# Rings, Modules, and Algebras in Stable Homotopy Theory

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 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
 LVII 155 LIX 159 LX 163 LXI 165 LXII 167 LXIII 168 LXIV 172 LXV 176 LXVI 179 LXVIII 182 LXIX 186 LXX 188 LXXI 191 LXXII 197 LXXIV 200 LXXV 205 LXXVI 209 LXXVIII 211 LXXIX 215 LXXXI 219 LXXXII 225 LXXXIV 226 LXXXV 227 LXXXVI 229 LXXXVII 231 LXXXVIII 234 LXXXIX 237 XC 239 XCI 241 XCII 243 XCIII 247 Copyright