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  "body": "\n## Statement\n\nLet $X$ be a [[Stein space|Z4AMJS5KA6]] and let $\\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.\nThen $\\mathcal{F}$ is generated by its global sections.\nThat is, for every $x \\in X$, the germs at $x$ of the elements of $\\Gamma(X, \\mathcal{F})$ generate the stalk $\\mathcal{F}_x$ as a module over $\\mathcal{O}_{X,x}$.\n\n## Remarks\n\nHenri Cartan proved theorems A and B in his seminar of 1951–52, building on the work of Oka.\nIn the usual proof, $X$ is exhausted by compact holomorphically convex sets.\nOn each of them, $\\mathcal{F}$ is generated by finitely many sections, by a local argument.\nCartan's theorem B on these pieces and a Runge-type approximation theorem then pass to the limit.\nExample: let $\\mathcal{I}$ be the ideal sheaf of a closed complex submanifold $Y$ of a Stein manifold $X$.\nTheorem A says that near each point, $Y$ is cut out by global holomorphic functions on $X$ that vanish on $Y$.\nFor $X = \\C^n$ and $\\mathcal{F} = \\mathcal{O}$ the theorem is trivial, because the constant function $1$ generates every stalk.\nThe Stein hypothesis is needed: on the projective line $\\mathbb{P}^1$, the sheaf $\\mathcal{O}(-1)$ is coherent but has no nonzero global sections, so it is not generated by them.\n\n## Explanations\n\n### L1 human\n\nA coherent sheaf describes local data, such as the holomorphic functions that vanish on a given analytic set near each point.\nTheorem A says that on a Stein space all this local data comes from global data.\nEverything one sees near a point is produced by objects defined on the whole space.\nIt is the complex-analytic counterpart of the fact that every quasi-coherent sheaf on an affine scheme is generated by its global sections.\n\n### L2 human\n\nTheorem A is usually proved together with [[Cartan's theorem B|R7VKRV8AGQ]], and each is used in the proof of the other on exhausting compact sets.\nA typical consequence: global holomorphic functions give local coordinates at every point of a Stein manifold.\nAnother one: every analytic subset of a Stein space is the common zero set of global holomorphic functions.\n",
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