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  "body": "\n## Definition\n\nLet $X$ be a [[scheme|5A35TYZD02]] and $n \\ge 0$ an integer.\nA *vector bundle of rank $n$* on $X$ is a [[sheaf of $\\mathcal{O}_X$-modules|S33B2H3MYD]] $\\mathcal{E}$ with the following property.\nEvery point of $X$ has an open neighbourhood $U$ such that $\\mathcal{E}|_U \\cong \\mathcal{O}_U^{\\oplus n}$ as $\\mathcal{O}_U$-modules.\nIt is also called a *locally free sheaf of rank $n$*.\n\n## Remarks\n\nThe free sheaf $\\mathcal{O}_X^{\\oplus n}$ is a vector bundle of rank $n$, with $U = X$ for every point.\nOn $\\mathbb{A}^1_k = \\operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\\widetilde{M}$ with $M = k[t]/(t)$ is not a vector bundle.\nIts stalk at the origin is $k$, which is killed by $t$ and so is not a free module over the local ring $k[t]_{(t)}$; its stalks elsewhere are $0$.\nA vector bundle is a [[quasi-coherent sheaf|KPKND0K7W6]], and it is coherent when $X$ is locally Noetherian.\nTo $\\mathcal{E}$ corresponds the geometric vector bundle $\\mathbb{V}(\\mathcal{E}^\\vee) = \\operatorname{Spec}_X \\operatorname{Sym}(\\mathcal{E}^\\vee)$, whose sheaf of sections is $\\mathcal{E}$.\nHartshorne writes $\\mathbb{V}(\\mathcal{E}) = \\operatorname{Spec}_X \\operatorname{Sym}(\\mathcal{E})$, whose sheaf of sections is $\\mathcal{E}^\\vee$, so the placement of the dual differs between sources.\n",
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