{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$, and let $I$ be an [[$\\mathfrak m$-primary ideal|TMTPWVETQK]].\nLet $P$ be the polynomial that agrees with the [[Hilbert–Samuel function|BNQVS83GZV]] $n \\mapsto \\ell(R/I^n)$ for all large $n$.\nIt has degree $d$, and its leading coefficient has the form $e/d!$ for a positive integer $e$.\nThe integer $e = e(I, R)$ is the *multiplicity* of $I$.\nThe *multiplicity of $R$* is $e(R) = e(\\mathfrak m, R)$.\n\n## Remarks\n\nExample: let $R = k[x, y]_{(x, y)}/(y^2 - x^3)$, with $k$ a field.\nFor $n \\ge 2$, $\\ell(R/\\mathfrak m^n) = 2n - 1$, since the monomials of degree $\u003c n$ in $k[x, y]$ number $n(n+1)/2$ and those divisible by $y^2$ number $(n-2)(n-1)/2$.\nSo $d = 1$, the leading coefficient is $2 = e/1!$, and $e(R) = 2$.\nNon-example: the function $n \\mapsto \\ell(R/I^n)$ for an ideal $I$ that is not $\\mathfrak m$-primary, such as $I = (y)$ in $k[x, y]_{(x, y)}$, takes infinite values, so it has no multiplicity in this sense.\nA regular local ring has multiplicity $1$, and an Artinian local ring $R$ has $e(R) = \\ell(R)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "JX20MNDDXT",
    "kind": "definition",
    "msc": [
      "13H15",
      "13D40"
    ],
    "name": "Multiplicity of a local ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert–Samuel_function"
      }
    ],
    "tag": "B7CV5DZPRK",
    "uses": [
      "MZ02BKPHJA",
      "BNQVS83GZV",
      "MVGBS3J1P7",
      "TMTPWVETQK",
      "0670T59HGE"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/B7CV5DZPRK/",
  "links": [
    {
      "field": "defines",
      "target": "JX20MNDDXT"
    },
    {
      "field": "uses",
      "target": "MZ02BKPHJA"
    },
    {
      "field": "uses",
      "target": "BNQVS83GZV"
    },
    {
      "field": "uses",
      "target": "MVGBS3J1P7"
    },
    {
      "field": "uses",
      "target": "TMTPWVETQK"
    },
    {
      "field": "uses",
      "target": "0670T59HGE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/B7CV5DZPRK.md",
  "msc": [
    "13H15",
    "13D40"
  ],
  "name": "Multiplicity of a local ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B7C/B7CV5DZPRK.md",
  "status": null,
  "tag": "B7CV5DZPRK",
  "uri": "m2://B7CV5DZPRK",
  "url": "https://math2.tamnd.com/wiki/multiplicity-of-a-local-ring-definition/",
  "writing": {
    "tag": "B7CV5DZPRK",
    "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
      }
    ]
  }
}