{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "BP0ZYHGD0S"
    },
    {
      "field": "uses",
      "target": "BP0ZYHGD0S"
    },
    {
      "field": "uses",
      "target": "J7K42A9F8A"
    },
    {
      "field": "uses",
      "target": "MB52NDA6WN"
    },
    {
      "field": "defines",
      "target": "VT9QWDTKA4"
    }
  ],
  "body": "\n## Summary\n\nThe normalization of an [[integral scheme|17NVR4XBHK]] $X$ is a [[normal scheme|EK261G1XP9]] $\\tilde{X}$ with a morphism $\\tilde{X} \\to X$.\nIt is built by taking the [[integral closure|G0V86SJBQW]] of each affine coordinate ring in its fraction field.\nIt is the best normal approximation of $X$: every dominant morphism from a normal integral scheme to $X$ factors through it.\nFor curves it resolves singularities, which makes it a basic tool in the study of singular curves.\n\n## Explanations\n\n### L1 human\n\nNormalization repairs the mildest singularities of a space by adding the functions that \"should\" be there.\nThe cusp $y^2 = x^3$ is traced by $t \\mapsto (t^2, t^3)$.\nThe function $t = y/x$ is missing on the cusp, and adding it gives back the ordinary line with coordinate $t$.\nSo the normalization of the cusp is the line, mapped onto the cusp by $t \\mapsto (t^2, t^3)$.\nA curve that crosses itself is pulled apart at the crossing in the same way.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14A15",
      "13B22",
      "14B05"
    ],
    "name": "Normalization",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Normal_scheme"
      }
    ],
    "tag": "M2FAMD9HZN"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/M2FAMD9HZN/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/M2FAMD9HZN.md",
  "msc": [
    "14A15",
    "13B22",
    "14B05"
  ],
  "name": "Normalization",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M2F/M2FAMD9HZN.md",
  "status": null,
  "tag": "M2FAMD9HZN",
  "uri": "m2://M2FAMD9HZN",
  "url": "https://tamnd.github.io/math2/wiki/normalization/",
  "writing": {
    "tag": "M2FAMD9HZN",
    "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
      }
    ]
  }
}