{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field, and let $Z_r(X)$ be the group of $r$-dimensional [[algebraic cycles|9VFWPZEVGN]] on $X$.\nLet $W \\subset X$ be an irreducible closed subvariety of dimension $r + 1$, and let $f$ be a nonzero element of its [[function field|2D5SS6WM14]] $k(W)$.\nFor each irreducible closed subvariety $V \\subset W$ of dimension $r$, the local ring $A = \\mathcal{O}_{W, V}$ is a one-dimensional Noetherian local domain with fraction field $k(W)$.\nWrite $f = a/b$ with $a, b \\in A \\setminus \\set{0}$.\nSet $\\operatorname{ord}_V(f) = \\ell_A(A/aA) - \\ell_A(A/bA)$, where $\\ell_A$ is the [[length|V6JBKT168T]] of an $A$-module.\nThis does not depend on the choice of $a$ and $b$.\nThe *divisor* of $f$ is the $r$-cycle $\\operatorname{div}(f) = \\sum_V \\operatorname{ord}_V(f) \\, [V]$, a finite sum.\nLet $\\operatorname{Rat}_r(X) \\subset Z_r(X)$ be the subgroup generated by all such $\\operatorname{div}(f)$, for all $W$ and $f$.\nTwo $r$-cycles $Z, Z'$ are *rationally equivalent* if $Z - Z' \\in \\operatorname{Rat}_r(X)$.\n\n## Remarks\n\nWhen $W$ is normal along $V$, $\\operatorname{ord}_V$ is the discrete valuation of $k(W)$ at $V$.\nExample: on $X = \\mathbb{P}^1$ over a field $k$, take $W = X$ and $f = (x - a)/(x - b)$ with $a \\ne b$ in $k$.\nThen $\\operatorname{div}(f) = [a] - [b]$, so the points $a$ and $b$ are rationally equivalent.\nNon-example: on an elliptic curve $E$ over an algebraically closed field, two distinct points $P$ and $Q$ are not rationally equivalent.\nIf they were, then $[P] - [Q] = \\operatorname{div}(f)$ for some $f \\in k(E)^{\\times}$, since $W = E$ is the only choice on a curve.\nThen $f$ would be a rational function of degree $1$, hence an isomorphism $E \\to \\mathbb{P}^1$, which is impossible because $E$ has genus $1$.\nEquivalently, $Z - Z' = \\sum_i \\big([V_i(0)] - [V_i(\\infty)]\\big)$ for finitely many subvarieties $V_i \\subset X \\times \\mathbb{P}^1$ of dimension $r + 1$ that map onto $\\mathbb{P}^1$, where $V_i(a)$ is the fiber of $V_i$ over $a$, viewed in $X$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "K76N026JFK",
    "kind": "definition",
    "msc": [
      "14C15",
      "14C25"
    ],
    "name": "Rational equivalence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chow_group"
      }
    ],
    "tag": "95N3QVHAQN",
    "uses": [
      "9VFWPZEVGN",
      "HV6YBS0YDH",
      "2D5SS6WM14",
      "V6JBKT168T"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/95N3QVHAQN/",
  "links": [
    {
      "field": "defines",
      "target": "K76N026JFK"
    },
    {
      "field": "uses",
      "target": "9VFWPZEVGN"
    },
    {
      "field": "uses",
      "target": "HV6YBS0YDH"
    },
    {
      "field": "uses",
      "target": "2D5SS6WM14"
    },
    {
      "field": "uses",
      "target": "V6JBKT168T"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/95N3QVHAQN.md",
  "msc": [
    "14C15",
    "14C25"
  ],
  "name": "Rational equivalence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/95N/95N3QVHAQN.md",
  "status": null,
  "tag": "95N3QVHAQN",
  "uri": "m2://95N3QVHAQN",
  "url": "https://tamnd.github.io/math2/wiki/rational-equivalence-definition/",
  "writing": {
    "tag": "95N3QVHAQN",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}