On exotic equivalences and a theorem of Franke

Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum 푅 whose graded homotopy ring 휋∗푅 is concentrated in dimensions divisible by a natural number 푁⩾5 and has homological dimension at most three, the homotopy category of 푅 ‐modules is equivalent to the derived category of 휋∗푅 . The Johnson–Wilson spectrum 퐸(3) and the truncated Brown–Peterson spectrum 퐵푃⟨2⟩ for any prime 푝⩾5 are our main examples. If additionally the homological dimension of 휋∗푅 is equal to two, then the homotopy category of 푅 ‐modules and the derived category of 휋∗푅 are triangulated equivalent. Here the main examples are 퐸(2) and 퐵푃⟨1⟩ at 푝⩾5 . The last part of the paper discusses a triangulated equivalence between the homotopy category of 퐸(1) ‐local spectra at a prime 푝⩾5 and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof.