{
  "id": "1311.1661",
  "version": "v2",
  "published": "2013-11-07T12:48:46.000Z",
  "updated": "2014-10-27T04:45:05.000Z",
  "title": "Chow groups of ind-schemes and extensions of Saito's filtration",
  "authors": [
    "Abhishek Banerjee"
  ],
  "comment": "Version 2 (corrects some errors in previous version; several new results added)",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $K$ be a field of characteristic zero and let $Sm/K$ be the category of smooth and separated schemes over $K$. For an ind-scheme $\\mathcal X$ (and more generally for any presheaf of sets on $Sm/K$), we define its Chow groups $\\{CH^p(\\mathcal X)\\}_{p\\in \\mathbb Z}$. We also introduce Chow groups $\\{\\mathcal{CH}^p(\\mathcal G)\\}_{p\\in \\mathbb Z}$ for a presheaf with transfers $\\mathcal G$ on $Sm/K$. Then, we show that we have natural isomorphisms of Chow groups $$ CH^p(\\mathcal X)\\cong \\mathcal{CH}^p(Cor(\\mathcal X))\\qquad\\forall\\textrm{ }p \\in \\mathbb Z$$ where $Cor(\\mathcal X)$ is the presheaf with transfers that associates to any $Y\\in Sm/K$ the collection of finite correspondences from $Y$ to $\\mathcal X$. Additionally, when $K=\\mathbb C$, we show that Saito's filtration on the Chow groups of a smooth projective scheme can be extended to the Chow groups $CH^p(\\mathcal X)$ and more generally, to the Chow groups of an arbitrary presheaf of sets on $Sm/\\mathbb C$. Similarly, there exists an extension of Saito's filtration to the Chow groups of a presheaf with transfers on $Sm/\\mathbb C$. Finally, when the ind-scheme $\\mathcal X$ is ind-proper, we show that the isomorphism $CH^p(\\mathcal X)\\cong \\mathcal{CH}^p(Cor(\\mathcal X))$ is actually a filtered isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-07T12:48:46.000Z",
      "title": "The motive of a sheaf on the category of smooth schemes",
      "abstract": "We define the motive of an \\'{e}tale sheaf on the category $Sm/K$ of smooth and separated schemes over a field $K$ of characteristic zero. This enables us to construct motives that capture information about a wider class of \"morphisms\" between schemes, such as morphisms between corresponding analytic spaces (when $K=\\mathbb C$) or morphisms at the level of cohomology. Further, we are also able to define motives of objects such as algebraic spaces and simplicial schemes. Our theory naturally leads us to consider Chow groups of presheaves and presheaves with transfers and these are defined in the spirit of bivariant Chow groups. Additionally, we extend Saito's filtration on the Chow groups of a smooth projective scheme over $\\mathbb C$ to Chow groups of an arbitrary presheaf on $Sm/\\mathbb C$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-27T04:45:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth schemes",
      "bivariant chow groups",
      "extend saitos filtration",
      "construct motives",
      "define motives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.1661B"
    }
  }
}