= Construction of a closed subscheme from a quasi-coherent ideal
A <quasi-coherent sheaf> of ideals $\mathcal I\subseteq\mathcal O_X$ gives a <closed subscheme> by gluing $\operatorname{Spec}(A/J)$ on <affine open subschemes> where $\mathcal I=\widetilde J$. Compatibility follows from <exactness of localization>. On the closed set $Z=\{x:\mathcal I_x\ne\mathcal O_{X,x}\}$, the <structure sheaf> is $i^{-1}(\mathcal O_X/\mathcal I)$, where $i:Z\hookrightarrow X$ is the inclusion. Its pushforward is $\mathcal O_X/\mathcal I$.
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