**Mathematical Surveys and Monographs**

Volume: 47;
1997;
249 pp;
Softcover

MSC: Primary 19; 55;

Print ISBN: 978-0-8218-4303-1

Product Code: SURV/47.S

List Price: $75.00

AMS Member Price: $60.00

MAA member Price: $67.50

**Electronic ISBN: 978-1-4704-1278-4
Product Code: SURV/47.S.E**

List Price: $75.00

AMS Member Price: $60.00

MAA member Price: $67.50

# Rings, Modules, and Algebras in Stable Homotopy Theory

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*A. D. Elmendorf; I. Kriz; M. A. Mandell; J. P. May; M. Cole*

This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum \(S\), the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of “\(S\)-modules” whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of “\(S\)-algebras” and “commutative \(S\)-algebras” in terms of associative, or associative and commutative, products \(R\wedge _SR \longrightarrow R\). These notions are essentially equivalent to the earlier notions of \(A_{\infty }\) and \(E_{\infty }\) ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of \(R\)-modules in terms of maps \(R\wedge _SM\longrightarrow M\). When \(R\) is commutative, the category of \(R\)-modules also has an associative, commutative, and unital smash product, and its derived category has properties just like the stable homotopy category. These constructions allow the importation into stable homotopy theory of a great deal of point-set level algebra.

#### Readership

Graduate students and research mathematicians interested in algebraic topology.

#### Reviews & Endorsements

Very well organized … The exposition is quite clear, with just the right amount of motivational comments. All algebraic topologists should obtain some familiarity with the contents of this book.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Rings, Modules, and Algebras in Stable Homotopy Theory

- Contents ix10 free
- Introduction 114 free
- Chapter I. Prologue: the category of L-spectra 922 free
- 1. Background on spectra and the stable homotopy category 922
- 2. External smash products and twisted half-smash products 1124
- 3. The linear isometries operad and internal smash products 1326
- 4. The category of L-spectra 1730
- 5. The smash product of L-spectra 2033
- 6. The equivalence of the old and new smash products 2235
- 7. Function L-spectra 2538
- 8. Unital properties of the smash product of L-spectra 2841

- Chapter II. Structured ring and module spectra 3144
- 1. The category of S-modules 3144
- 2. The mirror image to the category of S-modules 3548
- 3. S-algebras and their modules 3750
- 4. Free A[sub(∞)] and E[sub(∞)] ring spectra; comparisons of definitions 3952
- 5. Free modules over A[sub(∞)] and E[sub(∞)] ring spectra 4255
- 6. Composites of monads and monadic tensor products 4457
- 7. Limits and colimits of S-algebras 4760

- Chapter III. The homotopy theory of R-modules 5164
- 1. The category of R-modules; free and cofree R-modules 5164
- 2. Cell and CW R-modules; the derived category of R-modules 5467
- 3. The smash product of R-modules 5871
- 4. Change of S-algebras; q-cofibrant S-algebras 6174
- 5. Symmetric and extended powers of R-modules 6477
- 6. Function R-modules 6578
- 7. Commutative S-algebras and duality theory 6982

- Chapter IV. The algebraic theory of R-modules 7184
- 1. Tor and Ext; homology and cohomology; duality 7184
- 2. Eilenberg-Mac Lane spectra and derived categories 7487
- 3. The Atiyah-Hirzebruch spectral sequence 7891
- 4. Universal coefficient and Kunneth spectral sequences 8194
- 5. The construction of the spectral sequences 8396
- 6. Eilenberg-Moore type spectral sequences 8699
- 7. The bar constructions B(M,R,N) and B(X,G,Y) 88101

- Chapter V. R-ring spectra and the specialization to MU 91104
- Chapter VI. Algebraic K-theory of S-algebras 103116
- 1. Waldhausen categories and algebraic K-theory 103116
- 2. Cylinders, homotopies, and approximation theorems 106119
- 3. Application to categories of R-modules 110123
- 4. Homotopy invariance and Quillen's algebraic K-theory of rings 113126
- 5. Morita equivalence 115128
- 6. Multiplicative structure in the commutative case 119132
- 7. The plus construction description of KR 121134
- 8. Comparison with Waldhausen's K-theory of spaces 125138

- Chapter VII. R-algebras and topological model categories 127140
- 1. R-algebras and their modules 128141
- 2. Tensored and cotensored categories of structured spectra 130143
- 3. Geometric realization and calculations of tensors 135148
- 4. Model categories of ring, module, and algebra spectra 140153
- 5. The proofs of the model structure theorems 144157
- 6. The underlying R-modules of q-cofibrant R-algebras 148161
- 7. q-cofibrations and weak equivalences; cofibrations 151164

- Chapter VIII. Bousfield localizations of R-modules and algebras 155168
- Chapter IX. Topological Hochschild homology and cohomology 167180
- Chapter X. Some basic constructions on spectra 179192
- Chapter XI. Spaces of linear isometries and technical theorems 197210
- Chapter XII. The monadic bar construction 209222
- Chapter XIII. Epilogue: The category of L-spectra under S 215228
- Appendix A. Twisted half-smash products and function spectra 225238
- 1. Introduction 225238
- 2. The category J(U';U) 226239
- 3. Smash products and function spectra 227240
- 4. The object M α ∈ J(U';U) 229242
- 5. Twisted half-smash products and function spectra 231244
- 6. Formal properties of twisted half-smash products 234247
- 7. Homotopical properties of α [omitted] E and F[α,E') 237250
- 8. The cofibration theorem 239252
- 9. Equivariant twisted half-smash products 241254

- Bibliography 243256
- Index 247260