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  "body": "\n## Definition\n\nLet $k$ be a field and let $S = k[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] graded by total degree.\nFor an integer $j$, let $S(-j)$ be $S$ with grading shifted so that $S(-j)_d = S_{d - j}$.\nLet $M \\ne 0$ be a [[finitely generated|05QY1VEBYD]] [[graded|8KB4SQGC0C]] $S$-module.\nLet\n$$0 \\to F_p \\to \\cdots \\to F_1 \\to F_0 \\to M \\to 0$$\nbe a minimal graded free [[resolution|HK2AA1YSQ2]] of $M$, with $F_i = \\bigoplus_j S(-j)^{\\beta_{i,j}}$.\nThe numbers $\\beta_{i,j}$ are the graded [[Betti numbers|3B1743XS7S]] of $M$.\nThe *Castelnuovo–Mumford regularity* of $M$ is\n$$\\operatorname{reg} M = \\max \\set{ j - i : \\beta_{i,j} \\ne 0 }.$$\n\n## Remarks\n\nLet $f \\in S$ be homogeneous of degree $d \\ge 1$ and $M = S/(f)$.\nThe minimal resolution is $0 \\to S(-d) \\to S \\to M \\to 0$, so $\\beta_{0,0} = 1$ and $\\beta_{1,d} = 1$.\nHence $\\operatorname{reg} M = \\max(0, d - 1) = d - 1$.\n\nMinimality of the resolution is needed.\nTake $S = k[x]$ and $M = S/(x)$, so that $\\operatorname{reg} M = 0$.\nAdd the exact summand $S(-5) \\to S(-5)$ in homological degrees $1$ and $0$.\nThe result is a non-minimal resolution with the pair $(i, j) = (0, 5)$, and it would give the wrong value $5$.\n\nAn equivalent form uses [[local cohomology|6JB30BXDA2]] with respect to $\\mathfrak{m} = (x_1, \\ldots, x_n)$: $\\operatorname{reg} M = \\max_i \\set{ a_i(M) + i }$, where $a_i(M)$ is the largest degree $d$ with $H^i_{\\mathfrak{m}}(M)_d \\ne 0$, and the maximum is over the $i$ with $H^i_{\\mathfrak{m}}(M) \\ne 0$.\nFor a nonzero proper homogeneous ideal $I$, one has $\\operatorname{reg} I = \\operatorname{reg}(S/I) + 1$.\n",
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