{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "BRXX5Y929C"
    },
    {
      "field": "uses",
      "target": "KTQC33ZJTN"
    }
  ],
  "body": "\n## Summary\n\nThe embedding dimension of a Noetherian [[local ring|MZ02BKPHJA]] $(R, \\mathfrak m, k)$ is the dimension of its [[cotangent space|Z1CGB483J6]] $\\mathfrak m / \\mathfrak m^2$ over $k$.\nBy [[Nakayama's lemma|15JHBYR6KH]] it equals the minimal number of generators of $\\mathfrak m$.\nIt is always at least the [[Krull dimension|MVGBS3J1P7]] of $R$, and equality defines a [[regular local ring|63XXTF6B5M]].\n\n## Explanations\n\n### L1 human\n\nThe embedding dimension counts how many coordinates you need to describe a space near a point.\nA smooth curve needs one coordinate, but the cusp $y^2 = x^3$ at the origin needs two, even though it is still a curve.\nThe difference between the embedding dimension and the actual dimension measures how singular the point is.\nThe [[definition|BRXX5Y929C]] states it.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13H05",
      "13H99"
    ],
    "name": "Embedding dimension",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_local_ring"
      }
    ],
    "tag": "DP3KSCRKFP"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/DP3KSCRKFP/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/DP3KSCRKFP.md",
  "msc": [
    "13H05",
    "13H99"
  ],
  "name": "Embedding dimension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DP3/DP3KSCRKFP.md",
  "status": null,
  "tag": "DP3KSCRKFP",
  "uri": "m2://DP3KSCRKFP",
  "url": "https://math2.tamnd.com/wiki/embedding-dimension/",
  "writing": {
    "tag": "DP3KSCRKFP",
    "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
      }
    ]
  }
}