Rings, Modules and Algebras in Stable Homotopy Theory (Mathematical surveys and Monographs 47)

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 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.