Closed immersions into affine schemes are quotient spectra

Write $E=i(Z)=V(J)\subseteq X$. By [F1], $E$ and hence $Z$ are quasi-compact. Choose a finite affine open cover $W_j=\mathrm{Spec}{C}_j$ of $Z$. Since $Z$ has the subspace topology on $E$, the principal-open basis in [F2] and quasi-compactness supply finitely many labelled functions $f_i\in A$ whose $E\cap D(f_i)$ cover $E$, each lying in one $W_j$. The inverse image of $D(f_i)$ is the principal open defined by $f_i$ in that affine chart, so is affine. Empty covers are allowed.

The cover implies $V(J+(f_i))=\mathrm{\varnothing }$, hence $J+(f_i)=A$ by [F2]: a proper ideal in a nonzero ring would lie in a maximal ideal. For the zero ring the equality is immediate. Thus $1={\sum }_i{a}_i{f}_i+{\sum }_{\mathrm{\ell }}{b}_{\mathrm{\ell }}{g}_{\mathrm{\ell }}$ for finitely many $g_{\mathrm{\ell }}\in J$. On each affine $W_j$, every image of $g_{\mathrm{\ell }}$ lies in all primes, since every point maps into $V(J)$; it is nilpotent by [F2]. Finiteness of the cover supplies a common exponent, so each restriction $g_{\mathrm{\ell }}{\mathrm{\mid }}_Z$ is nilpotent globally by the sheaf axiom. The finite sum $h={\sum }_{\mathrm{\ell }}{b}_{\mathrm{\ell }}{g}_{\mathrm{\ell }}{\mathrm{\mid }}_Z$ is nilpotent by the multinomial expansion. Consequently ${\sum }_i{a}_i{f}_i{\mathrm{\mid }}_Z=1-h$ is a unit, with inverse a finite geometric sum. The $f_i{\mathrm{\mid }}_Z$ generate the unit ideal.

Put $B=\mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$ and write $f_i$ also for the pulled-back global sections. The affine opens $U_i=Z_{f_i}$ cover $Z$ by step 1.1. If this list is empty, $Z$ is empty and $B=0$, giving $Z=\mathrm{Spec}0$. Otherwise put $C_i=\mathrm{\Gamma }(U_i,{\mathcal{O}}_Z)$ and $C_{ij}=\mathrm{\Gamma }(U_i\cap U_j,{\mathcal{O}}_Z)$. Each intersection is the principal open defined by $f_j$ in the affine $U_i$, so is affine by [F2]. By [F5], $B$ is the kernel of the difference of restriction maps ${\prod }_i{C}_i\to {\prod }_{i,j}{C}_{ij}$, a map of $B$-modules. Fix $k$. Exact localization in [F4] preserves this kernel, and localization commutes with finite products by taking a common denominator in any finite tuple. Basic-open sections in [F4] identify $(C_i{)}_{f_k}$ and $(C_{ij}{)}_{f_k}$ with sections on $U_i\cap U_k$ and $U_i\cap U_j\cap U_k$, respectively, respecting restrictions. Their kernel is therefore $\mathrm{\Gamma }(U_k,{\mathcal{O}}_Z)=C_k$ by [F5], since the $U_i\cap U_k$ cover $U_k$. Thus restriction gives a ring isomorphism $B_{f_k}\cong C_k$: all maps come from restriction and fraction multiplication. By [F3] the identity of $B$ defines a canonical map $j:Z\to \mathrm{Spec}B$. Its inverse image of $D_B(f_k)$ is $U_k$, and on that open its comorphism is precisely this ring isomorphism. The sections $f_i$ generate the unit ideal by step 2.1, so the $D_B(f_i)$ cover $\mathrm{Spec}B$. These local inverse morphisms agree on overlaps because each inverts $j$, and glue to an inverse of $j$. Hence $Z=\mathrm{Spec}B$ in every case.

Let $\phi :A\to B$ correspond to $i$ by [F3]. For any prime $\mathfrak{p}\subset A$, sections of $i_{\ast }{\mathcal{O}}_Z$ on $D(f)$ are $B_{\phi (f)}$. Taking the colimit over $f\mathrm{\notin }\mathfrak{p}$ identifies its stalk with $(A\setminus \mathfrak{p}{)}^{-1}B$. The surjective sheaf map in [F1] therefore gives a surjection $A_{\mathfrak{p}}\to (A\setminus \mathfrak{p}{)}^{-1}B$. Exact localization in [F4] implies $M_{\mathfrak{p}}=0$ for $M=\mathrm{coker}\phi$. If $m\in M$ were nonzero, [F2] would give a maximal ideal $\mathfrak{p}$ containing its proper annihilator. Then $m\mathrm{/}1\mathrm{\ne }0$ in $M_{\mathfrak{p}}$: vanishing would require a denominator outside $\mathfrak{p}$ annihilating $m$, contrary to that containment. Thus $M=0$ and $\phi$ is onto. For $A=0$, unitality already forces $B=0$. Setting $I=\ker \phi$ gives $B\cong A\mathrm{/}I$ and the required isomorphism over $X$.

Conversely, the spectrum map of $A\twoheadrightarrow A\mathrm{/}I$ is a homeomorphism onto $V(I)$ by [F4]: contraction identifies its primes and all their closed vanishing sets. At a prime containing $I$, its direct-image stalk map is the surjection $A_{\mathfrak{p}}\to (A\mathrm{/}I{)}_{\mathfrak{p}\mathrm{/}I}$; at a prime not containing $I$, the target stalk is zero, since an element of $I$ is inverted. Hence its structure-sheaf map is surjective by [F4], proving it is a closed immersion. The kernel of $A\to \mathrm{\Gamma }(Z,{\mathcal{O}}_Z)$ recovers $I$ uniquely. Any isomorphism of quotient spectra over $X$ is induced by the unique compatible quotient-ring map, so is unique. In particular $I=A$ gives the empty immersion, also when $A=0$.