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

9 | |

11 | |

13 | |

17 | |

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 |

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247 | |

### Common terms and phrases

adjunction analogous apply bar construction based spaces Bousfield localizations category of R-modules category of spectra cell R-module cellular cocomplete cofibration colimits commutative R-algebras composite Corollary CW spectrum define definition denote derived category diagrams commute E-acyclic E-local Eilenberg-Mac Lane equivalence of spectra equivariant Ex ring spectra exact functor fibration finite fl-modules following diagram following result forgetful functor full subcategory g-cofibrant g-fibrations geometric realization give given Hochschild homology homeomorphism homology and cohomology homotopy equivalence homotopy groups homotopy type implies inclusion indexed internal smash products L-spectra L-spectrum left R-module Lemma linear isometries map of R-modules map of spectra model category monad morphisms natural isomorphism natural map objects obtain operad pair point-set level prespectrum PROOF properties Proposition pushout R-ring relative cell smash product spectral sequence stable homotopy category structure maps subcomplex tensor Theorem thhR(A twisted half-smash products Waldhausen category weak equivalence weakly wedge summand

### 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.