On exotic equivalences and a theorem of Franke

TL;DR

The paper constructs new exotic equivalences between homotopy categories of ring spectra and derived categories, extending Franke's methods to spectra like $E(3)$ and $BP extless 2 extgreater$ for primes $p extgreater 5$, and clarifies a key proof gap.

Contribution

It extends Franke's methods to establish new equivalences for specific ring spectra and fills a gap in the proof of a known theorem about $E(1)$-local spectra.

Findings

Homotopy category of $R$-modules is equivalent to the derived category of $ ext{pi}_* R$ for certain spectra.

Triangulated equivalences are established for $E(2)$ and $BP extless 1 extgreater$ at primes $p extgreater 5$.

Fills a gap in the proof of Franke's theorem on $E(1)$-local spectra at primes $p extgreater 5$.

Abstract

Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ is concentrated in dimensions divisible by a natural number $N \geq 5$ and has homological dimension at most three, the homotopy category of $R$-modules is equivalent to the derived category of $\pi_*R$. The Johnson-Wilson spectrum $E(3)$ and the truncated Brown-Peterson spectrum $BP\langle 2 \rangle$ for any prime $p \geq 5$ are our main examples. If additionally the homological dimension of $\pi_*R$ is equal to two, then the homotopy category of $R$-modules and the derived category of $\pi_*R$ are triangulated equivalent. Here the main examples are $E(2)$ and $BP \langle 1 \rangle$ at $p \geq 5$. The last part of the paper discusses a triangulated equivalence between the homotopy category of $E(1)$-local spectra at a prime $p \geq 5$ and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof.