{
  "id": "1612.03732",
  "version": "v1",
  "published": "2016-12-12T15:08:42.000Z",
  "updated": "2016-12-12T15:08:42.000Z",
  "title": "On exotic equivalences and a theorem of Franke",
  "authors": [
    "Irakli Patchkoria"
  ],
  "categories": [
    "math.AT",
    "math.KT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-12T15:08:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P42",
      "18G55",
      "18E30"
    ],
    "keywords": [
      "exotic equivalences",
      "homotopy category",
      "derived category",
      "main examples",
      "homological dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}