On exotic equivalences and a theorem of Franke

Irakli Patchkoria* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)1085-1099
Number of pages15
JournalBulletin of the London Mathematical Society
Volume49
Issue number6
Early online date27 Oct 2017
Publication statusPublished - Dec 2017

Keywords

  • 55P42
  • 18E30 (primary)
  • 18G55 (secondary)

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