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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]] and let $\\mathcal{F}$ be a [[sheaf of $\\mathcal{O}_X$-modules|S33B2H3MYD]].\nThe sheaf $\\mathcal{F}$ is of *finite type* if each point of $X$ has an open neighbourhood $U$ with a surjective morphism $\\mathcal{O}_U^p \\to \\mathcal{F}|_U$ for some $p \\ge 0$.\nThe sheaf $\\mathcal{F}$ is a *coherent analytic sheaf* if:\n\n1. $\\mathcal{F}$ is of finite type;\n2. for every open $U \\subseteq X$, every integer $q \\ge 0$ and every morphism $\\varphi \\colon \\mathcal{O}_U^q \\to \\mathcal{F}|_U$ of sheaves of modules, the kernel of $\\varphi$ is of finite type.\n\n## Remarks\n\nExample: the sheaf $\\mathcal{O}_X^p$ is coherent for every complex space $X$; for $X = \\C^n$ this is the Oka coherence theorem, and the general case follows from it.\nGiven it, a sheaf is coherent if and only if it has, locally, a presentation $\\mathcal{O}_U^q \\to \\mathcal{O}_U^p \\to \\mathcal{F}|_U \\to 0$.\n\nNon-example: let $X = \\C$, $U = \\C \\setminus \\set{0}$, and let $\\mathcal{F}$ be the sheaf of holomorphic functions on $U$ extended by zero to $\\C$.\nIts stalk at $0$ is $0$ and its stalks at other points are $\\mathcal{O}_{\\C,x}$.\nA section of $\\mathcal{F}$ on a connected open set $V \\ni 0$ is a holomorphic function on $V \\setminus \\set{0}$ that vanishes near $0$, so it is zero by the identity theorem.\nSo no finite set of sections over a neighbourhood of $0$ generates the nonzero stalks near $0$, and clause 1 fails.\n",
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