Quasi-coherent ideals and closed subschemes

Let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be quasi-coherent and let $U=\mathrm{Spec}A$ be affine. By [F2], $J=\mathrm{\Gamma }(U,\mathcal{I})$ is an ideal of $A$ and the natural comparison identifies $\mathcal{I}{\mathrm{\mid }}_U$ with $\tilde{J}$ as an ideal subsheaf of ${\mathcal{O}}_U=\tilde{A}$. At $\mathfrak{p}\in U$ one has ${\mathcal{I}}_{\mathfrak{p}}\mathrm{\ne }{\mathcal{O}}_{U,\mathfrak{p}}$ exactly when $J\subseteq \mathfrak{p}$, so $V(\mathcal{I})\cap U=V(J)$; these chartwise closed sets make $V(\mathcal{I})$ closed in $X$.

Conversely, let $i:Z\hookrightarrow X$ be closed and let $K=\ker ({\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z)$. For every affine $U=\mathrm{Spec}A\subseteq X$, the current affine quotient supplier [F1] gives a unique ideal $J\subseteq A$ and an isomorphism $i^{-1}(U)\cong \mathrm{Spec}(A\mathrm{/}J)$ over $U$. By [F3], the kernel of the local quotient map $\tilde{A}\to \widetilde{A\mathrm{/}J}$ has stalks $J_{\mathfrak{p}}$, hence equals $\tilde{J}$; therefore $K{\mathrm{\mid }}_U=\tilde{J}$. These local descriptions prove $K$ is a quasi-coherent ideal sheaf, including when $i^{-1}(U)$ is empty and $J=A$.

Form $Z_{\mathcal{I}}=(V(\mathcal{I}),({\mathcal{O}}_X\mathrm{/}\mathcal{I}){\mathrm{\mid }}_{V(\mathcal{I})})$. On $U$ as in step 1.1, [F3] identifies its underlying closed set with $\mathrm{Spec}(A\mathrm{/}J)$ and its stalk at $\mathfrak{p}\supseteq J$ with $A_{\mathfrak{p}}\mathrm{/}{J}_{\mathfrak{p}}$, the same local ring as the quotient scheme. The canonical sheaf map from $\widetilde{A\mathrm{/}J}$ to the restricted quotient sheaf is an isomorphism on these stalks, hence an isomorphism; this is also the exact chart identification of [F5]. Thus $Z_{\mathcal{I}}$ is locally $\mathrm{Spec}(A\mathrm{/}J)$, hence a scheme, and its inclusion into $X$ is a closed immersion: it is a homeomorphism onto $V(\mathcal{I})$ and ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_{Z_{\mathcal{I}}}$ is stalkwise the quotient ${\mathcal{O}}_{X,x}\to {\mathcal{O}}_{X,x}\mathrm{/}{\mathcal{I}}_x$ on that set and the map to zero outside it, so is surjective by [F4]. Its kernel is $\mathcal{I}$.

Starting with $\mathcal{I}$, step 2.1 gives $K_{Z_{\mathcal{I}}}=\mathcal{I}$. Starting with $i:Z\hookrightarrow X$, step 1.2 gives $K$ and, on every affine $U$, both $Z_K\cap U$ and $Z\cap U$ are canonically $\mathrm{Spec}(A\mathrm{/}J)$ over $U$; these local isomorphisms agree on overlaps because both are induced by ${\mathcal{O}}_X\mathrm{/}K\cong i_{\ast }{\mathcal{O}}_Z$, so $Z_K\cong Z$ over $X$. If $\mathcal{I}={\mathcal{O}}_X$, then $V(\mathcal{I})=\mathrm{\varnothing }$ and $Z_{\mathcal{I}}$ is empty; the empty immersion has kernel ${\mathcal{O}}_X$. These are the mutually inverse assignments in the Statement. The Axiom of Choice is used through [F1], [F2] and [F5]; no earlier published affine-quotient or correspondence theorem is a premise.