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.
Author(s): A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, and M. Cole
Series: Mathematical Surveys and Monographs
Publisher: American Mathematical Society
Year: 2007
Language: English
Commentary: Vector PDF with bookmarks, cover and pagination. Missing Appendix
Pages: 269
Cover
RINGS, MODULES, AND ALGEBRAS IN STABLE HOMOTOPY THEORY
Abstract
Contents
Introduction
Prologue: the category of L-spectra
1. Background on spectra and the stable homotopy category
2. External smash products and twisted half-smash products
3. The linear isometries operad and internal smash products
4. The category of L-spectra
5. The smash product of L-spectra
6. The equivalence of the old and new smash products
7. Function L-spectra
8. Unital properties of the smash product of L-spectra
Structured ring and module spectra
1. The category of S-modules
2. The mirror image to the category of S-modules
3. S-algebras and their modules
4. Free A1 and E1 ring spectra and comparisons of definitions
5. Free modules over A1 and E1 ring spectra
6. Composites of monads and monadic tensor products
7. Limits and colimits of S-algebras
The homotopy theory of R-modules
1. The category of R-modules; free and cofree R-modules
2. Cell and CW R-modules; the derived category of R-modules
3. The smash product of R-modules
4. Change of S-algebras; q-cofibrant S-algebras
5. Symmetric and extended powers of R-modules
6. Function R-modules
7. Commutative S-algebras and duality theory
The algebraic theory of R-modules
1. Tor and Ext; homology and cohomology; duality
2. Eilenberg-Mac Lane spectra and derived categories
3. The Atiyah-Hirzebruch spectral sequence
4. Universal coefficient and Knneth spectral sequences
5. The construction of the spectral sequences
6. Eilenberg-Moore type spectral sequences
7. The bar constructions B(M; R; N) and B(X; G; Y )
R-ring spectra and the specialization to MU
1. Quotients by ideals and localizations
2. Localizations and quotients of R-ring spectra
3. The associativity and commutativity of R-ring spectra
4. The specialization to MU-modules and algebras
Algebraic K-theory of S-algebras
1. Waldhausen categories and algebraic K-theory
2. Cylinders, homotopies, and approximation theorems
3. Application to categories of R-modules
4. Homotopy invariance and Quillen's algebraic K-theory of rings
5. Morita equivalence
6. Multiplicative structure in the commutative case
7. The plus construction description of KR
8. Comparison with Waldhausen's K-theoryofspaces
R-algebras and topological model categories
1. R-algebras and their modules
2. Tensored and cotensored categories of structured spectra
3. Geometric realization and calculations of tensors
4. Model categories of ring, module, and algebra spectra
5. The proofs of the model structure theorems
6. The underlying R-modules of q-cofibrant R-algebras
Bousfield localizations of R-modules and algebras
1. Bousfield localizations of R-modules
2. Bousfield localizations of R-algebras
3. Categories of local modules
4. Periodicity and K-theory
Topological Hochschild homology and cohomology
1. Topological Hochschild homology: rst definition
2. Topological Hochschild homology: second definition
3. The isomorphism between thhR(A) and A S1
Some basic constructions on spectra
1. The geometric realization of simplicial spectra
2. Homotopical and homological properties of realization
3. Homotopy colimits and limits
4. Sigma-cofibrant, LEC, and CW prespectra
5. The cylinder construction
Spaces of linear isometries and technical theorems
1. Spaces of linear isometries
2. Fine structure of the linear isometries operad
3. The unit equivalence for the smash product of L-spectra
4. Twisted half-smash products and shift desuspension
5. Twisted half-smash products and cofibrations
The monadic bar construction
1. The bar construction and two deferred proofs
2. Cofibrations and the bar construction
Epilogue: The category of L-spectra under S
1. The modified smash products CL, BL,and?L
2. The modified smash products CR, BR,and?R
Bibliography
