{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "FPK5ZH7FKC"
    },
    {
      "field": "uses",
      "target": "HVMYKZ2PB5"
    },
    {
      "field": "uses",
      "target": "JK0S0WYN01"
    },
    {
      "field": "uses",
      "target": "QZN6P7AG33"
    },
    {
      "field": "uses",
      "target": "TGN01264C4"
    },
    {
      "field": "uses",
      "target": "W9DXVGAE3R"
    },
    {
      "field": "uses",
      "target": "Y60D7BN1BN"
    },
    {
      "field": "specializes",
      "target": "N9C4X7XKE4"
    },
    {
      "field": "defines",
      "target": "T09DXZAT6R"
    },
    {
      "field": "about",
      "target": "TGN01264C4"
    }
  ],
  "body": "\n## Summary\n\nA morphism of finite type $X \\to Y$ is a [[morphism of schemes|JXRKH6P1Z4]] described by finitely many coordinates over $Y$.\nOver each affine piece $\\operatorname{Spec} B$ of $Y$, the source is covered by finitely many spectra of finitely generated $B$-algebras.\nIt is the relative version of a [[finitely generated algebra|AC5TFFQT9M]].\nVarieties over a field $k$ are schemes of finite type over $\\operatorname{Spec} k$.\n\n## Explanations\n\n### L1 human\n\nA scheme of finite type over a field is one that can be described by finitely many coordinates and equations on each of finitely many pieces.\nThe plane curve $y^2 = x^3 - x$ over $\\Q$ is of finite type: it is $\\operatorname{Spec} \\Q[x, y]/(y^2 - x^3 + x)$.\nThe spectrum of the field $\\Q(x)$ of rational functions is not of finite type over $\\Q$, because no finite set of elements generates $\\Q(x)$ as an algebra.\nMost of classical algebraic geometry deals with schemes of finite type over a field.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14A15"
    ],
    "name": "Morphism of finite type",
    "same_as": {
      "wikidata": "Q112108702",
      "wikipedia": "Morphism of finite type"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Morphism_of_finite_type"
      }
    ],
    "specializes": [
      "41009J4M5D"
    ],
    "tag": "AHTE5DVX61"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AHTE5DVX61/",
  "links": [
    {
      "field": "specializes",
      "target": "41009J4M5D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AHTE5DVX61.md",
  "msc": [
    "14A15"
  ],
  "name": "Morphism of finite type",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AHT/AHTE5DVX61.md",
  "status": null,
  "tag": "AHTE5DVX61",
  "uri": "m2://AHTE5DVX61",
  "url": "https://tamnd.github.io/math2/wiki/morphism-of-finite-type/",
  "writing": {
    "tag": "AHTE5DVX61",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}