Stable homotopy over the Steenrod algebra

133 Pages·2001·0.866 MB·English

by  Palmieri, John

Stable homotopy over the Steenrod algebra
John H. Palmieri
Author address:
Department of Mathematics, University of Notre Dame, Notre
Dame, IN 46556
E-mail address: [email protected]
1991 Mathematics Subject Classi(cid:12)cation. 55S10, 55U15, 18G35, 55U35, 55T15,
55P42, 55Q10,55Q45,18G15, 16W30, 18E30, 20J99.
Researchpartiallysupportedby NationalScience FoundationgrantDMS-9407459.
Abstract. Weapplythetoolsofstablehomotopytheorytothestudyofmod-
ules over the Steenrod algebra A(cid:3); in particular, we study the (triangulated)
category Stable(A) of unbounded cochain complexes of injective comodules
overA,thedualofA(cid:3),inwhichthemorphismsarecochainhomotopyclasses
ofmaps. Thiscategorysatis(cid:12)estheaxiomsofastablehomotopycategory(as
given in [HPS97]); so we can use Brown representability, Bous(cid:12)eld localiza-
tion, Brown-Comenetz duality, and other homotopy-theoretic tools to study
Ext(cid:3)A(cid:3)(Fp;Fp),whichplaystheroleofthestablehomotopygroupsofspheres.
We also have nilpotence theorems, periodicity theorems, a convergent chro-
matictower,andanumberofotherresults.
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(Γ) 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. Bous(cid:12)eld classes and Brown-Comenetz 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. Quasi-elementary quotients of A 28
2.2. Ps-homology 29
t
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 necessarywhen p is odd 40
2.4. Self-maps via vanishing lines 42
2.5. Further discussion 44
Chapter 3. Quillen strati(cid:12)cation and nilpotence 47
3.1. Statements of theorems 48
3.1.1. Quillen strati(cid:12)cation 48
3.1.2. Nilpotence 50
3.2. Nilpotence and F-isomorphism via the Hopf algebra D 51
3.2.1. Nilpotence: Proof of Theorem 3.1.5 54
3.2.2. F-isomorphism: Proof of Theorem 3.1.2 55
3.3. Nilpotence and F-isomorphism via quasi-elementary quotients 56
3.3.1. Nilpotence: Proof of Theorem 3.1.6 56
3.3.2. F-isomorphism: 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 y-maps 65
4.3. The proof of the periodicity theorem 67
4.4. Computation of some invariants in HD(cid:3)(cid:3) 69
4.5. Computation of a few Bous(cid:12)eld 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 speci(cid:12)ed 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
(cid:3)(cid:3)
Appendix B. Steenrod operations and nilpotence in Ext (k;k) 109
Γ
B.1. Steenrod operations in Hopf algebra cohomology 109
(cid:3)(cid:3)
B.2. Nilpotence in HB(cid:3)(cid:3) =ExtB(F2;F2) 110
(cid:3)(cid:3)
B.3. Nilpotence in HB(cid:3)(cid:3) =ExtB(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 Pro(cid:12)le function for A(n). 26
2.1.C Pro(cid:12)le functions for maximal elementary quotients of A, p=2. 28
2.3.A Vanishing line at the prime 2. 34
3.1.A Pro(cid:12)le function for D. 48
3.2.B Pro(cid:12)le function for D(n). 52
3.3.C Pro(cid:12)le functions for D and D . 57
r r;q
4.4.A Graphical depiction of coaction of A on limHE(cid:3)(cid:3) 72
0
4.8.A T(t;s) and T(m) as subsets of Slopes. 85
5.1.A Vanishing curve for (cid:25) (CfS0). 93
ij n
b
5.2.A The coe(cid:14)cients of HA(1) and HA(1). 96
v
vi LIST OF FIGURES
Preface
TheobjectofstudyforthisbookisthemodpSteenrodalgebraAanditscoho-
mology Ext . Various people (including the author) have approached this subject
A
by taking results in stable homotopy theory and then trying to prove analogous
resultsforA-modules. Thishasproventobesuccessful,buttheanalogieswerejust
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
homotopytheory." Inparticular,wegaveaxiomsforastable homotopy category; in
anysuchcategory,onehasavailablemanyofthetoolsofclassicalandmodernstable
homotopy theory|tools like Brown representability and Bous(cid:12)eld localization. It
turns out that a category Stable(A) (de(cid:12)ned in the next paragraph)related to the
category of A-modules is such a category; as one might expect, the trivial module
F plays the role of the sphere spectrum S0, and Ext(cid:3)(cid:3)(−;−) plays the role of
p A
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
(cid:3)(cid:3)
Ext (F;F) (and related groups). In this book we apply some of these tools
A p p
(nilpotence theorems,periodicity theorems, chromatictowers,etc.) to the study of
ExtovertheSteenrodalgebra. Itisourhopethatthisbookwillservetwopurposes:
(cid:12)rst,to providea referencesourcefor anumber ofresults aboutthe cohomologyof
the Steenrod algebra, and second, to provide an example of an in-depth use of the
language and tools of axiomatic stable homotopy theory in an algebraic setting.
First we describe the category in which we work. We (cid:12)x a prime p, let A(cid:3) be
the mod p Steenrod algebra, and let A = Hom (A(cid:3);F) be the (graded) dual of
Fp p
the Steenrod algebra. We let Stable(A) be the category whose objects are cochain
complexes of injective left A-comodules, 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
ofresults inStable(A); someoftheseareanaloguesofresultsin the ordinarystable
homotopy category, and some are not. Some of these are new, and some already
known,at leastin the setting of A(cid:3)-modules; the old results often need new proofs
to apply in the more general setting we discuss here.
Note. This work arose from the study of the abelian category of (left) A(cid:3)-
modules; to apply stable homotopy theoretic techniques, though, it is most con-
venient to work in a triangulated category. One’s (cid:12)rst guess for an appropriate
category might have objects which are chain complexes of projectiveA(cid:3)-modules;
it turns out that this category has some technical di(cid:14)culties (see Remark 1.2.1).
It is much more convenient to work with A-comodules instead of A(cid:3)-modules, and
fortunately,onedoesnotlosemuchbydoingthis. MostA(cid:3)-modulesofinterestcan
vii
viii PREFACE
be viewed as A-comodules; the main e(cid:11)ects of using comodules are things of the
(cid:3)(cid:3)
following sort: various arrows go the \wrong" way, Ext (k;k) is covariant in A,
A
and one studies A by means of its quotient Hopf algebras (because those are dual
to the sub-Hopf algebras of A(cid:3)).
Each chapter is divided into a number of sections; at the beginning of each
chapter, we give a brief description of its contents, section by section. In this
introduction, we give a brief overview of each chapter. We note that each chapter
has at least one \Further discussion" section, in which we discuss issues auxiliary
to the general discussion.
In Chapter 1, we set up notation and discuss results that hold in the category
Stable(Γ) for any graded commutative Hopf algebra Γ over a (cid:12)eld k, e.g., the dual
of a group algebra, the dual of an enveloping algebra, or the dual of the Steenrod
algebra. Aside from setting up notation for use throughout this book, the main
topics of this chapter include: construction of cellular and Postnikov towers, an
examination of the Adams spectral sequence associated to particular homology
theorieson Stable(Γ), andsomeremarksonBous(cid:12)eldclassesandBrown-Comenetz
duality.
Note. While some of Chapter 1 may be well-known, we recommend that the
readerlookoverSection1.2andthe (cid:12)rstpartofSection 1.3(at leastthe de(cid:12)nition
of the functor H) before reading later parts of the book. These sections introduce
notation that gets used throughout the book.
In Chapter 2 wespecialize to the casein whichp is a prime, k =F is the (cid:12)eld
p
with p elements, and A is the dual of the mod p Steenrod algebra. Recall from
[Mil58] that as algebras, we have
(
(cid:24) F2[(cid:24)1;(cid:24)2;(cid:24)3;:::]; if p=2,
A=
F[(cid:24) ;(cid:24) ;(cid:24) ;:::]⊗E[(cid:28) ;(cid:28) ;(cid:28) ;:::]; if p is odd.
p 1 2 3 0 1 2
The coproduct on A is determined by
Xn
(cid:1): (cid:24) 7−! (cid:24)pi ⊗(cid:24) ;
n n−i i
i=0
Xn
(cid:1): (cid:28) 7−! (cid:24)pi ⊗(cid:28) +(cid:28) ⊗1:
n n−i i n
i=0
In this chapter, we discuss two tools with which to study Stable(A): quotient Hopf
algebras of A and Ps-homology. We use these tools to prove two theorems: the
t
(cid:12)rstisavanishinglinetheorem(givenconditionsonthePs-homologygroupsofX,
t
thenExts;t(F;X)=0whens>mt−c,forsomenumbersmandc). Thesecondis
A p
a \self-map" theorem: given a (cid:12)nite A-comodule M, we construct a non-nilpotent
(cid:3)(cid:3)
element of Ext (M;M) satisfying certain properties.
A
Let p=2. In Chapter 3 we developanalogues in the categoryStable(A) of the
nilpotence theorem of Devinatz, Hopkins, and Smith, as well as the strati(cid:12)cation
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 is a
quotient Hopf algebra D of A so that, if M is a (cid:12)nite-dimensional A-comodule,
an element z 2 Ext(cid:3)(cid:3)(M;M) is nilpotent under Yoneda composition if and only
A
(cid:3)(cid:3)
if its restriction to Ext (M;M) is nilpotent. The second nilpotence theorem is
D
similar, but uses a family of ring objects (somewhat like the Morava K-theories)
to detect nilpotence. These are versions in Stable(A) of the nilpotence theorems
PREFACE ix
(cid:3)(cid:3)
of [DHS88] and [HSb]. We strengthen these results when studying Ext (F;F),
A 2 2
by \identifying" the image of Ext(cid:3)(cid:3)(F;F)−!Ext(cid:3)(cid:3)(F;F) (and similarly for the
A 2 2 D 2 2
other nilpotence theorem). One can view this as an analogue of Quillen’s theorem
[Qui71, 6.2], in which he identi(cid:12)es the cohomology of a compact Lie group up to
F-isomorphism.
Again, let p = 2. In Chapter 4, we discuss applications of the theorems from
thepreviouschapter. Inordinarystablehomotopytheory,thenilpotencetheorems
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 periodicity
theorem, and only a conjecture as to a classi(cid:12)cation of the thick subcategories of
(cid:12)niteobjectsinStable(A). Moreprecisely,ifM isa(cid:12)nite-dimensionalA-comodule,
(cid:3)(cid:3)
then we produce a number of central non-nilpotent elements in Ext (M;M) by
A
using the \variety of M": the kernel of Ext(cid:3)(cid:3)(F;F)−!Ext(cid:3)(cid:3)(M;M).
D 2 2 D
One of our analogues of Quillen’s theorem says that the elements of the kernel
of Ext(cid:3)(cid:3)(F;F) −! Ext(cid:3)(cid:3)(F;F) are nilpotent, and it identi(cid:12)es the image. This
A 2 2 D 2 2
identi(cid:12)cation 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 smaller
Bous(cid:12)eld class than the sphere.
Let p be any prime. In Chapter 5 we consider Steenrod algebra analogues
of chromatic theory and the functors L and Lf. The latter turns out be more
n n
tractable; in fact, it is a generalization (from the module setting) of Margolis’
killing construction [Mar83, Chapter 21]. We show that L 6= Lf if n > 1, at
n n
least at the prime 2. We compute Lf on some particular ring spectra, and show
n
that,atleastfortheserings,itturns\groupcohomology"into\Tatecohomology."
Weusethis resulttoshowthatthechromatictowerconstructedusingthe functors
Lf converges for any (cid:12)nite object. (This is an extension of a theorem of Margolis
n
[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 section
aredue to Hovey. In Appendix A.2 we provea theorem due to Hopkins and Smith
[HSa], that the property of having a vanishing line with given slope, at some term
oftheAdamsspectralsequenceandwithsomeintercept,isgeneric. (Weprovethis
inthecontextofAdamsspectralsequencesinStable(A),whicharetrigraded;hence
we actually discuss vanishing planes.) In Appendix B, wediscuss the nilpotence of
(cid:3)(cid:3)
certainclassesinExt (F;F)whenAistheSteenrodalgebra;weusetheseresults
A p p
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 co(cid:12)bration 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 complete
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.