Rings, Modules, and Algebras in Stable Homotopy Theory

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 $Rwedge _SR longrightarrow R$. 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 $Rwedge _SMlongrightarrow M$. 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.