{
  "id": "0906.3109",
  "version": "v2",
  "published": "2009-06-17T08:42:18.000Z",
  "updated": "2016-11-30T10:49:14.000Z",
  "title": "Equivariant K-theory and Higher Chow Groups of Schemes",
  "authors": [
    "Amalendu Krishna"
  ],
  "comment": "25 pages. The title and abstract changed. The new results added. This is the final version. To appear in Proc. London Math. Soc",
  "categories": [
    "math.AG"
  ],
  "abstract": "For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary higher Chow groups. We obtain several applications to algebraic $K$-theory. We show that for a reductive group $G$ acting on a smooth projective scheme $X$, the forgetful map $K^G_i(X) \\to K_i(X)$ induces an isomorphism $K^G_i(X)/{I_G K^G_i(X)} \\cong K_i(X)$ with rational coefficients. This generalizes a result of Graham to higher $K$-theory of such schemes. We prove an equivariant Riemann-Roch theorem, leading to a generalization of a result of Edidin and Graham to higher $K$-theory. Similar techniques are used to prove the equivariant Quillen-Lichtenbaum conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-17T08:42:18.000Z",
      "title": "Equivariant K-theory and Higher Chow Groups of Smooth Varieties",
      "abstract": "For a quasi-projective variety $X$ over a field, with the action of a split torus, we construct a spectral sequence relating the equivariant and the ordinary higher Chow groups. We then completely describe the equivariant higher Chow groups of smooth projective varieties in terms of the ordinary higher Chow groups of certain subvarieties. As applications, we show that for a connected reductive group $G$ acting on a smooth variety $X$, the forgetful map from the rational equivariant higher $K$-theory to the ordinary $K$-theory is surjective and we describe its kernel. We also generalize the eqivariant Riemann-Roch theorem of Edididn and Graham to the higher K-theory of such varieties. We finally discuss the equivariant K-theory of these varieties with finite coefficients and prove the equivariant version of the Quillen-Licthenbaum conjecture as a simple application of the techniques involved in proving the above results.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-11-30T10:49:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C40",
      "14C35"
    ],
    "keywords": [
      "equivariant k-theory",
      "smooth variety",
      "ordinary higher chow groups",
      "equivariant higher chow groups",
      "rational equivariant higher"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.3109K"
    }
  }
}