra is quasielementary; Wilkerson [Wil81 , Section
6] gives several examples of quasielementary Hopf algebras which are not eleme*
*n
tary. In particular, one of his examples is a quotient of A, when p is odd. S*
*ee
[Wil81 , Counterexample 6.3] for the following.
Example 2.1.11.Suppose that p is odd, and let B be this quotient Hopf alge
bra of A:
2 p
B = Fp[1; 2; 3]=(p1; p2; 3):
Then B is quasielementary, but not elementary.
When p is 2, things are a bit nicer. See [Wil81 , Theorem 6.4] for the follo*
*wing.
Proposition 2.1.12.Suppose that p = 2. A quotient Hopf algebra B of A is
elementary if and only if it is quasielementary.
Note that we have classified the elementary quotients of A in Proposition 2.*
*1.7
and Corollary 2.1.8. We have also computed the coefficient rings of these Hopf
algebras in Proposition 2.1.9.
At odd primes, we do not have a classification of the quasielementary Hopf
algebras, so we content ourselves with the following. This would follow from Co*
*n
jecture 3.4.1.
Conjecture 2.1.13.Suppose that p is odd. Then every quasielementary quo
tient Hopf algebra of A is a quotient of
2 p3 pn
D = A=(p1; p2; 3 ; : :;:n ; : :)::
See [Pal97] for some results relating to quasielementary Hopf algebras.
2.2.Psthomology
In this section we discuss Psthomology; this tool has been used by many aut*
*hors
to study ExtA, so it is reasonable to expect it to be useful in the present set*
*ting.
[Mar83 , Section 19.1] is a good reference for basic results on Psthomology of
modules; in this section, we prove analogues of some of those results.
Let A* be the dual of A. Note that an Acomodule M with structure map
is naturally an A*module, via the map
A* M 1!A* A M ev1!M:
We remind the reader that if one dualizesswith respect to the monomial basis for
A, then Pstis the element of A* dual to pt, and (when p is odd) Qn is the eleme*
*nt
30 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
dual to on. If s < t, then (Pst)p = 0, so in this case one may define the Psth*
*omology
of an Acomodule M by
(Pst)p1
H(M; Pst) = ker(M_!_M)_Ps:
im(M !M) t
For all n we have Q2n= 0, so one may define H(M; Qn) similarly. One may do
these things, but that does not mean that one should do them when working in the
context of cochain complexes of injective Acomodules.
Definition 2.2.1.Given integers s and t with 0 s < t, we define the Pst
homology spectrum, Pst, to be the cochain complex with jth term
(kpps
t A; j = 2k;
(Pst)j= ps
(kp+1)tA; j = 2k + 1;
with differentials given as in the following diagram:
: : :! As ! A s! A s ! A ! : : :
pt 7! 1; (p1)pt7! 1; pt 7! 1
(A is a rank 1 cofree comodule, so a comodule map into A is determined by what
hits 1_see Lemma 1.1.11.) We define the connective Psthomology spectrum, pst, *
*to
be the cochain complex obtained by truncatingsthescomplex+for1Pstat homological
dimension zero; in other words, pst= HFp[pt]=(pt ). Similarly, when p is odd,
we define the Qnhomology spectrum, Qn, to be the cochain complex with jth term
(Qn)j= jonA, and with differentials given by
: : :! A ! A ! A ! A ! : : :
on 7! 1; on 7! 1; on 7! 1
We define the connective Qnhomology spectrum, qn, to be the truncation of the *
*Qn
complex below homological degree zero; it is equal to HE[on]. For any spectrum
X, we define its Psthomology to be (Pst)**X, and we define its Qnhomology to *
*be
(Qn)**X.
In the remainder of this section, we compute the coefficient rings of these *
*spec
tra and we prove a few results for later use.
s ps+1
Proposition 2.2.2. (a)If s < t, then Fp[pt]=(t ) is a quotient coalge
bra of A over which A is injective as a right comodule. For any n 0, E[on]
is a quotient Hopf algebra of A (and hence A is injective as a right comod*
*ule
over it).
(b)Hence,
(
(pst)**~= F2[hts]; p = 2;
Fp[bts] E[hts];p odd;
s ps
where hts = (1; pt) and bts = (2; pt ). Also,
(qn)**~=Fp[vn];
where vn = (1; on).
2.2. PstHOMOLOGY 31
(c)Hence
( 1
(Pst)**= F2[hts];1 p = 2;
Fp[bts] E[hts];p odd;
(Qn)**= Fp[v1n];
and for any connective X,
( 1 s
(Pst)**X = hts(pt)**X;p=12;
bts(pst)**X;p odd;
(Qn)**X = v1n(qn)**X:
Proof. Part (a) is wellknown. It is dual to the statement that A*, the dual
of A, is free over the subalgebra Fp[Pst]=(Pst)p; i.e., H(A*; Pst) = 0. (And si*
*milarly,
A* is free over E[Qn]; i.e., H(A*; Qn) = 0.) See [AM71 ], [MP72 ], or [Mar83 ,
Proposition 19.1].
Parts (b) and (c) are standard, trivial, or both (using Lemma 1.3.4, for in
stance). See [Mar83 , Propositions 19.23] for an alternate formulation_of part
(c). __
By the way, we note that for all primes, the spectrum p0t= HFp[t]=(pt) is a
ring spectrum, while P0tis a field spectrum for p = 2 (perhaps P0tis an example*
* of
an "Artinian ring spectrum" when p is odd). Similarly, when p is odd, qn = HE[o*
*n]
is a ring spectrum and Qn is a field spectrum. The spectra pstfor s > 0 are not
ring spectra, but are examples of the sort discussed in Proposition 1.3.2(b).
Remark 2.2.3.As one might expect, there is a relationship between the mod
ule definition of Psthomology (given just before Definition 2.2.1) and the hom*
*ology
functor represented by the Psthomology spectrum: if X is an injective resoluti*
*on
of a comodule M, then
(Pst)**X = H*(M; Pst) (Pst)**:
(Here we are viewing the singlygraded vector space H*(M; Pst) as doublygraded
by putting Hi(M; Pst) in bidegree (0; i).) In particular, (Pst)**X = 0 if and o*
*nly if
H*(M; Pst) = 0. Similarly, we have
(Qn)**X = H*(M; Qn) (Qn)**:
It is convenient to have alternate notation for the spectra Pstand Qn, based
on the "slopes" of the polynomial generators in their coefficient rings.
ps
Notation 2.2.4.We define the slope of Pstto be pt_2, and the slope of Qn
to be on. Note that these spectra all have distinct slopes. The set
{2st :s < t}; p = 2;
ps
{pt_2:s < t} [ {on :n 0};p odd;
is called the set of slopes of A; the phrase "fix a slope n" means "fix an elem*
*ent
n of this set." Given a slope n, we let Z(n) denote the corresponding Pstor Qn
spectrum. For example, when p = 2, we have
s s
Z(1) = P01; Z(3) = P02; Z(6) = P12; Z(7) = P03; : :;:Z(2t) = Pt; : :::
32 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
When p is odd, we have
Z(1) = Q0;Z(p  1) = Q1; Z(p2 p) = P01; Z(p2 1) = Q2; : :;:
ps
Z(pt__2) = Pst; : :;:Z(om ) = Qm ; : :::
We let z(n)sbe the connective cover of Z(n). We let yn denote the element of A,
either ptor om , with slope n.
See [Mar83 , 19.21] for the following at the prime 2.
Proposition 2.2.5.LetsB be a quotient Hopf algebra of A. Then (Pst)**HB 6=
0 if and only if pt6= 0 in B. For p odd, (Qn)**HB 6= 0 if and only if on 6= 0 i*
*n B.
We will use this result in Section 5.3.
Proof. We will give the proof for Pstand leave the Qn proofsfor the reader.
By Remark 2.2.3, we are interested in H(A 2BFp; Pst). If pt6= 0 in B, then
1 2 A 2BFpgenerates a nonzero homology class.
To prove the converse, first we reduce to the case when B is finite. We defi*
*ne
B(n) to be the quotient Hopf algebra of A defined by the following pushout diag*
*ram
of Hopf algebras:
A ! B:
?? ?
y ?y
A(n)! B(n)
(A(n) is defined in Example 2.1.4.) In other words, B(n) is the quotient of B
induced by the map A i A(n). Each B(n) is finite, and one can apply Proposi
~=
tion 2.1.5 to see that (Pst)**HB ! limn(Pst)**HB(n).
s
So it suffices to show that if B is a finite quotient of A in which pt= 0, t*
*hen
(Pst)**HB = 0. We do this by induction on dimFpB. The induction starts with
B = Fp, in which case HFp= A. Then
(Pst)**HFp= (HFp)**Pst;
the homology of the cochain complex Pst. This complex is acyclic, so its homolo*
*gy is
zero. (Alternatively, see the remark followingsthe proof.) This starts the indu*
*ction.
Now fix a finitedimensional B with pt= 0 in B, and assume that for every
proper quotient C of B, we have (Pst)**HC = 0. As noted in Remark 2.1.3, we have
a Hopf algebra quotient C of B with Hopf algebra kernel either Fp[pqr]=(pq+1r) *
*or
E[om ]. So by Lemma 1.3.10 this leads to a cofiber sequence (in which we neglect
suspensions)
HB f!HB !Z;
where either Z = HC or Z is the cofiber of a selfmap of HC. In either case,
(Pst)**HC = 0 ) (Pst)**Z = 0, so (Pst)**f is an isomorphism.
Now we argue essentially as in the proof of Lemma 2.3.11 to see that bound
edness of the comodule A 2BFpimplies that (Pst)**HB = 0. To be precise, we first
have to specify the degree of the map f. There are three cases:
q
(1)p = 2: then f :1;2rHB !HB.
(2)p odd, Hopf algebra kernel E[om ]: then f :1;omHB !HB.
2.2. PstHOMOLOGY 33
q
(3)p odd, Hopf algebra kernel Fp[pqr]=(pq+1r); then f :2;pprHB !HB.
We deal with cases (1) and (3); (2) is handled the same way. By our computations
in Proposition 2.2.2, we know that for all i and j, we have an isomorphism
(Pst)ijHB ~=(Pst)i+2;j+ppstHB:
In cases (1) and (3), f induces an isomorphism
(Pst)ijHB ~=(Pst)i+2;j+ppqrHB:
Combining these, for each integer k we get an isomorphism
(Pst)i;jHB ~=(Pst)i;j+pk(pstpqr)HB:
s q ps
Now, pqr6= 0 in B and pt = 0, so pr 6= t . In particular, these two elements
have different degrees. By Remark 2.2.3, for fixed i, (Pst)i;*HB is equal (up *
*to
suspension) to H*(A 2BFp; Pst); since A 2BFpis a bounded below comodule, then
this homology must be zero in small enough degrees. By the above isomorphism,_
we may then conclude that it is zero in all degrees. __
We remark that it is easy to show that (Pst)**A = 0_this follows by a result
of MilnorMoore [MM65 ], as in [Mar83 , Proposition 19.1]. It seems as though
there should be a similar proof of Proposition 2.2.5, but we have not been able*
* to
find one.
Recall from [HPS97 , A.2.4] that if X is any object in Stable(A), then DX
denotes the SpanierWhitehead dual of X. DX has the property that for any Y ,
[X ^ Y; S0] = [Y; DX]:
We mention the following easy fact.
Proposition 2.2.6.Let E and X be spectra with X finite. Then E**X = 0 if
and only if E**DX = 0.
Proof. If X is Eacyclic, then so is X ^Y for any Y . In particular, DX ^X ^*
* __
DX is Eacyclic. Since DX is a retract of this when X is finite, we are done. *
* __
Corollary 2.2.7.Let X be a finite spectrum. Then (Pst)**X = 0 if and only
if (Pst)**DX = 0; and for p odd, (Qn)**X = 0 if and only if (Qn)**DX = 0.
One should be able to get more precise information about the relationship
between (Pst)**X and (Pst)**DX, as in [Mar83 , 19.12], but we do not need it.
We end this section with a note on operations on Psthomology. The next resu*
*lt
follows from Remark 2.2.3.
Proposition 2.2.8.We have
(Pst)**Pst= H*(A 2D[pst]Fp; Pst) (Pst)**;
(Qn)**Qn = H*(A 2E[on]Fp; Qn) (Qn)**:
Margolis has calculated H*(A 2D[pst]Fp; Pst) for s = 0 at the prime 2 [Mar83*
* ,
19.26]; similar calculations work at odd primes for H*(A 2E[on]Fp; Qn).
34 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA


i


 ______slope__1_oe
 d1



__ __ ____ __ __ __ __ __ __ __ __
  i1

___________________________________________
j0 ff j0  j  i
  

  
__ __ ____ __ __ __ __ __ __ __ __
i0

ssijX
Figure 2.3.A. Vanishing line at the prime 2. In this picture,
ssijX = 0 above the slanted line (the vanishing line), to the left of
the vertical line j  i = j0, and below the horizontal line i = i0. ff
is a number depending only on the slope d.
2.3. Vanishing lines for homotopy groups
In this section we prove several theorems relating the vanishing of Pst and*
* Qn
homology groups of an object X to the homotopy groups of X; these are versions
in the cochain complex category of wellknown theorems about modules over the
Steenrod algebra. These results are at the heart of many of the other results o*
*f the
book.
Recall that if X is an injective resolution of a comodule M, then ss**(X) =
Ext**A(Fp; M).
We make heavy use of Notation 2.2.4 in this section. CLspectra are defined *
*in
Definition 1.4.6.
Here are our main theorems. In the setting of modules over the Steenrod
algebra, the first is due to AndersonDavis [AD73 ] (p = 2) and MillerWilkerson
[MW81 ] (p odd), and the second to AdamsMargolis [AM71 ] (p = 2) and Moore
Peterson [MP72 ] (p odd). Both of the results are proved in [MW81 ]; we follow
those proofs.
Theorem 2.3.1 (Vanishing line theorem).Let X be a CLspectrum. Suppose
that there is a number d so that Z(n)**X = 0 for all slopes n with n < d. Then
ss**X has a vanishing line of slope d: for some c, we have ssijX = 0 when j < d*
*ic.
Of course, this vanishing line has slope _1_d1in (j  i; i)coordinates. Se*
*e Fig
ure 2.3.A for a picture.
Theorem 2.3.2.Let X be a CLspectrum, and assume that Z(n)**X = 0 for
all slopes n; then ss**X has a "horizontal" vanishing line: for some c, we have
ssijX = 0 when i > c.
Corollary 2.3.3.Under the hypotheses of Theorem 2.3.2, X is in the local
izing subcategory generated by A.
Proof. This follows from Lemma 1.4.8. ___
2.3. VANISHING LINES FOR HOMOTOPY GROUPS 35
Remark 2.3.4. (a)When we speak of phenomena "above" a vanishing line,
we mean "above" in the grading of Figure 2.3.A_i.e., in the region in which
ssijmight be zero.
(b)We will see in the proof that the "intercept" c of the vanishing line in
Theorem 2.3.1 may be given by c = (d  1)i1 j0+ ff, where ff = ff(d) is a
number depending only on d. In the grading of Figure 2.3.A, the homotopy
ssijis zero above the line of slope _1_d1through the point (j0  ff; i1).*
* To
compute ff, we find the smallest integer n so that the map A ! A(n) is
an isomorphism through degree d (i.e., so that 2n+1 > d when p = 2, or
2(p  1)pn > d when p is odd); then
8 X s
>>> pt; if p = 2,
>>>s+tn+1
< psd
ff = > Xt ps X
>>> (d + (p  1)t ) + oi;if p is odd.
>>:s+tn in
pstd oid
For Theorem 2.3.2, the intercept is i1_the homotopy ssijis zero if i > i1.
(c)For both of these theorems (in fact, for all of the results of this sectio*
*n), we
can weaken condition (ii) of Definition 1.4.6 slightly_it suffices to assu*
*me
that for all i, there is a j0 = j0(i) so that ssijX = 0 if j  i < j0. Wit*
*h this
assumption, one replaces j0 in the above formulas for c with min{j0(i)  i0
i i1}.
Let M be a bounded below Acomodule with injective resolution X; then it is
easy to see that X is CL. Furthermore, we know from [Mar83 , Theorem 13.12]
that ssstX = ExtstA(Fp; M) has a "horizontal" vanishing line if and only if M is
injective. Hence our theorems do indeed provide generalizations of the previous*
*ly
cited ones.
One can generalize these results somewhat. The same proofs carry over essen
tially unchanged, so to keep the notation simple we prove Theorems 2.3.1 and 2.*
*3.2,
rather than the following.
Theorem 2.3.5.Let X be a CLspectrum, and fix a quotient Hopf algebra B
of A.
(a)Suppose that there is a number d so that Z(n)**X = 0 for all slopes n with
n < d and yn 6= 0 in B. Then HB**X has a vanishing line of slope d: for
some c, we have HBijX = 0 when j < di  c.
(b)If Z(n)**X = 0 for all slopes n with yn 6= 0 in B, then HB**X has a
"horizontal" vanishing line; hence HB ^ X is in the localizing subcategory
generated by A.
Of course, HB** relates to ExtB the same way ss** relates to ExtA, so one
should view this result as a vanishing line theorem over B. (One could also work
in Stable(B) and prove a result about ss**in that category, but it seems better*
* to
work in Stable(A) whenever possible.)
Remark 2.3.4 also applies here.
2.3.1. Proof of Theorems 2.3.1 and 2.3.2 for p = 2. We give the proofs
of Theorems 2.3.1 and 2.3.2 when p = 2. In the next subsection we indicate the
changes necessary when working at odd primes.
36 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
Recall from Lemmas 1.3.10 and 2.1.2, as well as Remark 2.1.3, that if B is a
finitedimensional quotient Hopf algebra of A, then for some s and t we have the
following conditions:
o2st6= 0 in B,
o2s+1t=s0 in B,
o2j= 0 in B for all j < t.
Hence there is a Hopf algebra extension
s
E[2t] !B !C;
hence an element hts2 HB1;2stand a cofibration
2s hts
1;tHB ! HB !HC:
Given this situation, we have the following result. (Given a spectrum Y and a
selfmap v :ijY ! Y , we say that v acts nilpotently on ss**Y if for all y 2 s*
*s**Y ,
some power of ss**(v) annihilates y.)
Lemma 2.3.6.Assume that B is finitedimensional, and fix s and t as above.
(a)If s t, then hts2 HB**is nilpotent; hence the selfmap hts:HB !HB
is nilpotent.
(b)Given an object X, consider the cofibration
2s hts^1X
1;tHB ^ X !HB ^ X !HC ^ X:
If hts^ 1X acts nilpotently on HB**X, and if HC**X has a vanishing line
of slope d, then HB ^ X has a vanishingsline of slope d. The difference in
intercepts depends only on d and 2t_it is independent of X.
Remark 2.3.7.In fact, the nilpotence of htsdoes not depend on B being
finitedimensional. See Theorem B.2.1 for a generalization, due to Lin.
Proof. Lin proved part (a) in [Lin, Corollary 3.2]; see also AndersonDavis
[AD73 ] and MillerWilkerson [MW81 , Proposition 4.1].
For part (b), we look at the long exact sequence in homotopy coming from the
given cofibration (we write h for ss**(hts^ 1X )):
: ::!HCi1;jX !HBi1;j2stX h!HBijX !HCijX !: :::
Fix a bidegree (i; j)sabove the vanishing line for HC**X. There are two cases:
suppose first that 2t > d. The exact sequence tells us that
HBi1;j2stX h!HBijX
is an epimorphism. Since 2st > d, then (i  1; j  2st) is above the vanish*
*ing line
for HC**X. Hence
HBik;jk2stX h!HBi(k1);j(k1)2stX
is an epimorphism for all ks> 0. Also since 2st > d, then one can see that for
k 0, (i  k; j  (k  1)2t) is a bidegree above the vanishing line for HC**X*
*, so
HBik;jk2stX h!HBi(k1);j(k1)2tX
is an isomorphism. Since h acts nilpotently on HB**X, though, these groups must
be zero for k large; since they surject onto HBijX, then HBijX = 0. Note that in
this case, the vanishing line for HC**X is the same as that for HB**X.
2.3. VANISHING LINES FOR HOMOTOPY GROUPS 37
Suppose, on the other hand, that 2st d. If (i; j) is a bidegree above the
vanishing line, then so is (i + 1; j), so the map
HBi;j2stX h!HBi+1;jX
is an isomorphism. Arguing as above, we see that the group HBi;j2stX must
be zero. In this case, the interceptsof the HB vanishing line changes_by 2st:
HBijX = 0 when j < di  c  2t. __
Lemma 2.3.8.Fix an object X satisfying condition (i) of Definition 1.4.6.
(a)Given an extension
s
E[2t] !B !C
where B and C are quotients of A, if 2st d and if HC**X has a vanishing
line of slope d, then HB**X has a vanishing line of slope d. In fact, HB**X
has the same vanishing line as HC**X.
(b)Given finitedimensional quotients B i C of A, if HC**X has a vanishing
line of slope d and if the map B i C is an isomorphism in dimensions less
than d, then HB**X has a vanishing line of slope d. In fact, HB**X has
the same vanishing line as HC**X.
Proof. Part (a): (This proof is based on that of [MW81 , Proposition 3.2].)
We assume that HCi;jX = 0 when j < dic, and we want to show that HBi;jX = 0
when j < di  c. We prove this by induction on i. By Lemma 1.3.10, we have a
cofibration
2s hts
1;tHB ^ X ! HB ^ X !HC ^ X:
This gives us a long exact sequence in homotopy:
(2.3.9) : ::!HBi1;j2stX !HBi;jX !HCi;jX !: :::
Since X satisfies condition (i) of Definition 1.4.6, then for i sufficiently sm*
*all and
for all j, we have
ssi;j(X) = 0 = HBi;jX = HCi;jX:
This starts the induction: if i0is the smallest value of i for which HCi;jX is *
*nonzero,
then we have an inclusion HBi0;jX ,! HCi0;jX (for all j).
Thesinductive step is also easy: if we have i and j with j < di  c, then
j  2t < d(i  1)  c; hence both HCi;jX and HBii;j2stX are zero. So we
apply exactness in the long exact sequence (2.3.9).
Part (b): Given B i C with B finitedimensional (or more generally, with C
of finite index in B), then there is a sequence of extensions
s1
E[2t1] !B !B1;
s2
E[2t2] !B1 !B2;
..
.
sn
E[2tn] !Bn1 !C:
(See [HSb , Lemma A.11] or [MW81 , Lemmas 3.43.5], for example.)sIf, further
more, B i C is an isomorphism through degree d  1, then each 2tihais degree_
at least d. So apply induction and part (a). __
38 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
Lemma 2.3.8 is a special case of the following; we have stated and proved th*
*em
separately because at odd primes we need a variant of the former, while the lat*
*ter
holds as stated.
Lemma 2.3.10.Fix an object X satisfying condition (i) of Definition 1.4.6.
Given a surjection of coalgebras f :A i B, if f is an isomorphism below degree *
*d,
and if HB**X has a vanishing line of slope d, then HA**X has a vanishing line of
slope d. In fact, they have the same vanishing line.
Proof. We have an injection of comodules A 2AF2! A 2BF2, and the cok
ernel is zero below dimension d. Taking injective resolutions gives a cofibrati*
*on
1;0Z !HA !HB !Z;
where Z is (0; d)connective (Definition 1.4.3). So to show that HAijX = ssijX *
*=_0_
if j < di  c, one argues by induction on i just as in Lemma 2.3.8(a). *
*__
Lemma 2.3.11.Let X be an object satisfying conditions (ii) and (iii) of Defi*
*ni
tion 1.4.6. Suppose that E is a finitedimensionalselementary quotient Hopf alg*
*ebra
of A. If (Pst)**X = 0 whenever 2t 6= 0 in E, then HE**X has a horizontal
vanishing line. In fact, given i1 as in condition (iii), then HEi*X = 0 if i > *
*i1.
Proof. Since the dual of E is an exterior algebra, we abuse notation and wri*
*te
E = E[x1;sx2; : :;:xn] where each xk is primitive; in other words, each xk is e*
*qual
to 2tfor some s and t. We point out that the xk's have distinct degrees.
Now HE**is a polynomial algebra on n generators; we write vk 2 HE1;xkfor
the generator corresponding to xk. We also use vk to denote the corresponding s*
*elf
map of HE, and we let HE=(vk) be its cofiber. Note that the selfmap vk induces
multiplication by vk on homotopy, so that
ss**(HE=(vk)) = (ss**HE)=(vk):
We define HE=(vk1; : :;:vkm) similarly (assuming that the numbers k1; : :;:km a*
*re
distinct). Hence if ` is an integer with 1 ` n and ` 62 {k1; : :;:km }, then
v`: HE !HE induces a selfmapsof HE=(vk1; : :;:vkm).
Also note that, if xk = 2t, then by a changeofcoalgebras, we have
(Pst)**X = v1k(HE=(v1; : :;:^vk; : :;:vn))**X:
We refer to this as the xkhomology of X.
We claim for any set {k1; : :;:km } {1; : :;:n},
(HE=(vk1; : :;:vkm))**(X)
has a horizontal vanishing line. We prove this by induction on n  m. (We will
have proved the lemma when n  m = n.)
We have to deal with the first few cases before we can apply the inductive s*
*tep.
If n  m = 0, then {k1; : :;:km } = {1; : :;:n}, so HE=(vk1; : :;:vkm) = HF2, a*
*nd by
assumption, HF2**X has a horizontal vanishing line with intercept i1. If nm = *
*1,
then we let k be the integer so that
{k} [ {k1; : :;:km } = {1; 2; : :;:n}:
To simplify the notation, we let HEm = HE=(vk1; : :;:vkm). Then we have a cofi
bration
HEm ^ X vk!HEm ^ X !HF2^ X:
2.3. VANISHING LINES FOR HOMOTOPY GROUPS 39
By hypothesis, HF2**X has a horizontal vanishing line, so the map vk induces an
isomorphism in ssijfor i > i1. On the other hand, the xkhomology of X is zero,*
* and
the xkhomology of X is equal to v1k(HEm )**X. Since vk induces an isomorphism
for i > i1, and since this localization is zero, then vk must be zero when i > *
*i1. So
(HEm )**X has a horizontal vanishing line with intercept i1.
Now fix m with n  m 2, and assume that
(HE=(v`1; : :;:v`t))**X
has a horizontal vanishing line whenever n  t < n  m. As above, let HEm =
HE=(vk1; : :;:vkm). Pick distinct integers k; ` n which are not in {k1; : :;:k*
*m },
and consider the following diagram (in which each row and column is a cofibrati*
*on):
2;xk+x`HEmvk!1;x`HEm!1;x`HEm =(vk)
? ? ?
v`?y v`?y ?y
1;xkHEm vk! HEm ! HEm =(vk)
?? ? ?
y ?y ?y
1;xkHEm =(v`)!HEm =(v`)! HEm =(vk; v`)
Now smash this diagram with X. By induction, all of the spectra HEm =(vk; v`)^X,
HEm =(vk)^X, and HEm =(v`)^X have horizontal vanishing lines with intercept i1.
Now we apply ssij(); for i > i1, the maps labeled vk and v` induce isomorphisms
on ssij, so we have
v1`Ovk
ssi;jHEm ^ X !~ssi;j+xkx`HEm ^ X:
=
This isomorphism, combined with the facts that xk 6= x` and that ssij(HEm ^*
*X)
is zero when j 0, implies that ssij(HEm ^ X) = 0 for all j; i.e., ss**(HEm ^ X)
has the predicted horizontal vanishing line. This completes the inductive_step,*
* and
hence the proof. __
Proof of Theorem 2.3.1.By Lemma 2.3.10, if we know that HB**X has a
vanishing line of slope d for all finitedimensional quotient Hopf algebras B o*
*f A,
then ss**(X) will, also. (For example, the map A !A(n) is an isomorphism throu*
*gh
degree 2n+1 1, so apply the lemma to the case B = A(n) where 2n+1 1 d.)
Now we show that HB**X has a vanishing line of slope d for all quotients B of
A with dimF2B < 1, by induction on dimF2B: The case where dimF2B = 1 (i.e.,
B = F2) is taken care of by condition (iii), so we move on to the inductive ste*
*p. By
Remark 2.1.3, there is a Hopf algebra extension
s
E[2t] !B !C:
By hypothesis, HC**X has a vanishing line of slope d, and we want to produce a
vanishing line for HB**X. There are several cases:
(1) If s t, then wesare done by Lemma 2.3.6.
(2) If s < t and 2t > d, then we are done by Lemma 2.3.8(a).
40 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
(3) If s < t and 2st d, then we have to prove something. We may assume
that j= 0 in B for j < t. Let
s+1 2n1 2n2 2n3
B = F2[t; t+1; t+2; : :]:=(2t ; t+1; t+2; t+3; : :):;
s+1 2m1 2m2 2m3
E = F2[t; t+1; t+2; : :]:=(2t ; t+1; t+2; t+3; : :):;
where mi = max (min(ni; s  i); 0). By Proposition 2.1.7,uE is an elementaryu
quotientsHopf algebra of A (and of B). Furthermore,uif 2v6= 0 in E, then 2v
2t. By hypothesis on X, then, (Puv)**X = 0 when 2v 6= 0 in E. So we apply
Lemma 2.3.11 to conclude that HE**X has a horizontal vanishing line with interc*
*ept
i1. Since X satisfies conditions(ii) of Definition 1.4.6, then HE**X also have*
* a
vanishing line of slope 2t + 1 (the line with this slope through the point (i*
*1; j0)).
We then apply Lemma 2.3.8(b)sto the case B i E to conclude that HB**X has a
vanishing line of slope 2t + 1 (andsin fact the same vanishing line).
Now, the map htsacts at slope 2t, and hence acts nilpotently on HB**X.
By Lemma 2.3.6(b), then, the vanishing line for HC**X, which has slope_d, gives
one for HB**X. __
Proof of Theorem 2.3.2.By Proposition 2.1.5, it suffices to show that there
is a uniform horizontal vanishing line for all of the groups HA(n)**X. Lemma 2.*
*3.11
and Lemma 2.3.8 together show that each HA(n)**X has a horizontal vanishing
line, and in fact these results identify the vanishing line: HA(n)i*X = 0 if i_*
*>_i1,
with i1 as in condition (iii) of Definition 1.4.6. _*
*_
2.3.2. Changes necessary when p is odd. The above proofs of Theo
rems 2.3.1 and 2.3.2 go through with a few changes when the prime is odd; we
indicate those changes in this subsection.
We start with the same setup as for the prime 2: we have an extension of Ho*
*pf
algebras in one of the following forms:
(E) E[on] !B !C;
s
(D ) D[pt] !B !C:
If case (E) arises, then Lemma 2.3.6(b) carries over as stated (writing vn for *
*the
homotopy element associated to the element on, rather than hts).
Otherwise extension (D ) arises, and we may assume that we have
s
opt6=s0+in1B,
opts = 0 in B,
opj= 0 in B for all j < t.
Lemma 1.3.10 then gives an element bts2 HB2;ppst, and hence a cofibration
ps bts 1;pps
(2.3.12) 2;ptHB !HB !gHC ! tHB;
where gHCis the cofiber of a selfmap of HC, as in Lemma 1.3.10. Hence if HC**X
has a vanishing line of slope d, then so does gHC**X. More precisely,sif HCijX *
*= 0
when j < di  c, then gHCijX = 0 when j < di  c + min(0; pt  d).
Here is the odd prime analogue of Lemma 2.3.6.
Lemma 2.3.13. (a)Given s and t as above so that we have extension (D ),
if s t, then bts2 HB**is nilpotent; hence the selfmap bts:HB !HB is
nilpotent.
2.3. VANISHING LINES FOR HOMOTOPY GROUPS 41
(b)Given extension (D ) and an object X, consider the cofibration
ps bts^1X
2;ptHB ^ X ! HB ^ X !gHC^ X:
If bts^ 1X acts nilpotently on HB**X, and if gHC**X has a vanishing line
of slope d, then HB ^ X has a vanishingsline of slope d. The difference in
intercepts depends only on d and pt_it is independent of X.
Proof. For part (a), see [MW81 , Proposition 4.1]. __
Part (b) is proved just as in the p = 2 case. __
The odd prime version of Lemma 2.3.8 is as follows.
Lemma 2.3.14.Fix an object X satisfying condition (i) of Definition 1.4.6.
(a)Given an extension
s
D[pt] !B !C;
2s
if pt_2 d and if HC**X has a vanishing line of slope d, then HB**X has
a vanishing line of slope d. In fact, HB**X has the same vanishing line as
HgC**X.
(b)Given an extension
E[on] !B !C;
if on d and if HC**X has a vanishing line of slope d, then HB**X has
a vanishing line of slope d. In fact, HB**X has the same vanishing line as
HC**X.
(c)Given finitedimensional quotients B i C of A, if HC**X has a vanishing
line of slope d and if the map B i C is an isomorphism in odd dimensions
less than d and in even dimensions less than 2d_p, then HB**X has a vanish*
*ing
line of slope d. Furthermore, the difference in intercept between the two
vanishing lines is independent of X.
Proof. Part (a) is proved just as is Lemma 2.3.8(a), but based on the cofibr*
*a
tion (2.3.12). Part (b) is the same as Lemma 2.3.8(a) (except for the character*
*istic
of the ground field, which is not relevant). Part (c) is proved, as in Lemma 2.*
*3.8(b),
by induction and parts (a) and (b). Since in (a), the vanishing line for gHC**X*
* may
have a different intercept than that for HC**X, the intercept for HB**X will ch*
*ange_
as the induction proceeds, but it will change by amounts dependent only on d. *
*__
Lemma 2.3.10 holds as stated, regardless of the prime involved.
Here is the analogue of Lemma 2.3.11.
Lemma 2.3.15.Let X be an object satisfying conditions (ii) and (iii) of Defi*
*ni
tion 1.4.6. Suppose that E is a finitedimensionalselementary quotient Hopf alg*
*ebra
of A. If (Pst)**X = 0 whenever 2t6= 0 in E, and (Qn)**X = 0 whenever on 6= 0
in E, then HE**X has a horizontal vanishing line. In fact, given i1 as in condi*
*tion
(iii), then HEi*X = 0 if i > i1.
The proof is the same as that for Lemma 2.3.11, using cofibrations of the fo*
*rm
ps
(2.3.12)repeatedly. We also need to point out that the slopes pt_2and on are*
* all
distinct. (Note also that in this case, we may choose d as large as we like, so*
* that
a horizontal vanishing line for HC**X induces the same one for gHC**X.)
42 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
Finally, given these modified tools, the proofs of the main theorems go thro*
*ugh
essentially unchanged.
2.4.Selfmaps via vanishing lines
In this section we use vanishing lines to construct selfmaps of finite spec*
*tra.
For each finite X, we construct one selfmap; we give many others in Theorem 4.*
*1.3.
We make heavy use of Notation 2.2.4 in this section.
Our approach for this section is based on some recent work in ordinary stable
homotopy theory, as described in work of Ravenel [Rav86 ] and Hopkins and Smith
[HSb ]. For now, we view the objects Pstand Qn (i.e., the Z(n)'s) as the analog*
*ues
of Morava Ktheories. This is not a perfect analogy, because the Morava Ktheor*
*ies
detect nilpotence, while the Z(n)'s do not.
The following definition is based on the definition of "vnmap" in ordinary
stable homotopy theory; see [HSb ] and [Rav92 ]. We will investigate and genera*
*lize
this definition in Section 4.2.
Notation 2.4.1.Fix a slope n. The ring z(n)** has a polynomial subalge
bra; we call the polynomial generator un. (In the notation of Remark 2.1.3 and
Proposition 2.2.2, un is one of hts, bts, or vn.)
Definition 2.4.2.Fix a spectrum X and a slope n.
(a)A selfmap f 2 [X; X]**is a unmap if for some j,
Z(n)**f = ujn^ 1X :
(b)We say that X is of type n if Z(d)**X = 0 for d < n, and Z(n)**X 6= 0.
Notice that if Z(n)**X = 0, then the zero map of X is a unmap.
We point out that in the ordinary stable homotopy category, Ravenel showed
in [Rav84 , 2.11] that for any finite spectrum X, K(n)*X 6= 0 ) K(n + 1)*X 6= 0.
The analogous statement here, with Z(n)**rather than K(n)*, does not hold. See
Proposition 4.8.1 (and also [Pal96b, Prop. 3.10 and Thm. A.1]) for the correct
statement when p = 2 (and for a guess in the odd prime case). We do know that
if X is a nonzero finite spectrum, then by Theorem 2.3.2 and Corollary 4.5.7, f*
*or
some n we have Z(n)**X 6= 0. (One could also use the AtiyahHirzebruch spectral
sequence to show this.)
The following first appeared (for modules) in [Pal92]. It is a slight genera*
*l
ization of a result of Hopkins and Smith [HSb ]. See Theorems 3.1.2 and 4.1.3 f*
*or
stronger results when p = 2.
Theorem 2.4.3.Fix a finite spectrum X of type n. Then for some k, there is
a nonnilpotent unmap
v :k;knX !X:
To prove this, we need a "relative vanishing line" result; this is a general*
*ization
of a standard result_see [Rav86 , 3.4.9], for instance.
Lemma 2.4.4.Fix a slope n. Suppose that X is a CL spectrum, and suppose
that X is of type at least n (hence ss**X has a vanishing line of slope n). Giv*
*en
m 0, let M be the number below which degree h: A !A(m) is an isomorphism.
Then the Hurewicz map h: ss**X ! HA(m)**X is an isomorphism above a line
of slope n: for some c independent of m, h is an isomorphism on ssijwhen j <
ni + M  c.
2.4. SELFMAPS VIA VANISHING LINES 43
Proof. This is a consequence of the form of the intercept in the vanishing l*
*ine
theorem_see Remark 2.3.4. Let W denote the fiber of the map S0 ! HA(m).
Since the kernel of the map A ! A(m) is zero below degree M, if we choose
numbers i0, i1, and j0 so that X satisfies Definition 1.4.6, then W ^ X satisfi*
*es_the
conditions with the numbers i0, i1, and j0+ M. __
We have the following result based on work of Lin [Lin] and Wilkerson [Wil81*
* ];
this is a corollary of Theorem 3.3.5.
Proposition 2.4.5.Fix m and consider the quotient Hopf algebra A(m) of A.
(a)Let p = 2. Fix integers s < t with 2stnonzero in A(m) (i.e., with s + t
m + 1). Then the restriction map
Ext**A(m)(F2; F2) !Ext**F2[t]=(2s+1t)(F2; F2) ~=F2[ht0; ht1; : :;:hts]
is surjective modulo nilpotence (i.e., the algebra cokernel consists entir*
*ely
of nilpotent elements). Hence for some i = i(m), there is a nonnilpotent
element
2s
w 2 Exti;itA(m)(F2; F2) = HA(m)i;i2st
which restricts to hits. s
(b)Let p be odd. Fix integers s < t with ptnonzero in A(m) (i.e., with s + t
m). Then the restriction map
Ext**A(m)(Fp;Fp)! Ext**Fp[t]=(ps+1(Fp; Fp)
fl t )
flfl
Fp[bt0; : :;:bts] E[ht0; : :;:hts]
is surjective modulo nilpotence. Hence for some j = j(m), there is a non
nilpotent element
ps
w 2 Ext2j;jptA(m)(Fp; Fp) = HA(m)2j;jppst
which restricts to bjts.
(c)Let p be odd. Fix an integer t with ot nonzero in A(m) (i.e., with t m).
Then the restriction map
Ext**A(m)(Fp; Fp) !Ext**E[ot](Fp; Fp) ~=Fp[vt]
is surjective mod nilpotence. Hence for some k = k(m), there is a non
nilpotent element
w 2 Extk;kotA(m)(Fp; Fp) = HA(m)k;kot
which restricts to vkt.
We need the following lemma. Recall from Notation 1.3.9 that D[y] is the Hopf
algebra Fp[y]=(yp) with y primitive.
Lemma 2.4.6.Fix a slope n, and choose an integer m large enough that yn is
nonzero in A(m) (and hence so that D[yn] or E[yn] is a quotient coalgebra of A(*
*m)
over which A(m) is injective).
(a)Then the map HA(m)** !z(n)** is an algebra map, and some power of
the polynomial generator un of z(n)**is in the image.
44 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
(b)For any object X, the following diagram commutes.
[X; HA(m) ^ X]**! [X; z(n) ^ X]**
x x
^X?? ??^X
HA(m)** ! z(n)**
(c)For any object X with Z(n)**X 6= 0, the map z(n)**! [X; z(n) ^ X]**is
an injection.
Proof. Part (a) follows from Proposition 1.3.2(b) and Proposition 2.4.5.
Part (b) follows from the fact that [X; z(n) ^ X]** is isomorphic to cochain
homotopy classes of D[yn]linear (resp., E[yn]linear) selfmaps of X, and the *
*hor
izontal maps are just restriction.
For part (c), we merely note that if Z(n)**X 6= 0, then the identity map on *
*X_
is not cochain homotopic to zero over D[yn] (resp., over E[yn]). *
*__
Proof of Theorem 2.4.3.By Lemma 2.4.4, if we choose m large enough,
then the map [X; X]** ![X; HA(m) ^ X]** is an isomorphism in the bidegrees
(k; kn), for all integers k. Now we apply Lemma 2.4.6 to find a nonnilpotent
element w 2 HA(m)k;kn(for some k) which maps nontrivially to z(n)** and to
[X; z(n) ^ X]**. The lift of w ^ 1X to [X; X]k;knclearly has the_desired_proper*
*ties.
__
We will need the following lemma in Section 4.7.
Lemma 2.4.7.Fix a finite type n spectrum X. Then for some k, the unmap
v 2 [X; X]** constructed in Theorem 2.4.3 is central in a band parallel to the
vanishing line. This band includes the origin.
Proof. Consider the element w 2 HA(m)k;kn, as given by Proposition 2.4.5.
Since HA(m)**is a commutative ring, w maps to a central element in [X; HA(m) ^
X]**. By Lemma 2.4.4, [X; X]**! [X; HA(m) ^ X]**is an isomorphism in a band
parallel to the vanishing line; by choice of m, that band includes the origin._*
*Hence
the lift of w to [X; X]**is central in that band. __
2.5.Further discussion
As mentioned in Subsection 2.1.1, one of the main gaps in this theory, when p
is odd, is the lack of a classification of the quasielementary quotient Hopf a*
*lgebras
of A. See Appendix B.3 for a discussion of conjectures and results related to t*
*his
issue.
We note that the vanishing line theorem 2.3.1 has been used many times in
many papers; it provides a convenient way to get valuable information about the
Adams E2term. Combined with newer results, such as Theorem A.2.6, it is even
more powerful.
Theorem 2.4.3, the result that ensures the existence of a nonnilpotent self*
*map
of any finite object, also has been used in topological applications. For examp*
*le,
Hopkins and Smith used it to prove the periodicity theorem [HSb ]: they had
constructed a particular spectrum X, and they used the theorem to find a vnmap
of X at the E2term of the Adams spectral sequence. Later, Theorem 2.4.3 was
used by Sadofsky and the author in [PS94 ] to give a new proof of the periodici*
*ty
2.5. FURTHER DISCUSSION 45
theorem: we used it not only to produce a vnmap, but also to construct the
spectrum X in question. This was done by taking iterated cofibers: when p = 2,
the cofiber of u1: S0 !S0 has a u3map (3 is the next slope after 1); the cofi*
*ber
of this map has a u6map, etc. This was done with modules over the Steenrod
algebra, and then realized at the spectrum level in the ordinary stable homotopy
category.
See the results of Section 4.7 for a similar application of Theorem 2.4.3, b*
*ut in
Stable(A).
46 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA
CHAPTER 3
Quillen stratification and nilpotence
The vanishing line theorem of Section 2.3 is a nilpotence theorem of a sort:*
* if
X is a finite spectrum, then any selfmap of X with slope smaller than that of *
*the
vanishing line of X must be nilpotent (because some power of it will lie above *
*the
vanishing line). We used the vanishing line theorem in Section 2.4 to construct*
* a
nonnilpotent selfmap of any finite spectrum.
In this chapter we give some related, but stronger, results; we work mostly *
*at
the prime 2. We let Q denote the category of quasielementary quotients of A,
with morphisms given by quotient maps. We can assemble the individual Hurewicz
(restriction) maps S0 !HE into
S0 !limQHE:
Since the maximal quasielementary quotients are conormal, there is an action of
A on limQHE**; we prove that
ss**S0 !(limQHE**)A
is an Fisomorphism_its kernel consists of nilpotent elements, and some pth pow*
*er
of every element in the target is actually in the image. One can view this as an
analogue of the Quillen stratification theorem [Qui71 , 6.2], which identifies *
*ss**S0
up to Fisomorphism in the category Stable(kG*), for G a finite group and k an
algebraically closed field of characteristic p.
Similarly, suppose that we write D for the following quotient of A:
2 p3 pn
D = A=(p1; p2; 3 ; : :;:n ; : :)::
Then D is conormal, and the Hurewicz map induces an Fisomorphism
ss**S0 !HDA**:
These Fisomorphism theorems first appeared in [Pal].
We have weaker results with nontrivial coefficients: if R is a ring spectrum,
then an element ff 2 ss**R is nilpotent if its image under the Hurewicz map in
HD**R is zero, or if its image in HE**R is zero for all quasielementary quotie*
*nts
E of A. (These nilpotence theorems are improvements on results of the author in
[Pal96a].) See also Theorem 4.2.2 for a statement about lifting invariants from
HD**R to ss**R.
We state these results more precisely in Section 3.1. In Section 3.2, we pro*
*ve
the two theorems_nilpotence and Fisomorphism_involving D; we then use these
these to prove the analogous results for quasielementary Hopf algebras in Sec
tion 3.3. We end the chapter with Sections 3.4 and 3.5; in the first of these, *
*we
indicate why our proofs only work at the prime 2; this leads us to a conjecture
about the nilpotence of particular classes in Ext over quotients of the dual of*
* the
47
48 3. QUILLEN STRATIFICATION AND NILPOTENCE
 
 ___  ___
 ___  ___
 
 ___  ___
 
 ___  ___
__ __
 
_________________________ __________________________
p = 2 __________p_odd__________
Figure 3.1.A. Profile function for D.
odd primary Steenrod algebra. In the second of these, we discuss a few possible
generalizations of the main results of this chapter.
Except for Sections 3.4 and 3.5, we work at the prime 2 in this chapter.
3.1.Statements of theorems
Let p = 2.
In this section we present two Fisomorphism results and two nilpotence resu*
*lts.
Let D be the following quotient Hopf algebra of A:
2 p3 pn
D = A=(p1; p2; 3 ; : :;:n ; : :)::
See Figure 3.1.A. Note that D and A have the same quasielementary quotients_
i.e., every quasielementary quotient map A ! E factors as A ! D ! E. As a
result, it turns out that HD plays a similar role in Stable(A) to that of BP in*
* the
ordinary stable homotopy category, at least as far as detection of nilpotence g*
*oes.
We write jD :S0 = HA ! HD for the unit map of the ring spectrum HD, and
similarly for jE :S0 !HE for E quasielementary.
3.1.1. Quillen stratification. We state our results describing ss**S0 up to
Fisomorphism. The results in this subsection first appeared in [Pal].
Definition 3.1.1. (a)Given a Hopf algebra B and a Bcomodule M, we
define the Binvariants of M to be
MB = HomB (Fp; M) M:
(This is the same as the primitives PM of M: if :M ! B M is the
coaction map, then we let PM = {m 2 M  (m) = 1 m}.)
(b)Following Quillen [Qui71 ], if ': R ! S is a map of graded commutative
Fpalgebras, we say that ' is an Fisomorphism if it satisfies the followi*
*ng
properties. p____
(i)Every x 2 ker' is nilpotent. (Hence ker' is the nilradicalnof R.)
(ii)For any element y 2 S, there is an integer n so that yp 2 im'.
In (ii), if one can choose the same n for every y, then we say that ' is a
uniform Fisomorphism.
Here is our first result. Theorem 2.1.1 tells us that D is conormal, so by
Remark 1.3.8, there is a coaction of A 2D F2on HD**.
3.1. STATEMENTS OF THEOREMS 49
Theorem 3.1.2 (Quillen stratification,TI).he Hurewicz map ss**S0 !HD**
factors through
': ss**S0 !HDA2DF2**;
and ' is an Fisomorphism.
Here is our second result, an analogue of Quillen's theorem [Qui71 , 6.2], w*
*hich
identifies group cohomology up to Fisomorphism. We have defined in Defini
tion 2.1.10 (see also Proposition 2.1.12) the notion of a quasielementary quot*
*ient
Hopf algebra of A; we let Q denote the category of quasielementary quotients of
A, with morphisms given by quotient maps. In Section 3.3, we construct a coacti*
*on
of A on limQHE**.
Theorem 3.1.3 (Quillen stratification,TII).he map ss**S0 !limQHE**fac
tors through
i jA
fl :ss**S0 ! limQHE** ;
and fl is an Fisomorphism.
We prove Theorem 3.1.2 in Section 3.2; we show that Theorem 3.1.3 follows
from Theorem 3.1.2 in Section 3.3.
Remark 3.1.4. (a)One can view Theorem 3.1.3, and hence Theorem 3.1.2,
as giving an analogue of the Quillen stratification theorem [Qui71 , 6.2]:
given a finite group G and an algebraically closed field k of characteris
tic p, we let A be the category whose objects are the elementary abelian
psubgroups of G, and whose morphisms are generated by inclusions and
compositions. Then the natural map
H*(G; k) !limE2AH*(E; k)
is an Fisomorphism. The role of the conjugation maps in the category A is
played, in our results, by the process of taking invariants.
(b)We have an explicit formula for the algebra limHE** (Proposition 4.4.1),
as well as for the coaction of A on limHE**(Proposition 4.4.4). One can
use this to predict the presence of large families of nonnilpotent elemen*
*ts
in ss**S0. See Section 4.4 for details.
(c)We do not expect the Fisomorphisms in these results to be uniform, al
though we do not have much evidence either way.
(d)One can also view Theorem 3.1.2 as an analogue of Nishida's theorem
[Nis73]_in the ordinary stable homotopy category, ss*S0 is isomorphic to Z,
mod nilpotence; however, while Nishida's theorem does not immediately lead
to guesses as to further structure in the ordinary stable homotopy categor*
*y,
our analogue does. For instance, see Section 4.6 for a suggested classific*
*ation
of thick subcategories of finite spectra in Stable(A).
(e)Of course, if A0is any quotient Hopf algebra0of A, then similar results ho*
*ld:
for instance, the map HA0**!(HD0**)A is an Fisomorphism. The proofs
for Theorems 3.1.3 and 3.1.2 carry over easily.
50 3. QUILLEN STRATIFICATION AND NILPOTENCE
3.1.2. Nilpotence. We move on to our nilpotence theorems. The first of these
is based on the nilpotence theorem of Devinatz, Hopkins, and Smith [DHS88 ]: t*
*he
ring spectrum BP detects nilpotence.
Theorem 3.1.5 (Nilpotence theorem,TI).he ring spectrum HD detects nilpo
tence:
(a)Fix a ring spectrum R and ff 2 ss**R. Then ff is nilpotent if and only if
jD ^ ff 2 HD**R is nilpotent.
(b)Fix a finite spectrum Y and a selfmap f :Y ! Y . Then f 2 [Y; Y ]** is
nilpotent if and only if jD ^ f 2 [Y; HD ^ Y ]**is nilpotent.
(c)Fix a finite spectrum F, an arbitrary spectrum X, and a map f :F ! X.
Then f 2 [F; X]**is smashnilpotent if jD ^ f 2 [F; HD ^ X]**is zero.
The corresponding result for modules is [Pal96a, Theorems 3.1 and 4.2]. In
[Pal96a] we prove this in enough generality so that the proof goes through here
without difficulty. We also give a (slightly different) proof in Section 3.2.
The second nilpotence theorem is, more or less, an analogue of the K(n) nilp*
*o
tence theorem in [HSb ]. The following appeared for bounded below modules in
[Pal96a, Theorems 1.1 and 4.3].
Theorem 3.1.6 (Nilpotence theorem,TII).he collection of ring spectra
{HE  E quasielementary}
detects nilpotence:
(a)Fix a ring spectrum R and ff 2 ss**R. Then ff is nilpotent if and only if
jE ^ ff 2 HE**R is nilpotent for all quasielementary quotients E of A.
(b)Fix a finite spectrum Y and a selfmap f :Y ! Y . Then f is nilpotent if
and only if jE ^ f is nilpotent for all quasielementary quotients E of A.
(c)Fix a finite spectrum F, an arbitrary spectrum X, and a map f :F ! X.
Then f is smashnilpotent if jE ^ f :F ! HE ^ X is zero, for all quasi
elementary quotients E of A.
The proof of the analogous result in [Pal96a] does not apply to nonconnective
situations, so we give a new proof below.
Remark 3.1.7. (a)Note that F2is a quasielementary quotient Hopf alge
bra of A, so HF2is included as one of the detecting spectra in Theorem 3.1*
*.6
(as compared to [HSb ], when HF2is included for parts (a) and (c), but not
for (b)).
(b)Fix a quotient Hopf algebra A0of A, and let D0be the quotient of A0induced
by A i D. One can generalize Theorem 3.1.5 in an obvious sort of way: if
Y is a finite spectrum and f :Y ! HA0^ Y is a "selfmap," then
f 2[Y; HA0^ Y ]**is nilpotent
, jD0^ f 2 [Y; HD0^ Y ]**is nilpotent
, jE0^ f 2 [Y; HE0^ Y ]**is nilpotent forEall0:
(Here E0ranges over all quasielementary quotients of A0.) There are simil*
*ar
versions of the ring spectrum and smashnilpotence results. The proofs are
straightforward generalizations of the ones below, so we omit them.
3.2. NILPOTENCE AND FISOMORPHISM VIA THE HOPF ALGEBRA D 51
(c)The quotient Hopf algebra D is "best possible," in the sense that if B is
quotient of A which does not map onto D, then there are nonnilpotent
elements in ss**S0 which are in the kernel of ss**S0 !HB**. One can see
this from Theorem 3.1.2.
(d)Theorem 3.1.5 identifies a single ring spectrum HD which detects nilpotenc*
*e,
but we do not know its coefficient ring completely. See Propositions 3.3.4
and 4.4.1 for partial information. On the other hand, we have computed the
coefficient rings of HE for E quasielementary in Proposition 2.1.9_they
are polynomial rings.
(e)We also mention that one does not need to use all of the quasielementary
quotient Hopf algebras of A to detect nilpotence; for instance, one can use
only the maximal quasielementary quotients, or only the finitedimensional
ones.
(f)In fact, one can see from the proof of Theorem 3.1.6 that one only needs *
*the
following spectra to detect nilpotence:
j+1
{h1m+1;jH(E(m)=(2m+1))  m 0; 0 j m}:
3.2. Nilpotence and Fisomorphism via the Hopf algebra D
In this section we show that the spectrum HD detects a lot of information: we
prove that it detects ss**S0 modulo nilpotent elements (Theorem 3.1.2), and tha*
*t it
detects nonnilpotent elements of ss**R for any ring spectrum R (Theorem 3.1.5).
The Hopf algebra D is a conormal quotient of A, by Theorem 2.1.1. So there
is a Hopf algebra extension
A 2D Fp! A !D;
and a spectral sequence (as in Section 1.5) with
Es;t2= ExtsA2DFp(Fp; ExttD(Fp;)Fp)) Exts+tA(Fp; Fp):
The restriction (Hurewicz) map factors through the edge homomorphism
ExttA(Fp; Fp) !E0;t2:
This gives the factorization of the Hurewicz map h: ss**S0 !HD**as advertised
in Theorem 3.1.2:
': ss**S0 !HDA2DF2**:
It remains to prove Theorem 3.1.5_the spectrum HD detects nilpotence_and
to verify conditions (i) and (ii) of Definition 3.1.1(b)_every element of ker' *
*is
nilpotent,mand for every element y 2 HDA2DFp**, there is an integer m so that
y2 2 im'.
Note that condition (i) follows from Theorem 3.1.5(a) with R = S0. Also, the
proof of Theorem 3.1.5 and the verification of (ii) are quite similar, so first*
* we lay
the groundwork for both.
For each integer n 1, we let
n
D(n) = A=(21; 42; : :;:2n):
We let D(0) = A. See Figure 3.2.B. Each D(n) is a conormal quotient Hopf algebra
of A, and we have a diagram of Hopf algebra surjections:
A = D(0) i D(1) i D(2) i : :::
52 3. QUILLEN STRATIFICATION AND NILPOTENCE

 
 
 
n______
n1



_________________________
p = 2
Figure 3.2.B. Profile function for D(n). As in Figure 2.1.B, the
diagonal line is an abbreviation for a staircase shape.
D is the colimit of this diagram.
First we discuss how to lift information from HD**to HD(n)**for some n. We
have the following lemma.
Lemma 3.2.1. (a)We have HD**= lim!HD(n)**.
(b)We have HDA2DFp**= lim!(HD(n)A2D(n)Fp**).
Proof. Since homotopy commutes with direct limits, part (a) is clear.
Part (b): The coaction of A on HD(n)**is defined in Remark 1.3.8. Note that
each restriction map HD(n)**! HD(n + 1)**is an Acomodule map (and in fact,
a map of comodule algebras.) We take injective resolutions of these comodules,
and apply ss0;*() (i.e., Hom *A(Fp; )); since homotopy commutes with colimits,
we have
Hom A(Fp; HD**) = HomA (Fp; lim!HD(n)**) = lim!HomA(Fp; HD(n)**):
Now, Hom A(Fp; HD(n)**) = HD(n)A**= HD(n)A2D(n)Fp**, by conormality, so_we
have the desired result. __
Now we discuss how to take information about HD(n)**and get information
about HD(n  1)**. (We want to know about ss**S0 = HD(0)**, so we will even
tually want to use downward induction on n.)
Not only is each D(n) a conormal quotient of A, it is also a conormal quotie*
*nt
of D(n  1). The Hopf algebra kernel of the quotient map is easy to identify: we
have an extension of Hopf algebras
n
F2[2n] !D(n  1) !D(n);
where 2nnis primitive in the kernel. So given any D(n  1)comodule M, there is
a changeofrings spectral sequence (see (1.5.2)) with
Es;t;u2(M) = Exts;uF2[2nn](F2; Extt;*D(n)(F2;)M)) Exts+t;uD(n1)(F2; M):
By Proposition 1.5.3, in the category Stable(D(n1)) of cochain complexes of in*
*jec
tive left D(n1)comodules, this spectral sequence is the same (up to regrading*
*) as
the HD(n)based Adams spectral sequence. Because of this, for parts of the proof
we will work in the category Stable(D(n1)). We also use the grading on the spe*
*c
tral sequence as given here; we do not use the Adams spectral sequence grading *
*from
Section 1.5. Throughout, we abuse notation somewhat, writing Ext**D(n)(F2; X) f*
*or
ss**(HD ^ X).
3.2. NILPOTENCE AND FISOMORPHISM VIA THE HOPF ALGEBRA D 53
We need to establish an important property of this spectral sequence, that it
has a nice vanishing plane at some Erterm. We write Es;t;ur(X) for the spectral
sequence converging to Exts+t;uD(n1)(F2; X).
Proposition 3.2.2.Fix a connective spectrum X and an integer m. For some
r and some c, we have Es;t;ur(X) = 0 when ms + t  u > c.
We prove this using Theorem 1.5.5, which says that vanishing planes in Adams
spectral sequences are generic (Definition 1.4.7). (And again, this is an Adams
spectral sequence, as long as we work in the category Stable(D(n  1)).)
We fix a connective spectrum X.
Lemma 3.2.3.For each integer m, there is a finite D(n  1)comodule W so
that
Es;t;u2(W ^ X) = 0
when ms + t  u > c, where c depends only on the connectivity of X.
(In Adams spectral sequence grading, this vanishing plane is of the form
Ep;q;v2(W ^ X) = 0
when mp  q > 0, i.e., when
p _1__m(p1+ v) + __1__m(q1+ v):
In particular, the coefficients _1_m1and __1_m1) satisfies the hypotheses of*
* Theo
rem 1.5.5.)
Proof. Choose an integer j n so that 2jn = 2j(2n  1) > m, and let
W = Wj= F2[2nn]=(2jn), with the apparent D(n  1) 2D(n)Fpcomodule structure.
Then Wj is a trivial D(n)comodule (by definition), and as coalgebras we have
n 2j
F2[2n] ~=Wj F2[n ]:
Hence the E2term of the spectral sequence for Wj^ X looks like
Es;t;u2~=Exts;uF2[2nn](F2; Extt;*D(n)(F2; Wj^)X )
~=Exts;uF(F n; W Extt;*(F ; X))
2[2n]2 j D(n) 2
~=Exts;u (F ; Extt;*(F ; X)):
F2[2jn]2 D(n) 2
The Hopf algebra F2[2jn] is 2jnconnected, so if L is a comodule which is zero
below degree t, then
Exts;uF2[2j(F2; L) = 0
n ]
when u < 2jns + t. Now note that Extt;*D(n)(F2; X) is zero below degree t + c*
* for
some c dependent only on the connectivity of X. ___
Next we need to show that X is in thick(Wj^ X). We start with the following
lemma, which describes how to build Wj out of F2in a nice way.
54 3. QUILLEN STRATIFICATION AND NILPOTENCE
Lemma 3.2.4.For j n, let Wj = F2[2nn]=(2jn). Then there is a short exact
sequence of D(n  1) 2D(n)F2comodules
2j
0 !Wj! Wj+1! n Wj! 0:
The connecting homomorphism in Ext**D(n1)(F2; ) is multiplication by
j **
hnj= [2n] 2 ExtD(n1)(F2; F2):
Replacing Wj by its injective resolution gives a cofibration sequence
2j hnj
1;nWj !Wj! Wj+1! Wj:
(We are abusing notation a bit here, by writing Wj for both the module and
its injective resolution. We will continute this practice for the remainder of *
*this
section.)
Proof. This follows from Lemma 1.3.10. ___
Suppose that j n; then Theorem B.2.1(a) tells us that the element hnj 2
HD(n  1)**is nilpotent. Hence we have the following:
Lemma 3.2.5.If j n, then X 2 thick(Wj^ X).
Proof. We show by downward induction on i that for j i n, Wi^ X is
in thick(Wj^ X); the lemma is proved when i = n, since Wn = S0.
The induction starts (trivially) with i = j. Suppose that i < j. By the
cofibration sequence in Lemma 3.2.4, together with the nilpotence of hni,_we_see
that Wi^ X 2 thick(Wi+1^ X), and hence in thick(Wj^ X) by induction. __
Proof of Proposition 3.2.2.This follows immediately from Lemma 3.2.3,_
Lemma 3.2.5, and Theorem 1.5.5. __
3.2.1. Nilpotence: Proof of Theorem 3.1.5. Now we prove Theorem 3.1.5,
and hence verify condition (i) of Definition 3.1.1(b).
Proof of Theorem 3.1.5.The basic idea of the proof is, of course, based on
that of the nilpotence theorem in [DHS88 ]. As in that proof, one can reduce to
the ring spectrum case_part (a)_in which the ring is connective. So we let R be
a connective ring spectrum, we fix ff 2 ss**R and assume that HD**ff is nilpote*
*nt.
By raising ff to a power, we may assume that HD**ff = 0. We want to show that
ff is nilpotent in ss**R = HD(0)**R.
By Lemma 3.2.1, since HD**y = 0, then we must have HD(n)**y = 0 for some
n. We want to show that if HD(n)**y = 0, then HD(n  1)**yj = 0 for some j;
the result will follow by downward induction on n.
Consider the changeofrings spectral sequence
Es;t;u2(R) = Exts;uF2[2nn](F2; Extt;*D(n)(F2;)R)) Exts+t;uD(n1)(F2; R):
Write z for HD(n  1)**y. Since HD(n)**y = 0, then z must be represented by a
class "z2 Ep;q;v2with p > 0.
Choose an integer m so that mp + q  v > 0. Then for any c, we can find a j
so that
mpj + qj  vj > c:
3.2. NILPOTENCE AND FISOMORPHISM VIA THE HOPF ALGEBRA D 55
By Proposition 3.2.2 (with X = R), for some r and c we have Es;t;ur= 0 when
ms+tu > c. As noted above, we can choose j so that "zjlies above this vanishing
plane, so at the Erterm for which we have the vanishing plane, "zjmust be zero.
Modulo terms of higher filtration, "zjis zero at E1 , but the higher filtration*
* pieces
are also above the vanishing plane, and therefore zero. So zj = 0 in the_abutme*
*nt,
HD(n  1)**, which is what we wanted to show. __
3.2.2. Fisomorphism: Proof of Theorem 3.1.2. We need to show that
the map
': ss**S0 !HDA2DF2**
is an Fisomorphism. By Theorem 3.1.5, it is a monomorphism mod nilpotents, so
we have to show that it is an epimorphism mod nilpotents; in other words, we ha*
*ve
to verify condition (ii) of Definition 3.1.1(b).
Verification of condition (ii).Fix y 2 HDA2DF2i;j. We show that there
is an integer m so that y2m 2 im'. By Lemma 3.2.1(b), there is an n so that
y lifts to HD(n)A2D(n)F2**. (Alternatively, one can use Lemma 3.2.1(a) to lift *
*y to
HD(n)**for some n, and then use Lemma 3.2.6 to show that some power of that
lift is invariant.) Now we show that some power of y lifts to HD(n  1)A2D(n1)*
*F2**;
since D(0) = A, then downward induction on n will finish the proof.
Since y is invariant under the A 2D(n)F2coaction, then it is also invariant*
* under
the coaction of D(n  1) 2D(n)F2(since the latter is a quotient Hopf algebra of*
* the
former). So y represents a class at the E2term of the changeofrings spectral
sequence
Es;t;u2(F2) = Exts;uF2[2nn](F2; Extt;*D(n)(F2;)F2)) Exts+t;uD(n1)(F2; F*
*2):
Also, by assumption, y lies in the (t; u)plane, say y 2 E0;q;v2. Choose m lar*
*ge
enough so that m + q  v  1 is positive. By Proposition 3.2.2 (with X = S0), we
know that for some c and r, we have Es;t;ur= 0 for c < ms + t  u (hence the sa*
*me
as true for Es;t;ur0, for all r0 r).
For each i 0, Proposition 1.5.4 tells us that the possible differentials on*
* y2i
are
i j+1;2iqj;2iv
dj+1(y2 ) 2 Ej+1 ;
for j 2i. Choose i so that
2i> max(r  1; ___c__m___m)+;q  v  1
and fix j 2i. Then we have a vanishing plane at the E2i+1term (and hence at t*
*he
Ej+1term); we claim that the element dj+1(y2i) lies above the vanishing plane,*
* and
so is zero. We just have to verify the inequality specified by the vanishing pl*
*ane:
m(j + 1) + 2iq  j =2iv(m  1)j + m + 2i(q  v)
(m  1)2i+ m + 2i(q  v)
= 2i(m + q  v  1) + m
> ___c__m___m(+mq+qvv1 1) + m
= c:
56 3. QUILLEN STRATIFICATION AND NILPOTENCE
Hence y2iis a permanent cycle. For degree reasons, it cannot be a boundary; hen*
*ce
it gives a nonzero element of E1 , and hence a nonzero element of HD(n  1)**.
It only remains to show that y2i, or at least some power y2i+j, is invariant*
* under
the Acoaction. Let ae: HD(n  1)**!HD(n)**denote the Hurewicz (restriction)
map. This map detects nilpotence: if x 2 kerae, then x is nilpotent. (This foll*
*ows
from Remark 3.1.7(b), for instance; alternatively, this is the main inductive s*
*tep in
proving Theorem 3.1.5.) Hence by Lemma 3.2.6 below, y2i+jis invariant for some
j.
This completes the verification of condition (ii) of Definition 3.1.1(b),_an*
*d hence
the proof of Theorem 3.1.2. __
We have used the following.
Lemma 3.2.6.Suppose that R and S are commutative Acomodule algebras,
with an Alinear map ae: R !S that detects nilpotence: every x 2 kerae isnnilp*
*otent.
Given z 2 R so that ae(z) 2 S is invariant under the Acoaction, then z2 is al*
*so
invariant, for some n.
Proof. Since ae(z) is invariant, then the coaction on z is of the form
X
z 7! 1 z + ai xi;
i
where each xiis in kerae, and hence is nilpotent. Since R is an Acomodule alge*
*bra,
then we see that
n 2n X 2n 2n
z2 7! 1 z + ai xi :
i
This is a finite sum (since these are comodules), so for n sufficiently_large, *
*z2n is
invariant. __
3.3.Nilpotence and Fisomorphism via quasielementary quotients
In this section we use our Fisomorphism and nilpotence theorems for HD_
Theorems 3.1.2 and 3.1.5_to prove analogous theorems for the quasielementary
quotients of A_Theorems 3.1.3 and 3.1.6.
3.3.1. Nilpotence: Proof of Theorem 3.1.6. We start with Theorem 3.1.6,
because we use it in the proof of Theorem 3.1.3. We need a few preliminary resu*
*lts.
Suppose we have a map f :S0 !X. We define X(1) to be the sequential colimit
of the following diagram:
S0 f!X f^1!X ^ X f^1^1!X ^ X ^ X !: :;:
and we let f(1): S0 !X(1) be the obvious map. We recall the following from
[HSb , Lemma 2.3].
Lemma 3.3.1.Given a map f :S0 !X and a ring spectrum E with unit map
j :S0 !E, the following are equivalent.
(a)E ^ X(1)= 0.
(b)j ^ f(1): S0 !E ^ X(1) is zero.
(c)j ^ f(n):S0 !E ^ X(n)is zero for n 0.
(d)1E ^ f(n):E = E ^ S0 !E ^ X(n)is zero for n 0.
3.3. NILPOTENCE VIA QUASIELEMENTARY QUOTIENTS 57
 
 ___  ___
 ___  ___
 
 ___  ___
  
 ___  :.: :
  ..
 ___  .:.: :
 ___  : :.: 
  .. 
 ___  __________.
   
 :.:.:  .:.: :
:.::D2 :.: :D2;6
   
_________________________ _________________________
Figure 3.3.C. Profile functions for Dr and Dr;q.
For integers q > r 0, we define the following quotient Hopf algebras of D:
Dr = D=(1; : :;:r);
r+1 2r+1
Dr;q= Dr=(2r+2; : :;:q ):
See Figure 3.3.C. Recall from Corollary 2.1.8 that the maximal quasielementary
quotients of A are called E(m), m 0.
Lemma 3.3.2.We have lim!rHDr = HF2and lim!qHDr;q= HE(r).
Proof. We leave this as an exercise. ___
Lemma 3.3.3.Let i, j, q, r, and R be integers.
(a)Suppose that q > r i 0 and q  r > j 0, and consider the Hopf
algebra B = Dr;q=(2i+1r+1; 2r+2+jq+1). By Lemma 2.1.2, there are Hopf alge*
*bra
extensions
i
E[2r+1] !B !C1;
r+1+j
E[2q+1 ] !B !C2;
leading to elements hr+1;iand hq+1;r+1+jin HB**. Then hr+1;ihq+1;r+1+j
is nilpotent.
(b)Whenever q > r and r i 0, there is a Bousfield equivalence
i+1 1 2i+1
=
:
(c)Whenever q > R, we have a Bousfield class decomposition
R_ _r i+1
= _
r=0i=0
R_ _r i+1
= _ :
r=0i=0
Proof. For (a), the nilpotence of hr+1;ihq+1;r+jis due to Lin [Lin]; we reca*
*ll
the statement of this result as Theorem B.2.1(b). (The nilpotence of this produ*
*ct
is also the content of [Wil81 , Theorem 6.4], as well as being essentially equi*
*valent
to the classification of quasielementary quotients of A in Propositions 2.1.7 *
*and
2.1.12.)
58 3. QUILLEN STRATIFICATION AND NILPOTENCE
Part (b) follows from part (a) and Corollary 1.6.2, by induction: one can get
from Dr=(2i+1r+1) to Dr;q=(2i+1r+1) by dividing out by one 2stat a time, where *
*s r+1.
In other words, one has a sequence of extensions of the form
s
E[2t] !B !C;
and hence cofibrations of the form
HB hts!HB !HC:
One inverts hr+1;iin each term; then by the nilpotence of hr+1;ihts, one gets an
equivalence of Bousfield classes
= :
Part (c) is similar to part (b). ___
Proof of Theorem 3.1.6.We imitate the proof of [HSb , Theorem 3]. As
in that proof, one can reduce to the smashnilpotence case, and using Spanier
Whitehead duality, one can reduce to the case where F = S0. Suppose we have a
map f :S0 !X so that jE ^ f = 0 for all quasielementary E. We want to show
that j ^ f(n):S0 ! HD ^ X(n)is zero for some n; then Theorem 3.1.5(c) will
tell us that f is smashnilpotent. By Lemma 3.3.1, this map is zero if and only*
* if
HD ^ X(1)= 0. We use a Bousfield class argument to show this.
By assumption, HE^X(1)= 0 for all quasielementary E. By Lemma 3.3.3(c),
we must show that
HDR+1 ^ X(1)= 0;
for some R, and then that
i+1 (1)
h1r+1;iH(Dr;q=(2r+1)) ^ X = 0;
for all r R, i r, and q 0.
First we show that HDR+1 ^ X(1) = 0. By Lemma 3.3.1, it is equivalent to
show that j ^ f(1): S0 !HDR+1 ^ X(1) is zero. Let R go to infinity; then by
Lemma 3.3.2, the map
S0 !lim!RHDR+1 ^ X(1)= HF2^ X(1)
is null. Since homotopy commutes with direct limits, then for some R, the map
S0 !HDR+1 ^ X(1) is null.
One uses the same argument to show that
i+1 (1)
h1r+1;iH(Dr;q=(2r+1)) ^ X = 0;
but using the second equality of Lemma 3.3.2, rather than the first. *
*___
3.3.2. Fisomorphism: Proof of Theorem 3.1.3. Now we work on Theo
rem 3.1.3: the map
i jA
ss**S0 ! limQHE**
is_an Fisomorphism. First we construct the coaction of A on limQHE**. We let
Q denote the full subcategory of Q consisting of the conormal quasielementary
quotient Hopf algebras_of_A. Since the maximal quasielementary quotients are
conormal, we see that Q is final in Q; hence we have
limQHE**= lim_QHE**:
3.3. NILPOTENCE VIA QUASIELEMENTARY QUOTIENTS 59
On the right we have an inverse limit of comodules over A; we give this the ind*
*uced
Acomodule structure. So we have (since taking invariants is an inverse limit)
i jA i Aj
limQHE** = lim_QHE**
A
= lim_QHE**
i j
= lim_QHEA==E**:
To finish the proof of Theorem 3.1.3, we show that we can compute HD**up
to Fisomorphism in terms of the coefficient rings HE**for E quasielementary.
Proposition 3.3.4.The natural map
HD**! limQHE**;
is an Fisomorphism, as is the induced map
i jA
HDA**! limQHE** :
See Proposition 4.4.1 for an explicit computation of the ring limQHE**.
Proof of Theorem 3.1.3.This follows immediately from Theorem 3.1.2_and
Proposition 3.3.4. __
Proof of Proposition 3.3.4.We need to show that the restriction map
ae: HD**! limHE**
E2Q
is an Fisomorphism, so we need to show two things: every element in the kernel
of f is nilpotent, and we can lift some 2nth power of any element in the range *
*of f.
By the nilpotence theorem 3.1.6 (or more precisely, by the generalization in
Remark 3.1.7(b)), we know that an element z 2 HD**is nilpotent if and only if
zn 2 kerae for some n.
The second statement follows, almost directly, from a result of Hopkins and
Smith [HSb , Theorem 4.12], stated as Theorem 3.3.5 below, combined with the
fact that the Hopf algebra D is a direct limit of finitedimensional Hopf algeb*
*ras.
We explain.
For each integer r 1, we let B(r) be the following quotient Hopf algebra of
D:
B(r) = D=(1; 2; : :;:r):
Then each B(r) is conormal in D, the kernel K(r) = D 2B(r)F2is finitedimension*
*al,
and we have D = lim!rK(r). Note that this colimit stabilizes in any given degr*
*ee.
Given a quasielementary quotient E of D, we define the quotient F(r) of E
to be the pushout of B(r)  D !E, and we let E(r) be the Hopf algebra kernel
of E ! F(r). In other words, we have the following diagram of Hopf algebra
extensions:
K(r) ! D ! B(r)
?? ? ?
y ?y ?y
E(r) ! E ! F(r):
60 3. QUILLEN STRATIFICATION AND NILPOTENCE
Since E = lim!rE(r), then given an element y 2 HE**, for r sufficiently large,
y is in the image of the inflation map Ext**E(r)(F2; F2) ! Ext**E(F2; F2). No*
*w,
K(r) !E(r) is a quotient map of finitedimensional graded connected commuta
tive Hopf algebras; let resK(r);E(r)denote the induced map on Ext. Then given
y 2 Ext**E(r)(F2; F2), for some n we have ypn 2 im(resK(r);E(r)), by Theorem 3.*
*3.5.
The following commutative diagram finishes the proof of the first statement:
Ext**K(r)(F2;F2)!Ext**D(F2; F2)
?? ?
y ?y
Ext**E(r)(F2;F2)!Ext**E(F2; F2):
Now we want to show that the natural map
HDA2DF2**!lim_QHEA2EF2**
is an Fisomorphism. We treat the A 2D F2coaction on HD** as an Acoaction
with trivial Dcoaction (and similarly for A 2EF2coacting on HE**). So we want
to compare HDA**with limHEA**. We write f and "ffor the maps
f :HD**! lim_QHE**;
"f:HDA**!lim__HEA :
 Q **
The maximal quasielementary quotients are conormal; hence the conormal quasi
elementary quotients are final in the inverse system of all quasielementary qu*
*o
tients; hence f is an Fisomorphism. So it is clear that if x 2 HDA**is in the *
*kernel
of "f, then x is nilpotent. Furthermore, given y 2 limHEA**, we know that y is*
* in
the image of f; hence by Lemma 3.2.6, some 2nth power of y must be_in_the image
of "f. __
We have used the following theorem. This first appeared in [HSb ], and is a
generalization of results in [Wil81 ]. In this theorem, Ext is taken in the cat*
*egory
of modules, rather than comodules; we have actually used the theorem which is
dual to this one.
Theorem 3.3.5 (Theorem 4.12 in [HSbS]).uppose that is an inclusion
of finitedimensional graded connected cocommutative Hopf algebras over a field*
*mk
of characteristic p > 0. For any 2 Ext**(k; k), there is a number m so that p
is in the image of the restriction map Ext**(k; k) !Ext**(k; k).
3.4.Further discussion: nilpotence at odd primes
There are two main obstructions to proving the above results at odd primes.
The first is clear: we do not have a classification of the quasielementary quo*
*tients
of A; hence we cannot prove the quasielementary versions of the theorems. (Equ*
*iv
alently, we do not have an odd primary analogue of Theorem B.2.1(b).) The second
is perhaps more important, since it affects the HD versions of the theorems, and
more surprising, because the corresponding result at the prime 2 seems rather s*
*tan
dard. This is the propertyjthat allows one to prove Lemma 3.2.5: at the prime 2,
the element hnj= [2n] 2 HD(n)**is nilpotent, if j n (see Theorem B.2.1).
Recall from Notation 1.3.9 and Lemma 1.3.10 that primitives y in a Hopf al
gebra B give rise to classes [y] 2 Ext1;yB(k; k). When p is odd, evendimensi*
*onal
primitives y in B give rise to classes fifP0[y] in Ext2;pyB(k; k).
3.5. FURTHER DISCUSSION: MISCELLANY 61
Conjecture 3.4.1.Fixsan odd prime p. Fix a quotient Hopf algebrasB of A,
and suppose that pt is primitive in B. If s t, then bts= fifP0[pt] is nilpotent
in HB**.
As at the prime 2, we define D as follows:
2 p3
D = A=(p1; p2; 3 ; : :)::
One can see from the proofs when p = 2 that the odd prime versions of Theo
rems 3.1.2 and 3.1.5 would follow from this conjecture. (One has to make a slig*
*ht
change in Lemma 3.2.4_as in the proof of Lemma 1.3.10, there are two related
short exact sequences of comodules, leading to a cofibration in which the conne*
*ct
ing homomorphism is a map of homological degree 2. Other than that, everything
goes through as written.)
We discuss the status of Conjecture 3.4.1 in Appendix B.3.
3.5. Further discussion: miscellany
The nilpotence theorem [DHS88 ] in stable homotopy theory has farreaching
implications for the global structure of the stable homotopy category_see [Hop8*
*7 ]
and [Rav92 ], for instance. We develop analogues of some of the structural resu*
*lts
in the next chapter, but there are still many gaps. We discuss those in Section*
*s 4.6
and 4.9 below.
If is a Hopf algebra, then the quasielementary quotients of give the right
homology functors to consider for detecting nilpotence, essentially by the defi*
*nition
of "quasielementary." Indeed, if is commutative and finitedimensional, it is*
* not
too hard to prove an analogue of Theorem 3.1.6, either directly (as in [Pal97])
or via Chouinard's theorem (as in [HPS97 , 9.6.109.6.11]). If, in addition, is
connected, then Theorem 3.3.5 allows one to prove an analogue of Theorem 3.1.3_
see [Pal97]. Actually, by Theorem 3.3.5, some power of every element in the inv*
*erse
limit is invariant, so one gets an Fisomorphism
Ext**(k; k) !limQExt**E(k; k);
where the inverse limit is over Q, the category of quasielementary quotients o*
*f .
Here are some related questions: one already has Quillen stratification for *
*group
algebras; can one prove it, as we do, with vanishing lines or planes? Can one u*
*se our
approach to Quillen stratification for the Steenrod algebra to study the cohomo*
*logy
of other Hopf algebras? Given a Hopf algebra , one should study the pullback
of the quasielementary quotients of , so one might want to assume that those
quotients are somewhat wellbehaved. Even more generally, in an arbitrary stable
homotopy category one could consider a subcategory of appropriately chosen ring
spectra, and look at the inverse limit of that.
Can one refine the description of ss**S0 as given by Theorem 3.1.2? In parti*
*c
ular, can one describe the nilpotence height of elements in the kernel, or say *
*which
powers of classes in the codomain of
': ss**S0 !HDA2DF2**
lift to the domain?
Since one has a version of Quillen stratification for the Steenrod algebra, *
*one
might look for analogues of other grouptheoretic results. Chouinard's theorem
[Cho76 ] is an example: a kGmodule M is projective if and only if it is projec*
*tive
upon restriction to kE for every elementary abelian subgroup E of G. An analogue
62 3. QUILLEN STRATIFICATION AND NILPOTENCE
in Stable(A) might be: for any spectrum X, X is in loc(A) if and only if HD ^ X*
* is
in loc(A) if and only if HE ^ X is in loc(A) for every quasielementary quotien*
*t E
of A. While one can prove things like this in Stable() for finitedimensional H*
*opf
algebras _see [Pal97]_the infinitedimensionality of A might cause a problem.
One should be able to prove something like this when X is a finite spectrum, but
one would like a version of Chouinard's theorem without any such restrictions.
CHAPTER 4
Periodicity and other applications of the
nilpotence theorems
The nilpotence theorem in ordinary stable homotopy theory [DHS88 ] has a
number of important consequences: the periodicity theorem and the thick subcat
egory theorem of [HSb ] are examples. In this chapter we study applications of *
*our
nilpotence and Quillen stratification theorems_Theorems 3.1.2, 3.1.3, 3.1.5, and
3.1.6.
One of our main results of this chapter is a version of the periodicity theo*
*rem:
if R is a finite ring spectrum, then we produce a number of central nonnilpote*
*nt
elements in ss**R via the "variety of R over D," which is essentially the kerne*
*l of
HD**j :HD**! HD**R. Equivalently, this gives families of central nonnilpotent
selfmaps of any finite spectrum X. This is our analogue of the periodicity the*
*orem
of Hopkins and Smith. We state this precisely in Section 4.1, and we prove it in
Sections 4.2 and 4.3.
Theorem 3.1.2 says that ss**S0 is Fisomorphic to the Ainvariants in HD**; *
*in
Section 4.4 we discuss some examples of invariant elements in HD**.
W In Section 4.5, we show that the objects that detect nilpotence_HD and
HE_have strictly smaller Bousfield classes than that of the sphere. The role
of D (as well as the action of A on D) has led us to a conjectured thick subcat
egory theorem, which we give in Section 4.6. We also discuss a few properties of
"varieties" of spectra over D in this section.
In Section 4.7 we construct finite spectra (analogues of generalized Toda V *
*(n)'s)
with vanishing lines of various slopes, using Theorem 2.4.3, and we examine some
properties of these spectra. These will be used in studying chromatic phenomena
in Chapter 5. We end the chapter with two sections of miscellany_one on slope
supports of finite spectra, and a very brief section with a few additional ques*
*tions
and remarks.
As in the previous chapter, we work at the prime 2 (unless otherwise stated).
4.1.The periodicity theorem
We start this chapter by giving our version of the periodicity theorem. This*
* is
a weak analogue of Theorem 3.1.2, with nontrivial coefficients.
The following definition was motivated by work in modular group represen
tation theory of Alperin, Benson, Carlson, and the rest of the group theoretic
alphabet.
Definition 4.1.1.Given a finite spectrum X, we define the ideal of X, I(X),
to be the radical of i j
kerHD**^X![X; HD ^ X]**:
63
64 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
Since X is finite, this is the same as the radical of the annihilator ideal *
*in HD**
of HD**X via the action by composition (see Proposition 4.6.2). The ideal I(X) *
*is
also invariant under the Acoaction, so that the Acoaction on HD**induces one
on HD**=I(X).
Definition 4.1.2.Given an element y 2 Ext**D(F2; F2) = HD**and an object
X in Stable(B), a map z :X ! X is a ymap if HD**z :HD**X ! HD**X is
multiplication by yn for some n.
Theorem 4.1.3 (Periodicity theorem).Let X be a finite spectrum. For every
i jA
y 2 HD**=I(X) ;
X has a ymap which is central in the ring [X; X]**.
(There is an equivalent statement involving elements in the homotopy of a fi*
*nite
ring spectrum R, which we give as Theorem 4.2.2 below.)
We conjecture that (HD**=I(X))Ais Fisomorphic to the center of [X; X]**.
See Section 4.6 for this and related ideas.
As an application, we have Corollary 4.1.5 below, which first appeared as
[Pal96b, Theorem 4.1]. We need a bit of notation to state it.
Definition 4.1.4.Let Slopesdenote the set of slopes of A (Notation 2.2.4).
Given a spectrum X, we define its slope support to be the set
{n  Z(n)**X 6= 0} Slopes:
Since we are working at the prime 2, we have a bijection
Slopes! {(t; s)  t > s 0};
s
2t! (t; s):
Let Slopes0denote the righthand side of this bijection. We say that a subset T*
* of
Slopes0is admissible if T satisfies the following conditions:
(t; s) 2 T ) (t + 1; s) 2 T;
(t; s) 2 T ) (t + 1; s  1) 2 T;
card(Slopes0\ T) < 1:
(We call such sets "admissible" because those are the possible slope supports
of finite spectra_see Section 4.8 and [Pal96b].)
Recall from Notations 2.2.4 and 2.4.1 that for each slope n, there is a spec*
*trum
Z(n) and a nonnilpotent element un 2 Z(n)**. There is a ring map HD** !
Z(n)**, and we show in Section 4.4 that some power of un lifts to HD**. Hence we
can consider the existence of unmaps,a la Definition 4.1.2.
Corollary 4.1.5.Let X be a finite spectrum and let T Slopesbe the slope
support of X. If n 2 T and T \ {n} is admissible (viewed as a subset of Slopes0*
*),
then there is a (nonnilpotent) unmap in [X; X]**.
Proof. One can see that I(X) contains {ht;s (t; s) 2 T Slopes0}, and_
hence un is invariant in HD**=I(X). __
4.2. PROPERTIES OF yMAPS 65
4.2.Properties of ymaps
In this section we lay the groundwork for proving Theorem 4.1.3. In particul*
*ar,
we study analogues of vnmaps [Hop87 , HSb ]. In ordinary stable homotopy the
ory, vnmaps are defined via the Morava Ktheories_these are field spectra, and
hence have various convenient properties, such as K"unneth isomorphisms. Our an*
*a
logues are defined via HD, and hence are not quite as easy to work with. Noneth*
*e
less, our versions of vnmaps, called "ymaps," share many of the same properti*
*es as
vnmaps in ordinary stable homotopy theory; in particular, we show in this sect*
*ion
that the property of having a ymap is generic in our setting.
Fix a quotient Hopf algebra B of A which maps onto D. We work in the
category Stable(B). We start by expanding Definition 4.1.2 a bit:
Definition 4.2.1.Fix y 2 Ext**D(Fp; Fp) = HD**, and let X be an object
in Stable(B). A map z :X ! X is a ymap if HD**z :HD**X ! HD**X is
multiplication by yn for some n. Similarly, if R is a ring object in Stable(B),*
* then
an element ff 2 ss**R is a yelement if for some n, we have HD**ff = yn as maps
HD**! HD**R.
Note that the set of ymaps from X to itself is in bijection with the set of
yelements in the ring spectrum X ^ DX, by SpanierWhitehead duality. So the
following is equivalent to Theorem 4.1.3.
Theorem 4.2.2 (Periodicity theorem for ringpspectra).Let_R be a finite ring
spectrum with unit map j :S0 !R. Let I = kerHD**j. For every
i jA
y 2 HD**=I ;
there is a yelement which is central in ss**R.
The main tool for proving Theorems 4.1.3 and 4.2.2 is the following. See Def*
*i
nition 1.4.7 for the definition of "thick subcategory."
Theorem 4.2.3.Suppose that B is a quotient Hopf algebra of A with A i
B i D. Fix y 2 HD**. The full subcategory C consisting of finite objects of
Stable(B) having a ymap is thick.
In other words, the property of having a ymap is generic.
The proof is a simple modification of the proof in [HSb ] that having a vnm*
*ap
is a generic property. We devote most of this section to the details. We start *
*with
a variant of the notion of ymap, and a general lemma.
Suppose that we have B i D. Since a changeofcoalgebras isomorphism (as
in Lemma 1.3.4) gives
Ext**B(M; (B 2D Fp) M) ~=Ext**D(M; M);
it is sometimes useful to consider [X; HD ^ X]**. Note that the ring HD** =
[S0; HD]**acts on this, via the smash product. We have the following.
Definition 4.2.4.Fix y = HD**, and let X be an object in Stable(B). Write
j :S0 !HD for the unit map. A map z :X ! X is a strong ymap if j ^ z =
yn ^ 1 2 [X; HD ^ X]**for some n.
For instance, the map produced in Theorem 2.4.3 is a strong ymap (where
y = un), and hence a ymap by the following lemma.
66 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
Lemma 4.2.5.Let X be an object in Stable(B). If z :X ! X is a strong
ymap, then it is a ymap.
Proof. Since j is the unit map of the ring spectrum HD, then the following
composite equals 1 ^ z:
HD ^ X 1^j^z!HD ^ HD ^ X ^1!HD ^ X:
By assumption, 1 ^ j ^ z = 1 ^ yn ^ 1, so this composite also equals yn ^ 1. We
compute the induced map on ss**by
n^1
S0 !HD ^ X y!HD ^ X:
Hence z is a ymap. ___
We move on to the proof of Theorem 4.2.3. Here is an easy lemma.
Lemma 4.2.6.Let p be a prime. Fix elements y1 andny2nof an Fpalgebra. If
y1 and y2 commute and y1 y2 is nilpotent, then yp1= yp2 for some n 0.
For the remainder of the section, we consider a fixed element y of HD**. We
work at the prime 2.
Lemma 4.2.7.Let R be a ring spectrum in Stable(B), and fix yelements ff; fi*
* 2
ss**R. If ff and fi commute, then there exist positive integers i and j so that*
* ffi= fij.
Proof. We may assume that HD**(ff  fi) = 0 by raising ff and fi to suitable
powers; hence fffi is nilpotent. Since ff and fi commute, then Lemma_4.2.6 fin*
*ishes
the proof. __
Lemma 4.2.8.Let R be a finite ring object in Stable(B), and fix a yelement
ff 2 ss**R. For some i > 0, the element ffiis central in ss**R.
Proof. Let `(ff); r(ff) 2 End(ss**R) denote left and right multiplication by*
* ff,
respectively. More precisely, `(ff) is induced by the following selfmap of R:
R ff^1!R ^ R !R;
and similarly for r(ff). Since HD**ff is central in HD**R, then `(ff)  r(ff) m*
*aps
to zero in End(HD**R), and so is nilpotent by Theorem 3.1.5. By Lemma 4.2.6,_
we conclude that ff is central in ss**R. __
Corollary 4.2.9.Let R be a finite ring spectrum in Stable(B). For any y
elements ff; fi 2 ss**R, there exist positive integers i and j so that ffi= fij.
Corollary 4.2.10.Let X be a finite spectrum in Stable(B), and let f and g
be two ymaps of X. Then fi = gj for some positive integers i and j.
Corollary 4.2.11.Suppose that X1 and X2 have ymaps y1 and y2. Then
there are positive integers i and j so that for every Z and every f : Z ^ X1 !*
*X2,
the following diagram commutes:
Z ^ X1 ! X2
? f ?
1^yi1?y ?yyj2
Z ^ X1 ! X2
f
4.3. THE PROOF OF THE PERIODICITY THEOREM 67
Proof. DX1 ^ X2 has two ymaps: Dy1^ 1 and 1 ^ y2. Now we apply Corol_
lary 4.2.10 and SpanierWhitehead duality. __
Proof of Theorem 4.2.3.If Y has a (central) ymap gY and if X is a retract
of Y , then the induced selfmap of X
X !Y gY!Y ! X
is easily seen to be a ymap.
Suppose that X1 ! X2 ! X3 is a cofibration, and that X1 and X2 have
ymaps y1 and y2, respectively. We may assume that HD**y1 and HD**y2 both
are multiplication by yn. By Corollary 4.2.11, we can find a map y3 so that this
diagram commutes:
X1 ! X2 ! X3
?? ? ?
yy1 ?yy2 ?yy3
X1 ! X2 ! X3
We claim that some power of y3 is a ymap. So we compare HD**(y3) and yn in
End(HD**X3). We have the following commutative diagram:
HD ^ X2 ! HD ^ X3 ! HD ^ X1:
? ? ?
1^y2yn^1?y ?y1^y3yn^1 ?y1^y1yn^1
HD ^ X2 ! HD ^ X3 ! HD ^ X1
Since y1 and y2 are ymaps, the left and righthand vertical maps induce zero *
*on
ss**, so by a simple diagram chase, one can see that ss**(1^y3yn^1)2_= 0. Hence
y23is a ymap. __
4.3.The proof of the periodicity theorem
In this section we prove Theorem 4.1.3. Fix a finite object X in Stable(A). *
*For
each element y 2 (HD*=I(X))A, we want to show that X has a ymap (Defini
tion 4.1.2).
The basic pattern of the proof is the same as that of Theorem 3.1.2: we indu*
*c
tively work our way from D to A via the Hopf algebras D(n) (defined in Section *
*3.2).
As in Lemma 3.2.1, since X is finite, then [X; HD ^ X]** is the colimit of
[X; HD(n)^X]**. So for some n sufficiently large, the element j^y 2 [X; HD^X]**
lifts to a strong ymap in [X; HD(n) ^ X]**= [X; X]D(n)**. By Lemma 4.2.5, this
gives a ymap in [X; X]D(n)**, and by Lemma 4.2.8, we may assume that this ymap
is central. This starts the induction.
Now, assume that X has a ymap when viewed as an object in Stable(D(n)).
We want to show that X still has a ymap, but when viewed as an object in
Stable(D(n  1)). In the latter category, for any object Y we have the Adams
spectral sequence based on HD(n):
Exts;uD(n1)2D(n)F2(F2; Extt;*D(n)(F2;)Y))Exts+t;uD(n1)(F2; Y ):
68 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
In particular, if we let Wj be defined as in Lemmas 3.2.3 and 3.2.4, then we ha*
*ve
(4.3.1) Exts;uD(n1)2D(n)F2(F2; Extt;*D(n)(F2; X ^ Wj^ DX ^)DWj)
) Exts+t;uD(n1)(F2; X ^ Wj^ DX ^ DWj)
= [X ^ Wj; X ^ Wj]D(n1)**:
Lemma 3.2.5 tells us that thick(S0) = thick(Wj) for any j n, so thick(X) =
thick(X ^ Wj). Hence by Theorem 4.2.3, it suffices to show that X ^ Wj has a
ymap for some j.
The idea is that for j sufficiently large, the endomorphisms of X ^ Wj over
D(n  1) should be more or less the same as the endomorphisms of X over D(n).
Since X has a ymap over D(n), then X ^ Wj should have one over D(n  1), and
hence X should have one, by genericity.
The details are as follows. Since X is finite, then there is a number a so t*
*hat
Exts;uD(n)(F2; X ^ DX) is zero when ua < s. The comodule Wjis nonzero between
degrees 0 and 2jn, so
Exts;uD(n)(F2; X ^ Wj^ DX ^ DWj)
is zero when u  (a  2jn) < s. By Lemmas 3.2.5 and 3.2.3, then, we see that *
*the
E2term for the Adams spectral sequence (4.3.1)has the following vanishing plan*
*e:
Es;t;u2(X ^ Wj^ DX ^ DWj ) = 0 when 2jns + t  u  (2jn  a) > 0:
The inductive hypothesis tells us that we have a ymap in Ext**D(n)(X; X). We u*
*se
"yto denote this element, as well as its image in Ext**D(n)(X ^ Wj; X ^ Wj). No*
*w,
Ext**D(F2; F2)=I(X) !Ext**D(n)(X ^ Wj; X ^ Wj)
is a map of D(n1) 2D(n)F2comodules, so since y is assumed to be in the invari*
*ants
of Ext**D(F2; F2)=I(X), then "yis invariant under the D(n  1) 2D(n)F2coaction.
Hence "yrepresents an element at the E2term of the spectral sequence. We claim
that, when j is large enough, "yis a permanent cycle.
Suppose that "y2 Extp;qD(n)(X; X); then it gives a class in E0;p;q2. The rth
differential on this class would lie in Er;pr+1;qr. So we only have to check t*
*hat for
all r 2, this group is above the vanishing plane, and hence zero. We check our
inequality:
?
2jnr + (p  r + 1)  q  (2jn  a) > 0:
Whatever p, q, and a are, we can choose j large enough so that this holds for
all r 2. Hence "yis a permanent cycle in the spectral sequence for X ^ Wj; it
obviously cannot support a differential, so it survives to give a nonzero class*
* at E1 .
Since the resulting selfmap of X over D(n  1) restricts to the ymap "yover D*
*(n),
then one can check that it is a ymap over D(n  1). This completes the inducti*
*ve
step, and with it, the proof of Theorem 4.1.3.
Remark 4.3.2.We have used the nilpotence part of the "Quillen stratification"
theorem 3.1.2 (i.e., we have used Theorem 3.1.5) in the proof of Theorem 4.2.3,*
* and
hence in the proof of the periodicity theorem 4.1.3. We have not used the other
part of Theorem 3.1.2_that some power of any invariant element in HD**lifts to
ss**S0. Indeed, this follows from the periodicity theorem, so it gives us an al*
*ternate
proof of Theorem 3.1.2.
4.4. COMPUTATION OF SOME INVARIANTS IN HD** 69
4.4.Computation of some invariants in HD**
Theorem 3.1.3 gives an Fisomorphism
ss**S0 !(limQHE**)A:
We compute limHE** in Proposition 4.4.1 below; in Proposition 4.4.4 we give a
formula for the coaction of A on this inverse limit, and then we give a few exa*
*mples
of invariant elements.
The maximal quasielementary quotients of A when p = 2 are the Hopf algebras
E(m), m 0. We recall from Corollary 2.1.8 and Proposition 2.1.9 their definiti*
*on
and the computation of their coefficient rings:
m+1 2m+1 2m+1
E(m) = A=(1; : :;:m ; 2m+1; m+2 ; m+3 ; : :):;
HE(m)**= F2[hts t m + 1; s m]:
The bidegrees of the polynomial generators are given by hts = (1; 2st). Sin*
*ce it
is easy to see the effects of the maps in Q on the coefficients, we immediately*
* have
the following.
Proposition 4.4.1.There is an isomorphism
~=
limHE**! F2[hts s < t] = (htshvu  u t);
where hts = (1; 2st).
Proposition 3.3.4 gives us an Fisomorphism HD**! limHE**, which we can
compose with this isomorphism; hence, to compute HD** up to Fisomorphism,
one does the following:
oone takes the coefficient rings HE**, as E ranges over the maximal quasi
elementary quotients of A,
otensors them together,
oidentifies polynomial generators from different E's if they have the same
name, and
odivides out by the product of two polynomial generators if they do not come
from the same E.
We want to describe the coaction of A on limHE**. The coaction of A 2E(m)F2
on HE(m)** is determined by the coaction on the polynomial generators hts.
Proposition 4.4.2.Fix m 0. Let O: A ! A denote the conjugation map
of A, and for n 1 let in = O(n). Let i0 = 1. Under the coaction map
HE(m)**! (A 2E(m)F2) HE(m)**;
we have
msXtjX s i+j+s
hts7! i2j2tij hi;j+s:
j=0i=m+1
(The ij part comes, essentially, from the right coaction of A on itself, whi*
*le the
tijcomes from the left coaction.)
Proof. (We assume that t > m + 1 and that s < m; the special cases when
t = m + 1 or s = m are even easier to deal with.) First we find allnof the term*
*s in
the coaction htsof the form ahijfor a 2 A primitive, i.e., a = 21for some n. For
70 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
degree reasons, the only possible such terms are 2s1ht1;s+1and 2s+t11ht1;s.
By computations as in the proofs of Lemmas A.3 and A.5 of [Pal96b], we see that
both of these terms do appear; that is, we can see that
s 2s+t1
hts7! 1 hts+ i21 ht1;s+1+ 1 ht1;s+ other terms;
where the "other terms" are of the form b hij, with b 2 A nonprimitive. So
the formula given in the proposition is "correct on the primitives"; once we ha*
*ve
verified that the formula is coassociative, we will have finished the proof. *
*This
verification is a straightforward (although slightly messy) computation; it co*
*uld
be left to the diligent reader, but due to lack of space constraints,_we includ*
*e it in
Lemma 4.4.3 below. __
Lemma 4.4.3.The formula
msXtjX s i+j+s
hts7! i2j2tij hi;j+s:
j=0i=m+1
defines a coassociative coaction of A on HE(m)**.
Proof. We write for the coproduct on A, and we write for the coaction
map of A on HE(m)**. We have to verify that ( 1) O = (1 ) O , so we
compute both of these on hts. First, we have ( 1) O (hts):
0 1
msXXtj s i+j+s
( 1) ( (hts)) = ( 1) @ i2j2tij hi;j+sA
j=0i=m+1
msXtjX s i+j+s
= (i2j)(2tij) hi;j+s
j=0i=m+1
msXtjX Xj s s+n! tijXi+j+s+q i+j+s!
= i2n i2jn 2tijq 2q hi;j+s
j=0i=m+1 n=0 q=0
msXtjXXj tijXs i+j+s+q s+n i+j+s
= i2n2tijq i2jn2q hi;j+s:
j=0i=m+1n=0 q=0
Note that we can read off the "coefficient" of ha;b+s: it is
XbtabX s a+b+s+q s+n a+b+s
i2n2tabq i2bn2q :
n=0 q=0
4.4. COMPUTATION OF SOME INVARIANTS IN HD** 71
On the other hand, for (1 ) O (hts) we have
msX tjX s i+j+s
(1 )( (hts)) = (1 )( i2j2tij hi;j+s)
j=0i=m+1
msXtjX s i+j+s
= i2j2tij (hi;j+s)
j=0i=m+1
msXtjX s i+j+s mjsXi`X j+sk+`+j+s
= i2j2tij i2` 2ik` hk;`+j+s
j=0i=m+1 `=0 k=m+1
msXtjXmjsXi`X s i+j+s j+sk+`+j+s
= i2j2tij i2` 2ik` hk;`+j+s:
j=0i=m+1 `=0 k=m+1
So here the coefficient of ha;b+sis (upon setting k = a and ` + j = b, so that j
ranges from 0 to b, and i ranges from a + b  j to t  j):
Xb Xtj s i+j+s j+s a+b+s
i2j2tij i2bj2iab+j:
j=0i=a+bj
Hence this formula is indeed coassociative. ___
Since the E(m)'s are maximal quasielementary quotients, the following is an
immediate corollary.
Proposition 4.4.4.Under the coaction map
limQHE**! A limQHE**;
we have
bs+t1_2cXtjX
s2i+j+s
hts7! i2jtij hi;j+s:
j=0 i=j+s+1
Dualizing, we have this formula for the action of A* on limHE**:
8
>ht1;s+1if s + 1 < t  1 and k = s,
:0 otherwise:
We give a graphical depiction of the (co)action in Figure 4.4.A, in which we in*
*dicate
the coaction by the primitives (i.e., the 2nth powers of 1).
We end the section with a few examples.
Example 4.4.5.Let R denote the ring limHE**.
(a)The element ht;t12 R in bidegree (1; 2t1t) = (1; 2t1(2t 1)) is an
invariant, for t 1. Indeed, we know that h10 lifts to an element of the
same name in ss1;1S0; also h421lifts to an element in ss4;24S0 (the element
known as g or ___see [Zac67]). We do not know which power of ht;t1
survives for t 3.
72  4. APPLICATIONS OF THE NILPOTENCE THEOREMS




 _____.o.e.7
 h43
 @I
 2
 @
 @
 _____oe5 _____.o.e.6
 h32 h42
 @I @I
 1 1
 @ @
 @ @
 _____oe3_____oe4 _____.o.e.5
 h21 h31 h41
 @I @I @I
 0 0 0
 @ @ @
 @ @ @
 _____oe1_____oe2_____oe3 _____.o.e.4
 h10 h20 h30 h40

_________________________________________________________
Figure 4.4.A. Graphical depiction of coaction of A on limHE**.
k
An arrow labeled by k representskan action by Sq2 2 A*, or
equivalently a "coaction" by 21 2 A_in other words, a term of
the form 2k1 (target) in the coaction on the source.
(b)We have some families of invariants. h202 R is not invariant: we have
h207! 1 h20+ 21 h10:
But since h10h21= 0 in R, then hi20hj21is invariant for all i 0 and j
1. It turns out that more of these elements lift to ss**S0 than one might
expect from Theorem 3.1.2: the elements in the "MahowaldTangora wedge"
[MT68 ] are lifts of the elements hi20hj21for all i 0 and j 8. (See
[MPT71 ]; Zachariou [Zac67] first verified this for elements of the form
h2i20h2(i+j)21for i; j 0.) These elements are distributed over a wedge be*
*tween
lines of slope 1_2and 1_5(in the Adams spectral sequence (t  s; s) gradin*
*g).
(c)Similarly, while h30and h31are not invariant, the monomials
{hi30hj31hk32 i 0; j 0; k 1}
are invariant elements. Hence some powers of them lift to ss**S0. We do not
know what powers of them lift, but they will be distributed over a wedge
between lines of slope 1_6and 1_27. Continuing in this pattern, we find th*
*at for
n 1, we have sets of invariant elements
{hi0n0hi1n1:h:i:n1n;n1 i0; : :;:in2 0; in1 1}:
The lifts of these elements lie in a wedge between lines of slope __1_2n2*
*and
____1_____
2n1(2n1)1. Hence the family of elements in the MahowaldTangora wedge
is not a unique phenomenon_we have infinitely many such families, and
when n 3 they give more than a lattice of points in ss**S0.
(d)Margolis, Priddy, and Tangora [MPT71 , p. 46] have found some other non
nilpotent elements, such as x 2 ss10;63S0 and B212 ss10;69S0. These both
4.5. COMPUTATION OF A FEW BOUSFIELD CLASSES 73
come from the invariant
z = h240h321+ h220h221h41+ h230h21h231+ h220h331
in R. While we do not know which power of z lifts to ss**S0, we do know
that B21maps to the product h320h221z, and x maps to h520z. In other words,
we have at least found some elements in the ideal
(z) (R)A
which lift to ss**S0.
(e)Computer calculations have lead us to a few other such "sporadic" invariant
elements (i.e., invariant elements that do not belong to any family_any
family that we know of, anyway):
h820h431+ h830h421+ h1121h31
in bidegree (12; 80), an element in bidegree (9; 104) (a sum of 8 monomial*
*s in
the variables hi;0and hi;1, 2 i 5), and an element in bidegree (13; 104)*
* (a
sum of 12 monomials in the same variables). We do not know what powers
of these elements lift to ss**S0, nor are we aware of any elements in the *
*ideal
that they generate which are in the image of the restriction map from ss***
*S0.
4.5. Computation of a few Bousfield classes
In [HPS97 , Section 5.1] we show that if a spectrum E is Bousfield equivalent
to the sphere, then E detects nilpotence. This is true in any stable homotopy
category, and it is not very deep. While we do not vouch for the depth of the
nilpotence theorems of Chapter 3, we at least point out that they are not examp*
*les
of this generic nilpotence theorem.
Many of the results in this section hold at all primes; some only hold at the
prime 2. Unless otherwise indicated, fix an arbitrary prime p.
W
Theorem 4.5.1.We have > , and when p = 2, > E,
where the wedge is taken over all quasielementary quotients E of A.
The proof is quite similar to that of analogous results in [Rav84 ]. We need*
* a
few lemmas, first.
Lemma 4.5.2.Suppose that B and C are quotient Hopf algebras of A that fit
into a Hopf algebra extension
B 2CFp,! B i C:
(a)Suppose that dimFpB 2CFp= 1. If B and C are conormal quotient Hopf
algebras of A, then [HC; HB]**= 0. Hence HC ^ IHB = 0; hence >
.
(b)Suppose that dimFpB 2CFp< 1. s
(i)If B 2CFp contains some on or some pt with s < t, then >
.
(ii)Otherwise, if p = 2, then = .
Proof. Part (a): The statement that [HC; HB]**= 0 implies the rest of the
lemma, by Proposition 1.6.1, so we only have to verify that. For that verificat*
*ion, we
prove the corresponding statement about A*modules, and let the reader translate
back to Acomodules and to Stable(A).
74 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
Let B* be the dual of B, C* the dual of C, and B*==C* the dual of B 2CFp.
We show that
Ext**A*(A*==B*; A*==C*) ~=Ext**B*(Fp; A*==C*) = 0:
By Lemma 4.5.3 below, as a B*module, A*==C* is a direct sum of copies of B*==C*
**,
so it suffices to show that Ext**B*(Fp; B*==C*) = 0. From the Hopf algebra exte*
*nsion
C* !B* !B*==C*;
we get a spectral sequence (as in Section 1.5)
Ep;q2= ExtpB*==C*(Fp; ExtqC*(Fp; B*==C*)) ) Ext**B*(Fp; B*==C*):
We claim that E**2= 0. Since B*==C* is a trivial C*module, then
Ext**C*(Fp; B*==C*) ~=B*==C* Ext**C*(Fp; Fp)
as B*==C*modules, and hence is free, and hence injective, over B*==C*. So Ep;**
*2=
0 if p > 0, and
M
E0;*2= HomB*==C*(Fp; nffB*==C*):
ff
But Hom B*==C*(Fp; B*==C*) = 0 (see Lemma 4.5.4); this finishes the calculation.
Part (b)(i) follows from Corollary 1.6.2 and induction, using the nonnilpot*
*ence
of the classes hts(when s < t), bts(when s < t), and vn. Part (b)(ii) is simila*
*r,_but
uses the nilpotence of htswhen s t: see Theorem B.2.1. __
We have used the following two lemmas in the proof of Lemma 4.5.2.
Lemma 4.5.3.Suppose B* and C* are normal subHopf algebras of A*, with
C* B*. Then as a B*module, A*==C* is a direct sum of suspensions of B*==C*.
Proof. We have isomorphisms of B*modules (with B* acting on the left):
A*C* Fp ~= A*B* B* C* Fp
~= A*==B* B*==C*:
Now by normality, A*==B* is a trivial B*module, so this tensor product is a di*
*rect_
sum of copies of B*==C*, indexed by a vector space basis of A*==B*. __
Lemma 4.5.4.Suppose B* and C* are normal subHopf algebras of A*, with
C* B*. Then
ExtsB*==C*(Fp; B*==C*) = 0
for all s > 0. If dimFpB*==C* = 1, then
HomB*==C*(Fp; B*==C*) = 0:
Proof. B*==C* is selfinjective, so the Ext group in question is zero if s >
0. So we need to show that Hom B*==C*(Fp; B*==C*) = 0, if B*==C* is infinite
dimensional. An element of this Hom group corresponds to an element x 2 B*==C*
which supports no operations by elements of B*==C*; we want to show that any
such x must be zero. Fix such an x, and assume that x 6= 0.
Using the classification of (normal) subHopf algebras of A* in Theorem 2.1.*
*1,
we see that there are two possibilities: either (1) B*==C* contains Pst's for a*
*rbi
trarily large values of t, or (2) for some fixed t, B*==C* contains Pstfor all *
*s 0.
(By "B*==C* contains Pst," we mean that Pstis in B* and not in C*, so that it
represents a nonzero class in the quotient.) In case (1), we choose Pstin B*==C*
4.5. COMPUTATION OF A FEW BOUSFIELD CLASSES 75
so that 2t> x. Then it is an easy exercise in multiplication with the Milnor *
*basis
to show that xPst6= 0. In case (2), we argue similarly to show that Pstx_6=_0 f*
*or
s 0. We leave the details to the reader. __
(Perhaps Ext**B(k; B) should be zero for any graded connected Hopf algebra B
over a field k, if B is nonzero in infinitely many degrees. We have not been ab*
*le to
show this, though.)
Here are some consequences of Lemma 4.5.2(a).
Corollary 4.5.5.[HD; S0]**= 0.
Proof. Apply part (a) of the lemma with B = A and C = D. ___
Corollary 4.5.6.Let p = 2. Then [HE; HD]**= 0 for any quasielementary
quotient E of A.
Proof. This follows immediately from Lemma 4.5.2(a), since the maximal
quasielementary quotient Hopf algebras of A are conormal quotients_of both A
and D. __
Corollary 4.5.7.[A; HD]**= 0. Hence if X is a finite object in loc(A), then
X = 0.
Proof. For the first statement, we apply the lemma with B = D and C = Fp.
This then implies that [Y; HD]** = 0 for all Y in loc(A). If X is finite with
SpanierWhitehead dual DX, then [X; HD]**= HD**(DX). So it suffices to show
that if X is finite and nontrivial, then HD**X 6= 0. Well, HD**X = 0 if and only
if X is contractible when viewed as a cochain complex of comodules over D, in
which case the homology of X must be zero. On the other hand, by the Hurewicz *
* __
Theorem 1.4.4, if X is finite and nontrivial, then it has nonzero homology. *
* __
Proof of Theorem 4.5.1.It is clear (by Proposition 1.6.1(c), for instance)
that we have _
E quasi
elem.
(where the second inequality is only known to be valid when p = 2). By Corol
lary 4.5.5 and Proposition 1.6.1(f), we see that > ; in particular, HD*
* ^
IS0 = 0. Hence the first inequality is strict.
When p = 2, by Corollary 4.5.6 and Proposition 1.6.1(c)(e), we see that
HD ^ IHD 6= 0, while HE ^ IHD = 0 for all quasielementary E. Hence the_second
inequality is strict. __
W
Remark 4.5.8.Let p = 2 and let X = HE. One can in fact show a stronger
result_that X has no complement in HD. Suppose otherwise: suppose that there
were an object G so that = _ and 0 = X ^ G. Smashing the former
equality with IHD gives
= _ ;
But HE ^ IHD = 0 for all quasielementary E by Corollary 4.5.6 and Proposi
tion 1.6.1(b), so we have
= :
76 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
Also, by Proposition 1.6.1(a), so we have
= = <0>:
But 6= 0.
4.6. Ideals and thick subcategories
Let X be a finite spectrum; as in Section 4.1, we define the ideal I(X) HD**
to be the radical of the kernel of
HD**^X![X; HD ^ X]**:
The periodicity theorem 4.1.3 tells us that for any element y in (HD**=I(X))A, *
*we
can lift some power of y to a central element in [X; X]**; therefore, the ideal*
* I(X)
is worth studying. In this section, we establish some properties of I(X)_finite
generation and invariance_and we examine a possible relation with a classificat*
*ion
of thick subcategories of finite objects in Stable(A).
4.6.1. Ideals. Let p = 2.
We can define the ideal I(X) for any object X in Stable(D); as we impose
conditions on X_first finiteness, then X being defined over A rather than D_we
find more properties of I(X). One of those properties is invariance:
Definition 4.6.1.Suppose that R is a commutative comodule algebra over A
(i.e., a commutative algebra and a left Acomodule, compatibly); we write for
the comodule structure map. We say that an ideal IPof R is invariant under the
Acoaction if for all x 2 I, (x) is of the form aj xj 2 A R, where each xj
lies in I.
Proposition 4.6.2. (a)Let X be a finite object of Stable(D). Then
p_______________________ p ______________
ker(HD**! [X; HD ^ X]**)= annHD**(HD**X):
(b)Let X be a finite object of Stable(D). Then I(X) is a finitely generated i*
*deal.
(c)Let X be a finite object of Stable(A). Then I(X) is invariant under the
Acoaction.
The proof of part (a) is based on similar work in [Ben91b , Section 5.7].
Proof. Part (a): Let
p _______________________
I = ker(HD**! [X; HD ^ X]**);
p ______________
J = annHD**(HD**X):
The Yoneda action of HD** on HD**X factors through the composition action
(since HD**is commutative). Hence I J. On the other hand, since X is finite,
then it is in thick(S0). So any element of HD** that annihilates [S0; HD ^ X]**
will also annihilate [X; HD ^ X]**. Hence J I.
Part (b) [Sketch of proof]: To see that I(X) is finitely generated, note that
since X is finite, then "most" of D acts trivially on X: we let B(n) denote the
following quotient Hopf algebra of D:
n+1 2n+2
B(n) = D=(1; 2; : :;:n) = F2[n+1; n+2; : :]:=(2n+1; n+2 ; : :)::
4.6. IDEALS AND THICK SUBCATEGORIES 77
Then B(n) should "act trivially on X" for n large enough. More precisely, there*
* is
an AtiyahHirzebruch spectral sequence with
E2 = HFp**X HB(n)**) HB(n)**X:
Since X is finite, then its homology HFp**X is bounded. So if n is large enough,
then for degree reasons, the elements in the image of the edge homomorphism
HFp**X !E2 = HFp**X HB(n)00
are all permanent cycles. This is a spectral sequence of modules over HB(n)**, *
*so
everything must be a permanent cycle, and we find that
HB(n)**X ~=HFp**X HB(n)**:
Consider the ring map HD**! HB(n)**. The annihilator in HD**of HD**X is
contained in the annihilator in HD**of HB(n)**X. Since
annB(n)**(HB(n)**X) = (0);
then I(X) is contained in the radical of the kernel of HD**! HB(n)**. We can
calculate this kernel, up to radical, by imitating the arguments in Section 4.4*
*; we
find that we have an Fisomorphism
HB(n)**!F2[hts s < t; n < t] = (htshvu  u t);
and hence the radical of the kernel of HD**! HB(n)**is equal to
K = (hts s < t n):
If we view K as an ideal of the Noetherian ring
F2[hts s < t n]=(htshvu  u t) HD**;
then we see that not only is K finitely generated, but so is any subideal of it.
Part (c): Now we show that I(X) is invariant under the Acoaction. By Re
mark 1.3.8, both HD**and HD**X are A 2D Fpcomodules, and one can check that
the action map
HD** HD**X !HD**X
is a map of A 2D Fpcomodules.P Suppose that y 2 I(X), and that under the
A 2D Fpcoaction, y maps to iai yi. (We may assume that the ai's are lin
early independent elements of A.) We want to show that each yi is in I(X); i.e.,
that some power of yiannihilates HD**X.k k
Fix x 2 HD**X. If we assume that y2 x = 0, then we claim that y2ix = 0 for
each i. Suppose that under the A 2D Fpcoaction, we have
nX
(4.6.3) x 7! 1 x + bj xj:
j=1
We prove, by induction on n, that y2kix = 0 for all i. When n = 0 (i.e., when x*
* is
primitive in HD**X), then we have
HD**X !(A 2D Fp) HD**X;
k X 2k X 2k 2k
0 = y2 x7! ( ai yi) (1 x) = ai yi x:
i i
Since the ai's are linearly independent, we conclude that y2kix = 0 for all i.
78 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
Suppose that the 2kth power of each yiannihilates every element of HD**which
has at most n terms in its diagonal, and fix x with diagonal as in (4.6.3). The*
*n by
coassociativity, each xj has n terms or fewer in its diagonal, so we have
k X 2k X
0 = y2 x7! ( ai yi) (1 x + bj xj)
i j
X k k X k k
= (a2i y2ix + a2ibj y2ixj)
i j
X k k
= a2i y2ix:
i
Hence y2kix = 0 for each i. ___
4.6.2. A thick subcategory conjecture. Since this subsection consists pri
marily of conjectures, we may as well let p be an arbitrary prime (although we *
*have
more evidence for the conjectures when p = 2).
Theorems 3.1.2 and 4.1.3 provide support for several conjectures about the
"global structure" of the category Stable(A). For instance, we have the followi*
*ng
suggested analogue of the result of Hopkins and Smith [HSb , Theorem 11], in
which they identify the center of [X; X]* up to Fisomorphism, for any finite p
local spectrum X. Given a ring R, we let Z(R) denote the center of R.
Conjecture 4.6.4.For any finite spectrum X, there is an Fisomorphism
Z[X; X]**! (HD**=I(X))A:
We should point out there is not even an obvious map between these two rings.
Theorem 4.1.3 also suggests a conjectured classification of thick subcategor*
*ies of
finite spectra in Stable(A); we spend most of this section discussing this conj*
*ecture
and related ideas.
Conjecture 4.6.5.The thick subcategories of finite spectra in Stable(A) are *
*in
onetoone correspondence with the finitely generated radical ideals of HD**whi*
*ch
are invariant under the coaction of A 2D Fp.
As above, given a finite spectrum X, we let I(X) denote the radical of the k*
*ernel
of HD**! [X; HD^X]**. The conjectured bijection should send an invariant ideal
I to the full subcategory D(I) with objects
{X finite I(X) I}:
This is clearly a thick subcategory. The other arrow in the bijection should se*
*nd a
thick subcategory D of the finite objects in Stable(A) to the ideal I(D), defin*
*ed by
"
I(D) = I(X):
X2obD
This is a finitely generated radical invariant ideal by Proposition 4.6.2.
Example 4.6.6.Here is a bit more evidence for Conjecture 4.6.5.
(a)By Example 4.4.5(a), we have maps
h10:S1;1!S0;
h421:S4;24!S0:
4.6. IDEALS AND THICK SUBCATEGORIES 79
We let S0=h10 and S0=h421denote the cofibers of these. One can easily
compute the ideals of these cofibers:
p ____
I(S0=h10) = (h10);
p ____
I(S0=h421) = (h21):
Since h10h21= 0 in HD**, then Conjecture 4.6.5 would tell us that
thick(S0=h10; S0=h421) = thick(S0):
Let C = thick(S0=h10; S0=h421). One can show directly that S0 is contained
in C, using the octahedral axiom: the cofiber of the map h10h421:S5;25!S0
fits into a cofibration with S0=h10and S0=h421, and hence is in C. But sin*
*ce
h10h421is zero, then this cofiber is just S0 _ S4;25; hence S0 is in C.
(b)By Example 4.4.5(b), we have nonnilpotent selfmaps of the sphere spec
trum called hi20hj21, for certain exponentspi_and j. If i and j are both
positive, then I(S0=(hi20hj21)) = (h20h21), so Conjecture 4.6.5 would im
ply that thick(S0=(hi20hj21)) is independent of i and j. Arguing as in part
(a), one can see that this is true.
(c)Lastly, we point out that thick(S0) 6= thick(S0=(hi20hj21)), even though
the two spectra S0 and S0=(hi20hj21) have the same slope supports (Defi
nition 4.1.4). For instance, if we let d0 = h220h221, then S0=d0^ d10S0 =*
* 0,
and hence X ^ d10S0 = 0 for every X in thick(S0=d0).
Here is a sketch of part of the proof of Conjecture 4.6.5.
Conjecture 4.6.7.Given any invariant finitely generated radical ideal I of
HD**, there is a finite spectrum X so that I(X) = I.
Idea of proof.Suppose that I is generated by classes y1, y2, : :,:yn, with
yi 2 HDsi;ti. We order these so that s1 s2 . . .sn; then (y1; : :;:yi) is
invariant for each i n. For each i 0, we define spectra Xiinductivelypso_that
I(Xi) = (y1; : :;:yi). We start by letting X0 = S0; then I(X0 ) = (0). Given
Xi1with I(Xi1) = (y1; : :;:yi1), Theorem 4.1.3 tells us that then Xi1has a
yimap; we let Xi be the cofiber of this map. Clearly I(Xi) (y1; : :;:yi);_if *
*we
could prove equality, we would be done. __
This would show that the composite I 7! D(I) 7! I(D(I)) is the identity.
For the composite D 7! I(D) 7! D(I(D)), one needs to show that every thick
subcategory D is of the form D = D(I) for some invariant ideal I.
One might also conjecture that there is a bijection between the set of local*
*izing
subcategories of Stable(A) and the set of all radical ideals of HD**, but that *
*seems
a bit much to expect without any evidence.
We end this section with one other body of ideas, based on work of Nakano and
the author [NP ] (and this was based, in turn, on work of Friedlander and Parsh*
*all
[FP86 , FP87 ], among others). Let p = 2. We let W be the vector space
W = SpanF2(Pst s < t):
We view W as being an inhomogeneous subvector space of the Steenrod algebra
A*, and we let VD (F2) be the following subset of W:
VD (F2) = {y 2 W  y2 = 0}:
80 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
We do not require that the elements y be homogeneous. One can show (as in
the proof of [NP , 1.7]) that VD (F2) consists precisely of linear combinations*
* of
commuting Pst's_i.e., it is a union of affine spaces, one such space for each m*
*aximal
elementary quotient of A. Therefore it is equal to the prime ideal spectrum of
limQHE**, and hence is homeomorphic to the prime ideal spectrum of HD**. At
odd primes, we define
Wev= SpanFp(Pst s < t);
Wodd= Span(Qn  n 0);
and then let VD (Fp) be the set of all elements y = (y1; y2) in Wev Woddwith
yp1= 0 and y22= 0. We do not have as nice a description of VD (Fp) at odd prime*
*s;
because of this, and for other technical reasons, it might be best to restrict *
*the
following discussion to the case p = 2.
Given an element y 2 VD (Fp), we can construct the yhomology spectrum H(y),
just as we did the Psthomology spectrum in Definition 2.2.1 . Then given a spe*
*c
trum X, we define its rank variety, VD (X), to be the following subset of VD (F*
*p):
VD (X) = {y 2 VD (Fp)  H(y)**X 6= 0}:
We say that a subset V of VD (Fp) is realizable if V = VD (X) for some finite
spectrum X.
Conjecture 4.6.8.Let X be a finite spectrum. The rank variety VD (X) deter
mines the thick subcategory generated by X, and hence the ring of central self*
*maps
of X, up to Fisomorphism. In other words, there is a bijection between the inv*
*ari
ant finitely generated radical ideals of HD**and the realizable subsets of VD (*
*Fp).
This bijection should come about by the following: if X is a finite spec
trum, then there should be a homeomorphism (actually, an "inseparable isogeny"
[Ben91b , p. 172]) between the prime ideal spectrum of HD**=I(X) and VD (X).
4.7.Construction of spectra of specified type
Let p be a prime.
In this section we construct certain objects for later use; these are analog*
*ues
of the "generalized Toda V (n) spectra," as used by MahowaldSadofsky [MS95 ],
among others. Some of the basic ideas are standard; most of the rest are due to
them. A few of the details are different in our setting.
We make heavy use of Notations 2.2.4 and 2.4.1 in this section.
The results of this section follow from Theorem 2.4.3 and Lemma 2.4.7, but
they have the flavor of other results in this chapter; hence we include them he*
*re.
(In particular, the results holds at all primes, not just when p = 2.)
We start by constructing the relevant spectra. See Notation 2.2.4 and 2.4.1 *
*for
the terminology used here.
Proposition 4.7.1.Let p be a prime, and fix a slope n. Let 1 = d1 < d2 <
. .<.dm be the slopes less than n. For any integers k1, : :,:km , there are int*
*egers
j1, : :,:jm with ki jifor each i, so that there is a spectrum F = F(uj1d1; : :;*
*:ujmdm)
satisfying the following.
(a)When n = 1, F = S0.
(b)F is Z(d)**acyclic for d < n, and Z(n)**F 6= 0.
4.7. CONSTRUCTION OF SPECTRA OF SPECIFIED TYPE 81
(c)Hence F has a unmap; call it u. If Z(n)**(u) is multiplication by ujm+1n,
then F(uj1d1; : :;:ujm+1n) is the fiber of u. More precisely, there is a *
*fiber
sequence
jm+1
F(uj1d1; : :;:ujm+1n) !F(uj1d1; : :;:ujmdm) un!jn;njnF(uj1d1; : :;*
*:ujmdm):
(d)F is selfdual, as is its unmap u. That is, for some number q, we have
DF ~=qF, and u maps to itself under the chain of isomorphisms
[F; F] ~=[DF; DF] ~=[qF; qF] ~=[F; F]:
Proof. The statements of (a)(c) indicate how the spectra are constructed.
Starting with S0, one applies Theorem 2.4.3 to find a u1map uj11of it, and one
lets F(uj11) be the fiber. By definition, essentially, uj11induces an isomorphi*
*sm on
Z(1)**, so F(uj11) is Z(1)**acyclic; uj11induces zero on Z(d)**for d > 1, so F*
*(uj11)
is not Z(d)**acyclic for any larger value of d. One proceeds inductively. Th*
*is
proves (a)(c).
Part (d) is also straightforward; it is proved by induction on m. We_leave t*
*he
details to the reader. __
Up to suspension, the object F(uj1d1; : :;:ujmdm) is the analogue in Stable(*
*A)
of Mahowald and Sadofsky's spectrum (in the usual stable homotopy category)
M(pj0; vj11; : :;:vjn1n1). We have used the letter F rather than M since our *
*spectra
are iterated fibers rather than cofibers, as in [MS95 ]. Also, the letter M can*
* get
somewhat overworked; in particular, we want to avoid confusion with the functor
Mnfof Section 5.3.
Now we establish the main properties of the spectra F(uj1d1; : :;:ujmdm).
Theorem 4.7.2.Fix a prime p, a slope n, and other notation as in Proposi
tion 4.7.1, and let F = F(uj1d1; : :;:ujmdm) be as in that result. Then F satis*
*fies the
following properties.
(a)For any finite spectrum W which is Z(d)**acyclic for all d < n, W is in
thick(F).
(b)Hence the Bousfield class of F is independent of the choice of exponents j*
*i.
(c)Let u denote the unmap of F. Suppose that `1, : :,:`m are integers so
that W = F(u`1d1; : :;:u`mdm) exists, and let v denote its unmap as given*
* by
Proposition 4.7.1(b). Then there are integers i and j so that ui^1W = 1F^vj
as selfmaps of F ^ W.
(d)Suppose that `1, : :,:`m are integers so that F(u`1d1; : :;:u`mdm) exists,*
* and so
that ji `ifor each i. Then there is a map
F(uj1d1; : :;:ujmdm) !F(u`1d1; : :;:u`mdm)
commuting with "projection to the top cell"_i.e., the map F(: :):!S0.
Proof. (a): For each i m, let F(Ui) = F(uj1d1; : :;:ujidi). We have cofiber
sequences
ujidi
F(Ui) !F(Ui1) ! F(Ui1):
We claim that 1W ^ujidiis a nilpotent self map of W ^F(Ui1). Once we know this,
then we see that W ^ F(Ui1) is in the thick subcategory generated by W ^ F(Ui)
for each i. By induction, W ^ F(U0) = W is in the thick subcategory generated
82 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
by W ^ F(Um ), which is a subcategory of the thick subcategory generated by
F(Um ) = F.
The claim that 1W ^ujidiis nilpotent follows by application of the vanishing*
* line
theorem 2.3.1: the group
[W ^ F(Ui1); W ^ F(Ui1)]**= ss**(W ^ DW ^ F(Ui1) ^ DF(Ui1))
has a vanishing line of slope n, and 1W ^ ujidiacts along a line of slope di< n.
(b): This follows immediately from (a).
(c): We would like to use the properties of ymaps discussed in Section 4.2,*
* but
these depend on the nilpotence theorem 3.1.5, and hence would force us to work *
*at
the prime 2. With a bit of care, we can avoid this dependence.
By Proposition 4.7.1(b), F ^ W is Z(d)**acyclic for d < n; hence [F ^ W; F ^
W]** has a vanishing line of slope at least n. If the slope is larger than n, t*
*hen
the powers of both u ^ 1W and 1F ^ v would eventually lie above the vanishing
line, and hence would both be zero. So we may assume that the vanishing line has
slope equal to n. By our choice of unmap u, Lemma 2.4.7 tells us that u ^ 1W
is central in a band parallel to the vanishing line; 1F ^ v lies in that band, *
*on the
line of slope n through the origin. Therefore u ^ 1W and 1F ^ v commute. By our
choice of maps u and v, after raising them to powers, we may assume that both of
our unmaps agree when restricted to
[F ^ W; HA(m) ^ F ^ W]**;
for large enough m. But this ring is isomorphic to [F ^ W; F ^ W]**in the bideg*
*ree
of interest; hence our selfmaps agree in [F ^ W; F ^ W]**.
(d): This follows from (c). We prove it by induction on m. When m = 1, then
the following diagram commutes:
uj11 0
F(uj11)!S0 ! S :
flfl ?
fl ?yu`1j11
u`11 0
F(u`11)!S0 ! S
Hence there is an induced map F(uj11) !F(u`11).
Assume that we have a map
F(uj1d1; : :;:ujm1dm1)f!F(u`1d1; : :;:u`m1dm1):
flfl fl
fl flfl
F W
4.7. CONSTRUCTION OF SPECTRA OF SPECIFIED TYPE 83
We abbreviate these spectra as F and W, as indicated. Consider the following
diagram, in which the rows are fiber sequences:
ujmdm
F(ujmdm)!F ! F:
flfl ?
fl ?yu`mjmdm
u`mdm
F(u`mdm)!F ! F
?? ?
yf ?yf
u`mdm
W(u`mdm)!W ! W
Clearly the top right square commutes. We claim that part (c) implies that the
lower right square commutes. Given this, we get maps
F(ujmdm) !F(u`mdm) !W(u`mdm):
The composite is the desired map.
To see that the lower right square commutes, we imitate the proof of Corol
lary 4.2.11. The key is to use SpanierWhitehead duality so that the question is
whether 1 ^ u`mdmand Du`mdm^ 1 agree as selfmaps of the spectrum DF ^ W. By
part (c), it suffices to show that F is selfdual (up to suspension), as is_its*
* unmap
u. But this is Proposition 4.7.1(d). __
We point out that since F = F(uj1d1; : :;:ujmdm) is welldefined up to Bousf*
*ield
class, and since its unmap is essentially unique, then the telescope u1nF is *
*well
defined up to Bousfield class. But the telescope of a unmap of an arbitrary fi*
*nite
spectrum of type n could have a different Bousfield class. For example, at the *
*prime
2, there is a nonnilpotent selfmap of the sphere called d0:
d0: S4;18!S0:
(Under the map ss**S0 ! HD**, the element d0 maps to h220h221_see Exam
ple 4.4.5(b).) Although S0 and S0=d0 are both type 0, they are not Bousfield
equivalent_S0 sees d10S0, while S0=d0 does not. Hence the telescopes h10S0 and
h10(S0=d0) probably have distinct Bousfield classes.
We also point out that this theorem has the following obvious generalization,
at least at the prime 2. See Conjecture 4.6.7 for related work.
Theorem 4.7.3.Let p = 2, and fix a finitely generated invariant radical ideal
I HD**. Write I = (y1; : :;:ym ), and order the generators so that (y1; : :;:y*
*i)
is invariant for each i m. For any integers k1, : :,:km , there are integers j*
*1,
: :,:jm with ki ji for each i, so that there is a spectrum F = F(yj11; : :;:yj*
*mm)
satisfying thepfollowing._
(a)When I = (0), then F = S0.
(b)I is contained in the ideal of F.
(c)Hence if y 2 HD**=I is invariant, then F has a ymap, u.
(d)F is selfdual, as are its ymaps.
(e)For any finite spectrum W with I I(W), then W is in thick(F).
(f)Hence the Bousfield class of F is independent of the choice of exponents *
*ji.
(g)Fix y and u as in (c). For any finite spectrum W and any ymap v on W,
there are integers i and j so that ui^ 1W = 1F ^ vj as selfmaps of F ^ W.
84 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
(h)If `1, : :`:mare integers with `i ji and such that F(y`11; : :;:y`mm) exis*
*ts,
then there is a map F(y`11; : :;:y`mm) !F(yj11; : :;:yjmm).
4.8. Further discussion: slope supports
Let p be any prime. In this section, we discuss slope supports (Definition 4*
*.1.4).
This material was originally introduced in [Pal96b] as an approach to the perio*
*d
icity theorem and thick subcategory conjecture. Since those topics now seem more
closely related to I(X), the ideal of X, slope supports appear to be more perip*
*h
eral. On the other hand, since Psthomology groups do determine vanishing lines
and other useful information, studying slope supports may be worthwhile.
Our main result in this section is Proposition 4.8.1, which gives a classifi*
*cation,
when p = 2, of the possible supports of finite spectra.
Recall from Definition 4.1.4 that Slopesis the set of slopes of A, whereas S*
*lopes0
is the set (
Slopes0= {(t; s)  t > s 0}; p = 2;
{(t; s)  t > s 0} [ {n  np 0};odd.
There is, of course, a (meaningful) bijection between Slopesand Slopes0, with t*
*he
ps 0
slope pt_22 Slopescorresponding to (t; s) 2 Slopes, and when p is odd, on 2
Slopescorresponding to n 2 Slopes0.
As in Definition 4.1.4, we say that the slope support of a spectrum X is the*
* set
supp(X) = {n  Z(n)**X 6= 0} Slopes:
(See Notation 2.2.4 for the definition of Z(n).) We may also view supp(X) as be*
*ing
a subset of Slopes0, using the bijection above. We say that a subset T of Slope*
*s0is
admissible if T satisfies the following conditions:
(i)When p = 2:
(t; s) 2 T ) (t + 1; s) 2 T;
(t; s) 2 T ) (t + 1; s  1) 2 T;
card(Slopes0\ T) < 1:
(ii)When p is odd: the above conditions, as well as:
n 2 T ) n + 1 2 T:
We provide a bit of justification for the term "admissible" in the following.
Proposition 4.8.1.[Pal96b, Prop. 3.10 and Thm. A.1] If T Slopes0is
admissible, then T is the slope support of some finite spectrum. If p = 2, then*
* the
converse holds: if X is any finite spectrum, then suppX is admissible.
We conjecture that the converse is also true when p is odd. We can at least *
*prove
the following: for X finite, if (Qn)**X = 0, then (Qn1)**X = 0 (see [Pal96b,
A.8]). We also note that there are no restrictions on the slope support of an
arbitrary spectrum: the connective Psthomology spectrum psthas slope support
equal to {(s; t)}, so any subset of Slopes0may be realized as the support of a *
*wedge
of pst's (and similarly at odd primes, using the pst's and the qn's).
4.8. FURTHER DISCUSSION: SLOPE SUPPORTS 85







s @
 ________@
 T(t; s)
 _________________________
 T(m)
_________________________ _________________________
t m
Figure 4.8.A. T(t; s) and T(m) as subsets of Slopes0.
Proof. We include a sketch of the proof that every admissible T is the slope
support of some finite spectrum, since the corresponding result in [Pal96b] ass*
*umed
that p = 2. We focus on the case when p is odd; the reader can imitate this pro*
*of
or refer to [Pal96b] for the p = 2 case.
Every admissible T Slopes0can be written as
T = T(t1; s1) \ . .\.T(tn; sn) \ T(m)
for some numbers ti; si; m, where T(t; s) is the largest admissible set not con*
*taining
(t; s)_i.e., it is the complement of
{(t; s); (t  1; s); (t  1; s + 1); (t  2; s); (t  2; s + 1); (t  2; s +*
* 2); : :}:;
and T(m) is the complement of {0; : :;:m}. See Figure 4.8.A. If we can find
sufficiently nice finite spectra X(t; s) and X(m) so that suppX(t; s) = T(t; s)*
* and
suppX(m) = T(m), then we can realize any slope support T by the spectrum
X(t1; s1)^: :^:X(tn; sn)^X(m). (One might worry that although Z(d)**X(ti; si) 6=
0 and Z(d)**X(tj; sj) 6= 0 for some d, one might have Z(d)**X(ti; si) ^ X(tj; s*
*j) =
0. This does not happen for injective resolutions of finite comodules_see [NP ,
1.5]_and it certainly does not happen for the examples we construct in the next
paragraph. This is what we mean by "sufficiently nice.")
We recall from [Mit85 ] that the quotient Hopf algebra
n+1 pn p
P(n) = A=(p1 ; 2 ; : :;:n+1; n+2; n+3; : :;:o0; o1; o2; : :):
has the structure of an Acomodule, extending the P(n)comodule structure. The
same clearly holds for
V (m) = E[o0; : :;:om ]:
We will use P(n) and V (m) to refer to the Hopf algebras, the comodules, or the*
*ir
injective resolutions,sdepending on the context. It is wellknown that H(P(n); *
*Pst)
is zero ifspt 6= 0 in P(n)_see Proposition 2.2.2 or [MP72 ]. On the other
hand, if pt= 0 in P(n), then one can see by examining the Poincare series that
H(P(n); Pst) 6= 0. So Z(d)**P(n) = 0 if and only if yd 6= 0 in P(n), and simila*
*rly
for V (m). Hence as subsets of Slopes0, we have
suppP(n) = T(n + 1; 0);
suppV (m) = T(m):
86 4. APPLICATIONS OF THE NILPOTENCE THEOREMS
Also, there is a "pth power" functor on the category of Acomodules; if M
is evenly graded, then M is defined to be
(
(M)n = Mr n = pr;
0 n not divisiblepby;
with comodule structure M :M !A M given by
M (x) = (F i) O M (x):
Here, F is the Frobenius map on A defined by F(a) = ap, M is the comodule
structure map on M, and i: M !M is the map (which multiplies degrees by p)
that sends y to y. One can readily see that if s < t, then
suppsP(t  1) = T(t; s):
Hence an injective resolution of the comodule
s1P(t1 1) . . .snP(tn  1) V (m)
has slope support
T = T(t1; s1) \ . .\.T(tn; sn) \ T(m):
See [Pal96b, A.1] for the proof of the converse when p = 2. ___
4.9. Further discussion: miscellany
We have already mentioned several conjectures related to global structure of*
* the
category Stable(A) when p = 2_see Section 4.6. As far as working at odd primes,
one has to prove analogues of Theorems 3.1.2 and 3.1.5 first; see Section 3.4 f*
*or a
discussion of those issues.
We presented some preliminary computations of Ainvariants in HD**in Sec
tion 4.4; it would be nice to have further results. Extensive computer calculat*
*ions
could be useful, and obviously it would be nice to find new families of invaria*
*nts.
Along similar lines, we could use more information about invariant ideals of HD*
***_
basic properties, examples, and of course a classification would be helpful.
It is also natural to wonder if one can prove a version of the periodicity t*
*heo
rem 4.1.3 which uses quasielementary quotients of A instead of D.
CHAPTER 5
Chromatic structure
In this chapter we discuss "chromatic" results in Stable(A). We start in Sec
tion 5.1 by discussing Margolis' killing construction [Mar83 , Chapter 21]. Th*
*is
is the analogue, in our setting, of the functor Lfnin the ordinary stable homot*
*opy
category. We give several different constructions of the functor, and we prove *
*var
ious properties (e.g., for X with nice connectivity properties, if X has finite*
* type
homotopy, then so does LfnX). We also define an analogue of the functor Ln, and
we show that Ln 6= Lfnif n > 1, at least at the prime 2.
We have been using the functor H heavily throughout this book; the homotopy
groups of HB, for B a quotient of A, are the cohomology groups of the Hopf alge*
*bra
B. In Section 5.2 we construct bH(), a version of this functor whose homotopy
groups are the Tate cohomology groups of B. The spectra bHA(m)turn out to be
equal to LfnHA(m), for n sufficiently large compared to m; we use this result in
Section 5.3 to prove that the "chromatic tower," the tower
Lf0X  Lf1X  Lf2X  Lf3X  : :;:
converges to X, if X is finite. (This is an extension of a theorem of Margolis
[Mar83 , Theorem 22.1].)
In Section 5.4 we discuss some other questions related to chromatic issues, *
*such
as constructing chromatic towers in different orders, and relating the chromatic
tower construction to the multiple complex construction of Benson and Carlson.
5.1.Margolis' killing construction
In this section we present Margolis' killing construction. Thissis a (smashi*
*ng)
localization functor that kills off Pst and Qnhomology for ptand on of large *
*slope.
We make use of Notation 2.2.4 and 2.4.1 in this section.
Warning. Note that un is an element of either Z(n)1;nor Z(n)2;2n, depending
on the prime and the form of yn_see Proposition 2.2.2. So as to avoid dividing
all of our arguments into cases, we will abuse notation and write uinfor the po*
*wer
of un in Z(n)i;*(and similarly for unmaps: a selfmap uin:X !X has bidegree
(i; in)). Hence when yn = Pstand p is odd, only even powers of un make sense.
We defined the notions of "thick" and "localizing" subcategories in Defini
tion 1.4.7.
Definition 5.1.1.Recall from [Mil92] and [HPS97 , 3.3.3] that given any
thick subcategory C of finite spectra, there is a functor LfC:Stable(A) !Stabl*
*e(A),
called finite localization away from C, with the properties:
(i)LfCis exact; when viewed as a functor LfC:Stable(A) !LfCStable(A) to the
category of LfClocal spectra, it has a right adjoint.
87
88 5. CHROMATIC STRUCTURE
(ii)There is a natural transformation 1 !LfC.
(iii)LfCis idempotent_for any X, the map LfCX ! LfCLfCX induced by the
natural transformation in (ii) is an equivalence.
(iv)LfCis Bousfield localization with respect to the spectrum LfCS0.
(v)For any X, LfCX = X ^ LfCS0.
(vi)For any finite X, LfCX = 0 if and only if X 2 C; for any X, LfCX = 0 if a*
*nd
only if X is in the localizing subcategory generated by C.
Properties (i)(iii) say that LfCis a localization functor [HPS97 , 3.1.1]; *
*prop
erty (v) says that LfCis smashing [HPS97 , 3.3.2]. We write CfCfor the corre
sponding acyclization functor; that is, CfCX is the fiber of X !LfCX. Since Lf*
*Cis
smashing, then CfCX = X ^ CfCS0.
Definition 5.1.2.For any slope n, let LfndenoteWfinite localization away from
the thick subcategory of finite spectra which are dn Z(d)acyclic, and write C*
*fn
for the corresponding acyclization.
In Section 4.7, we constructed a finite spectrum F(uj11; : :;:ujnn) for an a*
*ppro
priately chosen set of exponents j1, : :,:jn, indexed by slopes. We will also w*
*rite
F(Un) for this spectrum. If d is the next largest slope after n, then by constr*
*uction,
F(Un) has a udmap. Let Tel(d) denote its telescope. Theorem 4.7.2 tells us that
the Bousfield class is independent of the exponents j1, : :,:jn; as not*
*ed
after that theorem, the same is true of .
The following is based on Mahowald and Sadofsky's analysis in [MS95 ] of the
functor Lfnin the ordinary stable homotopy category.
Proposition 5.1.3.The functor Lfncan be described in any of the following
ways:
(a)as finite localization away from the thickWsubcategory generated by F(Un);
(b)as Bousfield localization with respect to dn Tel(d);
(c)and as a colimit:_if we let F(ui11; : :;:uinn) !S0 be projection to the t*
*op cell
with cofiber F(ui11; : :;:uinn), then the map X !LfnX is given by
__ i i
X !lim!i1;:::;inX ^ F(u11; : :;:unn):
(The maps in the direct system will be defined in the proof.)
W On the other hand, for n large, Lfnis not Bousfield localization with respec*
*t to
dn Z(d). See Proposition 5.1.10.
W
Proof. By Theorem 4.7.2(a), the thick subcategory of finite dn Z(d)acyclic
spectra is equal to the thick subcategory generated by F(ui11; : :;:uinn) for a*
*ny choice
of exponents i1; : :;:in. This proves part (a).
Part (b) is essentially a Bousfield class computation. Repeated application *
*of
Proposition 1.6.1(a) gives us
(5.1.4) = _ _ . ._.;
as well as pairwise orthogonalityWof the Bousfield classes on the right. We nee*
*d to
show thatWa spectrum X is dn Tel(d)acyclic if and only if it is Lfnacyclic. *
*Assume
that dn Tel(d)**X = 0. By the decomposition of Bousfield classes (5.1.4), we
see that = , so LfnX = 0 if and only if LfnX ^ F(Un) = 0. But
5.1. MARGOLIS' KILLING CONSTRUCTION 89
certainly LfnF(Un) = 0, so since Lfnis smashing,Wthen LfnF(Un) ^ X = 0. To show
the converse, it suffices to show that F(Un) is dn Tel(d)acyclic; this follow*
*s by
the orthogonality of the above Bousfield classes.
To prove (c), we first need to construct the direct system. By "projection to
the top cell," we mean the composite F(Un) ! : ::!F(U1) ! F(U0) = S0.
Using properties of the spectra F(Uk) and their ujmaps (Theorem 4.7.2), we can
choose the ujmaps compatibly; i.e., if we have exponents ij and `j so that ij *
*`j
and F(ui11; : :;:uinn) and F(u`11; : :;:u`nn) are defined, then there is a map
F(ui11; : :;:uinn) !F(u`11; : :;:u`nn)
__ i
which commutes with projection to the top cell. We let F(u11; : :;:uinn) denote
the cofiber of F(ui11; : :;:uinn) ! S0, and we define the spectrum L0nX to be *
*the
following colimit:
__ i
X ! lim!F(u11; : :;:uinn) ^ X:
fl i1;:::;in fl
flfl flfl
0
X g! L0nX
0
For example, if n = 1, then we have S0 g!u11S0, where u11S0 is the mapping
telescope of u1: S0 !S0, and fiber(g0) = F(u11) is the analogue of the mod p1
Moore spectrum.
We need to verify that L0nagrees with Lfn. To do this, we note that L0nis
smashing, and we show that g0:S0 ! L0nS0 is an LfnS0equivalence and that
L0nS0 is LfnS0local. By construction, the fiber of g0is in the localizing subc*
*ategory
generated by the spectra F(ui11; : :;:uinn), and hence g0is an LfnS0equivalenc*
*e.
To show that L0nS0 is local, we have to show that [W; L0nS0]**= 0 for any Lf*
*n
acyclic W. For any finite localization, the acyclics are the localizing subcate*
*gory
generated by the finite acyclics, so it suffices to show this when W is a finit*
*e acyclic.
Note that if W is a finite acyclic, then so is DW, the SpanierWhitehead dual of
W. Since
[W; L0nS0]**= [S0; DW ^ L0nS0]**= ss**L0nDW;
it suffices to show that L0nW = 0 for any finite acyclic W. This is easy: if W *
*is
an acyclic, then for each slope j n, if k is the next largest slope after j, t*
*hen
by vanishing lines, every ukmap of F(ui11; : :;:uijj) ^ W is nilpotent. Theref*
*ore,_a
cofinal set of maps in the direct system defining L0nW is zero. *
*__
W
Recall from Definition 2.4.2 that a spectrum is of type n if it is d n.
In other words,
(i)0Z(d)**CfnX = 0 if d n, and
(ii)0Z(d)**LfnX = 0 if d > n.
(b)Suppose that X is a CLspectrum (Definition 1.4.6). Then LfnX is "bounded
to the left": for each i, ssijLfn= 0 for j 0. If, in addition, X has fini*
*te
type homotopy, then so does LfnX.
Proof. Part (a): By the theory of finite localization in [HPS97W], the spect*
*rum
CfnX is in theWlocalizing subcategory generated by the finite dn Z(d)acyclic*
*s,
and hence is dn Z(d)acyclic itself. This proves (i).
Since homology commutesWwith direct limits,Wwe see that Z(d)**Tel(k) is zero
if d 6= k; in particular, kn Tel(k) is d>n Z(d)acyclic. Since Lfnis Bousfi*
*eld
5.1. MARGOLIS' KILLING CONSTRUCTION 91
W W
localization with respect to kn Tel(k), then we conclude that LfnS0 is d>nZ(d*
*)
acyclic. Since Lfnis smashing, this proves (ii).
Part (b): Given X as in the statement, we show that the maps in the direct
system defining LfnX (Proposition 5.1.3(c)) are isomorphisms on ss** in a range
increasing with the exponents.
As above, we let F(Uk) = F(ui11; : :;:uikk) and we assume that d is the next
largest slope after k. We write F(Uk)(uidd) = F(ui11; : :;:uidd) for the fiber *
*of the
udmap on F(Uk). Consider the following diagram:
uidd
: ::! id+1;iddF(Uk)! F(Uk)(uidd)!F(Uk)! : : :
?? ? fl
yuidd ?yf1 flfl
u2idd
: ::!2id+1;2iddF(Uk)!F(Uk)(u2idd)!F(Uk)! : : :
?? ? fl
yuidd ?yf2 flfl
u3idd
: ::!3id+1;3iddF(Uk)!F(Uk)(u3idd)!F(Uk)! : : :
?? ? fl
yuidd ?yf3 flfl
.. . .
. .. ..
It suffices to show that the maps fr^ 1X are isomorphisms in a range increasing
with r. Well, fr^ 1X is an isomorphism whenever the map
uidd^1X(r+1)i +1;(r+1)i d
(5.1.7) rid+1;riddF(Uk) ^ X ! d F(Udk) ^ X
from the left column is. The fiber is rid+1;riddF(Uk)(uidd) ^ X; this spectrum
has a vanishing line with some slope ` > d, and we can use Remark 2.3.4 to comp*
*ute
its intercept. To do this, we compute (HFp)**F(Uk)(uidd)_each ujmap induces the
zero map on (HFp)**, so this computation is easy. We find that
(i)(HFp)i*= 0 for i 0,
(ii)(HFp)i*= 0 for i > 0, and
(iii)(HFp)*j= 0 for j < N for some fixed number N.
By the Hurewicz Theorem 1.4.4, (i) and (iii) also hold for ss**; hence we can c*
*ompute
the equation of the vanishing line for F(Uk)(uidd). Smashing with X moves the
intercept a bit, so we find that ssijF(Uk)(uidd) ^ X = 0 when j < `i + N0 for s*
*ome
N0 which is independent of r. Therefore the map (5.1.7)is an isomorphism on ssij
when j + ridd < `(i + rid 1) + N0, i.e., when j < `i + rid(`  d) + (N0 `). S*
*ince
` > d, then as r increases, this range increases.
So we see that for each s, then the graded groups sss*of the direct system h*
*ave
a uniform bound to the left; hence the same is true of the direct limit. If X a*
*lso
has finite type homotopy, then since the homotopy of each F(Uk) is of finite ty*
*pe,
the same goes for F(Uk) ^ X; since the homotopy of the direct system stabilizes*
*_at
each bidegree, we are done. __
Remark 5.1.8. (a)Much of this was already known in the module setting.
For p = 2, this is due to Margolis [Mar83 , Theorem 21.1]; part (a) (and t*
*he
connectivity in part (b)) for arbitrary primes can be found in [Pal92, The
orem 3.1]. Given an Acomodule M and a slope n, then the moduleversion
92 5. CHROMATIC STRUCTURE
of the killing construction (dualized to the category of comodules) gives *
*co
modules M and M<1; n>, welldefined up to injective summands,
and an injective comodule J so that the following is short exact:
0 !M !M J !M<1; n> !0:
Being welldefined up to injective summands translates in our setting to
being welldefined up to an object of loc(A) (cf. Lemma 5.1.9). So if X is
an injective resolution of M, then an injective resolution for M<1; n> is a
connective cover for LfnX.
(b)By studying the proof of Theorem 5.1.6(b), one can show that CfnS0 has a
nice vanishing curve: for i 0, ssijCfnS0 has a vanishing line of slope n,*
* and
if d is the slope preceding n, then for i < 0, ssijCfnS0 has a vanishing l*
*ine of
slope d. For example, when n = 1, we have
Cf1S0 ! S0 ! Lf1S0:
flfl fl fl
fl flfl flfl
F(u11)! S0 ! u11S0
So
ss**Lf1S0 = u11ss**S0 = h110ss**S0 = F2[h110]:
Hence ssijCf1S0 has a vanishing line of slope 2 for positive i, and a vani*
*shing
line of slope 1 (a vertical line, in (i; j  i)coordinates) for i negativ*
*e. See
also Figure 5.1.A.
(c)One can generalize this construction. We set p = 2 and use the periodicity
theorem 4.1.3 and its corollary Theorem 4.7.3. Let I be a finitely generat*
*ed
invariant ideal of HD**, and let C be the thick subcategory of finite spec*
*tra
X with I(X) I. Then we have a finite localization functor LfIand a
cofibration
CfIX f!X g!LfIX:
If the ideal I is generated by classes udi, then Z(d)**f is an isomorphism
for d 62 {di}, and Z(d)**g is an isomorphism if d 2 {di}. The analogue
of Proposition 5.1.3 holds. Our proof of Theorem 5.1.6(b), on the other
hand, does not work in this situation; it may be that these finite type and
connectivity results only hold when I is as in the theorem.
We need the following property of Lfnin Section 5.2.
Lemma 5.1.9.For any slope n, LfnA = 0. Hence any spectrum X in the local
izing subcategory generated by A is Lfnacyclic.
Proof. Since the category of Lfnacyclics contains a finite spectrum_F(Un),
then it contains A by Lemma 1.4.9. __
We close this section with a few remarks about the telescope conjecture (see
[Rav84W] and [HPS97 , 3.3.8]). Let Ln denote Bousfield localization with respect
to dn Z(d).
Proposition 5.1.10.Let p = 2. We have L1 = Lf1. If n > 1, then Ln 6= Lfn.
5.1. MARGOLIS' KILLING CONSTRUCTION 93

i slope _1_
 n1
  ii
  i i i
 ii?
 i i
___________________________________________ii
 j  i
_____slopeo_1_e
 d1




Figure 5.1.A. "Vanishing curve" for ssij(CfnS0): ssij(CfnS0) is
zero above the two indicated lines. Here, d is the slope preceding
n, and slopes are labeled in (i; j  i)coordinates. These lines are
drawn through the origin for convenience, but their intercepts may
actually be nonzero.
One can interpret the statement "Ln = Lfn" as an analogue of the telescope
conjecture, so this result would say that the telescope conjecture fails except*
* when
n = 1. On the other hand, it seems more proper to refer to the statement "every
smashing localization functor is a finite localization" as the telescope conjec*
*ture
[HPS97 , 3.3.8]. Since we do not know whether Ln is smashing, this result does
not necessarily present a counterexample to this version of the telescope conje*
*cture.
Proof. We know that the sphere has a u1map; a simple computation shows
that u11S0 = Z(1), so that Bousfield localization with respect to Lf1S0 = u11*
*S0 is
the same as that with respect to Z(1).
There is a nonnilpotent element d0 2 ss4;18S0 = Ext4;18A(F2; F2). We claim *
*that
d10S0 is Lnacyclic for all n, and Lfnlocal for n 3. Since 3 is the next slo*
*pe after
1, this covers all of the bases.
For degree reasons, d0 induces the zero map on Z(d)**for all d. (If an eleme*
*nt
ff 2 ssijS0 is nonzero on Z(d)**S0 = Z(d)**, then we must have d = j=i.) Hence
Z(d)**(d10S0) = 0 for all d, and d10S0 is Lnacyclic for allWn. To show that *
*d10S0
is Lfnlocal, we show that [F; d10S0]**= 0 for any finite dn Z(d)acyclic F. *
*We
compute:
[F; d10S0]**= [S0; DF ^ d10S0]**
= [S0; d10DF]**
= d10ss**DF:
W
Since F is a finite dn Z(d)**acyclic, so is DF (by Proposition 2.2.6). By The
orem 2.3.1, ss**DF has a vanishing line of some slope m > n. Since n 3, we
know that m 6, so the slope m is larger than 9_2, the slope of d0. Hence d0 ac*
*ts
nilpotently on ss**DF; i.e., d10ss**DF = 0. ___
When p is odd, one can again show that L1 = Lf1. There is every reason to
expect that there are nonnilpotent elements in Ext**A(Fp; Fp) which are not de*
*tected
by any single Z(d); once one knew this, one could conclude that Ln 6= Lfnfor n
large.
94 5. CHROMATIC STRUCTURE
5.2.A Tate version of the functor H
In this section we introduce a "Tate" version of the functor H, and we relate
it to the functor Lfn, at least when applied to the quotient Hopf algebra A(m).*
* We
will use our computations here to prove chromatic convergence in Section 5.3.
Let B be a quotient Hopf algebra of A. Note that practically all of our resu*
*lts
hold in the category Stable(B); the main exceptions are the strict inequalities*
* in
Theorem 4.5.1. Let resA;B:Stable(A) ! Stable(B) denote the forgetful functor
(also known as restriction); resA;Bhas a right adjoint, induction, written indB*
*;A,
and defined by X 7! A 2BX (cf. Lemma 1.3.4(a)). These functors are exact (i.e.,
they take cofibrations to cofibrations), and restriction preserves the smash pr*
*oduct
and the sphere object. (In the language of [HPS97 , Section 3.4], the restricti*
*on
functor is a stable morphism.)
When B is a finitedimensional Hopf algebra, we may consider another sta
ble homotopy category associated to it, called the stable category of Bcomodul*
*es,
written StComod(B). We provide a brief review of the relevant results here; see
[HPS97 , Section 9.6] for a few more details. The objects of StComod(B) are B
comodules, and the morphisms Hom_*Bare defined as follows: define the morphisms
of degree zero to be Hom_B(X; Y ) = Hom B(X; Y )= ', where f ' g :X ! Y if
f  g factors through an injective comodule. Hence two comodules are equiva
lent in StComod(B) if they differ by injective summands. Define the desuspension
functor 1 by the short exact sequence
0 !X !B X !1X !0;
where B X is the cofree comodule on X (Definition 1.1.9). This functor is
invertible: X is any comodule which fits into a short exact sequence
0 !X !I !X !0;
where I is injective. (Since B is finite, then injectives and projectives are t*
*he same,
and there are enough projectives.) We let
Hom_iB(X; Y ) = Hom_B(iX; Y ):
StComod(B) is a stable homotopy category. Indeed, on Stable(B) one has fi
nite localization (Definition 5.1.1) away from the thick subcategory generated *
*by
the finite spectrum B = HFp; StComod(B) is equivalent to the full subcategory
of Stable(B) of LfBlocal objects (see [HPS97 , 9.6.34]). So we have a functor
LfB:Stable(B) !StComod(B) with a right adjoint J :StComod(B) !Stable(B).
(Every localization functor has a right adjoint, namely inclusion of the local *
*objects
into the category.)
Definition 5.2.1.We note that the object HB is indB;A(S0), and we define
the object bHB to be indB;A(J(S0)).
Fix a slope n. On each of the categories Stable(A), Stable(B), and StComod(B*
*),
we can define the functor Lfnto be smashing with LfnS0, or with its image under
5.2. A TATE VERSION OF THE FUNCTOR H 95
resor LfBO res. This gives us the commuting diagram of functors
LfB
Stable(A) res!Stable(B)! StComod(B);
?? ? ?
y Lfn ?yLfn ?yLfn
LfB f
LfnStable(A)res!LfnStable(B)!LnStComod(B)
along with the commuting diagram of their right adjoints
Stable(A) indStable(B)J StComod(B):
x? x x
? RStable(A) ??RStable(B) ??RSt(B)
LfnStable(A)indLfnStable(B)JLfnStComod(B)
(The first diagram commutes by the definition of the vertical functors, and the
second diagram commutes as a result_given a commuting square of left adjoints,
their right adjointssalso commute.) Recall from Notation 2.2.4 that ydis the el*
*ement
of A_either ptor on_"with slope d."
Proposition 5.2.2.Fix a quotient Hopf algebra B of A, and let
N = max{d  yd 6= 0 inB}:
Then if n is a slope larger than N, we have
LfnHB = bHB:
Proof. We claim that the two maps
ff: HB !LfnHB;
fi :HB !HbB;
are the same. To show this, we show that the spectrum bHB is Lfnlocal, and that
the map fi :HB !HbB is an Lfnequivalence. By Lemma 5.2.4 below, the fiber of
fi has homotopy concentrated in the third quadrant, so by Lemma 1.4.8, it is in
loc(A). In particular, it is Lfnacyclic by Lemma 5.1.9; hence fi is an Lfnequ*
*ivalence.
To show that bHB is local, we show that it is in the image of the right adjo*
*int
RStable(A)of Lfn:Stable(A) ! LfnStable(A). We claim, in fact, that RSt(B)is the
identity functor. Given this claim, then we use the commutativity of the diagram
of right adjoints: we have
HbB = ind(JS0)
= ind(J(RSt(B)S0))
= RStable(A)(ind(JS0)):
Hence bHB is in the image of RStable(A), and so it is local.
It remains to verify the claim that RSt(B)is the identity functor, or what i*
*s the
same, that
Lfn:StComod(B) !StComod(B)
is the identity functor. So it suffices to show that the cofiber of S0 !LfnS0 *
*is zero
in the category StComod(B). This cofibration is obtained by applying the functor
96 5. CHROMATIC STRUCTURE
 
s_   s_  
_   _  
_   _  
_   _  
_  _ 
__ __
__________________________________*
*_
__ t  s  __ t  s
_  _
_  _
_   _
_   _
   
HA(1)st bHA(1)st
Figure 5.2.A. The coefficients of HA(1)and bHA(1)when p = 2.
Since the top cell of A(1) is in degree 6, the third quadrant of
HbA(1)**is the same as the first quadrant, reflected across the
origin and then translated by (1; 6).
LfBO resto this cofibration in Stable(A):
CfnS0 !S0 !LfnS0:
So the cofiber under consideration is LfB(resCfnS0). By Proposition 5.1.3, LfnS*
*0 is
finitelocalization away from the thick subcategory generated by F(Un); but F(U*
*n)
has no Z(d)homology for any d with yd 2 B, so res(F(Un)) is in loc(B). Since
res(F(Un)) is finite, it is in fact in thick(B). By fundamental properties of f*
*inite
localizations (Definition 5.1.1), res(CfnS0) is in loc(F(Un)), and hence in loc*
*(B).
Hence LfB(resCfnS0) = 0, as desired. ___
We are particularly interested in the case B = A(m) (this quotient of A is
defined in Example 2.1.4).
Corollary 5.2.3.Fix an integer m 0. For n sufficiently large, we have
LfnHA(m) = bHA(m):
In particular, n should be larger than
( m+1
max{d  yd 6= 0 inA(m)} = m+1p=_2m+1  1if p = 2,
2m  = p  pif p is odd.
Now we "compute" the homotopy groups of HbB. See Figure 5.2.A for an
example.
Lemma 5.2.4.Let B be a finitedimensional quotient Hopf algebra of A, and
consider ssijbHB.
(a)When ij < 0, then ssijbHB = 0. (In other words, the homotopy is concen
trated in the first and third quadrants.)
(b)When i and j are nonnegative, the map HB !HbB induces an isomorphism
ssijbHB ~=ssijHB.
(c)Let d be the maximal degree in which B is nonzero. When i and j are
negative, we have an isomorphism ssijbHB ~=ss1i;djHB.
5.3. CHROMATIC CONVERGENCE 97
Proof. We work in the category Stable(B). We have adjoint functors
L: Stable(B) !StComod(B);
J :StComod(B) !Stable(B);
and we want to compute the homotopy of JS0 by comparing it to that of S0 2
Stable(B). To do this, we need to recall the description of the functor J from
[HPS97 , 9.6.7]. Let I denote an injective resolution of the Bcomodule Fp(i.e.,
I ~=S0 in Stable(B)); then Hom Fp(I; Fp) is a projective resolution of Fp. We s*
*plice
these resolutions together to get the "Tate complex" tB(Fp):
: ::!Hom(I1; Fp) !Hom (I0; Fp) !I0 !I1 !: :::
Then the functor J is defined by J(M) = tB(Fp) M. It is clear that the map
S0 ! J(S0) induces an isomorphism on ssi*for i 0. If we take I to be a
minimal injective resolution, then Hom (I; Fp) is a minimal projective resoluti*
*on;
by minimality, [S0; JS0]i*= [Fp; JS0]i*gives the primitives in JS0 in degree i,*
* and
there is one primitive for each summand isomorphic to B. Since Hom (B; Fp) ~=__
dB, we get the reflection and translation as described in (c). _*
*_
5.3. Chromatic convergence
It is easy to see that given a spectrum X and slopes n1 and n2 with n1 < n2,
then the map X ! Lfn1X factors through X ! Lfn2X. Hence we get a tower of
cofibrations:
.. . .
. .. ..:
?? fl ?
y flfl ?y
Cf3X ! X ! Lf3X
?? fl ?
y flfl ?y
Cf2X ! X ! Lf2X
?? fl ?
y flfl ?y
Cf1X ! X ! Lf1X
(Strictly speaking, we have defined Lfnonly when n is a slope. For a general n,
we define Lfn, as inWDefinition 5.1.2, to be finite localization away from the *
*finite
spectra which are dn Z(d)acyclic. So if n is not a slope, then Lfn= Lfn1.) We
may as well just focus on the right hand column, giving the following diagram:
0 Lf1X  Lf2X  Lf3X  : :::
x x x
=?? ?? ??
M1fX M2fX M3fX
Here, MnfX = X ^ MnfS0 is defined to be the fiber of LfnX !Lfn1X. We call this
diagram the chromatic tower for X. (Clearly if n is not a slope, then MnfX = 0.)
The following theorem, in the module setting for p = 2, is due to Margolis
[Mar83 , Theorem 22.4] (see also [Pal94]).
98 5. CHROMATIC STRUCTURE
Theorem 5.3.1 (Chromatic convergence).If X is finite, then X = limnLfnX.
Indeed, the proof shows that the tower of groups ss**LfnX is proconstant.
In order to prove Theorem 5.3.1, we need the following proposition. This
proposition may hold for general spectra, not just for finite spectra and rings*
*, but
we do not know how to prove that. Z(n) is defined in Notation 2.2.4.
Proposition 5.3.2.Suppose that n is a slope. If R is either a finite spectrum
or a ring spectrum, and if Z(n)**R = 0, then MnfR = 0.
Proof. If X is a finite spectrum, then X ^ DX is a ring spectrum; arguing as
in Proposition 2.2.6, we have Z(n)**X = 0 , Z(n)**(X ^ DX) = 0 and MnfX =
0 , Mnf(X ^ DX) = 0. So the finite case follows from the ring spectrum case.
Suppose that m is the slope preceding n. Then we have a cofiber sequence
__
MnfR !LfnR H!LfmR:
By the octahedral axiom, we also have a cofiber sequence
CfnR H!CfmR !MnfR:
We start by looking at the map H, or rather, the maps related to it in the dire*
*ct limit
defining LfnR. Consider the following diagram. (As in Section 5.1, F(Um ) denot*
*es
F(ui11; : :;:uimm), and F(Um )(uinn) denotes the fiber of the uinnmap on F(Um *
*).)
in^1R
F(Um ) ^ R! F(Um )(uinn) ^Rh!F(Um ) ^ Run!F(Um ) ^ R:
?? ? fl ?
y uinn^1R ?y flfl uinn^1R?y
2in^1R
F(Um ) ^ R!F(Um )(u2inn) ^Rh!F(Um ) ^Run!F(Um ) ^ R:
The unmap uinnon F(Um ) is adjoint to a unelement ff 2 ss**F(Um ) ^ DF(Um ).
So there are two unelements in the ring spectrum (F(Um ) ^ DF(Um )) ^ R: ff ^ *
*1R
and 1F(Um)^DF(Um)^ 0 = 0. They obviously commute, so Lemma 4.2.7 tells us
that some power of ff ^ 1R is null. If we write j :S0 !R for the unit map of R,
then ff ^ 1R is adjoint to uinn^ j :F(Um ) !F(Um ) ^ R, and the selfmap uinn^*
* 1R
factors through this map; hence some iterate of it is null as well.
So for a large enough exponent in, the map h is a map onto a wedge summand.
Furthermore, the vertical maps uinn^ 1R in the above diagram are null.
To get information about H :LfnR !LfmR, we map everything into R and take
cofibers; we see that the cofiber of h is still a map onto a direct summand, an*
*d the__
vertical maps are still null. Taking direct limits gives the desired isomorphis*
*m. __
Corollary 5.3.3.Fix m. For n sufficiently large, MnfHA(m) = 0; hence the
chromatic tower for HA(m) stabilizes.
Proof. This follows from the previous result and Proposition 2.2.5. __*
*_
In particular, at the prime 2, MnfHA(m) = 0 if n > m+1; at odd primes,
MnfHA(m) = 0 if n > pm_2. Note that by Corollary 5.2.3, the chromatic tower f*
*or
HA(m) converges to bHA(m), not to HA(m). So while we can compute the limit of
the chromatic tower here, we also see that we do not have chromatic convergence
in this case.
5.4. FURTHER DISCUSSION 99
Proof of Theorem 5.3.1.If X is finite, then smashing with X commutes
with inverse limits. Since LfnX = X ^ LfnS0, then chromatic convergence for S0
implies chromatic convergence for all finite spectra.
By Theorem 5.1.6(b), LfnS0 has finite type homotopy bounded to the left, so
by Proposition 2.1.5, we see that
LfnS0 = limmHA(m) ^ LfnS0 = limmLfnHA(m):
Hence we have
limnLfnS0= limn(limmLfnHA(m))
= limm(limnLfnHA(m))
= limmbHA(m):
Now we use Lemma 5.2.4 to compute the homotopy groups of limmbHA(m): in
the first quadrant, the inverse limit stabilizes in each bidegree to the homoto*
*py
of limmHA(m) = S0, by Proposition 2.1.5. In the second and fourth quadrants,
there is no homotopy at any stage in the inverse system. In the third quadrant,
for each bidegree (i; j) there is an m0 so that ssijbHA(m)= 0 for all m m0; so
the inverse limit has no homotopy in the third quadrant, either. Hence_the_map
S0 !limmbHA(m)is an isomorphism in homotopy. __
One could probably get another proof of chromatic convergence by making
precise, and then using, the result mentioned in Remark 5.1.8(b).
By the way, Margolis gets chromatic convergence in his setting for all bound*
*ed
below modules; since we do not have convergence for HA(n)(which is an injective
resolution of a bounded below comodule), we cannot expect our result to general
ize, precisely as stated, to other spectra. On the other hand, perhaps one could
modify the proof of Theorem 5.3.1 to show that for any CLspectrum X, then
X !limLfnX is a connective cover. For this, the alternate formulation of CL a*
*fter
Definition 1.4.6 might be useful_X is CL if and only if there are numbers i0, i*
*1,
and j0 so that X has a cellular tower built of spheres Si;jwith i0 i i1 and
j  i j0.
5.4.Further discussion
It seems most convenient to consider the chromatic tower as above, in which
we kill off the Psthomology groups of X in order of slope. This ordering is wh*
*at
allows us to prove chromatic convergence, via the "connectivity result" of Theo
rem 5.1.6(b). But at the prime 2, at least, we can kill off the generators of H*
*D**
in many different orders: we can start with any invariant element in HD**, take
its cofiber, and continue (as in Theorem 4.7.3 and Remark 5.1.8). Can one still
prove convergence? Does other structure reveal itself when one works with other
generators of HD**or with other orderings of the same generators?
As noted after the statement of Proposition 5.1.10, the validity of the stro*
*ng
form of the telescope conjecture is not knownWin Stable(A). If one could show
that Bousfield localization with respect to dn Z(d) were smashing, that would
disprove it; we would be interested in any progress along these lines.
100 5. CHROMATIC STRUCTURE
It is conceivable that the finitetype result of Theorem 5.1.6(b) could have
bearing on the convergence (in the ordinary stable homotopy category) of the to*
*wer
: ::!Lf2X !Lf1X !Lf0X:
After all, the vnlocalized Adams spectral sequence in [MS95 ] has an E2term w*
*ith
some relation to what we call ss**LfnS0, so whether one can prove convergence t*
*his
way, computations and other information about ss**LfnS0 could have applications
to the finite localization functor Lfnin ordinary stable homotopy theory.
Work of Mahowald and others has led to calculations similar to that of bHA(m)
in Lemma 5.2.4. This may provide more connections between our functor Lfnand
topology.
Chromatic convergence for the sphere gives a convergent filtration of ss**S0*
* =
Ext**A(Fp; Fp). When p = 2, Mahowald and Shick [MS87 ] give another convergent
filtration; are these filtrations the same? The slope of the element n+1 is 2n+*
*1 1;
this element corresponds to the periodicity operator vn = hn+1;0, which has slo*
*pe
2n+11. Mahowald and Shick also construct something they call v1nExt**A(F2; F2*
*).
How does this compare to ss**(Lf2n+11S0)? (Shick [Shi88] has done similar work
at odd primes, and one can naturally ask the same questions about that.)
In parallel with the study of chromatic phenomena in the ordinary stable hom*
*o
topy category, one should try to understand the filtration pieces in the chroma*
*tic
tower, and other "monochromatic" objects: objects X so that Mnf= X. Other
than the trivial case of M1fX = h110X, we have no information about these obje*
*cts.
Lastly, it seems possible that the chromatic tower constructed here (as well
as those with other orderings) are the Steenrod algebra analogues of the multip*
*le
complexes of Benson and Carlson [BC87 ]. Is this a good analogy? If so, does
it give any new insight into Stable(A), Stable(kG), or Stable() for an arbitrary
commutative Hopf algebra ? (Evens and Siegel [ES96 ] have extended the multi
ple complex construction to modules over finitedimensional cocommutative Hopf
algebras, or equivalently to comodules over finitedimensional commutative Hopf
algebras, so their work is relevant here.)
APPENDIX A
Two technical results
A.1. An underlying model category
Let be a graded commutative Hopf algebra over a field k. In this section we
(briefly) describe a model category whose associated homotopy category is equiv
alent to Stable(). The main results here (Theorems A.1.3 and A.1.4) are due to
Hovey; see [Hov97 ] for the details.
Recall that Quillen [Qui67 ] defined the notion of a closed model category; *
*this
is a category, like the category of topological spaces, in which one has a noti*
*on
of a wellbehaved homotopy relation between maps. This allows one to define a
new category, the associated homotopy category, which has the same objects as t*
*he
original one, but where the morphisms are the homotopy classes of morphisms in
the original category. (Briefly, a closed model structure is determined by spec*
*ifying
three classes of maps_weak equivalences, fibrations, and cofibrations_satisfying
certain properties. See [Qui67 ], as well as [DS95 ] and [Hov97 ], for details.)
These days, by the way, one often says "model category" rather than "closed
model category."
Model categories are useful because, while one can do many constructions wor*
*k
ing entirely in a homotopy category, for certain more delicate operations one n*
*eeds
to work at the "pointset" level_i.e., in the model category. For example, while
Adams' definition in [Ada74 ] of the homotopy category of spectra is extremely
useful, there are constructions one cannot do unless one has a model category u*
*n
derlying it. Hence today one has various definitions of model categories of spe*
*ctra,
each with its own advantages and disadvantages.
In our work with the Steenrod algebra, we have not needed a model category
underlying Stable(). Nonetheless, it seems like a good idea, pedagogically and *
*for
future applications, to set up such a model category.
We assume that is a graded commutative Hopf algebra over a field k. We
let Ch() denote the category whose objects are cochain complexes of comodules
(not necessarily injective ones), and whose morphisms are cochain maps. We put
a model category structure on Ch(); to do this, we need to specify the weak
equivalences, the fibrations, and the cofibrations in Ch().
Notation A.1.1.Given cochain complexes X and Y , let [X; Y ] denote the set
of cochain homotopy classes of maps from X to Y . Given a comodule M, let
SiM denote the cochain complex which is M in degree i, and zero elsewhere. Let
L(k) denote an injective resolution of the trivial module k, and let S denote t*
*he
set of simple comodules of .
For i an integer, M 2 S , and X any object of Ch(), we define ssi(X; M) (the
"ith homotopy group of X with coefficients in M") by
ssi(X; M) = [SiM; L(k) X]:
101
102 A. TWO TECHNICAL RESULTS
We say that a map f :X !Y in Ch() is a weak equivalence if and only ssi(f; M)
is an isomorphism for all integers i and all simple comodules M. We say that a *
*map
f :X ! Y is a fibration if and only if each component fn: Xn !Yn of f is an
epimorphism with injective kernel. We say that a map f :X !Y is a cofibration
if and only if each component fn: Xn !Yn is a monomorphism.
Remark A.1.2.One can show (see [Hov97 ]) that a map f is a weak equiva
lence if and only if 1L(k) f is a cochain homotopy equivalence. Also, note that
ssi(X; k) = [Sik; L(k) X] is the ith homology group of the cochain complex of
primitives of L(k) X; this is a useful alternate description of ss*(; k).
Theorem A.1.3.[Hov97 ] With weak equivalences, fibrations, and cofibrations
defined as above, Ch() is a model category. Its associated homotopy category is
equivalent to Stable().
(Note that with this model structure, every object of Ch() is cofibrant.)
One can in fact give Ch() the structure of a cofibrantly generated model cat
egory, as follows; we refer to [Hov97 ] for the proofs. Given a comodule M, we
let DnM denote the (contractible) cochain complex which is M in degrees n and
n + 1, zero elsewhere, with differential given by the identity map:
: ::!0 !0 !M 1M!M !0 !0 !: :::
Both Sn (defined in Notation A.1.1) and Dn are functors from Ch() to itself. We
let J denote the following set of maps in Ch():
J = {Dnf  n 2 Z; f :M ,! N an inclusion of finitedimensional}comodules:
We define I by
I = J [ {Sn+1M ,! DnM  n 2 Z; M simple}:
(Here the map Sn+1M ,! DnM is the obvious inclusion.)
Theorem A.1.4.[Hov97 ] With I and J defined as above, Ch() has the struc
ture of a cofibrantly generated model category, in which I is the set of genera*
*ting
cofibrations and J is the set of generating trivial cofibrations.
A.2. Vanishing planes in Adams spectral sequences
In this appendix, we describe a result of Hopkins and Smith [HSa ]: for any
nice ring spectrum E and any number m, the collection of spectra X so that the
Ebased Adams spectral sequence converging to ss*X has a vanishing line of slope
m at some Erterm forms a thick subcategory. We also give a convergence result
for the Adams spectral sequence in Stable(A).
We work in a stable homotopy category like Stable(A)_one in which homotopy
groups are bigraded, and cofibrations look like this:
: ::!1;0Z !X !Y ! Z !1;0X !: :::
Hence the Adams spectral sequence is trigraded, so we discuss vanishing planes
rather than lines. (In a short subsection below, we also give the original stat*
*ement
due to Hopkins and Smith_the statement of the corresponding theorem in the
ordinary stable homotopy category.)
A.2. VANISHING PLANES IN ADAMS SPECTRAL SEQUENCES 103
Recall that we presented the Adams spectral sequence in this setting in Theo
rem 1.5.1. We defined "generic" in Definition 1.4.7. If a spectrum W is (w1; w2*
*)
connective (Definition 1.4.3) but neither (w1+ 1; w2)connective nor (w1; w2+ 1*
*)
connective, we write kWk = (w1; w2).
Theorem A.2.1.Suppose that E is a spectrum satisfying the conditions given
before Theorem 1.5.5, and consider the Ebased Adams spectral sequence
E****(X) ) ss***(X):
Fix numbers m 0 and n. The following properties on a spectrum X are each
generic.
(i)There exist numbers r and b so that for all s, t, and u with
s m(s + u) + n(t + u) + b;
we have Es;t;ur(X) = 0.
(ii)There exist numbers r and b so that for all finite spectra W with kWk =
(w1; w2) and for all s, t, and u with
s m(s + u  w1) + n(t + u + w2) + b;
we have Es;t;ur(X ^ W) = 0.
Note that the "slope" (m; n) of the vanishing plane is fixed, but the interc*
*ept
b and term r of the spectral sequence may vary in these generic conditions.
Remark A.2.2. (a)We want E to be a nice ring spectrum so we can iden
tify the E2term and so we have some convergence information. For the
proof of the theorem, convergence is important, but the form of the E2term
is not. Hence, if we can guarantee convergence by some other means, then
we can discard the assumptions on E. For example, in our application of
this genericity result in Chapter 3, we know that the Adams spectral se
quence coincides with the spectral sequence associated to a Hopf algebra
extension (Proposition 1.5.3), and hence has good convergence properties.
(In this application, it is also easy to verify the conditions mentioned i*
*n the
theorem.)
(b)We assume that m 0 for convenience in stating and proving Lemma A.2.4
below. It is certainly possible that the result holds regardless of the va*
*lue
of m.
One proves this theorem by showing that the purported generic conditions are
equivalent to conditions on composites of maps in the Adams tower; then one sho*
*ws
that those conditions are generic.
We start by describing a_construction of the Adams spectral sequence. Given
a ring spectrum E, we let E denote the fiber of the unit map S0 !E. For any
integer s 0, we let
__^s
FsX = E ^ X;
__^s
KsX = E ^ E ^ X:
104 A. TWO TECHNICAL RESULTS
We use these to construct the following diagram of cofibrations, which we call *
*the
Adams tower for X:
X _______F0X g F1X g F2X g : :::
?? ? ?
y ?y ?y
K0X K1X K2X
This construction satisfies the definition of an "E*Adams resolution" for X, as
given in [Rav86 , 2.2.1]_see [Rav86 , 2.2.9]. Note also that FsX = X ^ FsS0, and
the same holds for KsX_the Adams tower is functorial and exact.
Given the Adams tower for X, if we apply ss**, we get an exact couple and
hence a spectral sequence. This is called the Ebased Adams spectral sequence.
More precisely, we let
Ds;t;u1= sss+u;t+uFsX;
Es;t;u1= sss+u;t+uKsX:
If we let g :Fs+1X ! FsX denote the natural map, then g* = sss+u;t+u(g) is the
map Ds+1;t+1;u11!Ds;t;u1. Then we have the following exact couple (the triples
of numbers indicate the tridegrees of the maps):
(1; 1; 1)
Ds;t;u1_________________Ds+1;t+1;u11oe
@
@
(0; 0;@0)@ (1; 0; 0)
@@R
Es;t;u1
This leads to the following rth derived exact couple, where Ds;t;uris the image*
* of
gr1*, and the map Ds+1;t+1;u1r!Ds;t;uris the restriction of g*:
(1; 1; 1)
Ds;t;ur_________________Ds+1;t+1;u1roe
@
@
(r  1; r  1; r@+@1) (1; 0; 0)
@@R
Es+r1;t+r1;ur+1r
Unfolding this exact couple leads to the following exact sequence:
(A.2.3)
: ::!Es;t+1;u1r!Ds+1;t+1;u1r!Ds;t;ur!Es+r1;t+r1;ur+1r!::: :
Fix numbers m 0 and n. With respect to the Ebased Adams spectral
sequence E****(), we have the following conditions on a spectrum X:
(1)There exist numbers r and b so that for all s, t, and u with s m(s + u) +
n(t + u) + b, the map gr1*:sss+u;t+u(Fs+r1X) !sss+u;t+u(FsX) is zero.
A.2. VANISHING PLANES IN ADAMS SPECTRAL SEQUENCES 105
(2)There exist numbers r and b so that for all s, t, and u with s m(s + u) +
n(t + u) + b, we have Es;t;ur(X) = 0.
(3)There exist numbers r and b so that for all finite spectra W with kDWk =
(w1; w2) and for all s with s mw1 + nw2 + b, then the composite
W !Fs+r1X !FsX is null. (Here, DW denotes the SpanierWhitehead
dual of W.)
(4)There exist numbers r and b so that for all finite spectra W with kWk =
(w1; w2) and for all s, t, and u with s m(s + u  w1) + n(t + u  w2) + b,
we have Es;t;ur(X ^ W) = 0.
Each condition depends on a pair of numbers r and b, and we write (1)r;bto mean
that condition (1) holds with the numbers specified, and so forth.
Lemma A.2.4.Fix numbers m 0, n, r 1, and b. We have the following
implications:
(a)(1)r;b) (2)r;b+r1.
(b)(2)r;b) (1)r;bm.
(c)(3)r;b) (4)r;b+r1.
(d)(4)r;b) (3)r;bm
(Obviously, (3)r;b) (1)r;band (4)r;b) (2)r;b, but we do not need these facts*
*.)
We assume that r 1 so that we have the inequality r 1 + m. Since r is
the term of an Adams spectral sequence, assuming that r 1 is not much of a
restriction.
Proof. As above, we write g for the map Fs+1X !FsX and g* for the map
Ds+1;t+1;u11!Ds;t;u1, so that Ds;t;uris the image of
gr1*:sss+u;t+uFs+r1X !sss+u;t+uFsX:
(a): Assume that if s m(s + u) + n(t + u) + b, then
gr1*:sss+u;t+u(Fs+r1X) !sss+u;t+u(FsX)
is zero; i.e., that Dstur= 0. If s m(s + u) + n(t + u) + b, then since r m, we
have s + r m((s + r) + (u  r + 1)) + n((t + r  1) + (u  r + 1)) + b. So we *
*see
that Ds+r;t+r1;ur+1r= 0. By the long exact sequence (A.2.3), we conclude that
Es+r1;t+r1;ur+1r= 0 when s m(s + u) + n(t + u) + b. Reindexing, we find
that Ep;q;vr= 0 when p  r + 1 m(p + v) + n(q + v) + b; i.e., condition (2)r;b*
*+r1
holds.
(b): If Es;t;ur(X) = 0 when s m(s + u) + n(t + u) + b, then (since r  1 m)
Es+r1;t+r2;ur+2r(X) = 0 also. So by the exact sequence (A.2.3), we see that
Ds+1;t;ur!Ds;t1;u+1ris an isomorphism under the same condition. This map is
induced by g*: sss+1+u;t+uFs+1X ! sss+u+1;t+uFsX, so we conclude that when
s m(s + u) + n(t + u) + b, we have
limqsss+u+1;t+uFqX = Ds;t1;u+1r;
lim1qsss+u+1;t+uFqX = 0:
But by convergence of the spectral sequence, we know that limqsss+u+1;t+uFqX =*
* 0,
so Ds;t1;u+1r= imgr1*= 0. Reindexing gives Dp;q;vr= 0 when p m(p + v  1) +
n(q + 1 + v  1) + b; i.e., (2)r;bimplies (1)r;bm.
Parts (c) and (d) are similar:
106 A. TWO TECHNICAL RESULTS
(c): Fix a finite spectrum W with kWk = kD(DW) k = (w1; w2). By condition
(3), we see that the composite
s+u;t+uDW !Fs+r1X !FsX
is null when s m(s + u  w1) + n(t + u  w2) + b. This is adjoint to
Ss+u;t+u!Fs+r1(X ^ W) !Fs(X ^ W):
Hence Dstur(X ^ W) = 0 when s m(s + u  w1) + n(t + u  w2) + b, in which
case Ds+r;t+r1;ur+1r(X ^ W) is also zero. So by the long exact sequence (A.2.*
*3),
Es+r1;t+r1;ur+1r(X ^ W) = 0 when s m(s + u  w1) + n(t + u  w2) + b.
Reindexing as in part (a) gives us condition (4)r;b+r1.
(d): Fix a finite spectrum W with kDWk = (w1; w2). Condition (4) tells
us that Es;t;ur(X ^ DW) = 0 when s m(s + u + w1) + n(t + u + w2) + b. By the
exact sequence (A.2.3), we see that Ds;t;ur(X ^ DW) = 0 under the same condition
(as in part (b)), so the composite
Ss+u;t+u!Fs+r1(X ^ DW) !Fs(X ^ DW)
is null when s m(s + u + w1) + n(t + u + w2) + b  m. Hence the composite
S0 !Fs+r1(X ^ DW) !Fs(X ^ DW)
is null when s mw1+ nw2+ b  m, as is
W !Fs+r1X !FsX;
by SpanierWhitehead duality. So (4)r;bimplies (3)r;bm. ___
It is easy to prove Theorem A.2.1, once we have the lemma.
Proof of Theorem A.2.1.The proofs of the genericity of the two statements
are similar, so we only prove that condition (i) is generic.
We know by Lemma A.2.4 that condition (i) is equivalent, up to a reindexing,
to
(*)There exist numbers r and b so that for all s, t, and u with s m(s + u) +
n(t + u) + b, the map gr1:Fs+r1X !FsX is zero on sss+u;t+u.
This is generic, by the usual sort of argument: since the Adams tower is functo*
*rial,
if Y is a retract of X, then the Adams tower for Y is a retract of the Adams to*
*wer
for X. So if Fs+r1X ! FsX is zero on sss+u;t+u, then so is Fs+r1Y ! FsY .
(Given Ss+u;t+u!Fs+r1Y , then consider
Ss+t;t+u! Fs+r1Y ! FsY
?? ?
yi ?yi
Fs+r1X ! FsX
?? ?
yj ?yj
Fs+r1Y ! FsY
Since sss+u;t+uFs+r1X ! sss+u;t+uFsX is 0, then the map Ss+u;t+u!FsX is
null. But Ss+u;t+u!FsY factors through this map, and hence is also null.)
A.2. VANISHING PLANES IN ADAMS SPECTRAL SEQUENCES 107
Given a cofibration sequence X !Y ! Z in which X and Z satisfy conditions
(*)r;band (*)r0;b0, respectively, we show that Y satisfies (*)r+r01;max(b;b0*
*r+1).
Consider the following commutative diagram, in which the rows are cofibrations:
Fs+r+r02X! Fs+r+r02Y ! Fs+r+r02Z
?? ? ?
y ?yff ?yfi
Fs+r1X ! Fs+r1Y ! Fs+r1Z
?? ? ?
yfl ?yffi ?y
FsX ! FsY ! FsZ
We assume that s m(s + u) + n(t + u) + max(b; b0 r + 1), so that we have
s m(s + u) + n(t + u) + b;
s + r  1 m(s + u) + n(t + u) + b0:
If we apply sss+u;t+uto this diagram, then sss+u;t+ufi = 0; hence sss+u;t+uff f*
*actors
through sss+u;t+uFs+r1X. Since sss+u;t+ufl = 0, though, then sss+u;t+u(ffi O f*
*f)_= 0.
This shows that condition (*), and hence condition (i), is generic. *
*__
We end this section with a convergence result. We say that a spectrum X is
Ecomplete if the inverse limit of the Adams tower for X is contractible.
Proposition A.2.5.We work in Stable(A). Suppose that E is a ring spectrum
as in Theorem A.2.1. Then every connective spectrum X is Ecomplete.
Proof. This is an easy connectivity result: our conditions on E ensure that
if X is connective, then FsX is (is; js)connective, where is increases with s.*
* In
particular, for all i and j, ssijFsX = 0 for s large enough; hence the inverse_*
*limit
of the Adams tower will have no homotopy, and the lim1term will be zero. __
Note that this applies when E = HB, for B any quotient Hopf algebra of A.
A.2.1. Vanishing lines in ordinary stable homotopy. We give the state
ment of the original HopkinsSmith result in ordinary stable homotopy theory.
Given a connective spectrum W, we write W for its connectivity.
Theorem A.2.6 ([HSa ]).Suppose that E is a ring spectrum as in [Rav86 ,
2.2.5], and consider the Ebased Adams spectral sequence E***(X) ) ss*(X). Fix a
number m 0. The following properties on a spectrum X are each generic.
(i)There exist numbers r and b so that for all s and t with s m(t  s) + b,
we have Es;tr(X) = 0.
(ii)There exist numbers r and b so that for all finite spectra W with W = w
and for all s and t with s m(t  s  w) + b, we have Es;tr(X ^ W) = 0.
The proof above can be easily modified to apply here.
The same proof (in the case m = 0) also shows the following (using the langu*
*age
of [Chr97 ]).
Corollary A.2.7.If I is an ideal of maps that is part of a projective class,
then the following property is generic:
108 A. TWO TECHNICAL RESULTS
oThere exist numbers r and b so that for all s b, the composite
gr1:Fs+r1X !FsX
is in I.
APPENDIX B
Steenrod operations and nilpotence in Ext **(k; k)
Let A be the dual of the mod p Steenrod algebra. In this appendix we recall
a few results about Steenrod operations in the cohomology of any Hopf algebra
, and we focus in particular on the quotient Hopf algebras B of A. Then we
discuss the nilpotence of certain classes in Ext**B(Fp; Fp), which we need to p*
*rove
the nilpotence theorems of Chapter 3.
B.1.Steenrod operations in Hopf algebra cohomology
In this short section we recall a few facts about Steenrod operations in Hopf
algebra cohomology. May's paper [May70 , Section 11] is our basic reference; see
also [Sin73], [Wil81 ], [Saw82 ], [BMMS86 ], and [Rav86 ] for related results.
Suppose that is a Hopf algebra over the field Fp. Then there are Steenrod
operations acting on Ext**(Fp; Fp):
(a)If p = 2, then for each n 0 there are the following operations:
fSqn:Exts;t(F2; F2) !Exts+n;2t(F2; F2):
(b)If p is odd, then for each n 0 there are the following operations (where
q = 2p  2):
fPn: Exts;2t(Fp; Fp) !Exts+qn;2pt(Fp; Fp);
fifPn: Exts;2t(Fp; Fp) !Exts+qn+1;2pt(Fp; Fp):
Note that fifPn must be treated as a single operation, not the composite of
two operations.
These satisfy the usual properties of Steenrod operations: the Cartan formula, *
*the
Adem relations, and an instability condition: if x 2 Exts;t(Fp; Fp) (with s and*
* t
even if p is odd), then
fSqs(x) = x2;p = 2;
fP s_2(x) =pxp;odd:
Note that at odd primes, the operations are zero on classes in Exts;t(Fp; Fp)
when t is odd. This is an artifact of the grading conventions on the operations.
To remedy this, one can either use a different grading convention (see [May70 ]
and [BMMS86 , IV.2] for two different conventions), or one can define operati*
*ons
indexed by halfintegers, as in [Rav86 , 1.5.2].
Example B.1.1. (a)Fix a Hopf algebra . Recall from Lemma 1.1.15 that
Ext1 is isomorphic to the vector space of primitives in . If x is primitiv*
*e,
then we write [x] for the corresponding element of Ext. We have the follow*
*ing
0
Steenrod operation: fP0[x] = [xp]. (At the prime 2, fSq[x] = [x2].) One
109
110 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k)
0
can compute fP0and fSqon general Ext classes by a similar formula_these
operations are induced by the following map on the cobar complex:
[x1x2 : ::xn] 7! [xp1xp2 : ::xpn]:
See [May70 ] for a proof.
(b)If = Fp[x]=(x2) with x primitive, then Ext**(Fp; Fp) ~=Fp[h], where h =
1 n
[x]. If p = 2, then fSq(h) = h2, and fSq(h) = 0 for n 6= 1. If p is odd (in
which case x must be odd), then all operations vanish on h (because of o*
*ur
grading conventions).
(c)If p is odd and = Fp[x]=(xp) with x primitive, then Ext**(Fp; Fp) ~=E[h]
Fp[b], where h = [x] and b is the pfold Massey product of h with itself. *
*We
have fifP0(h) = b, and all other operations on h are zero; also, fP1(b) = *
*bp,
and all other operations on b are zero.
B.2. Nilpotence in HB**= Ext**B(F2; F2)
In this section we recall a result of Lin which we use in the proofs of the *
*main
results of Chapter 3. Lin's result first appeared as [Lin, 3.1 and 3.2]. Wilker*
*son
[Wil81 , 6.4] has also proved a related result.
Let A be the dual of thesmod 2 Steenrod algebra. Recall from Notation 1.3.9
and Remark 2.1.3sthat if 2tis primitive in a quotient Hopf algebra B of A, then
we let hts= [2t] denote the corresponding element of HB1;*.
Theorem B.2.1.Suppose that
n1 2n2
B = A=(21 ; 2 ; : :):
is a quotient Hopf algebra of A so that, for some integer m, we have ni< 1 for
i = 1; 2; : :;:m  1. Fix so that 2m is primitive in B.
(a)If m, then hm; is nilpotent in HB**.
(b)Fix an integer ` m, and suppose that for some , 2` is primitive in B.
If ` , then h`;hm; is nilpotent in HB**.
(Part (a) is a corollary of part (b)_just set ` = m and = .)
For example, the class h1 = h112 Ext1;2A(F2; F2) is nilpotent. This is easy *
*to
show directly: the (reduced) diagonal 2 7! 211in A gives the relation h11h10= 0
1
in Ext**A(F2; F2). Applying the Steenrod operation fSqgives
h12h210+ h311= 0;
so multiplying through by h11 and using h11h10 = 0 yields h411= 0. Note that
0
applying fSqto this gives h41n= 0 for all n 1.
For our purposes, part (a) is one of the key ingredients in the proof that r*
*e
striction to the quotient Hopf algebra D detects nilpotence. Part (b) is used *
*in
the classification of quasielementary quotients of A, essentially, and is also*
* used in
showing that restricting to the quasielementary quotients detects nilpotence.
B.3. NILPOTENCE IN HB**= Ext**B(Fp;Fp) WHEN p IS ODD 111
B.3. Nilpotence in HB**= Ext**B(Fp; Fp) when p is odd
In this section we discuss the oddprimary analogue of Theorem B.2.1.
Fix an odd prime p, and let A besthe dual of the mod p Steenrod algebra.
Recall from Remarks2.1.3 that if ptis primitive in a quotient Hopf algebra B of
A, then hts= [pt] is the corresponding element of HB1;*= Ext1;*B(Fp; Fp), and
bts2 HB2;*is defined to be fifP0(hts). (Alternatively, btsis equal to the pfold
Massey product of htswith itself.) If on is primitive in B, we let vn = [on] be*
* the
corresponding element of HB1;*.
For convenience, we restate Conjecture 3.4.1. This would be the oddprimary
analogue of Theorem B.2.1(a).
Conjecture B.3.1.Fix integers s and t. Suppose that
n1 pn2 e e
B = A=(p1 ; 2 ; : :;:o00; o11; : :):
s
is a quotient Hopf algebra of A in which pt is nonzero and primitive. If s t,
then btsis nilpotent in HB**.
By the way, one could consider a partial analogue of Theorem B.2.1(b): under
conditions on s, t, and n, the product btsvn is nilpotent.
Proposition B.3.2.Fix integersss, t, and n. Suppose that B is a quotients
Hopf algebra of A in which pi= 0 for i < t, and oj = 0 for j < n; hence ptand
on are primitive in B. If n s, then btsvn is nilpotent in Ext**B(Fp; Fp).
Proof. The coproduct
n+tXi
on+t7! pn+ti oi+ on+t 1;
i=0
together with the conditions on B, gives the following relation in the cobar co*
*mplex
for B:
n+t1X
hn+ti;ivi= 0:
i=n
Pn+t1
Hence htnvn =  i=n+1hn+ti;ivi. Applying Steenrod operations gives the fol
lowing (here one needs to use halfinteger indexed operations fP n_2, or one ne*
*eds to
use operations Pk which are indexed differently):
n+t1X
btnvpn= bn+ti;ivpi;
i=n+1
2 n+t1X p2
bt;n+1vpn= bn+ti;i+1vi ;
i=n+1
..
.
s+n1 n+t1X ps+n1
bt;svpn = bn+ti;i+snvi :
i=n+1
s s+n1 __
Since pi= 0 for i < t, then we see that btsvpn = 0. __
112 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k)
Another analogue of Theorem B.2.1(b) would be that under conditions on s,
t, v, and u, the product btsbvu is nilpotent. Conjecture B.3.1 should be a spec*
*ial
case of such a result, and since we do not know how to prove this conjecture, we
will not be able to prove such an analogue of Theorem B.2.1(b).
For the remainder of this section, we provide more evidence and partial resu*
*lts
towards Conjecture B.3.1. s+1
We point out that if s < t, then Fp[t]=(pt ) is a quotient Hopf algebra of B;
the cohomology of this quotient is
E[ht0; : :;:hts] Fp[bt0; : :;:bts]:
The element btsis nonnilpotent when restricted to this quotient, and hence non
nilpotent in ExtB. When s t, though, surprisingly little seems to be known abo*
*ut
the nilpotence (or lack thereof) of bts. For example, while it is easy to verif*
*y that
h411= 0 in Ext**A(F2; F2) at the prime 2 (see Section B.2), we have not been ab*
*le
to locate or prove a similar result for b11at an arbitrary odd prime. Working at
the prime 3, Nakamura [Nak75 ] proved that b211= h11z for some z. Since h11is in
odd total degree, then h211=s0; hence b411= 0. s
Note that if an element pt0is primitive in a Hopf algebra B, then so is ptfor
any s s0.
s Lemma B.3.3.Fix a Hopf algebra B, and fix integers s0 t 1. Suppose that
pt0is primitive in B.
(a)If bt;s0is nilpotent, then so is bt;s0+1.
s0t ps0t
(b)Conversely, assume that p1 = . .=.t1 = 0 in B, and that s0 t. If
bt;s0+1is nilpotent, then so is bt;s0.
As an application of (a), Nakamura's calculation implies that b41;n= 0 for a*
*ll
n 1, when p = 3. One might conjecture (based on very little evidence) that at *
*any
odd prime p, bq1;n= 0 for all n 1, where q = 2(p1). As an application of (b),*
* b11
is nilpotent in ExtBif and only if b1nis nilpotent for n 1; as another applica*
*tion
of (b), if the high powers of t are zero in a quotient B of A_for instance if B*
* is
finite_then btsis nilpotent when s t.
Proof. Part (a) follows from the relation fP0(bits) = bit;s+1.
For part (b), we have the following coproduct in B:
2tX i
2t7! p2ti i:
i=0
s0t
Since pi = 0 when i < t, this simplifies to
s0t ps0t ps0 ps0t ps0t
p2t 7! 2t 1 + t t + 1 2t :
s0t ps
Also since pi = 0 when i < t, then t is primitive in B for every s s0 t. So
the above coproduct in B translates to the relation ht;s0tht;s0= 0 in ExtB. We
apply Steenrod operations to this relation: applying fPpt: :f:Pp(fifP1)(fifP0) *
*gives
t+1 pt
bpt;s0= bt;s0tbt;s0+t
If bt;s0+1is nilpotent, then so is bt;s0+t, by part (a). Hence so is bt;s0. *
* ___
B.3. NILPOTENCE IN HB**= Ext**B(Fp;Fp) WHEN p IS ODD 113
We have the following conjecture, as a special case of Conjecture B.3.1. To
some extent, this would generalize Nakamura's result at the prime 3; on the oth*
*er
hand, he determines a much smaller nilpotence height than this would.
Conjecture B.3.4.Fix an odd prime p.
2(p1)
(a)The element b112 Ext2;2pA (Fp; Fp) is nilpotent.
(b)Fix t 1 and let j = p+1_2. Then bttis nilpotent in Ext**B(Fp; Fp), where
B = Fp[t; 2t; : :;:jt; jt+1; jt+2; jt+3; : :]::
We have a sketch of a proof (which contains a gap), but it is a bit lengthy,*
* and
so we relegate it to a subsection. We also include a few other technical result*
*s in
that subsection.
B.3.1. Sketch of proof of Conjecture B.3.4, and other results.
Sketch of proof. The proof involves some Massey product manipulations.
May's paper [May69 ] is the standard reference for Massey products; many of the
key results are reproduced in [Rav86 , A.1.4].
As remarkedsafter Lemma 1.3.10, the element btsis the pfold Massey product
of hts= [pt] with itself.
Part (a): In A, we have the coproduct
: 2 7! 2 1 + p1 1+ 1 2:
This produces the relation h10h11= 0 in ExtA. Applying the Steenrod operation
fifP0 gives the relation
h11b11 b0h12= 0:
Then for any k 1, we apply fPpk1: :f:PpfP1to get
k pk
h1;k+1bp11 b10h1;k+2= 0:
Using induction gives the following formula, valid for all k 2:
2+...+pk2 1+p+p2+...+pk2
(B.3.5) h11b1+p+p11 = h1;kb10 :
So we let
N = 1 + 1 + (1 + p) + (1 + p + p2) + . .+.(1 + p + p2+ . .+.pp2)
p1 1
= pp______(p: 1)2
and we look at bN11. Recall that b11 is the Massey product . This
p
Massey product has no indeterminacy.
Lemma B.3.6.The element bN11is contained in the pfold Massey product
p2
:
Proof. By definition definition of Massey products, if y = (wi*
*th
no indeterminacy), then for any elements x1, : :,:xn, we have
yx1: :x:n= x1: :x:n
2 :
So we apply this to bN11= bN111. __*
*_
114 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k)
By our computations with Steenrod operations (i.e., equation (B.3.5)), we ha*
*ve
p2
p2
= :
Lemma B.3.7.The Massey product
p2
contains bN110= 0.
We let in = O(n), where O: A ! A is the canonical antiautomorphism.
Hence i1 = 1, and
Xn i
(in) = ii ipni:
i=0
Proof. We have to prove two things: that the Massey product contains
bN110, and that = 0. To pro*
*ve the first
of these, one follows the proof of Lemma B.3.6. To prove the second, we show th*
*at
for all n 2, the nfold Massey product = 0; this *
*goes by
induction on n. Once we know this, then applying the Steenrod operation (fP0)i(*
*to
either the Massey product or to the proof) gives =
0 for any i.
Indeed, we show that for each n, the cobar element d[in] is equal to
;
hence this Massey product is cohomologousito zero. We also show that this Massey
product has no indeterminacy. (Hence d[ipn] kills .)
When n = 2, the coproduct
i2 7! i1 ip1= 1 p1
gives have the relation h10h11= 0. This starts the induction. When n = 3, we
have
2 p
i3 7! i2 ip1+ i1 i2:
Since d[i2] = h10h11, then d[ip2] = h11h12, so we have
= [i2]h12+ h10[ip2] = d[i3] = 0:
nq
Since h1n 2 Ext1;pA(Fp; Fp), then the indeterminacy of this Massey product is of
the form
h10x + yh12;
2) 1;q(1+p)
where x 2 Ext1;q(p+pA(Fp; Fp) and y 2 ExtA (Fp; Fp). Since we know that
Ext1;*is in onetoone correspondence with the primitives of A, then Ext1;iis
nonzero only when i is of the form pjq for some j. Hence both x and y must be
zero, and there is no indeterminacy. *
* j
Now, we assume that = 0 via ii, for all i < n. Then *
*ipi
kills for all i < n and all j, and so
n1X i
= [iiipni]:
i=1
B.3. NILPOTENCE IN HB**= Ext**B(Fp;Fp) WHEN p IS ODD 115
Hence the coproduct on in shows that this is zero. There is no indeterminacy_for
the same reason as when n = 3. __
So the Massey product in Lemma B.3.6 contains both bN11and 0; hence b11is
an element of the indeterminacy. Therefore we need some information about that
indeterminacy. The indeterminacy of a pfold Massey product is contained in the
union of certain (p  1)fold matric Massey products (see [May69 , 2.3]); if ev*
*ery
entry in the pfold Massey product has odd total degree, the same is true of ea*
*ch
entry in each matrix in the shorter Massey products.
Here is a general conjecture about "short" Massey products at an odd charac
teristic. This is the gap in our proof.
Conjecture B.3.8.Fix an odd prime p, and fix n < p. Consider an nfold
matric Massey product , in which each entry of each matrix Vihas o*
*dd
total degree. Then any element of this matric Massey product is nilpotent.
(As in [Rav86 , A.1.4], whenever we discuss matric Massey products, we as
sume that the matrices involved have entries with compatible degrees, so that t*
*heir
products have homogeneous degrees, etc. See [May69 , 1.1] for details.)
The conjecture is trivially true when n < 3, by graded commutativity; in
particular, it is true when p = 3. Indeed, when p = 3, we see that every elemen*
*t of
the indeterminacy is nilpotent of height p. One can also show that the conjectu*
*re is
true when taking the Massey product of onedimensional classes in the cohomology
of a space [Dwy ]; otherwise, we do not have much evidence for it. Meanwhile, it
has the following consequence.
Conjecture B.3.9.In particular, if n < p and if ai has odd total degree for
i = 1; : :;:n, then for any element x contained in is nilpotent.
Since bN11is an element of the indeterminacy of a pfold Massey product of
odddimensional classes, we may conclude that bN11is nilpotent. This would fini*
*sh
the proof of Conjecture B.3.4(a).
Part (b) of the conjecture would be proved similarly. One starts with the
relation ht0htt= 0 in ExtB and applies Steenrod operations to get the following
replacement for (B.3.5):
t1+p2t1+...+pkt1 pt1+p2t1+...+pkt1
httbptt = ht;(k+1)tbt0 :
If we set
M = 1 + pt1+ (pt1+ p2t1) + . .+.(pt1+ p2t1+ . .+.p(p1)t1);
then we get
bMtt= bM1tt
t1 pt1+p2t1 pt1+...+p(p1)t1
2
t1 pt1+p2t1 pt1+...+p(p1)t1
=
3 bM1t0
= 0:
So Conjecture B.3.9 implies that bMttis nilpotent. ___
As remarked above, the gap in the proof_Conjecture B.3.8_is not a gap when
p = 3, so Conjecture B.3.4 holds at the prime 3:
116 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k)
Proposition B.3.10.Fix p = 3.
(a)The element b112 Ext2;36A(F3; F3) is nilpotent. Indeed, b1511= 0.
(b)Fix t 1. Then bttis nilpotent in Ext**B(F3; F3), where
B = F3[t; 2t; 2t+1; 2t+2; 2t+3; 2t+4; : :]::
Arguing similarly, we see:
Proposition B.3.11.Fix p = 3. The element b222 Ext2;432B(F3; F3) is nilpo
tent, where B = A=(1).
2
Proof. The coproduct 4 7! p2 2 gives the relation h20h22= 0 in ExtB.
Hence h2;ih2;i+2= 0 for all i.
As in the "proof" of Conjecture B.3.4, we find that b1622is contained in the
Massey product ; this Massey product also contains the*
* ele
ment b1520. We claim that the threefold Massey product
is nilpotent. i
This Massey product is defined because the product h2;ih2;i+2is killed by 34.
So we consider the diagonal on 6:
2 33 34
6 7! 34+ 3 3+ 2 4
= + h30h33:
We have used the fact that 3i3is primitive in A=(1), and hence gives rise to a
1dimensional Ext class h1;i. Hence we have
h30h33= :
By graded commutativity, (h30h33)2 = 0; hence the same is true of the Massey
product.
We find that b1622= b1510+ x for some class x in the indeterm*
*inacy
of a threefold Massey product. Both x and are nilpotent,_and h*
*ence
so is b22. __
Corollary B.3.12.Fix p = 3. The element b222 Ext2;432C(F3; F3) is nilpo
tent, where C = A=(31).
Proof. If we know that b22 is nilpotent in ExtA=(1)and we want to know
about its nilpotency in ExtA=(31), then by Lemma 1.3.10 it suffices to determine
whether b10b22is nilpotent in ExtA=(31): b22is nilpotent over C if and only if *
*b10b22
is nilpotent over C.
This is easy, though: the coproduct of 3 reduces in C to
3 7! 32 1;
giving the relation h10h21= 0 in Ext**C(F3; F3). Applying fifP0 gives
h11b21 b10h22= 0;
and so b10h22= 0 (h11= 0 over C, since 31= 0 in C). Applying fifP3 then gives
b310b22= 0:
___
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Index
A, 23 complex
act nilpotently, see nilpotent actioncobar, 6
Adams spectral sequence, see spectralcseomultiplication, 2
quence, Adams conjugation, 2, 69
Adams tower, 104 connective, 14, 18
admissible, 64, 8486 weakly, 14
A(n), 26, 35, 39, 42, 94, 96 conormal, 16, see Hopf algebra, quotient,
conormal
bicommutative, 3 coproduct, 7
Bousfield class, 18, 57, 73, 81 cotensor product, 8
Bousfield equivalence, 18, 57, 73
BP, 48 D, 48, 51
fifP0, 11, 60 degree, 3, 11
fifPn, 109 derived category, see category, derived
Brown category, 7 detect nilpotence, 50, 51, 73
BrownComenetz dual, see dual, BrownComenetzD(n), 51, 67
bts, 25, 40, 42, 43, 61, 111 Dr, 57
nilpotence of, 60, 111116 Dr;q, 57
dual
category BrownComenetz, 18, 73
derived, 7 SpanierWhitehead, 33, 65, 105
model, 101102 DX, see dual, SpanierWhitehead
cofibrantly generated, 102 D[x], 11
stable homotopy, 6 D[ps], 25, 40
connective, 13 t
cellular tower, 13 Ecomplete, 107
Chouinard's theorem, 20, 61 elementary, see Hopf algebra, elementary
chromatic convergence, 9799 E[on], 40
chromatic tower, 97, 98 E[on], 25
CL, 14, 34, 42, 90 E[x], 11
closed model category, see category,Emodelxt, 6, 7
coaction, 4, 11, 48, 58, 69, 71 ExtA, 34
diagonal, 4 extension, see Hopf algebra extension; group
coalgebra, 2 extension
quotient, 810, 43
cobar complex, see complex, cobar Fisomorphism, 48, 49, 59, 64, 69, 78
cocommutative, 3 uniform, 48
cofiber sequence, 6, 8, 12, 19 fibration, 102
cofibration, 102, see also cofiber sequencefield spectrum, see spectrum, field
cofree, 5, 94 finite spectrum, see spectrum, finite
cohomology functor, see functor, cohomolfinitejtype,118jm
ogy F(ud1; : :;:udm), 8084, 8892
coideal F(Un), see F(uj1d1; : :;:ujmdm)
coaugmentation, 11 functor
comodule, 4 cohomology, 8
simple, 6, 13, 19, 101 exact, 94
comodulelike, see CL homology, 8
121
122 INDEX
2Bk, 11 Margolis' killing construction, 87, 90
==C, 11 Massey product, 12, 110, 111, 113116
generated, 6 M, 92
generic, 15, 17, 53, 65, 103, 107 Mnf, 97, 98
group algebra, 19 monogenic, 7, 13, 18
group extension, 19 Morava Ktheories, 42, 65
grouplike, 4 morphism, 6, 94
grading, 7
H, 812, 35 stable, 94
Tate version of, 9497
bH, see H, Tate version of nilpotence theorem, 50
HD, 48, 50, 51, 59, 65, 69, 83, 92 nilpotent action, 36, 41
Hom_, 94 Nishida's theorem, 49
homology functor, see functor, homologynormal, see Hopf algebra, sub, normal
homotopy group, 8
Hopf algebra, 2, 101 fP, 109
connected, 3, 6, 14 Palgebra, 26
elementary, 27, 29, 38, 41 periodicity theorem, 64, 65, 76, 92
quasielementary, 29, 4850, 58, 73ss**, 8, 34
quotient, 8, 9, 24, 36, 48, 73 PM, 48
conormal, 11, 16, 24, 48, 51, 58,P73ostnikov tower, 13
maximal elementary, 27 primitive, 4, 6, 11, 43, 48, 60, 102, 109, 1*
*11,
maximal quasielementary, 57, 69, 71112
sub profile function, 24
normal, 11, 74 projective, 5
Hopf algebra extension, 10, 19, 25,p36,r37,ojectivesclass, 107
40, 41, 51, 73 pt, 30,s84
horizontal vanishing line, see vanishingPline,t,s29, 30, 38, 42, 79
horizontal Pthomology, 30
hts, 25, 36, 42, 43, 54, 69, 71, 110,m111odule,s30, 31
nilpotence of, 36, 54, 57, 60, 110Pthomology spectrum, 30, 31, 80
Hurewicz map, 14, 42, 49, 51 Q, 49, 58, 69
Hurewicz theorem, 14 __Q, 58
ideal, 63, 7678 qn, 30
augmentation, 11 Qn, 29, 30, 42, 80
invariant, 76, 78, 83, 92 Qnhomology, 30
induction, 94 Qnhomology spectrum, 30
injective, 5, 6, 911, 30, 43, 94 quasielementary, see Hopf algebra, quasi
intercept, 35 elementary
invariants, 48, 69 Quillen stratification, 49, 61
IX, see dual, BrownComenetz rank variety, see variety, rank
I(X), see ideal relative vanishing line, see vanishing line,
kG, 19 relative
resolution
LfC, see localization, finite injective, 6, 101
Lfn, see localization, finite restriction, 9, 94
Lfn, 95 restriction map, 9, 43, 51
LfnA(m), 96 ring spectrum, see spectrum, ring
loc(A), 75, 92 selfmap, 42, 50
loc(), 15 central, 44, 64
loc(Y ), 15 Shapiro's lemma, 10
localization shearing isomorphism, 10
Bousfield, 88, 92 i;j, see suspension
finite, 87, 88, 94, 97 slope, 31, 34, 41, 42, 64, 80, 8486, 88, 95,
smashing, 88, 93 97
localizing subcategory, see subcategory,slolope support, 64, 8486
calizing Slopes, 64, 8486
INDEX 123
Slopes0, 64, 8486 x, see degree
small, 15 Xacyclic, 18
smash product, 7 n, 23
SpanierWhitehead dual, see dual, Spanier
Whitehead [y], 11, 60
spectral sequence yelement, see ymap
Adams, 16, 17, 52, 102108 ymap, 6467, 83
construction, 103104 strong, 65, 66
convergence, 107 yn, 32, 43, 95
AtiyahHirzebruch, 42, 77 Z(n), 31, 34, 42, 64, 8083, 88, 90, 97, 98
changeofrings, see spectral sequence,zex(n), 32, 42, 43
tension in, 69
extension, 16, 51, 52
LyndonHochschildSerre, 16
spectrum, 6
field, 10, 65
finite, 15, 50, 64, 75, 76, 78, 80, 83, 84, 87,
89, 98
ring, 8, 9, 18, 50, 65, 66, 98
sphere, 7
Sqn, 71
fSqn, 109, 110
stable homotopy category, see category, sta
ble homotopy
Stable(A), 24
Stable(), 6, 101
StComod(B), 94
Steenrod algebra, 23
Steenrod operation, 11, 17, 109110
subcategory
localizing, 15, 34, 88
thick, 15, 54, 65, 7879, 81, 83, 87, 94, 103
suspension, 7, 94
telescope, 18, 83, 88
telescope conjecture, 92
thick subcategory, see subcategory, thick
thick(Y ), 15
on, 23
trivial comodule, 13
type n, 42, 83, 89
un, 42, 43, 64
unmap, 42, 8083, 88
unit map, 8
vanishing line, 3444, 92, 107108
horizontal, 34, 38, 41
relative, 42
vanishing plane, 17, 53, 102107
variety
rank, 7980
vn, 25, 40, 42, 43, 111
vnmap, 42, 65
weak equivalence, 102
Wj, 53, 54
, see Bousfield class
[x], 11, 109