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  "body": "\n## Definition\n\nLet $S = \\bigoplus_{d \\ge 0} S_d$ be a [[graded ring|CAPQBQDRA7]], and let $X = \\operatorname{Proj} S$ be given by the [[Proj construction|A39XQHCVNW]].\nFor an integer $n$, let $S(n)$ be the graded $S$-module with $S(n)_d = S_{n + d}$, where $S_e = 0$ for $e \u003c 0$.\nLet $M$ be a graded $S$-module, and let $f \\in S$ be homogeneous of positive degree.\nLet $\\widetilde{M}$ be the sheaf on $X$ with sections $M_{(f)}$ over $D_+(f)$, where $M_{(f)}$ is the degree zero part of $M_f$.\nThe *twisting sheaf* of degree $n$ is the [[quasi-coherent sheaf|KPKND0K7W6]] $\\mathcal{O}_X(n) = \\widetilde{S(n)}$.\n*Serre's twisting sheaf* is $\\mathcal{O}_X(1)$.\n\n## Remarks\n\nLet $S = k[x_0, \\dots, x_r]$ over a field $k$, with $r \\ge 1$, so that $X = \\mathbb{P}^r_k$.\nThen $\\mathcal{O}_X(n)$ is an [[invertible sheaf|7JCF2TE3V5]], and its global sections are the homogeneous polynomials of degree $n$ for $n \\ge 0$, and $0$ for $n \u003c 0$.\nOn $D_+(x_i)$, the section $x_i^n$ generates $\\mathcal{O}_X(n)$, which gives the local isomorphism with $\\mathcal{O}_X$.\nIf $S$ is generated by $S_1$ as an $S_0$-algebra, then each $\\mathcal{O}_X(n)$ is invertible and $\\mathcal{O}_X(n) \\otimes \\mathcal{O}_X(m) \\cong \\mathcal{O}_X(n + m)$.\nWithout that hypothesis $\\mathcal{O}_X(1)$ need not be invertible.\nTake $S = k[x_0, x_1, x_2]$ with $x_0, x_1$ of degree $1$ and $x_2$ of degree $2$, and the point $p$ where $x_0 = x_1 = 0$.\nOn $D_+(x_2)$, the sections of $\\mathcal{O}_X(1)$ need the two generators $x_0$ and $x_1$ near $p$, so $\\mathcal{O}_X(1)$ is not invertible at $p$.\nFor a quasi-coherent sheaf $\\mathcal{F}$, one writes $\\mathcal{F}(n) = \\mathcal{F} \\otimes \\mathcal{O}_X(n)$, the twist of $\\mathcal{F}$ by $n$.\n",
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    "name": "Serre's twisting sheaf",
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    "tag": "JR83AKA184",
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  "msc": [
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  "name": "Serre's twisting sheaf",
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  "url": "https://tamnd.github.io/math2/wiki/serres-twisting-sheaf-definition/",
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