{
  "id": "0811.4080",
  "version": "v1",
  "published": "2008-11-25T13:08:17.000Z",
  "updated": "2008-11-25T13:08:17.000Z",
  "title": "DG-methods for microlocalization",
  "authors": [
    "Stephane Guillermou"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "For a complex manifold $X$ the ring of microdifferential operators $\\E_X$ acts on the microlocalization $\\mu hom(F,\\O_X)$, for $F$ in the derived category of sheaves on $X$. Kashiwara, Schapira, Ivorra, Waschkies proved, as a byproduct of their new microlocalization functor for ind-sheaves, $\\mu_X$, that $\\mu hom(F,\\O_X)$ can in fact be defined as an object of the derived category of $\\E_X$-modules: this follows from the fact that $\\mu_X \\O_X$ is concentrated in one degree. In this paper we prove that the tempered microlocalization also is an object of the derived category of $\\E_X$-modules. Since we don't know whether the tempered version of $\\mu_X \\O_X$ is concentrated in one degree, we introduce a method to build suitable resolutions for which the action of $\\E_X$ is realized in the category of complexes. We define a version of the de Rham algebra on the subanalytic site which is quasi-injective and we work in the category of dg-modules over this de Rham algebra instead of the derived category of sheaves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-25T13:08:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A27",
      "32C38"
    ],
    "keywords": [
      "derived category",
      "dg-methods",
      "rham algebra",
      "microdifferential operators",
      "complex manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.4080G"
    }
  }
}