Rings, Modules, and Algebras in Stable Homotopy Theory

269 Pages·2007·1.96 MB·English

by  A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, and M. Cole

RINGS, MODULES, AND ALGEBRAS IN STABLE
HOMOTOPY THEORY
A.D.Elmendorf, I.Kriz, M.A. Mandell, and J.P.May
Author addresses:
Purdue University Calumet, Hammond IN 46323
E-mail address: [email protected]
The University of Michigan, Ann Arbor, MI 48109-1003
E-mail address: [email protected]
The University of Chicago, Chicago, IL 60637
E-mail address: [email protected]
The University of Chicago, Chicago, IL 60637
E-mail address: [email protected]
ii
iii
Abstract. Let S be the sphere spectrum. We construct an associative, com-
mutative, and unital smash product in a complete and cocomplete category
S
M
of \S-modules" whose derived category is equivalent to the classical stable
S
D
homotopycategory. Thisallowsasimpleandalgebraicallymanageablede(cid:12)nition
of \S-algebras" and \commutative S-algebras" in terms of associative, or asso-
ciative and commutative,products R^ R−!R. These notions areessentially
S
equivalent to the earlier notions of A1 and E1 ring spectra, and the older no-
tionsfeednaturallyintothenewframeworktoprovideplentifulexamples. There
is an equally simple de(cid:12)nition of R-modules in terms of maps R^ M −! M.
S
When R is commutative, the category of R-modules also has an associa-
R
M
tive, commutative, and unital smash product, and its derived category has
R
D
properties just like the stable homotopy category.
Workinginthederivedcategory ,weconstructspectralsequencesthatspe-
R
D
cialize to give generalized universal coe(cid:14)cient and Ku¨nneth spectral sequences.
Classicaltorsion products and Ext groups are obtained by specializing our con-
structions to Eilenberg-MacLane spectra and passing to homotopy groups,and
the derivedcategoryof a discreteringR is equivalent to thederived categoryof
its associated Eilenberg-Mac Lane S-algebra.
We also develop a homotopical theory of R-ring spectra in , analogous
R
D
to the classical theory of ring spectra in the stable homotopy category, and we
use it to give new constructions as MU-ring spectra of a host of fundamentally
important spectra whose earlier constructions were both more di(cid:14)cult and less
precise.
Working in the module category , we show that the category of (cid:12)nite
R
M
cell modules over an S-algebraR gives riseto an associatedalgebraicK-theory
spectrum KR. Specialized to the Eilenberg-Mac Lane spectra of discrete rings,
this recovers Quillen’s algebraic K-theory of rings. Specialized to suspension
spectra (cid:6)1(ΩX) of loop spaces, it recovers Waldhausen’s algebraic K-theory
+
of spaces.
Replacing our ground ring S by a commutative S-algebra R, we de(cid:12)ne R-
algebras and commutative R-algebras in terms of maps A^ A −! A, and we
R
showthatthecategoriesofR-modules,R-algebras,andcommutativeR-algebras
arealltopologicalmodelcategories. WeusethemodelstructurestostudyBous-
(cid:12)eld localizations of R-modules and R-algebras. In particular, we prove that
KO and KU are commutative ko and ku-algebras and therefore commutative
S-algebras.
We de(cid:12)ne the topological Hochschild homology R-module THHR(A;M) of
A with coe(cid:14)cients in an(A;A)-bimodule M andgivespectralsequences for the
calculation of its homotopy and homology groups. Again, classical Hochschild
homologyandcohomologygroupsareobtainedbyspecializingtheconstructions
to Eilenberg-MacLane spectra and passing to homotopy groups.
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Contents
Introduction 1
Chapter I. Prologue: the category of L-spectra 9
1. Background on spectra and the stable homotopy category 9
2. External smash products and twisted half-smash products 11
3. The linear isometries operad and internal smash products 14
4. The category of L-spectra 18
5. The smash product of L-spectra 21
6. The equivalence of the old and new smash products 24
7. Function L-spectra 27
8. Unital properties of the smash product of L-spectra 30
Chapter II. Structured ring and module spectra 35
1. The category of S-modules 35
2. The mirror image to the category of S-modules 39
3. S-algebras and their modules 41
4. Free A and E ring spectra and comparisons of de(cid:12)nitions 44
1 1
5. Free modules over A and E ring spectra 47
1 1
6. Composites of monads and monadic tensor products 50
7. Limits and colimits of S-algebras 52
Chapter III. The homotopy theory of R-modules 57
1. The category of R-modules; free and cofree R-modules 57
v
vi CONTENTS
2. Cell and CW R-modules; the derived category of R-modules 60
3. The smash product of R-modules 65
4. Change of S-algebras; q-co(cid:12)brant S-algebras 68
5. Symmetric and extended powers of R-modules 71
6. Function R-modules 73
7. Commutative S-algebras and duality theory 77
Chapter IV. The algebraic theory of R-modules 81
1. Tor and Ext; homology and cohomology; duality 82
2. Eilenberg-MacLane spectra and derived categories 85
3. The Atiyah-Hirzebruch spectral sequence 89
4. Universal coe(cid:14)cient and Ku¨nneth spectral sequences 92
5. The construction of the spectral sequences 94
6. Eilenberg-Moore type spectral sequences 97
7. The bar constructions B(M;R;N) and B(X;G;Y) 99
Chapter V. R-ring spectra and the specialization to MU 103
1. Quotients by ideals and localizations 103
2. Localizations and quotients of R-ring spectra 107
3. The associativity and commutativity of R-ring spectra 111
4. The specialization to MU-modules and algebras 114
Chapter VI. Algebraic K-theory of S-algebras 117
1. Waldhausen categories and algebraic K-theory 117
2. Cylinders, homotopies, and approximation theorems 121
3. Application to categories of R-modules 124
4. Homotopy invariance and Quillen’s algebraic K-theory of rings 128
5. Morita equivalence 130
6. Multiplicative structure in the commutative case 134
7. The plus construction description of KR 136
8. Comparison with Waldhausen’s K-theory of spaces 141
Chapter VII. R-algebras and topological model categories 145
CONTENTS vii
1. R-algebras and their modules 146
2. Tensored and cotensored categories of structured spectra 149
3. Geometric realization and calculations of tensors 153
4. Model categories of ring, module, and algebra spectra 159
5. The proofs of the model structure theorems 163
6. The underlying R-modules of q-co(cid:12)brant R-algebras 167
Chapter VIII. Bous(cid:12)eld localizations of R-modules and algebras 173
1. Bous(cid:12)eld localizations of R-modules 174
2. Bous(cid:12)eld localizations of R-algebras 178
3. Categories of local modules 181
4. Periodicity and K-theory 184
Chapter IX. Topological Hochschild homology and cohomology 187
1. Topological Hochschild homology: (cid:12)rst de(cid:12)nition 188
2. Topological Hochschild homology: second de(cid:12)nition 192
3. The isomorphism between thhR(A) and A⊗S1 196
Chapter X. Some basic constructions on spectra 201
1. The geometric realization of simplicial spectra 201
2. Homotopical and homological properties of realization 204
3. Homotopy colimits and limits 209
4. (cid:6)-co(cid:12)brant, LEC, and CW prespectra 211
5. The cylinder construction 214
Chapter XI. Spaces of linear isometries and technical theorems 221
1. Spaces of linear isometries 221
2. Fine structure of the linear isometries operad 224
3. The unit equivalence for the smash product of L-spectra 230
4. Twisted half-smash products and shift desuspension 232
5. Twisted half-smash products and co(cid:12)brations 235
Chapter XII. The monadic bar construction 239
1. The bar construction and two deferred proofs 239
viii CONTENTS
2. Co(cid:12)brations and the bar construction 242
Chapter XIII. Epilogue: The category of L-spectra under S 247
1. The modi(cid:12)ed smash products C , B , and ? 247
L L L
2. The modi(cid:12)ed smash products C , B , and ? 251
R R R
Bibliography 257
Introduction
The last thirty years have seen the importation of more and more algebraic tech-
niques into stable homotopy theory. Throughout this period, most work in stable
homotopy theory has taken place in Boardman’s stable homotopy category [6], or
in Adams’ variant of it [2], or, more recently, in Lewis and May’s variant [37].
That category is analogous to the derived category obtained from the category of
chain complexes over a commutative ring k by inverting the quasi-isomorphisms.
The sphere spectrum S plays the role of k, the smash product ^ plays the role
of the tensor product, and weak equivalences play the role of quasi-isomorphisms.
A fundamental di(cid:11)erence between the two situations is that the smash product on
the underlying category of spectra is not associative and commutative, whereas the
tensor product between chain complexes of k-modules is associative and commu-
tative. For this reason, topologists generally work with rings and modules in the
stable homotopy category, with their products and actions de(cid:12)ned only up to ho-
motopy. In contrast, of course, algebraists generally work with di(cid:11)erential graded
k-algebras that have associative point-set level multiplications.
We here introduce a new approach to stable homotopy theory that allows one
to do point-set level algebra. We construct a new category of S-modules
S
that has an associative, commutative, and unital smash producMt ^ . Its derived
S
category is obtained by inverting the weak equivalences; is equivalent to the
S S
classical sDtable homotopy category, and the equivalence presDerves smash products.
This allows us to rethink all of stable homotopy theory: all previous work in the
subject might as well have been done in . Working on the point-set level,
S
in , we de(cid:12)ne an S-algebra to be an DS-module R with an associative and
S
uniMtal product R ^ R −! R; if the product is also commutative, we call R a
S
commutative S-algebra. Although the de(cid:12)nitions are now very simple, these are
not new notions: they are re(cid:12)nements of the A and E ring spectra that were
1 1
introduced over twenty years ago by May, Quinn, and Ray [47]. In general, the
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