On the Algebraic Classification of Module Spectra

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Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra (S-algebras). In particular, for any symmetric ring spectrum R whose graded homotopy ring π*R has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number N ≥ 4, we prove that the homotopy category of R-modules is equivalent to the derived category of the homotopy ring π*R. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of R-modules. The main examples of ring spectra to which our result applies are the p-local real connective K-theory spectrum ko₍ₚ₎, the Johnson-Wilson spectrum E(2), and the truncated Brown-Peterson spectrum BP<1>, for an odd prime p.