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  "body": "\n## Definition\n\nA *local model* is built as follows.\nLet $D \\subseteq \\C^n$ be open, let $\\mathcal{O}_D$ be the [[sheaf|TB6TM6R4MD]] of [[holomorphic functions|507C34W0ZT]] on $D$, and let $f_1, \\dots, f_k$ be holomorphic functions on $D$.\nLet $\\mathcal{I} \\subseteq \\mathcal{O}_D$ be the [[ideal sheaf|N3PGYJJQ7W]] generated by $f_1, \\dots, f_k$, and let $Z = \\set{z \\in D : f_1(z) = \\dots = f_k(z) = 0}$.\nThe local model is the ringed space $(Z, (\\mathcal{O}_D / \\mathcal{I})|_Z)$.\n\nA *complex analytic space* is a pair $(X, \\mathcal{O}_X)$ such that:\n\n1. $X$ is a [[Hausdorff space|0S1FJ8VEX9]];\n2. $\\mathcal{O}_X$ is a sheaf of $\\C$-algebras on $X$;\n3. each point of $X$ has an open neighbourhood $U$ such that $(U, \\mathcal{O}_X|_U)$ is isomorphic to a local model as a $\\C$-ringed space.\n\n## Remarks\n\nExample: $\\C^n$ with its sheaf of holomorphic functions is a complex space; take $D = \\C^n$ and $k = 0$.\nThe double point is the local model with $D = \\C$ and $f_1 = z^2$: the underlying space is $\\set{0}$ and the ring of functions is $\\C[z]/(z^2)$, which contains the nilpotent $z$.\n\nNon-example: $\\C$ with the sheaf of continuous complex functions satisfies clauses 1 and 2 but not 3.\nThe stalks of a local model are quotients of the [[ring of convergent power series|EVPZ7ZG1HB]], hence [[Noetherian|0670T59HGE]].\nLet $\\mathfrak{m}$ be the maximal ideal of the ring of germs of continuous functions at $0$.\nThen $\\mathfrak{m} = \\mathfrak{m}^2$, since $f = \\sqrt{\\abs{f}} \\cdot (f / \\sqrt{\\abs{f}})$.\nBy Nakayama's lemma $\\mathfrak{m}$ is not finitely generated, so this ring is not Noetherian.\n\nA complex space is a [[locally ringed space|B7FPT8JTXR]].\nMorphisms of complex spaces, also called holomorphic maps, are morphisms of locally ringed spaces of $\\C$-algebras.\nSome authors drop the Hausdorff condition.\n",
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