{
  "id": "1401.6685",
  "version": "v2",
  "published": "2014-01-26T20:19:16.000Z",
  "updated": "2018-03-12T08:20:30.000Z",
  "title": "Higher-dimensional study of extensions via torsors",
  "authors": [
    "Cristiana Bertolin",
    "Ahmet Emin Tatar"
  ],
  "comment": "We change the title and we add new results",
  "journal": "Ann. Mat. Pura Appl. 197 (2018), no. 2, pp. 433--468",
  "doi": "10.1007/s10231-017-0686-8",
  "categories": [
    "math.AG",
    "math.CT"
  ],
  "abstract": "Let S be a site. First we define the 3-category of torsors under a Picard S-2-stack and we compute its homotopy groups. Using calculus of fractions we define also a pure algebraic analogue of the 3-category of torsors under a Picard S-2-stack. Then we describe extensions of Picard S-2-stacks as torsors endowed with a group law on the fibers. As a consequence of such a description, we show that any Picard S-2-stack admits a canonical free partial left resolution that we compute explicitly. Moreover we get an explicit right resolution of the 3-category of extensions of Picard S-2-stacks in terms of 3-categories of torsors. Using the homological interpretation of Picard S-2-stacks, we rewrite this three categorical dimensions higher right resolution in the derived category of abelian sheaves on S.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-26T20:19:16.000Z",
      "title": "Resolution of extensions of Picard 2-stacks",
      "abstract": "Let S be a site. First we define the 3-category of torsors under a Picard S-2-stack and we furnish (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows and 3-arrows of the 3-category of torsors under a Picard S-2-stack by the cohomology groups of the derived functor of the functor of global sections, and (2) a geometrical description of the cohomology groups of the derived functor of the functor of global sections applied to length 3 complexes of abelian sheaves via torsors under a Picard S-2-stack. Then we describe extensions of Picard S-2-stacks in term of torsors under a Picard S-2-stack which are endowed with a group law on the fibers. As a consequence of such a description, we get an explicit right resolution of the 3-category of extensions of Picard S-2-stacks in terms of 3-categories of torsors under a Picard S-2-stack. Using the dictionary between the derived category of abelian sheaves on S and the category of Picard S-2-stacks, we rewrite this categorical right resolution in homological terms.",
      "comment": "29 pages. Continuation of arXiv:1205.6308 with new results",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-03-12T08:20:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18G15",
      "18D05"
    ],
    "keywords": [
      "extensions",
      "abelian sheaves",
      "cohomology groups",
      "global sections",
      "derived functor"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.6685B"
    }
  }
}