Rings, Modules, and Algebras in Stable Homotopy Theory (Google eBook)

Front Cover
American Mathematical Soc., 2007 - Mathematics - 249 pages
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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.
  

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Contents

Prologue the category of Lspectra
9
2 External smash products and twisted halfsmash products
11
3 The linear isometries operad and internal smash products
13
4 The category of Lspectra
17
5 The smash product of Lspectra
20
6 The equivalence of the old and new smash products
22
7 Function Lspectra
25
8 Unital properties of the smash product of Lspectra
28
Ralgebras and topological model categories
127
1 Ralgebras and their modules
128
2 Tensored and cotensored categories of structured spectra
130
3 Geometric realization and calculations of tensors
135
4 Model categories of ring module and algebra spectra
140
5 The proofs of the model structure theorems
144
6 The underlying Rmodules of qcofibrant Ralgebras
148
7 qCofibrations and weak equivalences cofibrations
151

Structured ring and module spectra
31
2 The mirror image to the category of Smodules
35
3 5algebras and their modules
37
4 Free Ax and Ex ring spectra comparisons of definitions
39
5 Free modules over A and E ring spectra
42
6 Composites of monads and monadic tensor products
44
7 Limits and colimits of Salgebras
47
The homotopy theory of Rmodules
51
2 Cell and CW Rmodules the derived category of Rmodules
54
3 The smash product of Rmodules
58
4 Change of Salgebras gcofibrant Salgebras
61
5 Symmetric and extended powers of Rmodules
64
6 Function Rmodules
65
7 Commutative Salgebras and duality theory
69
The algebraic theory of Rmodules
71
2 EilenbergMac Lane spectra and derived categories
74
3 The AtiyahHirzebruch spectral sequence
78
4 Universal coefficient and Kihmeth spectral sequences
81
5 The construction of the spectral sequences
83
6 EilenbergMoore type spectral sequences
86
7 The bar constructions BMRN and BXGY
88
Rring spectra and the specialization to MU
91
2 Localizations and quotients of Rring spectra
95
3 The associativity and commutativity of Rring spectra
98
4 The specialization to MUmodules and algebras
101
Algebraic Ktheory of Salgebras
103
2 Cylinders homotopies and approximation theorems
106
3 Application to categories of Rmodules
110
4 Homotopy invariance and Quillens algebraic Ktheory of rings
113
5 Morita equivalence
115
6 Multiplicative structure in the commutative case
119
7 The plus construction description of K R
121
8 Comparison with Waldhausens Ktheory of spaces
125
Bousfield localizations of Rmodules and algebras
155
2 Bousfield localizations of Ralgebras
159
3 Categories of local modules
163
4 Periodicity and Xtheory
165
Topological Hochschild homology and cohomology
167
first definition
168
second definition
172
3 The isomorphism between thhRA and A S
176
Some basic constructions on spectra
179
2 Homotopical and homological properties of realization
182
3 Homotopy colimits and limits
186
4 Ecofibrant LEG and CW prespectra
188
5 The cylinder construction
191
Spaces of linear isometrics and technical theorems
197
2 Fine structure of the linear isometries operad
200
3 The unit equivalence for the operadic smash product
205
The monadic bar construction
209
2 Cofibrations and the bar construction
211
Epilogue The category of Lspectra under S
215
2 The modified smash products R R and R
219
Twisted halfsmash products and function spectra
225
2 The category 𝒮U U
226
3 Smash products and function spectra
227
4 The object μα ϵ 𝒮U U
229
5 Twisted halfsmash products and function spectra
231
6 Formal properties of twisted halfsmash products
234
7 Homotopical properties of α E and Fα E
237
8 The cofibration theorem
239
9 Equivariant twisted halfsmash products
241
Bibliography
243
Index
247
Copyright

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Page 12 - Functors on prespectra that do not preserve spectra are extended to spectra by applying the functor L. For example, for a based space X and a prespectrum E, we have the prespectrum E/\X specified by (E/\X)(V) = EV/\X.
Page 13 - V') = EV A E'V. The structure maps fail to be homeomorphisms when E and E' are spectra, and we apply the spectrification functor L to obtain the desired spectrum level smash product.

References to this book

Model Categories
Mark Hovey
No preview available - 1999
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