On the Algebraic Classification of Module Spectra

TL;DR

This paper establishes algebraic classification results for modules over certain symmetric ring spectra, showing their homotopy categories are equivalent to derived categories of their homotopy rings, under specific conditions.

Contribution

It extends the algebraic classification of module spectra by proving equivalences for spectra with graded homotopy rings of global dimension 2 and specific degree concentration.

Findings

Homotopy category of modules over certain spectra is equivalent to derived category of homotopy ring.

Applicable to spectra like $ko_{(p)}$, E(2), and $BP<1>$ at odd primes.

Improves previous algebraic classification results.

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 $p$.