{
  "id": "1308.0135",
  "version": "v2",
  "published": "2013-08-01T09:55:00.000Z",
  "updated": "2017-06-19T12:48:27.000Z",
  "title": "Homotopy finiteness of some DG categories from algebraic geometry",
  "authors": [
    "Alexander I. Efimov"
  ],
  "comment": "65 pages, no figures; v2: numerous misprints corrected, the notation changed for clarification, proofs of several technical statements added, references added",
  "categories": [
    "math.AG",
    "math.CT",
    "math.RA"
  ],
  "abstract": "In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: $D^b_{coh}(Y)$ is equivalent to a DG quotient $D^b_{coh}(\\tilde{Y})/T,$ where $\\tilde{Y}$ is some smooth and proper variety, and the subcategory $T$ is generated by a single object. The proof uses categorical resolution of singularities of Kuznetsov and Lunts \\cite{KL}, and a theorem of Orlov \\cite{Or} stating that the class of geometric smooth and proper DG categories is stable under gluing. We also prove the analogous result for $\\Z/2$-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of $D^b_{coh}(\\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over $\\mathbb{A}_{\\mathrm{k}}^1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-01T09:55:00.000Z",
      "abstract": "In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary resolution of singularities. We believe that homotopy finiteness holds also over perfect fields of finite characteristic. We also prove the analogous result for $\\Z/2$-graded DG categories of coherent matrix factorizations on such schemes. In both cases, we represent our DG category as a DG quotient of a smooth and proper DG category by a subcategory generated by one object (we call this a smooth categorical compactification).",
      "comment": "58 pages, no figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-06-19T12:48:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic geometry",
      "coherent matrix factorizations",
      "homotopy finiteness holds",
      "proper dg category",
      "dg quotient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.0135E"
    }
  }
}