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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]].\nIt is *reduced* if, for every $x \\in X$, the [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is a [[reduced ring|SZHY975FFX]], that is, it has no nonzero nilpotent elements.\n\n## Remarks\n\nExample: the local model in $\\C^2$ given by $f = z_1 z_2$ is reduced.\nIts stalk at $0$ is $\\C\\{z_1, z_2\\}/(z_1 z_2)$, and if $g^k \\in (z_1 z_2)$ then $z_1 \\mid g^k$ and $z_2 \\mid g^k$ in the factorial ring $\\C\\{z_1, z_2\\}$, so $z_1 \\mid g$ and $z_2 \\mid g$, hence $z_1 z_2 \\mid g$.\nAt the other points the stalk is a ring of convergent power series, which is reduced.\n\nNon-example: the local model in $\\C$ given by $f = z^2$ is not reduced, since the class of $z$ in $\\C\\{z\\}/(z^2)$ is nonzero and has square $0$.\n\nEquivalently, $X$ is reduced if every section of $\\mathcal{O}_X$ whose value vanishes at every point of its domain is zero.\nFor any complex space $X$, the quotient of $\\mathcal{O}_X$ by its sheaf of nilpotent germs defines a reduced complex space $X_{\\mathrm{red}}$ on the same topological space.\n",
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