{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]] that is locally [[of finite type|AHTE5DVX61]].\nFor $x \\in X$ and $y = f(x)$, let $\\mathfrak{m}_x$ and $\\mathfrak{m}_y$ be the maximal ideals of the [[local rings|MZ02BKPHJA]] $\\mathcal{O}_{X,x}$ and $\\mathcal{O}_{Y,y}$.\nLet $\\kappa(x)$ and $\\kappa(y)$ be the [[residue fields|G51E8MW7J1]].\nThe morphism $f$ is *unramified at $x$* if the following two conditions hold.\n\n1. $\\mathfrak{m}_y \\mathcal{O}_{X,x} = \\mathfrak{m}_x$.\n2. $\\kappa(x)$ is a finite [[separable extension|0WCX7NB4N7]] of $\\kappa(y)$.\n\nThe morphism $f$ is *unramified* if it is unramified at every point of $X$.\n\n## Remarks\n\nAn open immersion or a closed immersion is unramified.\nFor a closed immersion, each $\\mathcal{O}_{X,x}$ is a quotient of $\\mathcal{O}_{Y,y}$, so both conditions hold with $\\kappa(x) = \\kappa(y)$.\nFor a field $k$ of characteristic not $2$, the map $\\operatorname{Spec} k[t] \\to \\operatorname{Spec} k[s]$ given by $s \\mapsto t^2$ is not unramified at $t = 0$.\nThere $\\mathfrak{m}_y \\mathcal{O}_{X,x} = (t^2)$, which is strictly smaller than $\\mathfrak{m}_x = (t)$, so condition 1 fails.\nFor $f$ locally of finite type, $f$ is unramified exactly when the sheaf of relative [[Kähler differentials|7HCFARHG7D]] $\\Omega_{X/Y}$ is zero.\nSome authors, as in EGA, also require $f$ to be locally of finite presentation.\nThe ring version of this notion is an [[unramified morphism of rings|GY3QA6NH0E]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "1JQX6HD8E5",
    "kind": "definition",
    "msc": [
      "14A15",
      "14E22"
    ],
    "name": "Unramified morphism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unramified_morphism"
      }
    ],
    "tag": "FPK5ZH7FKC",
    "uses": [
      "JXRKH6P1Z4",
      "AHTE5DVX61",
      "G51E8MW7J1",
      "MZ02BKPHJA",
      "0WCX7NB4N7",
      "7HCFARHG7D"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FPK5ZH7FKC/",
  "links": [
    {
      "field": "defines",
      "target": "1JQX6HD8E5"
    },
    {
      "field": "uses",
      "target": "JXRKH6P1Z4"
    },
    {
      "field": "uses",
      "target": "AHTE5DVX61"
    },
    {
      "field": "uses",
      "target": "G51E8MW7J1"
    },
    {
      "field": "uses",
      "target": "MZ02BKPHJA"
    },
    {
      "field": "uses",
      "target": "0WCX7NB4N7"
    },
    {
      "field": "uses",
      "target": "7HCFARHG7D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FPK5ZH7FKC.md",
  "msc": [
    "14A15",
    "14E22"
  ],
  "name": "Unramified morphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FPK/FPK5ZH7FKC.md",
  "status": null,
  "tag": "FPK5ZH7FKC",
  "uri": "m2://FPK5ZH7FKC",
  "url": "https://tamnd.github.io/math2/wiki/unramified-morphism-definition/",
  "writing": {
    "tag": "FPK5ZH7FKC",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}