{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p$ be a prime and let $R$ be a [[Noetherian ring|0670T59HGE]] of [[characteristic|0F6ES3BRYP]] $p$.\nLet $R^\\circ$ be the set of elements of $R$ that lie in no [[minimal prime ideal|S2D8823P8B]] of $R$.\nFor an [[ideal|096Z9MP2ME]] $I$ of $R$ and a power $q = p^e$ with $e \\ge 0$, let $I^{[q]}$ be the ideal generated by the elements $a^q$ with $a \\in I$.\nAn element $x \\in R$ is in the *tight closure* $I^*$ of $I$ if there is $c \\in R^\\circ$ such that $c x^q \\in I^{[q]}$ for all $q = p^e$ with $e$ large enough.\n\n## Remarks\n\nThe ideal $I^{[q]}$ is the ideal generated by the image of $I$ under the $e$-th power of the [[Frobenius endomorphism|0T3WVTD675]].\nIf $I = (a_1, \\ldots, a_m)$, then $I^{[q]} = (a_1^q, \\ldots, a_m^q)$.\n$I^*$ is an ideal, $I \\subseteq I^*$, and $(I^*)^* = I^*$.\nThe ideal $I$ is *tightly closed* if $I^* = I$.\n\nExample: let $k$ be a field of characteristic $p$, let $R = k[x, y]/(x^2)$, and let $I = (0)$.\nThe only minimal prime is $(x)$, so $1 \\in R^\\circ$, and $1 \\cdot x^q = 0 \\in I^{[q]}$ for $q \\ge 2$.\nSo $x \\in (0)^*$; more generally every [[nilpotent element|0ZB2JG1M9C]] lies in $(0)^*$.\n\nNon-example: let $R = k[x]$ and $I = (x^2)$; then $x \\notin I^*$.\nIndeed $c x^q \\in (x^{2q})$ means $x^q$ divides $c$, and if this holds for infinitely many $q$ then $c = 0$, which is not in $R^\\circ$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "KJ9MMP99G6",
    "kind": "definition",
    "msc": [
      "13A35"
    ],
    "name": "Tight closure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tight_closure"
      }
    ],
    "tag": "XZS36C6226",
    "uses": [
      "0670T59HGE",
      "0F6ES3BRYP",
      "096Z9MP2ME",
      "S2D8823P8B",
      "0T3WVTD675",
      "0ZB2JG1M9C"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/XZS36C6226/",
  "links": [
    {
      "field": "defines",
      "target": "KJ9MMP99G6"
    },
    {
      "field": "uses",
      "target": "0670T59HGE"
    },
    {
      "field": "uses",
      "target": "0F6ES3BRYP"
    },
    {
      "field": "uses",
      "target": "096Z9MP2ME"
    },
    {
      "field": "uses",
      "target": "S2D8823P8B"
    },
    {
      "field": "uses",
      "target": "0T3WVTD675"
    },
    {
      "field": "uses",
      "target": "0ZB2JG1M9C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/XZS36C6226.md",
  "msc": [
    "13A35"
  ],
  "name": "Tight closure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XZS/XZS36C6226.md",
  "status": null,
  "tag": "XZS36C6226",
  "uri": "m2://XZS36C6226",
  "url": "https://math2.tamnd.com/wiki/tight-closure-definition/",
  "writing": {
    "tag": "XZS36C6226",
    "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
      }
    ]
  }
}