{
  "id": "1007.4543",
  "version": "v6",
  "published": "2010-07-26T19:48:46.000Z",
  "updated": "2015-04-06T20:02:58.000Z",
  "title": "Weights for relative motives; relation with mixed complexes of sheaves",
  "authors": [
    "Mikhail V. Bondarko"
  ],
  "comment": "a few minor corrections made",
  "journal": "Int. Math. Res. Notices 2014 (17): 4715--4767",
  "doi": "10.1093/imrn/rnt088",
  "categories": [
    "math.AG",
    "math.KT"
  ],
  "abstract": "The main goal of this paper is to define the so-called Chow weight structure for the category of Beilinson motives over any 'reasonable' base scheme $S$ (this is the version of Voevodsky's motives over $S$ defined by Cisinski and Deglise). We also study the functoriality properties of the Chow weight structure (they are very similar to the well-known functoriality of weights for mixed complexes of sheaves). As shown in a preceding paper, the Chow weight structure automatically yields an exact conservative weight complex functor (with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of the Chow weight structure; it is 'generated' by motives of regular schemes that are projective over $S$. Besides, Grothendiek's group of $S$-motives is isomorphic to $K_0(Chow(S))$; we also define a certain 'motivic Euler characteristic' for $S$-schemes. We obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation to Beilinson's 'integral part' of motivic cohomology and to weights of mixed complexes of sheaves. For the study of the latter we introduce a new formalism of relative weight structures.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-02-25T20:50:04.000Z",
      "abstract": "The main goal of this paper is to define the so-called Chow weight structure for the category of Beilinson motives over any 'reasonable' base scheme $S$ (this is the version of Voevodsky's motives over $S$ defined by Cisinski and Deglise). We also study the functoriality properties of the Chow weight structure (they are very similar to the well-known functoriality of weights for mixed complexes of sheaves). As shown in a preceding paper, the Chow weight structure automatically yields an exact conservative weight complex functor (with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of the Chow weight structure; it is 'generated' by motives of regular schemes that are projective over $S$. Besides, Grothendiek's group of $S$-motives is isomorphic to $K_0(Chow(S))$; we also define a certain 'motivic Euler characteristic' for $S$-schemes. We obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation with Beilinson's 'integral part' of motivic cohomology and with weights of mixed complexes of sheaves. For the study of the latter we introduce a new formalism of relative weight structures.",
      "comment": "Minor corrections made; exposition in section 2.3 was improved. Also, an improvement in the current version of the relative motivic treatise of Cisinski and Deglise allowed me to generalize some of my (weight-exactness) results to a wider class of morphisms of schemes",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2015-04-06T20:02:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C15",
      "19E15",
      "14C25",
      "14F20",
      "14E18",
      "18E30",
      "13D15",
      "18G40"
    ],
    "keywords": [
      "mixed complexes",
      "relative motives",
      "exact conservative weight complex functor",
      "chow weight structure automatically yields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4543B"
    }
  }
}