{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2QSKZ59DB6"
    },
    {
      "field": "specializes",
      "target": "FTRW0679E2"
    },
    {
      "field": "defines",
      "target": "ZHBREAF3MJ"
    }
  ],
  "body": "\n## Summary\n\nA smooth morphism of rings is a map $R \\to A$ of commutative rings that makes $A$ a [[finitely presented|764GZHTGV8]] $R$-algebra and is [[formally smooth|8KPDNVKX7B]].\nIt is the algebraic counterpart of a submersion of manifolds: the fibres are nonsingular and vary nicely.\nA smooth algebra is flat over the base, and its module of [[Kähler differentials|7HCFARHG7D]] is finitely generated and projective.\nOver a field, smooth algebras are the coordinate rings of nonsingular varieties, in a sense that is stable under extension of the field.\n\n## Explanations\n\n### L1 human\n\nA polynomial ring $R[x_1, \\ldots, x_n]$ is the model of a smooth algebra: it is like flat space over $R$.\nA smooth algebra looks, near each point and to first order, like such a polynomial ring.\nThe circle $x^2 + y^2 = 1$ over the real numbers gives a smooth algebra $\\R[x, y]/(x^2 + y^2 - 1)$.\nThe cusp $y^2 = x^3$ does not, because of its sharp point at the origin.\nSmoothness is the property that lets one extend solutions step by step, as in Hensel's lemma.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13B40",
      "14B25",
      "13D10"
    ],
    "name": "Smooth morphism of rings",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Formally_smooth_map"
      }
    ],
    "specializes": [
      "8KPDNVKX7B"
    ],
    "tag": "286G0P1B69"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/286G0P1B69/",
  "links": [
    {
      "field": "specializes",
      "target": "8KPDNVKX7B"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/286G0P1B69.md",
  "msc": [
    "13B40",
    "14B25",
    "13D10"
  ],
  "name": "Smooth morphism of rings",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/286/286G0P1B69.md",
  "status": null,
  "tag": "286G0P1B69",
  "uri": "m2://286G0P1B69",
  "url": "https://math2.tamnd.com/wiki/smooth-morphism-of-rings/",
  "writing": {
    "tag": "286G0P1B69",
    "kind": "concept",
    "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
      }
    ]
  }
}