Affine-local graded algebras glue their Proj charts

Local graded model on $V$. Let $C=\mathrm{\Gamma }(V,{\mathcal{O}}_U)$, so that $V=\mathrm{Spec}C$ and the inclusion is induced by $R\to C$; by [F1] the graded algebra $\mathcal{A}{\mathrm{\mid }}_U$ is $\tilde{B}$ for the graded $R$-algebra $B={⨁}_d\mathrm{\Gamma }(U,{\mathcal{A}}_d)$, and by [F2] applied degreewise, $\mathrm{\Gamma }(V,{\mathcal{A}}_d)=C{\otimes }_R{B}_d$ with compatible products, so that $\mathrm{\Gamma }(V,\mathcal{A})=B{\otimes }_{R}C$ as a graded $C$-algebra; write $B_C=B{\otimes }_{R}C$ with $(B_C{)}_d=B_d{\otimes }_{R}C$.

Intersections of charts with $V$. Let $f\in B_{+}$ be homogeneous of positive degree, so $D_{+}(f)\subseteq \mathrm{Proj}B$ is an affine open chart [F4], and let $f_C$ denote its image in $B_C$. The inverse image of $V$ in $D_{+}(f)$ is the fibre product of the affine morphism $D_{+}(f)=\mathrm{Spec}{B}_{(f)}\to U=\mathrm{Spec}R$ with the open immersion $V=\mathrm{Spec}C\hookrightarrow U$. By [F3] it is affine with ring $B_{(f)}{\otimes }_{R}C$, which by [L1] is $(B_C{)}_{(f_C)}$; hence $D_{+}(f)\cap {\pi }^{-1}(V)=\mathrm{Spec}((B_C{)}_{(f_C)})$ is exactly a standard chart of $\mathrm{Proj}{B}_C=\mathrm{Proj}\mathrm{\Gamma }(V,\mathcal{A})$.

Chartwise isomorphism. The opens $D_{+}(f_C)$ for homogeneous $f\in B_{+}$ cover $\mathrm{Proj}{B}_C$: if a homogeneous prime contained all $f\otimes 1$, it would contain every positive-degree element ${\sum }_j{b}_j\otimes c_j$, contradicting its membership in Proj. The assignment $D_{+}(f)\cap V\mapsto \mathrm{Spec}(B_C{)}_{(f_C)}$ is the identity on rings, so for all $f$ it identifies the charts of $\mathrm{Proj}(B){\mathrm{\mid }}_V$ with the charts of $\mathrm{Proj}\mathrm{\Gamma }(V,\mathcal{A})$; the two transition systems are induced by the same canonical localisation maps $B_{(f)}\to B_{(fg)}$ and $B_{(g)}\to B_{(fg)}$ base changed along $R\to C$, hence agree, and by [F6] the chart identifications glue to an isomorphism $\mathrm{Proj}(B){\mathrm{\mid }}_V\to \mathrm{Proj}\mathrm{\Gamma }(V,\mathcal{A})$ over $V$, canonical because each chart identification is.

Compatibility with inclusions of affine opens. If $V^{\mathrm{\prime }}\subseteq V$ are affine open in $U$ with rings $C\to C^{\mathrm{\prime }}$, then the isomorphism of step 3.1 for $V^{\mathrm{\prime }}$ is the restriction of the one for $V$: on a chart, both are the base change of the identity map of $B_{(f)}$ along $R\to C\to C^{\mathrm{\prime }}$, and base change of localisations is compatible with composition by [L1].

Gluing over $S$ and triple overlaps. Let $U,U^{\mathrm{\prime }},U^{\mathrm{\prime }\mathrm{\prime }}$ be affine opens of $S$ and let $W\subseteq U\cap U^{\mathrm{\prime }}$ be an affine open; both restrictions $\mathrm{Proj}\mathrm{\Gamma }(U,\mathcal{A}){\mathrm{\mid }}_W$ and $\mathrm{Proj}\mathrm{\Gamma }(U^{\mathrm{\prime }},\mathcal{A}){\mathrm{\mid }}_W$ are identified with $\mathrm{Proj}\mathrm{\Gamma }(W,\mathcal{A})$ by step 3.1 (applied with $V=W$), and the resulting isomorphism over $W$ composes to the identity on triple overlaps $W\subseteq U\cap U^{\mathrm{\prime }}\cap U^{\mathrm{\prime }\mathrm{\prime }}$ by step 4.1, because all identifications are the canonical localisation isomorphisms for the graded rings of sections. Since the affine opens $W$ cover each intersection $U\cap U^{\mathrm{\prime }}$, the local isomorphisms glue; refining the local schemes to their standard affine charts and applying [F6], the local schemes $\mathrm{Proj}\mathrm{\Gamma }(U,\mathcal{A})$ therefore glue over the affine opens of $S$, and the identifications are canonical throughout.

Conclusion. Step 3.1 gives the canonical isomorphism $\mathrm{Proj}(B){\mathrm{\mid }}_V\cong \mathrm{Proj}\mathrm{\Gamma }(V,\mathcal{A})$ for every affine open $V\subseteq U$, step 4.1 its compatibility with inclusions, and step 5.1 the triple-overlap cocycle and the gluing conclusion. The Axiom of Choice [A1] is inherited only through the affine quasi-coherence equivalence [F1] and the associated-sheaf restriction [F2]; no additional choice is made. [A1, F1, F2, step 3.1, step 5.1] \qed