Classifying modules over K-theory spectra

Abstract

The category of modules over an S-algebra ( A ∞ or E ∞ ring spectrum) has many of the good properties of the category of spectra. When the homotopy groups of the S-algebra in question form a sufficiently nice ring, it is possible to see the deviation of the category of modules over an S-algebra from the corresponding algebraic module category. In particular, many algebraic modules are realized as homotopy groups of topological modules over S-algebras. Examples studied include real and complex K-theory, both connective and periodic. Further, Bousfield localization by a smashing spectrum is shown to yield a category of modules over the localized sphere. For periodic K-theory, these methods yield an algebraic criterion to determine when a local spectrum is a module over the K-theory S-algebra, real or complex.