{
  "id": "1408.1537",
  "version": "v2",
  "published": "2014-08-07T10:45:40.000Z",
  "updated": "2015-08-25T07:52:23.000Z",
  "title": "On rational equivalence in tropical geometry",
  "authors": [
    "Lars Allermann",
    "Simon Hampe",
    "Johannes Rau"
  ],
  "comment": "17 pages, 2 figures, updated to fit published version, Canadian Journal of Mathematics (2015)",
  "doi": "10.4153/CJM-2015-036-0",
  "categories": [
    "math.AG"
  ],
  "abstract": "This article discusses the concept of rational equivalence in tropical geometry (and replaces the older and imperfect version arXiv:0811.2860). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the \"bounded\" Chow groups of $\\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in $\\mathbb{R}^n$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-07T10:45:40.000Z",
      "abstract": "This article discusses the concept of rational equivalence in tropical geometry (and replaces the older and imperfect version arXiv:0811.2860). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the Chow groups of R^n by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).",
      "comment": "16 pages, 2 figures, updated version of arXiv:0811.2860",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-08-25T07:52:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14T05"
    ],
    "keywords": [
      "rational equivalence",
      "tropical geometry",
      "fan cycles",
      "tropical cycle",
      "independent interest"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.1537A"
    }
  }
}