{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $K$ be a field with [[absolute Galois group|PGKWBJDTKX]] $G_K$, and let $p$ be a [[prime number|0SN0MAEKYX]].\nFor a discrete torsion $G_K$-module $M$, let $H^q(K, M)(p)$ be the $p$-primary part of the [[Galois cohomology|EESNTXFWHZ]] group $H^q(K, M)$.\nThe *$p$-cohomological dimension* $\\operatorname{cd}_p(K)$ is the least integer $n \\ge 0$ with the following property.\nFor all $q \u003e n$ and all discrete torsion $G_K$-modules $M$, $H^q(K, M)(p) = 0$.\nIf there is no such $n$, then $\\operatorname{cd}_p(K) = \\infty$.\nThe *cohomological dimension* of $K$ is $\\operatorname{cd}(K) = \\sup_p \\operatorname{cd}_p(K)$.\n\n## Remarks\n\nThese are the [[cohomological dimensions|EF4ZWACHD8]] $\\operatorname{cd}_p(G_K)$ and $\\operatorname{cd}(G_K)$ of the profinite group $G_K$.\nIt suffices to test one degree and simple modules: $\\operatorname{cd}_p(K) \\le n$ if and only if $H^{n+1}(K, M) = 0$ for every simple discrete $G_K$-module $M$ with $pM = 0$.\nExample: a finite field $\\F_q$ has $\\operatorname{cd}(\\F_q) = 1$, since $G_{\\F_q}$ is the profinite completion $\\widehat{\\Z}$ of $\\Z$, whose cohomology with torsion coefficients vanishes above degree $1$, while $H^1(\\F_q, \\Z/p\\Z) = \\Hom(\\widehat{\\Z}, \\Z/p\\Z) \\ne 0$.\nExample: $\\operatorname{cd}(K) = 0$ if and only if $K$ is separably closed.\nExample: $\\R$ has $\\operatorname{cd}_2(\\R) = \\infty$, since $G_\\R$ has order $2$ and $H^q(\\Z/2\\Z, \\Z/2\\Z) \\cong \\Z/2\\Z$ for all $q \\ge 0$.\nMore generally, a field with a nontrivial finite absolute Galois group has infinite cohomological dimension.\nIf $K$ has characteristic $p \u003e 0$, then $\\operatorname{cd}_p(K) \\le 1$.\nA finite extension of $\\Q_p$ has cohomological dimension $2$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "QZMZXFFJFJ",
    "kind": "definition",
    "msc": [
      "12G10",
      "12G05"
    ],
    "name": "Cohomological dimension of a field",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cohomological_dimension"
      }
    ],
    "tag": "XKMBKAN1MR",
    "uses": [
      "EESNTXFWHZ",
      "PGKWBJDTKX",
      "EF4ZWACHD8",
      "0SN0MAEKYX"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/XKMBKAN1MR/",
  "links": [
    {
      "field": "defines",
      "target": "QZMZXFFJFJ"
    },
    {
      "field": "uses",
      "target": "EESNTXFWHZ"
    },
    {
      "field": "uses",
      "target": "PGKWBJDTKX"
    },
    {
      "field": "uses",
      "target": "EF4ZWACHD8"
    },
    {
      "field": "uses",
      "target": "0SN0MAEKYX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/XKMBKAN1MR.md",
  "msc": [
    "12G10",
    "12G05"
  ],
  "name": "Cohomological dimension of a field",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XKM/XKMBKAN1MR.md",
  "status": null,
  "tag": "XKMBKAN1MR",
  "uri": "m2://XKMBKAN1MR",
  "url": "https://math2.tamnd.com/wiki/cohomological-dimension-of-a-field-definition/",
  "writing": {
    "tag": "XKMBKAN1MR",
    "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
      }
    ]
  }
}