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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C^n$ be open and let $A \\subseteq \\Omega$ be an [[analytic subset|GB7M582M5Y]].\nFor each open $U \\subseteq \\Omega$ let $\\mathcal{I}_A(U)$ be the set of [[holomorphic functions|507C34W0ZT]] on $U$ that vanish at every point of $A \\cap U$.\nThen the [[ideal sheaf|N3PGYJJQ7W]] $\\mathcal{I}_A$ is a [[coherent analytic sheaf|DSJET02WPR]] on $\\Omega$.\n\n## Remarks\n\nThe question is local, and one may assume that the germ of $A$ at a point is irreducible.\nThe [[local parametrization theorem|K18PKZY5RS]] presents $A$ as a branched covering of a domain in $\\C^d$, and from this one writes down explicit holomorphic functions that generate $\\mathcal{I}_A$ near the point.\nCoherence then follows from the [[Oka coherence theorem|MSX3VKM9Z5]], since $\\mathcal{I}_A$ is a subsheaf of finite type of the coherent sheaf $\\mathcal{O}_\\Omega$.\n\nExample: for $A = \\set{z_1 = z_2 = 0} \\subseteq \\C^3$, the sheaf $\\mathcal{I}_A$ is generated by $z_1$ and $z_2$ at every point.\nFor the cusp $A = \\set{z_1^2 = z_2^3}$, it is generated by $z_1^2 - z_2^3$.\nIn contrast, the sheaf generated by $z_1^2$ has the same zero set $\\set{z_1 = 0}$ but is strictly smaller than the ideal sheaf of that set.\n\n## Explanations\n\n### L1 human\n\nAn analytic set is the common zero set of some holomorphic equations.\nThe theorem says that near each point, finitely many equations produce every holomorphic function that vanishes on the set.\nMoreover, the same equations keep doing so at all nearby points.\n\n### L2 human\n\nThe theorem gives every analytic set $A$ a canonical structure of [[reduced complex space|831N2Q4GCB]], with structure sheaf $\\mathcal{O}_\\Omega / \\mathcal{I}_A$ restricted to $A$.\nBy the Rückert Nullstellensatz, the nilradical of a local model $\\mathcal{O}_D / \\mathcal{I}$ is $\\mathcal{I}_Z / \\mathcal{I}$, where $Z$ is the zero set of $\\mathcal{I}$.\nSo the theorem shows that the nilradical of the structure sheaf of any complex space is coherent, and every complex space has a reduction.\nTogether with Oka's theorem it is one of the two pillars of the theory of coherent analytic sheaves.\n",
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