{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "1TNS604ZST"
    },
    {
      "field": "uses",
      "target": "1TNS604ZST"
    },
    {
      "field": "defines",
      "target": "XSZ9C68SPG"
    }
  ],
  "body": "\n## Summary\n\nA system of parameters of a Noetherian [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$ is a list of $d$ elements of the maximal ideal $\\mathfrak m$.\nThe ideal that they generate must be primary to $\\mathfrak m$.\nThe [[dimension theorem for local rings|KW3HY77V6D]] says that such a list always exists and that no shorter list can do the same job.\nSystems of parameters play the role of local coordinates, and they are the starting point of the theory of Cohen–Macaulay rings and multiplicities.\n\n## Explanations\n\n### L1 human\n\nNear a point of a $d$-dimensional space, $d$ equations are needed to cut the space down to that single point.\nA system of parameters is such a list of $d$ equations, read in the local ring of the point.\nFor the plane at the origin, the coordinates $x$ and $y$ work, and so do $x^2$ and $y^3$, since $x^2 = y^3 = 0$ still only allows the origin.\nThe [[definition|XSZ9C68SPG]] makes this precise.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13C15",
      "13H05"
    ],
    "name": "System of parameters",
    "same_as": {
      "wikidata": "Q7663733",
      "wikipedia": "System of parameters"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/System_of_parameters"
      }
    ],
    "tag": "95QTEPMDA4"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/95QTEPMDA4/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/95QTEPMDA4.md",
  "msc": [
    "13C15",
    "13H05"
  ],
  "name": "System of parameters",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/95Q/95QTEPMDA4.md",
  "status": null,
  "tag": "95QTEPMDA4",
  "uri": "m2://95QTEPMDA4",
  "url": "https://math2.tamnd.com/wiki/system-of-parameters/",
  "writing": {
    "tag": "95QTEPMDA4",
    "kind": "concept",
    "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
      }
    ]
  }
}