{
  "id": "math/0509684",
  "version": "v4",
  "published": "2005-09-29T09:55:14.000Z",
  "updated": "2007-10-05T10:05:20.000Z",
  "title": "Under Spec Z",
  "authors": [
    "Bertrand Toen",
    "Michel Vaquie"
  ],
  "comment": "45 pages, french. Several mistakes corrected.The central definition of scheme is slightly modified. Section 2.1 is new and contains more details about the fpqc topology",
  "categories": [
    "math.AG",
    "math.CT"
  ],
  "abstract": "We use techniques from relative algebraic geometry and homotopical algebraic geometry in order to construct several categories of schemes defined \"under Spec Z\". We define this way the categories of N-schemes, F_1-schemes, S-schemes, S_+-schemes, and S_1-schemes, where from a very intuitive point of view N is the semi-ring of natural numbers, F_1 is the field with one element, S is the sphere ring spectrum, S_+ is the semi-ring spectrum of natural numbers and S_1 is the ring spectrum with one element. These categories of schemes are related by several base change functors, and they all possess a base change functor to Z-schemes (in the usual sense). Finally, we show how the linear group Gl_n and toric varieties can be defined as objects in certain of these categories.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-10-05T10:05:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "base change functor",
      "categories",
      "natural numbers",
      "toric varieties",
      "relative algebraic geometry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9684T"
    }
  }
}