We define a (left) module M over an S-algebra R to be an S-module M with an action R ∧S M −→ M such that the standard diagrams commute. We obtain a category MR of (left) R-modules and a derived category DR. There is a smash product M ∧R N of a right R-module M and a left R-module N, which is an Smodule. For left R-modules M and N, there is a function S-module FR(M, N) that enjoys properties just like modules of homomorphisms in algebra. Each FR(M, M) is an S-algebra. If R is commutative, then M ∧R N and FR(M, N) are R-modules, and in this case MR and DR enjoy all of the properties of MS and DS. Thus each commutative S-algebra R determines a derived category of R-modules that has all of the structure that the stable homotopy category has. These new categories are of substantial intrinsic interest, and they give powerful new tools for the investigation of the classical stable homotopy category.

Upon restriction to Eilenberg-Mac Lane spectra, our topological theory subsumes a good deal of classical algebra. For a discrete ring R and R-modules M and N, we have TorR n (M, N) ∼= πn(HM ∧HR HN) and Extn R(M, N) ∼= π−nFHR(HM, HN). Here ∧R and FR must be interpreted in the derived category; that is, HM must be a CW HR-module. Moreover, the algebraic derived category DR is equivalent to the topological derived category DHR. In general, for an S-algebra R, approximation of R-modules M by weakly equivalent cell R-modules is roughly analogous to forming projective resolutions in algebra. There is a much more precise analogy that involves developing the derived INTRODUCTION 3 categories of modules over rings or, more generally, DGA’s in terms of cell modules. It is presented in [34], which gives an algebraic theory of A∞ and E∞ k-algebras that closely parallels the present topological theory. Upon restriction to the sphere spectrum S, the derived smash products M ∧S N and function spectra FS(M, N) have as their homotopy groups the homology and cohomology groups N∗(M) and N∗ (M). This suggests the alternative notations