On the algebraic classification of module spectra

Abstract

Using methods developed by Franke in [K-theory Preprint Archives 139 (1996)], 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 ${\pi }_{\ast }R$ has graded global homological dimension $2$ and is concentrated in degrees divisible by some natural number $N\ge 4$, we prove that the homotopy category of $R$–modules is equivalent to the derived category of the homotopy ring ${\pi }_{\ast }R$. 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 $k{o}_{\left(p\right)}$, the Johnson–Wilson spectrum $E\left(2\right)$, and the truncated Brown–Peterson spectrum $BP\langle 1\rangle$, all for an odd prime $p$. We also show that the equivalences for all these examples are exotic in the sense that they do not come from a zigzag of Quillen equivalences.