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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].\nAn *invertible sheaf* on $X$ is a [[sheaf of $\\mathcal{O}_X$-modules|S33B2H3MYD]] $\\mathcal{L}$ with the following property.\nEvery point of $X$ has an open neighbourhood $U$ such that $\\mathcal{L}|_U \\cong \\mathcal{O}_X|_U$ as $\\mathcal{O}_U$-modules.\n\n## Remarks\n\nThe structure sheaf $\\mathcal{O}_X$ is invertible, with $U = X$ for every point.\nOn $X = \\mathbb{A}^2_k = \\operatorname{Spec} k[x, y]$ over a field $k$, the ideal sheaf $\\mathcal{I}$ of the origin, given by the ideal $(x, y)$, is not invertible.\nIts stalk at the origin is the maximal ideal of $k[x, y]_{(x, y)}$, which needs two generators, so it is not free of rank $1$.\nIn other words, an invertible sheaf is a locally free sheaf of rank $1$.\nFor invertible sheaves $\\mathcal{L}$ and $\\mathcal{M}$, the tensor product $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{M}$ is invertible.\nThe dual $\\mathcal{L}^\\vee = \\mathcal{H}om(\\mathcal{L}, \\mathcal{O}_X)$ satisfies $\\mathcal{L} \\otimes \\mathcal{L}^\\vee \\cong \\mathcal{O}_X$, which explains the name.\n",
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