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  "body": "\n## Definition\n\nLet $X$ be a locally [[Noetherian scheme|ND6XEY68P4]].\nA [[quasi-coherent sheaf|KPKND0K7W6]] $\\mathcal{F}$ on $X$ is *coherent* if $X$ has a cover by affine open sets $U = \\operatorname{Spec} A$ with the following property.\nFor each such $U$, $\\mathcal{F}|_U \\cong \\widetilde{M}$ for a [[finitely generated|05QY1VEBYD]] $A$-module $M$.\n\n## Remarks\n\nOn $\\mathbb{A}^1_k = \\operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\\widetilde{M}$ with $M = k[t]/(t)$ is coherent, since $M$ is generated by one element.\nOn $\\operatorname{Spec} \\Z$, the sheaf $\\widetilde{\\Q}$ is quasi-coherent but not coherent.\nFor it to be coherent, $\\Q = \\widetilde{\\Q}(\\operatorname{Spec} \\Z)$ would have to be a finitely generated $\\Z$-module, and it is not.\nIf $\\mathcal{F}$ is coherent on a locally Noetherian scheme, then $\\mathcal{F}(U)$ is finitely generated over $A$ for every affine open $U = \\operatorname{Spec} A$.\nOn a general ringed space, Serre's definition asks that $\\mathcal{F}$ is locally finitely generated and that every finitely generated subsheaf over an open set is locally finitely presented.\nOn a locally Noetherian scheme the two definitions agree.\n",
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  "name": "Coherent sheaf",
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