{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "3HKREN3Y3F"
    },
    {
      "field": "uses",
      "target": "3HKREN3Y3F"
    },
    {
      "field": "uses",
      "target": "4GG7VRW0DY"
    },
    {
      "field": "uses",
      "target": "6GW1FHF9MB"
    },
    {
      "field": "uses",
      "target": "9E9T41A7XN"
    },
    {
      "field": "uses",
      "target": "B76G1KR3SB"
    },
    {
      "field": "defines",
      "target": "W6V340HYNK"
    }
  ],
  "body": "\n## Summary\n\nThe Chow groups of a variety $X$ are the groups of [[algebraic cycles|9VFWPZEVGN]] on $X$ modulo [[rational equivalence|K76N026JFK]], one for each dimension.\nThey are the algebraic analogue of homology groups.\nFor a smooth variety, the intersection product turns their sum into a graded ring, the Chow ring, which is the setting of [[intersection theory|4HV4WTAX9P]].\nIn codimension one they recover the divisor class group and the [[Picard group|3Z9DCWYXQZ]] of a smooth variety.\n\n## Explanations\n\n### L1 human\n\nA Chow group counts subvarieties of a given dimension, where two of them count as the same when one can be deformed into the other through a rational family.\nOn the projective plane, every curve of degree $d$ is equivalent to $d$ times a line, so the group of curves is $\\Z$, generated by the class of a line.\nOn an elliptic curve, the group of points is much bigger: it records the degree and also a point of the curve itself.\nThe Chow groups are thus simple for rational varieties and can be enormous in general.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14C15",
      "14C25"
    ],
    "name": "Chow group",
    "same_as": {
      "wikidata": "Q5105523",
      "wikipedia": "Chow group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chow_group"
      }
    ],
    "tag": "EBEF6PEK4N"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/EBEF6PEK4N/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/EBEF6PEK4N.md",
  "msc": [
    "14C15",
    "14C25"
  ],
  "name": "Chow group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EBE/EBEF6PEK4N.md",
  "status": null,
  "tag": "EBEF6PEK4N",
  "uri": "m2://EBEF6PEK4N",
  "url": "https://tamnd.github.io/math2/wiki/chow-group/",
  "writing": {
    "tag": "EBEF6PEK4N",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}