{
  "id": "1901.02109",
  "version": "v1",
  "published": "2019-01-08T00:11:36.000Z",
  "updated": "2019-01-08T00:11:36.000Z",
  "title": "Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra",
  "authors": [
    "Agnes Beaudry",
    "Irina Bobkova",
    "Michael Hill",
    "Vesna Stojanoska"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height 2 Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition to a cyclic subgroup of order 32 generated by $ E\\wedge S^1$ the Picard group contains a subgroup of order 2 generated by $E\\wedge S^{7+\\sigma}$, where $\\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO(C_4)$-graded Mackey functor homotopy fixed point spectral sequence for the $C_4$-spectrum $E$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-08T00:11:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55M05",
      "55P42",
      "20J06",
      "55Q91",
      "55Q51",
      "55P60"
    ],
    "keywords": [
      "homotopy fixed point spectral sequence",
      "functor homotopy fixed point spectral",
      "mackey functor homotopy fixed point",
      "picard group contains",
      "morava stabilizer group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}