{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\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 $k$.\nIs every [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module [[free|0DZ9DKAKZR]]?\n\n## Remarks\n\nThe problem is solved: the answer is yes, by the [[Quillen–Suslin theorem|7N50GA46W1]].\nFor $n = 1$, the ring $k[x]$ is a principal ideal domain, so the answer is yes by the structure theorem for modules over such rings.\nSerre showed that every finitely generated projective $R$-module $P$ is stably free: $P \\oplus R^m$ is free for some $m \\ge 0$.\n\n## History\n\nJean-Pierre Serre raised the question in his 1955 paper on coherent algebraic sheaves.\nC. S. Seshadri proved the case $n = 2$ in 1958.\nDaniel Quillen and Andrei Suslin solved the general case independently in 1976.\n\n## Explanations\n\n### L1 human\n\nSerre asked whether every vector bundle on affine space is trivial, in the language of algebra.\nThe question is easy to state: does every finitely generated projective module over a polynomial ring have a basis?\nIt was known to be true for one and two variables, but the general case stayed open for about twenty years.\nQuillen and Suslin both proved that the answer is yes, in 1976.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "BFFA03GE3V",
      "0SDC934R52"
    ],
    "history": {
      "first_stated": "1955",
      "posed_by": "Jean-Pierre Serre"
    },
    "kind": "problem",
    "msc": [
      "13C10",
      "19A13"
    ],
    "name": "Serre's problem on projective modules",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem"
      }
    ],
    "tag": "7GG0XYHB0T",
    "uses": [
      "0SDC934R52",
      "BFFA03GE3V",
      "05QY1VEBYD",
      "0DZ9DKAKZR"
    ]
  },
  "kind": "problem",
  "link": "https://math2.tamnd.com/node/7GG0XYHB0T/",
  "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"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7GG0XYHB0T.md",
  "msc": [
    "13C10",
    "19A13"
  ],
  "name": "Serre's problem on projective modules",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7GG/7GG0XYHB0T.md",
  "status": null,
  "tag": "7GG0XYHB0T",
  "uri": "m2://7GG0XYHB0T",
  "url": "https://math2.tamnd.com/wiki/serres-problem-on-projective-modules/",
  "writing": {
    "tag": "7GG0XYHB0T",
    "kind": "problem",
    "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
      }
    ]
  }
}