On the Algebraic Classification of Module Spectra

Abstract

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 $\pi_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $\pi_*R$. 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 $p$.