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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 the [[sheaf cohomology|PQ6Q143ARJ]] groups vanish in positive degree:\n$$H^q(X, \\mathcal{F}) = 0 \\quad \\text{for all } q \\ge 1.$$\n\n## Remarks\n\nThe converse holds: a complex space on which $H^1(X, \\mathcal{F}) = 0$ for every coherent analytic sheaf $\\mathcal{F}$ is Stein, so theorem B characterizes Stein spaces.\nThe proof first treats compact polydiscs and analytic polyhedra.\nThere, vanishing follows from Cousin-type gluing and from the local theory of coherent sheaves.\nThen it passes to the limit over an exhaustion of $X$ by holomorphically convex compact sets, using approximation to control the inverse limit.\nExample: on a Stein manifold, $H^1(X, \\mathcal{O}) = 0$, so the first Cousin problem is always solvable.\nExample: for an analytic subset $Y$ with ideal sheaf $\\mathcal{I}$, the exact sequence $0 \\to \\mathcal{I} \\to \\mathcal{O}_X \\to \\mathcal{O}_X / \\mathcal{I} \\to 0$ and $H^1(X, \\mathcal{I}) = 0$ show that every holomorphic function on $Y$ extends to a holomorphic function on $X$.\nThe Stein hypothesis is needed: on the projective line, $H^1(\\mathbb{P}^1, \\mathcal{O}(-2)) \\cong \\C$ is not zero; and on $\\C^2 \\setminus \\set{0}$, the group $H^1(X, \\mathcal{O})$ is infinite-dimensional.\n\n## Explanations\n\n### L1 human\n\nMany problems in complex analysis ask to glue local holomorphic solutions into a global one.\nThe obstruction to such gluing lies in a cohomology group.\nTheorem B says that on a Stein space these obstructions vanish, for all problems described by coherent sheaves.\nSo on Stein spaces, whatever can be solved locally can be solved globally.\n\n### L2 human\n\nTheorem B, with the exponential sequence, solves the Cousin problems: the additive one always, and the multiplicative one exactly when a topological obstruction in $H^2(X, \\Z)$ vanishes.\nIt gives extension of holomorphic functions from closed analytic subsets, interpolation on discrete sets, and the embedding of Stein manifolds into $\\C^N$.\nIt is the analytic analogue of Serre's vanishing theorem for quasi-coherent sheaves on affine schemes.\n",
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