Proj carries a scheme structure

The localisation map $\psi ={\psi }_{f,g}:S_f\to S_{fg}$ exists and is unique, because $f$ becomes a unit in $S_{fg}$ (its inverse is $g\mathrm{/}(fg)$); it is graded, hence restricts to a ring homomorphism $S_{(f)}\to S_{(fg)}$ mapping $a\mathrm{/}{f}^i$ to $a{g}^i\mathrm{/}(fg{)}^i$, and the element ${\tau }_{fg}=g^d\mathrm{/}{f}^e\in S_{(f)}$ maps to $g^d\mathrm{/}{f}^e\in S_{(fg)}$, which is a unit there with inverse $f^e\mathrm{/}{g}^d$; symmetrically ${\psi }_{g,f}({\tau }_{gf})$ is a unit with inverse $g^d\mathrm{/}{f}^e$.

The induced map $\tilde{\psi }:S_{(f)}[{\tau }_{fg}^{-1}]\to S_{(fg)}$ of step 1.1 is surjective: an element of $S_{(fg)}$ has the form $y=c\mathrm{/}(fg{)}^k$ with $c\in S$ homogeneous of degree $k(d+e)$, and putting $i=k(e+1)$ and $a=c{g}^{k(d-1)}\in S$, which is homogeneous of degree $k(d+e)+k(d-1)e=kd(e+1)=id$, the element $x=a\mathrm{/}{f}^i$ lies in $S_{(f)}$ and $\tilde{\psi }(x\mathrm{/}{\tau }_{fg}^k)=\left(c{g}^{k(d-1)}\mathrm{/}{f}^{k(e+1)}\right)\left(f^{ek}\mathrm{/}{g}^{dk}\right)=c\mathrm{/}(fg{)}^k=y$.

The map $\tilde{\psi }$ of step 1.1 is injective: if $\tilde{\psi }(x\mathrm{/}{\tau }_{fg}^j)=0$, then multiplying by ${\tau }_{fg}^j$ gives $\psi (x)=0$, so for $x=a\mathrm{/}{f}^i$ one has $a\mathrm{/}{f}^i=0$ in $S_{fg}$ and hence $(fg{)}^{N}a=0$ in $S$ for some $N$; choosing $M$ with $dM\ge N$ gives $f^N{g}^{dM}a=g^{dM-N}(fg{)}^{N}a=0$, so ${\tau }_{fg}^{M}x=g^{dM}a\mathrm{/}(f^{eM}{f}^i)=0$ in $S_{(f)}$, whence $x\mathrm{/}{\tau }_{fg}^j=0$ in the localisation.

The symmetric map ${\tilde{\psi }}_{g,f}:S_{(g)}[{\tau }_{gf}^{-1}]\to S_{(fg)}$, ${\tau }_{gf}=f^e\mathrm{/}{g}^d$, is likewise an isomorphism, by the same two arguments with the roles of $f$ and $g$ exchanged, so the two isomorphisms are the canonical transition isomorphisms between the charts $D({\tau }_{fg})\subseteq \mathrm{Spec}{S}_{(f)}$ and $D({\tau }_{gf})\subseteq \mathrm{Spec}{S}_{(g)}$ required in clause (2) of the Statement.

The transition isomorphisms of step 3.1 satisfy the identity and cocycle conditions: for a triple $f,g,h$ every one of the pairwise transitions, transported to $S_{(fgh)}$, is the canonical map of the localisation $S_{fgh}$ induced by $S\to S_{fgh}$, which is unique by [F7]; hence the composite around each triangle of charts is the identity on the triple overlap, and the transition of a pair with itself is the identity. By [F3] the overlap identifications fit the intersections $D_{+}(fg)=D_{+}(f)\cap D_{+}(g)$ and $D_{+}(fgh)=D_{+}(f)\cap D_{+}(g)\cap D_{+}(h)$.

By step 4.1 the affine schemes $\mathrm{Spec}{S}_{(f)}$, indexed by homogeneous $f\in S_{+}$, with the overlap isomorphisms of step 3.1 are gluing data satisfying the identity and cocycle conditions; the gluing theorem produces a scheme $X$ with an open affine cover by the images of the $\mathrm{Spec}{S}_{(f)}$, identified with the charts, uniquely up to a unique chart-compatible isomorphism. For $f$ nilpotent the chart is $\mathrm{Spec}0=\mathrm{\varnothing }$ by [F3], so the corresponding open subscheme is empty.

The underlying topological space of $X$ is $\mathrm{Proj}S$: by [F3] each chart is in bijection with $D_{+}(f)\subseteq \mathrm{Proj}S$, and step 4.1 says the bijections agree on overlaps, so the chartwise bijections glue to a well-defined bijection $\mathrm{\mid }X\mathrm{\mid }\to \mathrm{Proj}S$. The bijection is a homeomorphism: distinguished opens form a basis inside each affine chart of $X$, and each is a member of the family in [F3]. Indeed, for $u=a\mathrm{/}{f}^k\in S_{(f)}$ with $k>0$, $D(u)=D(u^d)=D(a^d\mathrm{/}{f}^{kd})$, which is the chart open corresponding to $D_{+}(fa)$; for $k=0$, $a\in S_0$ and $D(a)=D(a^d)=D((af{)}^d\mathrm{/}{f}^d)$ corresponds to $D_{+}(a{f}^2)$. Thus the chart opens $D_{+}(fg)$ form a basis for $X$. On $\mathrm{Proj}S$ the same family is a basis by [F2], and the chart prime correspondences of [F3] match both bases. In the empty cases both sides are empty: if $S=0$ then $\mathrm{Proj}S=\mathrm{\varnothing }$ by [F1] and every chart is $\mathrm{Spec}0$; if $\mathrm{Proj}S=\mathrm{\varnothing }$ then each $D_{+}(f)=\mathrm{\varnothing }$ and $S_{(f)}=0$ by [F3] together with [F6], since a nonzero $S_{(f)}$ has a prime, contradicting the bijection with the empty set $D_{+}(f)$.

Sections: on the chart $D_{+}(f)$ the identification ${\phi }_f$ exhibits the structure sheaf of $X$ as that of $\mathrm{Spec}{S}_{(f)}$, so ${\mathcal{O}}_X(D_{+}(f))=\mathrm{\Gamma }(\mathrm{Spec}{S}_{(f)},\mathcal{O})=S_{(f)}$ by [F5]; in particular the identifications are isomorphisms of locally ringed spaces respecting the structure sheaves, and clause (1) of the Statement holds.

Clauses (1)-(4) of the Statement are exactly steps 5.1 (existence, cover, uniqueness), 6.1 (sections), 4.1 (overlap agreement) and 5.2 (underlying space). The Axiom of Choice [A1] is inherited through the affine-scheme and prime-existence interfaces [F5, F6] and is used only in step 5.2, to know that a nonzero $S_{(f)}$ has a prime and hence a nonempty chart; the gluing and the transition isomorphisms are choice-free.