Quasi-coherent ideals and closed subschemes, complete route

An ideal sheaf on an affine chart is associated to its global sections: let $U=\mathrm{Spec}A$ be an affine open and let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be a quasi-coherent ideal sheaf; put $J=\mathrm{\Gamma }(U,\mathcal{I})$, which is an ideal of $A=\mathrm{\Gamma }(U,{\mathcal{O}}_X)$ because $\mathcal{I}$ is an ideal in every section ring. By [F1] applied to the quasi-coherent module $\mathcal{I}{\mathrm{\mid }}_U$, the canonical comparison $\tilde{J}\to \mathcal{I}{\mathrm{\mid }}_U$ is an isomorphism, and naturality of the comparison with respect to the inclusion $\mathcal{I}\subseteq {\mathcal{O}}_X$ shows that it is an isomorphism of subsheaves of ${\mathcal{O}}_U=\tilde{A}$; hence $\mathcal{I}{\mathrm{\mid }}_U=\tilde{J}$ and, by [F2], the stalk criterion ${\mathcal{I}}_{\mathfrak{p}}\mathrm{\ne }{\mathcal{O}}_{U,\mathfrak{p}}$ holds exactly for $\mathfrak{p}\in V(J)$, so $V(\mathcal{I})\cap U=V(J)$ and $V(\mathcal{I})$ is closed in $X$ because the affine charts cover $X$.

The kernel of a closed immersion is a quasi-coherent ideal sheaf: let $i:Z\to X$ be a closed immersion and put $K=\ker ({\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z)$, computed as a kernel of sheaves, so that $K$ is a subsheaf of ${\mathcal{O}}_X$ and an ideal in every section ring. Let $U=\mathrm{Spec}A$ be an affine open of $X$; by the batch-5 supplier [F6] there is an ideal $J\subseteq A$ with $i^{-1}(U)\cong \mathrm{Spec}(A\mathrm{/}J)$ and with $i{\mathrm{\mid }}_U$ the quotient map, so over $U$ the structure map is $\tilde{A}\to \widetilde{A\mathrm{/}J}$ and [F2] identifies its kernel with $\tilde{J}$; hence $K{\mathrm{\mid }}_U=\tilde{J}$ is associated, and because the affine charts cover $X$ the ideal sheaf $K$ satisfies the local definition of quasi-coherence. The uniqueness of $J$ in [F6] shows in addition that the local descriptions agree on overlaps, so $K$ is well defined independently of the charts. This is the exact point where the batch-5 supplier is consumed.

The affine model of the construction: with $J=\mathrm{\Gamma }(U,\mathcal{I})$ as in step 1.1, [F2] identifies the quotient ${\mathcal{O}}_U\mathrm{/}\mathcal{I}{\mathrm{\mid }}_U={\mathcal{O}}_U\mathrm{/}\tilde{J}$ with $\widetilde{A\mathrm{/}J}$, and [F3] identifies $\mathrm{Spec}(A\mathrm{/}J)$ with the closed subset $V(J)$; the sections of both structure sheaves on the distinguished open $D(\bar{f})\subseteq \mathrm{Spec}(A\mathrm{/}J)$ corresponding to $D(f)\cap V(J)$ are $(A\mathrm{/}J{)}_f=A_f\mathrm{/}{J}_f$, so the locally ringed space $(V(J),({\mathcal{O}}_U\mathrm{/}\tilde{J}){\mathrm{\mid }}_{V(J)})$ is isomorphic to the affine scheme $\mathrm{Spec}(A\mathrm{/}J)$. Consequently every point of $V(\mathcal{I})$ has an open neighbourhood in the closed ringed subspace $Z_{\mathcal{I}}=(V(\mathcal{I}),({\mathcal{O}}_X\mathrm{/}\mathcal{I}){\mathrm{\mid }}_{V(\mathcal{I})})$ isomorphic to an affine scheme, and by [F5] the space $Z_{\mathcal{I}}$ is a scheme.

The closed immersion is recovered from its kernel: with $K=K_Z$ as in step 1.2, the map ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective by the definition of a closed immersion, so it induces an isomorphism of sheaves of rings ${\mathcal{O}}_X\mathrm{/}K\cong i_{\ast }{\mathcal{O}}_Z$; moreover $V(K)=\{x:K_x\mathrm{\ne }{\mathcal{O}}_{X,x}\}=\{x:(i_{\ast }{\mathcal{O}}_Z{)}_x\mathrm{\ne }0\}$ because $K_x$ is the kernel of the surjection ${\mathcal{O}}_{X,x}\to (i_{\ast }{\mathcal{O}}_Z{)}_x$, and this set is the image of $i$: for $x=i(z)$ the stalk $(i_{\ast }{\mathcal{O}}_Z{)}_x$ is the local ring ${\mathcal{O}}_{Z,z}$, which is nonzero for every point of a scheme, while for $x$ outside the closed image the stalk is $0$. Hence the canonical map $Z\to Z_K$ (identity on the underlying spaces, the isomorphism ${\mathcal{O}}_X\mathrm{/}K\cong i_{\ast }{\mathcal{O}}_Z$ on structure sheaves) is an isomorphism of closed subschemes of $X$ over $X$.

$Z_{\mathcal{I}}$ is a closed subscheme with kernel $\mathcal{I}$: the inclusion $i:Z_{\mathcal{I}}\to X$ is a homeomorphism onto the closed subset $V(\mathcal{I})$ by construction. On each affine chart $U=\mathrm{Spec}A$, step 1.1 gives $\mathcal{I}{\mathrm{\mid }}_U=\tilde{J}$ and step 2.1 identifies $Z_{\mathcal{I}}\cap U$ with $\mathrm{Spec}(A\mathrm{/}J)$. On every distinguished open $D(f)\subseteq U$, both $({\mathcal{O}}_U\mathrm{/}\tilde{J})(D(f))$ and $(i_{\ast }{\mathcal{O}}_{Z_{\mathcal{I}}})(D(f))$ are $(A\mathrm{/}J{)}_f$ by [F2], [F4] and step 2.1. Since distinguished opens form a basis, the natural map ${\mathcal{O}}_X\mathrm{/}\mathcal{I}\to i_{\ast }{\mathcal{O}}_{Z_{\mathcal{I}}}$ is an isomorphism of sheaves. Thus ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_{Z_{\mathcal{I}}}$ is surjective and $i$ is a closed immersion by [F4], with kernel $\mathcal{I}$. In the extreme case $\mathcal{I}={\mathcal{O}}_X$ the quotient is the zero sheaf, whose support is $\mathrm{\varnothing }$, giving the empty subscheme; for $\mathcal{I}=0$ the immersion is the identity of $X$. This constructs $\mathcal{I}\mapsto Z_{\mathcal{I}}$ with the asserted properties.

The two constructions are inverse bijections: starting from a quasi-coherent ideal sheaf $\mathcal{I}$, step 3.1 gives $K_{Z_{\mathcal{I}}}=\ker ({\mathcal{O}}_X\to {\mathcal{O}}_X\mathrm{/}\mathcal{I})=\mathcal{I}$, including $\mathcal{I}={\mathcal{O}}_X$; starting from a closed immersion $i:Z\to X$, step 1.2 gives a quasi-coherent ideal sheaf $K_Z$ and step 2.2 gives $Z_{K_Z}\cong Z$ over $X$. Hence $\mathcal{I}\mapsto Z_{\mathcal{I}}$ and $Z\mapsto K_Z$ are mutually inverse bijections between quasi-coherent ideal sheaves and closed subschemes, and the empty subscheme is covered by the case $\mathcal{I}={\mathcal{O}}_X$ on the one side and by the empty closed immersion, whose kernel is ${\mathcal{O}}_X$, on the other.

Choice accounting: the forward construction is a globally defined ringed subspace, so it involves neither a choice of charts nor a gluing; the Axiom of Choice enters only through the associated-sheaf and affine-equivalence machinery in [F1] and [F2] and through the batch-5 supplier [F6] in step 1.2, and it is stated in the hypothesis. No use is made of the published correspondence theorem or the published affine-quotient theorem of the earlier run.