Affine morphisms are relative spectra

Suppose $f$ is affine and put $\mathcal{A}=f_{\ast }{\mathcal{O}}_X$, with its ${\mathcal{O}}_S$-algebra structure from [F3]. For each affine open $U=\mathrm{Spec}R\subseteq S$, the inverse image is affine by [F1]. The canonical identification in [F5] shows that $\mathcal{A}{\mathrm{\mid }}_U$ is module-associated and that its principal-open restrictions are the required localizations. Thus $\mathcal{A}$ satisfies [F2]. This argument includes an empty inverse image, whose coordinate ring is $0$.

Let $\pi :Y={\mathrm{Spec}}_S\mathcal{A}\to S$ be the relative spectrum. For every affine $U\subseteq S$, [F6] gives ${\pi }^{-1}(U)=\mathrm{Spec}\mathrm{\Gamma }(U,\mathcal{A})$. By [F4], $\mathrm{\Gamma }(U,\mathcal{A})=\mathrm{\Gamma }(U,f_{\ast }{\mathcal{O}}_X)=\mathrm{\Gamma }(f^{-1}(U),{\mathcal{O}}_X)\mathrm{.}$ Both $f^{-1}(U)$ and ${\pi }^{-1}(U)$ are affine, so [F7] gives a canonical isomorphism between them. Its maps to $U$ agree: the ring maps from $\mathrm{\Gamma }(U,{\mathcal{O}}_S)$ are the algebra structure maps of $f_{\ast }{\mathcal{O}}_X$.

These chart isomorphisms are compatible under restriction. Indeed, for an affine open $W\subseteq U$, the restriction $\mathrm{\Gamma }(f^{-1}(U),{\mathcal{O}}_X)\to \mathrm{\Gamma }(f^{-1}(W),{\mathcal{O}}_X)$ is exactly the restriction of $f_{\ast }{\mathcal{O}}_X$ in [F4], while the relative-spectrum chart transition in [F6] is induced by the same restriction of $\mathcal{A}$. Naturality in [F7] therefore identifies the restriction of the chart isomorphism over $U$ with the one over $W$. Every overlap of two affine opens of $S$ is covered by affine opens $W$ contained in it, so these equalities give compatible chart isomorphisms on all overlaps. Applying [F8] yields an $S$-isomorphism $X\cong Y$. The construction uses the full set of affine opens and canonical global-sections rings; it makes no simultaneous choice of affine presentations.

Conversely, let $\mathcal{A}$ be any algebra sheaf as in [F2], let $Y={\mathrm{Spec}}_S\mathcal{A}$, and suppose $e:X\cong Y$ is an $S$-isomorphism. For every affine open $U\subseteq S$, [F6] makes ${\pi }^{-1}(U)$ affine, so $f^{-1}(U)$ is affine under $e$. By [F1], $f$ is affine.

On an affine $U=\mathrm{Spec}R$, the isomorphism $\mathcal{A}{\mathrm{\mid }}_U\cong \widetilde{B_U}$ and [F6] identify ${\pi }^{-1}(D(r))$ with $\mathrm{Spec}(B_U[{\phi }_U(r{)}^{-1}])$ for every principal open $D(r)\subseteq U$. Global sections of this affine spectrum recover $B_U[{\phi }_U(r{)}^{-1}]$ by [F7]. By [F4] applied to $\pi$, these identifications are exactly the sections of $({\pi }_{\ast }{\mathcal{O}}_Y){\mathrm{\mid }}_U$ on the principal-open basis, with the same restriction maps as $\mathcal{A}{\mathrm{\mid }}_U$. They give a canonical ${\mathcal{O}}_U$-algebra isomorphism $({\pi }_{\ast }{\mathcal{O}}_Y){\mathrm{\mid }}_U\cong \mathcal{A}{\mathrm{\mid }}_U$. The identifications agree on overlaps because they are induced by the same restrictions in [F6]; [F9] glues them to ${\pi }_{\ast }{\mathcal{O}}_Y\cong \mathcal{A}$ on $S$. Finally, $e$ and [F4] identify $f_{\ast }{\mathcal{O}}_X$ with ${\pi }_{\ast }{\mathcal{O}}_Y$, compatibly with the ${\mathcal{O}}_S$-algebra structures. Hence $f_{\ast }{\mathcal{O}}_X\cong \mathcal{A}$.