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  "body": "\n## Definition\n\nLet $A$ be a ring and $M$ an $A$-[[module|0NZ8JA25WK]].\nFor $f \\in A$, let $D(f)$ be the basic open set of $\\operatorname{Spec} A$ where $f$ does not vanish.\nThe sheaf $\\widetilde{M}$ on $\\operatorname{Spec} A$ is the unique sheaf of modules with $\\widetilde{M}(D(f)) = M_f$ for every $f \\in A$, with restriction maps given by localization.\nLet $X$ be a [[scheme|5A35TYZD02]].\nA [[sheaf of $\\mathcal{O}_X$-modules|S33B2H3MYD]] $\\mathcal{F}$ is *quasi-coherent* if $X$ has a cover by open sets $U$ with the following property.\nEach $U$ is an [[affine scheme|B3V3XJ2W8G]] $\\operatorname{Spec} A$, and $\\mathcal{F}|_U \\cong \\widetilde{M}$ for some $A$-module $M$.\n\n## Remarks\n\nOn $X = \\operatorname{Spec} A$, the sheaf $\\widetilde{I}$ of an ideal $I \\subseteq A$ is quasi-coherent, with the cover by $X$ itself.\nLet $R$ be a discrete valuation ring with fraction field $K$, and $X = \\operatorname{Spec} R = \\set{\\eta, s}$, with $\\eta$ the generic point.\nThe sheaf $\\mathcal{F}$ with $\\mathcal{F}(X) = 0$ and $\\mathcal{F}(\\set{\\eta}) = K$ is an $\\mathcal{O}_X$-module but is not quasi-coherent.\nThe only open set containing $s$ is $X$, so $\\mathcal{F} \\cong \\widetilde{M}$ would force $M = \\mathcal{F}(X) = 0$ and then $\\mathcal{F}(\\set{\\eta}) = 0$.\nIf $\\mathcal{F}$ is quasi-coherent, then $\\mathcal{F}|_U \\cong \\widetilde{\\mathcal{F}(U)}$ for every affine open $U$, not only for those of the chosen cover.\nOn a general ringed space, $\\mathcal{F}$ is called quasi-coherent if it is locally the cokernel of a map between free modules; on schemes the two definitions agree.\n",
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