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 whose graded homotopy ring has graded global homological dimension and is concentrated in degrees divisible by some natural number , we prove that the homotopy category of –modules is equivalent to the derived category of the homotopy ring . 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 –local real connective –theory spectrum , the Johnson–Wilson spectrum , and the truncated Brown–Peterson spectrum , all for an odd prime . 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.
Irakli Patchkoria. "On the algebraic classification of module spectra." Algebr. Geom. Topol. 12 (4) 2329 - 2388, 2012. https://doi.org/10.2140/agt.2012.12.2329