On exotic equivalences and a theorem of Franke

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 ${\pi }_{\ast }R$ is concentrated in dimensions divisible by a natural number $N\ge 5$ and has homological dimension at most three, the homotopy category of $R$-modules is equivalent to the derived category of ${\pi }_{\ast }R$. The Johnson-Wilson spectrum $E(3)$ and the truncated Brown-Peterson spectrum $BP⟨2⟩$ for any prime $p\ge 5$ are our main examples. If additionally the homological dimension of ${\pi }_{\ast }R$ is equal to two, then the homotopy category of $R$-modules and the derived category of ${\pi }_{\ast }R$ are triangulated equivalent. Here the main examples are $E(2)$ and $BP⟨1⟩$ at $p\ge 5$. The last part of the paper discusses a triangulated equivalence between the homotopy category of $E(1)$-local spectra at a prime $p\ge 5$ and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof.

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