Closed immersions are affine quotients and survive base change

Fix an affine open $U=\mathrm{Spec}A$ of $Y$. By F3, complete $U$ to an open cover of $Y$ with affine neighbourhoods of points outside $U$. Then F1 and F2 show that the restriction $i_U:i^{-1}(U)\to U$ is a closed immersion. It is therefore enough to prove the assertion when $Y=\mathrm{Spec}A$. Write $E=i(Z)$; F1 says that $E$ is closed and $i$ is a homeomorphism from $Z$ onto $E$.

The space $\mathrm{Spec}A$ is quasi-compact by F4. A closed subset of a quasi-compact space is quasi-compact: add its open complement to any open cover and take a finite subcover upstairs. Hence $E$ and $Z$ are quasi-compact. Since $Z$ is a scheme, its affine open neighbourhoods cover it by F3; compactness gives a finite affine-open cover $W_1,\dots ,W_m$. The empty cover is permitted when $Z=\mathrm{\varnothing }$.

Each $W_j$ is open in $E$, so the subspace topology gives an open $O_j\subseteq \mathrm{Spec}A$ with $O_j\cap E=W_j$. By F5 and F6, principal opens form a basis. Thus the family of pairs $(j,f)$ satisfying $E\cap D(f)\subseteq W_j$ covers $E$. Quasi-compactness gives finitely many such labelled pairs $(j_i,f_i)$ covering $E$, without choosing one function for every point.

The inverse image of each selected $D(f_i)$ lies in its labelled affine chart $W_{j_i}=\mathrm{Spec}{C}_{j_i}$. The morphism on that chart corresponds by F12 to a ring map $A\to C_{j_i}$, so the inverse image is the distinguished open defined by the image of $f_i$. It is affine by F7. The selected inverse images cover $Z$.

Since $E$ is closed in $\mathrm{Spec}A$, write $E=V(J)$ by F5. The opens $D(f_i)$ cover $V(J)$, so $V(J+(f_1,\dots ,f_n))=\mathrm{\varnothing }\mathrm{.}$ If $A\mathrm{\ne }0$ and this aggregate ideal were proper, F10 would put it in a maximal (hence prime) ideal, contradicting the cover. If $A=0$, the aggregate ideal already equals $A$. Therefore $J+(f_1,\dots ,f_n)=A$. By the definition of a generated ideal this gives a finite relation $1={\sum }_i{a}_i{f}_i+{\sum }_{\mathrm{\ell }=1}^r{b}_{\mathrm{\ell }}{g}_{\mathrm{\ell }},g_{\mathrm{\ell }}\in J\mathrm{.}$

Conversely, let $\pi :A\to A\mathrm{/}I$ be a quotient map. By F18 its spectrum map is a bijection onto $V(I)$. It is continuous because inverse images of vanishing sets are vanishing sets. If $K$ is an ideal of $A\mathrm{/}I$, the image of $V_{A\mathrm{/}I}(K)$ is $V_A({\pi }^{-1}K)$, hence is closed by F5. The map is therefore a closed continuous bijection onto $V(I)$, and is a homeomorphism onto that closed subset.

Apply the map $A\to \mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$ of F11 to the relation from step 1.5. On an affine chart $W_j=\mathrm{Spec}{C}_j$, every point maps into $E=V(J)$; by F12 the image of each $g_{\mathrm{\ell }}$ belongs to every prime of $C_j$. By F9 it is nilpotent. There are finitely many $j$ and $\mathrm{\ell }$, so a common positive exponent kills every restriction $g_{\mathrm{\ell }}{\mathrm{\mid }}_{W_j}$; the sheaf axiom then makes each $g_{\mathrm{\ell }}{\mathrm{\mid }}_Z$ nilpotent in $\mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$. Their finite linear combination $h={\sum }_{\mathrm{\ell }}{b}_{\mathrm{\ell }}{g}_{\mathrm{\ell }}$ is nilpotent: if it has $r$ terms and each term has $N$th power zero, then $h^{r(N-1)+1}=0$ by the multinomial expansion. It follows from the relation in step 1.5 that ${\sum }_i{a}_i{f}_i=1-h$ is a unit, with inverse $1+h+\cdots +h^{q-1}$ when $h^q=0$. Thus the $f_i$ generate the unit ideal in $\mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$.

By step 1.4 the principal opens defined by the $f_i$ are affine, and they cover $Z$. By step 2.1 they are defined by sections generating the unit ideal. F8 therefore makes $Z$ affine, say $Z=\mathrm{Spec}B$. This includes the empty case: if the list is empty, its unit-ideal condition forces $\mathrm{\Gamma }(Z,{\mathcal{O}}_Z)=0$, and the permitted empty case of F8 gives $Z=\mathrm{Spec}0$.

By step 3.1, $Z=\mathrm{Spec}B$, so the affine anti-equivalence F12 identifies $i$ with a ring map $\phi :A\to B$. For $\mathfrak{p}\in \mathrm{Spec}A$, the source stalk is $A_{\mathfrak{p}}$ by F15. By F13, the sections of $i_{\ast }{\mathcal{O}}_Z$ on $D(f)$ are $\mathrm{\Gamma }(D_B(\phi (f)),{\mathcal{O}}_Z)=B_{\phi (f)}$ by F14; the opens $D(f)$ with $f\mathrm{\notin }\mathfrak{p}$ are cofinal neighbourhoods of $\mathfrak{p}$. By F25 the stalk is the colimit of these section rings. Its canonical map from $B$ inverts every element of $S_{\mathfrak{p}}=\phi (A\setminus \mathfrak{p})$, so F24 gives a map $S_{\mathfrak{p}}^{-1}B$ to this colimit. Conversely, each $B_{\phi (f)}$ maps compatibly to $S_{\mathfrak{p}}^{-1}B$; the two maps are inverse by F24. Thus the stalk is $S_{\mathfrak{p}}^{-1}B$, and the stalk map is the localization $A_{\mathfrak{p}}\to S_{\mathfrak{p}}^{-1}B$. F1 makes the sheaf map surjective, so F16 makes every such localized map surjective.

Regard $M=\mathrm{coker}(A\stackrel{\phi }{\to }B)$ as an $A$-module, using F23. By F17, its localization $M_{\mathfrak{p}}$ is the cokernel of the localized map in step 4.1, hence is zero for every prime $\mathfrak{p}$. If $A=0$, unitality forces $B=0$, so $M=0$. Otherwise, if $M$ had a nonzero element $m$, its annihilator would be a proper ideal. By F10 and AC there would be a maximal ideal $\mathfrak{p}\supseteq \mathrm{Ann}(m)$. Then $m\mathrm{/}1\mathrm{\ne }0$ in $M_{\mathfrak{p}}$: F22 says $m\mathrm{/}1=0$ would require a denominator $s\mathrm{\notin }\mathfrak{p}$ with $sm=0$, but that puts $s\in \mathrm{Ann}(m)\subseteq \mathfrak{p}$. This contradicts $M_{\mathfrak{p}}=0$. Thus $M=0$ and $\phi$ is surjective. Its kernel $I=\ker \phi$ is unique, and the explicit map $A\mathrm{/}I\to B$, $a+I\mapsto \phi (a)$, is a ring isomorphism. By F12 it identifies $i$ over $\mathrm{Spec}A$ with the quotient immersion.

At $\mathfrak{p}\supseteq I$, the quotient induces a surjection $A_{\mathfrak{p}}\to (A\mathrm{/}I{)}_{\mathfrak{p}\mathrm{/}I}$: every localized quotient fraction has a numerator lifted from $A$. At $\mathfrak{p}\supseteq ̸I$, some $u\in I\setminus \mathfrak{p}$ becomes invertible while mapping to zero, so the target stalk of the direct image is zero. The stalk computation in step 4.1 and F16 show that ${\mathcal{O}}_{\mathrm{Spec}A}\to {\pi }_{\ast }{\mathcal{O}}_{\mathrm{Spec}(A\mathrm{/}I)}$ is surjective. Together with the result of step 1.6 and F1, this proves that every quotient map induces a closed immersion.

Let $Y^{\mathrm{\prime }}\to Y$ be any morphism. By F27 its pullback of $i$ is defined. The target $Y^{\mathrm{\prime }}$ has an affine-open cover by $V=\mathrm{Spec}{A}^{\mathrm{\prime }}$ whose maps factor through affine opens $U=\mathrm{Spec}A$ of $Y$: around each point, intersect an affine neighbourhood with the inverse image of an affine neighbourhood in $Y$, then refine inside the affine chart by a principal open using F5--F7. The pullback over each such $V$ is the affine fibre product $\mathrm{Spec}((A\mathrm{/}I){\otimes }_A{A}^{\mathrm{\prime }})\mathrm{.}$ By F26 and F20 it is canonically isomorphic to $\mathrm{Spec}(A^{\mathrm{\prime }}\mathrm{/}I{A}^{\mathrm{\prime }})$. This is a ring isomorphism: the map $\bar{a}\otimes a^{\mathrm{\prime }}\mapsto a^{\mathrm{\prime }}a+I{A}^{\mathrm{\prime }}$ is multiplicative, and its inverse sends $a^{\mathrm{\prime }}+I{A}^{\mathrm{\prime }}$ to $1\otimes a^{\mathrm{\prime }}$; this inverse kills $I{A}^{\mathrm{\prime }}$ since $1\otimes i{a}^{\mathrm{\prime }}=i\cdot (1\otimes a^{\mathrm{\prime }})=0$ for $i\in I$. By step 5.1 the restriction over $U$ is $\mathrm{Spec}(A\mathrm{/}I)\to U$, so the pullback over $V$ is exactly this quotient map and is a closed immersion by step 5.2. F2 glues these local closed immersions over the affine-open cover of $Y^{\mathrm{\prime }}$, proving stability under arbitrary base change.

If $Z=\mathrm{\varnothing }$, then $B=0$ and the unique kernel is $I=A$; after every base change the quotient ring remains zero. If $A=0$, both the target and every closed subscheme are empty and the same ideal conclusion holds. The extreme quotient ideals $I=0$ and $I=A$ give respectively the identity closed immersion and the empty immersion. No reducedness or finite-generation assumption was used, so nilpotents in the quotient are retained.