{
  "id": "1810.04493",
  "version": "v1",
  "published": "2018-10-10T12:59:54.000Z",
  "updated": "2018-10-10T12:59:54.000Z",
  "title": "Macaulayfication of Noetherian schemes",
  "authors": [
    "Kestutis Cesnavicius"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AG",
    "math.AC",
    "math.NT"
  ],
  "abstract": "To reduce to resolving Cohen-Macaulay singularities, Faltings initiated the program of \"Macaulayfying\" a given Noetherian scheme $X$. For a wide class of $X$, Kawasaki built the sought Cohen-Macaulay modifications, with a crucial drawback that his blowups did not preserve the locus $\\mathrm{CM}(X) \\subset X$ where $X$ is already Cohen-Macaulay. We extend Kawasaki's methods to show that every quasi-excellent, Noetherian scheme $X$ has a Cohen-Macaulay $\\widetilde{X}$ with a proper map $\\widetilde{X} \\rightarrow X$ that is an isomorphism over $\\mathrm{CM}(X)$. This completes Faltings' program, reduces the conjectural resolution of singularities to the Cohen-Macaulay case, and implies that every proper, smooth scheme over a number field has a proper, flat, Cohen-Macaulay model over the ring of integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-10T12:59:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E15",
      "13H10",
      "14B05",
      "14J17",
      "14M05"
    ],
    "keywords": [
      "noetherian scheme",
      "macaulayfication",
      "sought cohen-macaulay modifications",
      "extend kawasakis methods",
      "wide class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}