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  "body": "\n## Summary\n\nThe Koszul complex of elements $x_1, \\ldots, x_n$ of a commutative ring $R$ is a complex of free modules built from the [[exterior algebra|0MDMZ965VE]] of $R^n$.\nIts zeroth homology is $R/(x_1, \\ldots, x_n)$, and when the $x_i$ form a [[regular sequence|69R5R2FCS5]] it is a [[free resolution|HK2AA1YSQ2]] of this quotient.\nIt computes [[depth|M2RADBFW99]], gives the proof of [[Hilbert's syzygy theorem|R0DSZPRG6T]], and is a basic tool throughout commutative algebra.\n\n## Explanations\n\n### L1 human\n\nThe Koszul complex is the simplest free complex one can write down from a list of elements.\nFor one element $x$ it is $0 \\to R \\xrightarrow{x} R \\to 0$.\nFor two elements $x, y$ it is $0 \\to R \\to R^2 \\to R \\to 0$, where the last map sends $(a, b)$ to $ax + by$ and the first map sends $c$ to $(-cy, cx)$.\nThe composite is zero because $-cyx + cxy = 0$, and this simple cancellation is the heart of the construction.\n",
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