## 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 a |

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### Contents

II | 9 |

IV | 11 |

V | 13 |

VI | 17 |

VII | 20 |

VIII | 22 |

IX | 25 |

X | 28 |

XLIX | 127 |

L | 128 |

LI | 130 |

LII | 135 |

LIII | 140 |

LIV | 144 |

LV | 148 |

LVI | 151 |

XI | 31 |

XIII | 35 |

XIV | 37 |

XV | 39 |

XVI | 42 |

XVII | 44 |

XVIII | 47 |

XIX | 51 |

XXI | 54 |

XXII | 58 |

XXIII | 61 |

XXIV | 64 |

XXV | 65 |

XXVI | 69 |

XXVII | 71 |

XXIX | 74 |

XXX | 78 |

XXXI | 81 |

XXXII | 83 |

XXXIII | 86 |

XXXIV | 88 |

XXXV | 91 |

XXXVII | 95 |

XXXVIII | 98 |

XXXIX | 101 |

XL | 103 |

XLII | 106 |

XLIII | 110 |

XLIV | 113 |

XLV | 115 |

XLVI | 119 |

XLVII | 121 |

XLVIII | 125 |

### Common terms and phrases

adjunction analogous apply arrow bar construction based spaces category of R-modules category of spectra cell R-module cellular cocomplete cofiber sequence cofibration colim colimits commutative R-algebras composite Corollary CW spectrum cylinder functor 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 L-spectra L-spectrum left R-module Lemma map of R-modules map of spectra model category monad morphisms natural isomorphism natural map obtain operad pair point-set level prespectrum PROOF properties Proposition pushout R-ring relative cell right adjoint smash product spectral sequence sphere stable homotopy category structure maps subcomplex tensor Theorem thhR(A twisted half-smash products Waldhausen category weak equivalence weakly wedge summand