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

American Mathematical Soc., 2007 - Mathematics - 249 pages
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

### Popular passages

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.