Closed immersion from a quotient ring

By the converse clause of [F1] applied to the quotient map $\pi :A\to A\mathrm{/}I$, the induced morphism $i$ is a closed immersion. Now fix an affine open $U=\mathrm{Spec}{A}^{\mathrm{\prime }}\subseteq \mathrm{Spec}A$. By the first clause of [F1] applied to the closed immersion $i$ and this $U$, there is a unique ideal $I^{\mathrm{\prime }}\subseteq A^{\mathrm{\prime }}$ with $i^{-1}(U)\cong \mathrm{Spec}(A^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }})$ over $U$; in particular the inverse image $i^{-1}(U)$ is affine.

The quotient ring $A^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}$ is a cyclic $A^{\mathrm{\prime }}$-module: the class $1+I^{\mathrm{\prime }}$ generates it, because every $a+I^{\mathrm{\prime }}=a\cdot (1+I^{\mathrm{\prime }})$ lies in the submodule generated by $1+I^{\mathrm{\prime }}$, and that submodule is contained in $A^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}$. If $I^{\mathrm{\prime }}=A^{\mathrm{\prime }}$ then $A^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}$ is the zero module, which is generated by the empty family. Hence $A^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}$ is finitely generated, equivalently module-finite, over $A^{\mathrm{\prime }}$ by [F4] and [F5].

Steps 1.1 and 1.2 verify the condition of [F3] on the affine open $U$: the inverse image $i^{-1}(U)=\mathrm{Spec}(A^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }})$ is affine, and its coordinate algebra is module-finite over the coordinate algebra $A^{\mathrm{\prime }}$ of $U$. Since $U$ was an arbitrary affine open of $\mathrm{Spec}A$, the morphism $i$ is finite.

By step 1.1 the morphism $i$ is a closed immersion, so the AC-qualified [F2] shows that $i$ is proper. Combining with step 2.1, the morphism $\mathrm{Spec}(A\mathrm{/}I)\to \mathrm{Spec}A$ is both finite and proper.

Extreme cases. If $I=A$ then $A\mathrm{/}I=0$ and the source is $\mathrm{Spec}0=\mathrm{\varnothing }$, the empty closed immersion, to which [F1] and [F2] apply with the empty case included; if $I=0$ then $A\mathrm{/}I\cong A$ canonically and $i$ is the identity morphism, a closed immersion by [F1] and proper by [F2]; if $A=0$ then source and target are both empty and $i$ is again the identity of the empty scheme, covered by the same argument. The Axiom of Choice [F6] is assumed and is used only through the AC-qualified suppliers [F1] and [F2]. No Noetherian, field, reducedness or nonemptiness hypothesis is imposed.