{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $k$ be a field, let $n \\ge 1$, and let $R = k[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] in $n$ variables over $k$.\nThen every [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module is [[free|0DZ9DKAKZR]].\n\n## Remarks\n\nThe theorem answers [[Serre's problem|7GG0XYHB0T]] and was proved independently by Daniel Quillen and Andrei Suslin in 1976.\nQuillen's proof combines a local-global principle, which reduces the question to polynomial rings over local rings, with Horrocks' theorem on projective modules over $A[x]$ for a local ring $A$.\nSuslin's proof works with unimodular rows and shows that they can be completed to invertible matrices.\nQuillen's argument gives the result more generally for polynomial rings over a [[principal ideal domain|0CMD0BCH8T]].\n\nExample: if $f, g \\in R$ satisfy $af + bg = 1$ for some $a, b \\in R$, the kernel $P$ of $R^2 \\to R$, $(u, v) \\mapsto uf + vg$, is projective, because the map splits.\nThe theorem says that $P$ is free; here one can see it directly, since $P$ is generated by $(-g, f)$.\n\nThe hypothesis on the ring cannot be dropped.\nOver $A = \\R[x, y, z]/(x^2 + y^2 + z^2 - 1)$, the kernel of $A^3 \\to A$, $(u, v, w) \\mapsto ux + vy + wz$, is projective but not free; this module corresponds to the tangent bundle of the $2$-sphere.\n\n## Explanations\n\n### L1 human\n\nProjective modules are the algebraic version of vector bundles, families of vector spaces glued over a space.\nOver affine space, a space without holes, one expects every such family to be trivial.\nThe theorem confirms this: over a polynomial ring over a field, every finitely generated projective module has a basis.\nIt took about twenty years from the question to the answer.\n\n### L2 human\n\nThe theorem says that algebraic vector bundles on affine space $\\mathbb{A}^n_k$ are trivial, the algebraic analogue of the contractibility of $\\R^n$.\nIt gives $K_0(k[x_1, \\ldots, x_n]) \\cong \\Z$ in a strong form: every class is represented by a free module.\nIn matrix terms, every unimodular row over $k[x_1, \\ldots, x_n]$ is the first row of an invertible matrix.\nThe local-global principle in Quillen's proof is itself a widely used tool in the study of projective modules.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "BFFA03GE3V",
      "0SDC934R52"
    ],
    "kind": "statement",
    "msc": [
      "13C10",
      "19A13"
    ],
    "name": "Quillen–Suslin theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7272004",
      "wikipedia": "Quillen–Suslin theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem"
      }
    ],
    "tag": "7N50GA46W1",
    "uses": [
      "0SDC934R52",
      "BFFA03GE3V",
      "05QY1VEBYD",
      "0DZ9DKAKZR",
      "0CMD0BCH8T"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/7N50GA46W1/",
  "links": [
    {
      "field": "about",
      "target": "BFFA03GE3V"
    },
    {
      "field": "about",
      "target": "0SDC934R52"
    },
    {
      "field": "uses",
      "target": "0SDC934R52"
    },
    {
      "field": "uses",
      "target": "BFFA03GE3V"
    },
    {
      "field": "uses",
      "target": "05QY1VEBYD"
    },
    {
      "field": "uses",
      "target": "0DZ9DKAKZR"
    },
    {
      "field": "uses",
      "target": "0CMD0BCH8T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7N50GA46W1.md",
  "msc": [
    "13C10",
    "19A13"
  ],
  "name": "Quillen–Suslin theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7N5/7N50GA46W1.md",
  "status": null,
  "tag": "7N50GA46W1",
  "uri": "m2://7N50GA46W1",
  "url": "https://math2.tamnd.com/wiki/quillen-suslin-theorem/",
  "writing": {
    "tag": "7N50GA46W1",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}