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  "body": "\n## Summary\n\nThe Castelnuovo–Mumford regularity is an integer that bounds the degrees in the minimal free resolution of a finitely generated graded module over a polynomial ring.\nIt measures how complicated the module is from the point of view of syzygies.\nIt bounds the degree from which the Hilbert function agrees with the [[Hilbert polynomial|RMH4FBCFDA]].\nIt also bounds the complexity of [[Gröbner basis|C83KPT47R0]] computations.\n\n## Explanations\n\n### L1 human\n\nA module over a polynomial ring is described by generators, relations among the generators, relations among the relations, and so on.\nEach of these comes in certain degrees.\nThe regularity records how far the degrees drift upward as one goes down this chain, after a shift of one degree for each step.\nFor the quotient $k[x]/(x^d)$, the generator has degree $0$ and the single relation has degree $d$, so the regularity is $d - 1$.\nA small regularity means the module is simple in this sense.\n",
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