{
  "id": "math/0111049",
  "version": "v2",
  "published": "2001-11-05T17:04:05.000Z",
  "updated": "2002-05-28T11:57:55.000Z",
  "title": "Presheaves of triangulated categories and reconstruction of schemes",
  "authors": [
    "Paul Balmer"
  ],
  "comment": "18 pages; minor changes",
  "categories": [
    "math.AG",
    "math.CT"
  ],
  "abstract": "To any triangulated category with tensor product $(K,\\otimes)$, we associate a topological space $Spc(K,\\otimes)$, by means of thick subcategories of $K$, a la Hopkins-Neeman-Thomason. Moreover, to each open subset $U$ of $Spc(K,\\otimes)$, we associate a triangulated category $K(U)$, producing what could be thought of as a presheaf of triangulated categories. Applying this to the derived category $(K,\\otimes):=(D^{perf}(X),\\otimes^L)$ of perfect complexes on a noetherian scheme $X$, the topological space $Spc(K,\\otimes)$ turns out to be the underlying topological space of $X$; moreover, for each open $U\\subset X$, the category $K(U)$ is naturally equivalent to $D^{perf}(U)$. As an application, we give a method to reconstruct any reduced noetherian scheme $X$ from its derived category of perfect complexes $D^{perf}(X)$, considering the latter as a tensor triangulated category with $\\otimes^L$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-05-28T11:57:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E30"
    ],
    "keywords": [
      "reconstruction",
      "presheaves",
      "perfect complexes",
      "derived category",
      "tensor product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....11049B"
    }
  }
}