Ideal sheaf

Remarks

On $X = \operatorname{Spec} A$, an ideal $I \subseteq A$ gives the ideal sheaf $\widetilde{I}$, with $\widetilde{I}(D(f)) = I_f \subseteq A_f$. It is a sub-$\mathcal{O}_X$-module of $\widetilde{A} = \mathcal{O}_X$, since localization preserves ideals. On $X = \mathbb{A}^1_k = \operatorname{Spec} k[t]$, the subsheaf of $\mathcal{O}_X$ of constant functions is not an ideal sheaf. Its sections over $X$ form $k \subseteq k[t]$, which is not an ideal, since $t \cdot 1 = t$ is not constant. For 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 $\mathcal{I}_Z$. This gives a bijection between closed subschemes of $X$ and quasi-coherent ideal sheaves on $X$.