{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5FR6JX04RS"
    }
  ],
  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \\dots, x_r]$ be the [[polynomial ring|07GNX8GCYP]] with each $x_i$ of degree $1$.\nLet $M = \\bigoplus_{n \\in \\Z} M_n$ be a [[finitely generated|05QY1VEBYD]] [[graded|8KB4SQGC0C]] $S$-module, and let $H_M(n) = \\dim_k M_n$.\n\n1. The Hilbert series $\\sum_{n \\in \\Z} H_M(n)\\, t^n$ equals $Q(t)/(1 - t)^r$ for some Laurent polynomial $Q \\in \\Z[t, t^{-1}]$.\n2. There is a unique polynomial $P_M \\in \\Q[t]$ such that $H_M(n) = P_M(n)$ for all sufficiently large $n$, and $P_M$ is zero or has degree at most $r - 1$.\n\n## Remarks\n\nThe proof goes by induction on $r$.\nMultiplication by $x_r$ gives an exact sequence of graded modules $0 \\to K(-1) \\to M(-1) \\to M \\to C \\to 0$, where $K$ and $C$ are killed by $x_r$ and so are finitely generated graded modules over $k[x_1, \\dots, x_{r-1}]$.\nTaking dimensions degree by degree gives $(1 - t)$ times the Hilbert series of $M$ in terms of those of $C$ and $K$, and the induction hypothesis finishes the proof.\nPart 2 follows from part 1 by expanding $1/(1-t)^r = \\sum_{n \\ge 0} \\binom{n + r - 1}{r - 1} t^n$.\nExample: for $M = S$ one has $Q = 1$ and $H_S(n) = \\binom{n + r - 1}{r - 1}$ for $n \\ge 0$.\nWhen the Krull dimension of $M$ is at least $1$, the degree of $P_M$ is one less than this dimension.\n\n## Explanations\n\n### L1 human\n\nA graded module splits into pieces of degree $0, 1, 2, \\dots$, and each piece is a finite-dimensional vector space.\nThe theorem says that the dimensions of these pieces are eventually given by a polynomial in the degree.\nSo an infinite list of numbers is controlled by finitely many coefficients.\n\n### L2 human\n\nThe polynomial $P_M$ is the [[Hilbert polynomial|RMH4FBCFDA]] of $M$.\nIn projective geometry, applied to the coordinate ring of a projective variety, it gives the dimension and the degree of the variety.\nThe local analogue is the existence of the Hilbert–Samuel polynomial, which is proved by applying this theorem to the associated graded ring; it underlies the [[dimension theorem for local rings|KW3HY77V6D]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "RMH4FBCFDA"
    ],
    "kind": "statement",
    "msc": [
      "13D40",
      "13A02"
    ],
    "name": "Hilbert–Serre theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert_series_and_Hilbert_polynomial"
      }
    ],
    "tag": "HP56DJECMT",
    "uses": [
      "8KB4SQGC0C",
      "07GNX8GCYP",
      "05QY1VEBYD",
      "04C5ZBG1WT",
      "RMH4FBCFDA"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/HP56DJECMT/",
  "links": [
    {
      "field": "about",
      "target": "RMH4FBCFDA"
    },
    {
      "field": "uses",
      "target": "8KB4SQGC0C"
    },
    {
      "field": "uses",
      "target": "07GNX8GCYP"
    },
    {
      "field": "uses",
      "target": "05QY1VEBYD"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    },
    {
      "field": "uses",
      "target": "RMH4FBCFDA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HP56DJECMT.md",
  "msc": [
    "13D40",
    "13A02"
  ],
  "name": "Hilbert–Serre theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HP5/HP56DJECMT.md",
  "status": null,
  "tag": "HP56DJECMT",
  "uri": "m2://HP56DJECMT",
  "url": "https://math2.tamnd.com/wiki/hilbert-serre-theorem/",
  "writing": {
    "tag": "HP56DJECMT",
    "kind": "statement",
    "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
      }
    ]
  }
}