{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be an [[algebraic variety|HV6YBS0YDH]] over an algebraically closed field $k$, with its Zariski topology.\nThe *dimension* $\\dim X$ is the supremum of the integers $n$ such that there is a chain\n$$Z_0 \\subsetneq Z_1 \\subsetneq \\cdots \\subsetneq Z_n$$\nof [[irreducible|SY2MWC434Z]] [[closed subsets|0ZCK8DQYN1]] $Z_i$ of $X$.\n\n## Remarks\n\nFor every variety this supremum is finite.\nIf $X$ is an [[affine variety|JERVQT4C12]], then $\\dim X$ is the [[Krull dimension|MVGBS3J1P7]] of its [[coordinate ring|G9WB27VWP7]] $k[X]$.\nThe reason is that irreducible closed subsets of $X$ correspond to prime ideals of $k[X]$, with inclusions reversed.\nFor every variety, $\\dim X$ equals the [[transcendence degree|D7TR0YWR9H]] of the [[function field|2D5SS6WM14]] $k(X)$ over $k$.\nA nonempty open subset $U$ of $X$ has $\\dim U = \\dim X$.\n\nExample: $\\dim \\mathbb{A}^n = \\dim \\mathbb{P}^n = n$, since $k(x_1, \\dots, x_n)$ has transcendence degree $n$ over $k$.\nThe chain $\\set{0} \\subsetneq V(x_2, \\dots, x_n) \\subsetneq \\cdots \\subsetneq V(x_n) \\subsetneq \\mathbb{A}^n$ of linear subspaces has length $n$.\n\nExample: if $f \\in k[x_1, \\dots, x_n]$ is irreducible and not constant, the hypersurface $V(f) \\subseteq \\mathbb{A}^n$ has dimension $n - 1$.\n\nThe dimension of a reducible algebraic set is the largest dimension of its irreducible components.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "36X2WTHYCB",
    "kind": "definition",
    "msc": [
      "14A10",
      "13C15"
    ],
    "name": "Dimension of an algebraic variety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dimension_of_an_algebraic_variety"
      }
    ],
    "tag": "6Y0Y9JRXSK",
    "uses": [
      "HV6YBS0YDH",
      "SY2MWC434Z",
      "0ZCK8DQYN1",
      "MVGBS3J1P7",
      "G9WB27VWP7",
      "JERVQT4C12",
      "D7TR0YWR9H",
      "2D5SS6WM14"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6Y0Y9JRXSK/",
  "links": [
    {
      "field": "defines",
      "target": "36X2WTHYCB"
    },
    {
      "field": "uses",
      "target": "HV6YBS0YDH"
    },
    {
      "field": "uses",
      "target": "SY2MWC434Z"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    },
    {
      "field": "uses",
      "target": "MVGBS3J1P7"
    },
    {
      "field": "uses",
      "target": "G9WB27VWP7"
    },
    {
      "field": "uses",
      "target": "JERVQT4C12"
    },
    {
      "field": "uses",
      "target": "D7TR0YWR9H"
    },
    {
      "field": "uses",
      "target": "2D5SS6WM14"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6Y0Y9JRXSK.md",
  "msc": [
    "14A10",
    "13C15"
  ],
  "name": "Dimension of an algebraic variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6Y0/6Y0Y9JRXSK.md",
  "status": null,
  "tag": "6Y0Y9JRXSK",
  "uri": "m2://6Y0Y9JRXSK",
  "url": "https://tamnd.github.io/math2/wiki/dimension-of-an-algebraic-variety-definition/",
  "writing": {
    "tag": "6Y0Y9JRXSK",
    "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
      }
    ]
  }
}