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  "body": "\n## Summary\n\nIn algebraic geometry, a vector bundle of rank $n$ on a scheme $X$ is usually identified with a locally free sheaf of rank $n$.\nThis is a sheaf of modules locally isomorphic to $\\mathcal{O}_X^{\\oplus n}$.\nSuch a sheaf is the sheaf of sections of a geometric vector bundle.\nThis is a scheme over $X$ whose fibres are affine spaces with a linear structure.\nVector bundles of rank one are the invertible sheaves, and tangent and cotangent bundles of smooth varieties are the main examples of higher rank.\n\n## Explanations\n\n### L1 human\n\nA vector bundle attaches a vector space of fixed dimension to each point of a space, varying continuously.\nLocally it is a product of the space with a vector space, but globally it may twist.\nIn algebraic geometry one usually works with the sheaf of its sections, which is locally a direct sum of copies of the sheaf of functions.\nThe tangent bundle of a smooth variety is an example: at each point, it gives the tangent space.\n",
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