{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "CRFGFCNG3Q"
    },
    {
      "field": "uses",
      "target": "EGKVP4KK6V"
    },
    {
      "field": "uses",
      "target": "HVMYKZ2PB5"
    },
    {
      "field": "specializes",
      "target": "GH81SYAPPN"
    },
    {
      "field": "specializes",
      "target": "JDDXZ89N33"
    }
  ],
  "body": "\n## Summary\n\nA group scheme over a base scheme $S$ is a scheme over $S$ with multiplication, unit and inverse morphisms that satisfy the group axioms.\nEquivalently, its points with values in every $S$-scheme form a group, naturally in the $S$-scheme.\nGroup schemes include algebraic groups over fields, such as $\\mathrm{GL}_n$ and elliptic curves.\nThey also include infinitesimal objects such as $\\mu_p$ in characteristic $p$, which have no analogue among classical groups.\n\n## Explanations\n\n### L1 human\n\nA Lie group is a manifold that is also a group, with smooth operations.\nA group scheme is the same idea in algebraic geometry: a scheme whose points can be multiplied by polynomial formulas.\nThe simplest example is the multiplicative group $\\mathbb{G}_m$, whose points are the invertible numbers, multiplied as usual.\nOver a general base, a group scheme is a family of such groups that varies with a parameter.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "14L15"
    ],
    "name": "Group scheme",
    "same_as": {
      "wikidata": "Q5611270",
      "wikipedia": "Group scheme"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Group_scheme"
      }
    ],
    "tag": "TJ04VQP4NB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/TJ04VQP4NB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/TJ04VQP4NB.md",
  "msc": [
    "14L15"
  ],
  "name": "Group scheme",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TJ0/TJ04VQP4NB.md",
  "status": null,
  "tag": "TJ04VQP4NB",
  "uri": "m2://TJ04VQP4NB",
  "url": "https://tamnd.github.io/math2/wiki/group-scheme/",
  "writing": {
    "tag": "TJ04VQP4NB",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}