Rings, Modules, and Algebras in Stable Homotopy Theory

Introduces a fresh point-set level approach to stable homotopy theory that has had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $S$, this title constructs a smash product in a complete category of '$S$-modules' whose derived category is equivalent to the classical stable homotopy category.

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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 a