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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].\nAn *ideal sheaf* on $X$ is a sub-$\\mathcal{O}_X$-module $\\mathcal{I} \\subseteq \\mathcal{O}_X$ in the sense of [[sheaves of modules|S33B2H3MYD]].\nEquivalently, $\\mathcal{I}$ is a subsheaf of $\\mathcal{O}_X$ such that $\\mathcal{I}(U)$ is an [[ideal|096Z9MP2ME]] of $\\mathcal{O}_X(U)$ for every open set $U \\subseteq X$.\n\n## Remarks\n\nOn $X = \\operatorname{Spec} A$, an ideal $I \\subseteq A$ gives the ideal sheaf $\\widetilde{I}$, with $\\widetilde{I}(D(f)) = I_f \\subseteq A_f$.\nIt is a sub-$\\mathcal{O}_X$-module of $\\widetilde{A} = \\mathcal{O}_X$, since localization preserves ideals.\nOn $X = \\mathbb{A}^1_k = \\operatorname{Spec} k[t]$, the subsheaf of $\\mathcal{O}_X$ of constant functions is not an ideal sheaf.\nIts sections over $X$ form $k \\subseteq k[t]$, which is not an ideal, since $t \\cdot 1 = t$ is not constant.\nFor a closed subscheme $Z \\subseteq X$ of a scheme, the kernel of $\\mathcal{O}_X \\to \\mathcal{O}_Z$ is a quasi-coherent ideal sheaf $\\mathcal{I}_Z$.\nThis gives a bijection between closed subschemes of $X$ and quasi-coherent ideal sheaves on $X$.\n",
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