{
  "id": "1303.2318",
  "version": "v2",
  "published": "2013-03-10T13:04:19.000Z",
  "updated": "2013-03-12T15:05:44.000Z",
  "title": "Graded quiver varieties and derived categories",
  "authors": [
    "Bernhard Keller",
    "Sarah Scherotzke"
  ],
  "comment": "39 pages; v2: title in metadata corrected",
  "categories": [
    "math.RT",
    "math.AG",
    "math.RA"
  ],
  "abstract": "Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category S. We determine the quiver of S and the number of minimal relations between any two given vertices. We construct a delta-functor Phi taking each finite-dimensional representation of S to an object of the derived category of Q. We show that the functor Phi establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Phi. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of \"line bundles over the non commutative curve given by Q\". We show that the degeneration order between strata corresponds to Jensen-Su-Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category S is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-12T15:05:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "derived category",
      "graded quiver varieties",
      "nakajimas graded affine quiver varieties",
      "jensen-su-zimmermanns degeneration order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2318K"
    }
  }
}