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  "body": "\n## Statement\n\nLet $K$ be a [[field|04C5ZBG1WT]], let $n \\ge 0$, and let $S = K[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], [[graded|8KB4SQGC0C]] by degree.\nLet $M$ be a [[finitely generated|05QY1VEBYD]] graded $S$-module.\nThen $M$ has a graded [[free resolution|HK2AA1YSQ2]]\n$$0 \\to F_m \\to F_{m-1} \\to \\cdots \\to F_0 \\to M \\to 0$$\nby finitely generated graded [[free|0DZ9DKAKZR]] $S$-modules $F_i$, with homogeneous maps of degree $0$, of length $m \\le n$.\n\n## Remarks\n\nOne proof uses the [[Koszul complex|YZY11W08SD]] of $x_1, \\ldots, x_n$, which is a free resolution of $K = S/(x_1, \\ldots, x_n)$ of length $n$.\nIt shows that $\\operatorname{Tor}_i^S(M, K) = 0$ for $i \u003e n$, and the length of a minimal graded free resolution of $M$ is the largest $i$ with $\\operatorname{Tor}_i^S(M, K) \\ne 0$.\nThe bound $n$ is sharp: for $n = 2$, the resolution $0 \\to S \\xrightarrow{(-y,\\, x)^T} S^2 \\xrightarrow{(x \\;\\; y)} S \\to K \\to 0$ of $K = S/(x, y)$ has length $2$, and no shorter resolution exists.\nThe same method shows that the [[global dimension|GWA0T1WKCN]] of $K[x_1, \\ldots, x_n]$ is $n$.\n\n## Explanations\n\n### L1 human\n\nA syzygy is a relation among generators, and a second syzygy is a relation among relations.\nHilbert showed that for modules over a polynomial ring in $n$ variables this process stops after at most $n$ steps.\nSo the structure of such a module is described by finitely many finite lists of polynomials.\n\n### L2 human\n\nThe theorem says that the polynomial ring $K[x_1, \\ldots, x_n]$ has global dimension $n$.\nIt is closely related to [[Serre's theorem on regular local rings|93RQRG9R1P]], since the localizations of $K[x_1, \\ldots, x_n]$ at maximal ideals are regular.\nThe minimal free resolution gives the graded [[Betti numbers|3B1743XS7S]] of $M$, and Gröbner bases make it computable.\n",
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