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  "body": "\n## Statement\n\nLet $a \\in \\C^n$, let $\\mathcal{O}_{n,a}$ be the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $a$, and let $I \\subseteq \\mathcal{O}_{n,a}$ be an ideal.\nLet $V(I)$ be the [[germ of the analytic set|Q9B74BYZ8C]] at $a$ defined by $I$, and let $\\mathcal{I}(V(I))$ be the ideal of germs vanishing on $V(I)$.\nThen\n$$\\mathcal{I}(V(I)) = \\sqrt{I},$$\nthe [[radical|437YRCNYRJ]] of $I$.\nIn other words, if a germ $f \\in \\mathcal{O}_{n,a}$ vanishes on $V(I)$, then $f^m \\in I$ for some integer $m \\ge 1$.\n\n## Remarks\n\nThe inclusion $\\sqrt{I} \\subseteq \\mathcal{I}(V(I))$ is clear: if $f^m \\in I$, then $f^m$, and hence $f$, vanishes on $V(I)$.\nFor the other inclusion, write $\\sqrt{I}$ as a finite intersection of prime ideals, which is possible because $\\mathcal{O}_{n,a}$ is Noetherian, and reduce to the case of a prime ideal $\\mathfrak{p}$.\nFor $\\mathfrak{p}$ prime, the local parametrization of the irreducible germ $V(\\mathfrak{p})$ shows that a germ vanishing on $V(\\mathfrak{p})$ lies in $\\mathfrak{p}$.\nExample: for $I = (z_1^2) \\subseteq \\mathcal{O}_{2,0}$, the germ $V(I)$ is the line $\\set{z_1 = 0}$, and $\\mathcal{I}(V(I)) = (z_1) = \\sqrt{I}$; here $z_1^2 \\in I$.\nAs a consequence, $I \\mapsto V(I)$ and $A_a \\mapsto \\mathcal{I}(A_a)$ are mutually inverse bijections between radical ideals of $\\mathcal{O}_{n,a}$ and germs of analytic sets at $a$, and prime ideals correspond to irreducible germs.\nAs for Hilbert's Nullstellensatz, the base field matters: the statement fails for real analytic germs, since $x^2 + y^2$ vanishes only at $0 \\in \\R^2$ but $x \\notin \\sqrt{(x^2 + y^2)}$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the germs vanishing on the common zeros of some germs are exactly those of which some power can be built from them.\nIt is the local, holomorphic version of Hilbert's Nullstellensatz for polynomials.\nIt means that the shape of an analytic set near a point and the algebra of the ideal of germs determine each other.\n\n### L2 human\n\nThe Nullstellensatz is the dictionary of local analytic geometry.\nWith it, the decomposition of an analytic germ into irreducible components corresponds to the primary decomposition of its ideal.\nThe dimension of a germ $A_a$ equals the Krull dimension of $\\mathcal{O}_{n,a}/\\mathcal{I}(A_a)$.\nIt is also an ingredient in Cartan's coherence theorem for the ideal sheaf of an analytic set.\n",
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