Relative Proj of a graded quasi-coherent algebra

Assume the Axiom of Choice as inherited from the affine-local Proj construction (The Axiom of Choice). Let $S$ be a scheme and let $\mathcal{A}={⨁}_{d\ge 0}{\mathcal{A}}_d$ be a quasi-coherent graded ${\mathcal{O}}_S$-algebra (Quasi-coherent module on a scheme): each ${\mathcal{A}}_d$ is a quasi-coherent ${\mathcal{O}}_S$-module and the multiplication maps ${\mathcal{A}}_d{\otimes }_{{\mathcal{O}}_S}{\mathcal{A}}_e\to {\mathcal{A}}_{d+e}$ are ${\mathcal{O}}_S$-linear. No finite generation of $\mathcal{A}$ over ${\mathcal{A}}_0$ is assumed.

Construction. For an affine open $U=\mathrm{Spec}R\subseteq S$ put $B_U={⨁}_{d\ge 0}\mathrm{\Gamma }(U,{\mathcal{A}}_d)$, a graded $R$-algebra, and let $P_U=\mathrm{Proj}{B}_U$ be its Proj with its scheme structure (Proj carries a scheme structure). For affine opens $V\subseteq U$ the canonical isomorphisms $P_U{\mathrm{\mid }}_V\cong P_V$ of Affine-local graded algebras glue their Proj charts are compatible with inclusions and satisfy the triple-overlap cocycle, so the schemes $P_U$ glue along the affine opens of $S$ to a scheme ${\mathrm{Proj}}_S\mathcal{A},$ equipped with the open immersions $P_U\hookrightarrow {\mathrm{Proj}}_S\mathcal{A}$ identifying $P_U$ with the restriction to $U$. The relative twists are the ${\mathcal{O}}_{{\mathrm{Proj}}_S\mathcal{A}}$-modules $\mathcal{O}(n)$, $n\in \mathbb{Z}$, obtained by gluing, over each affine open $U$ with its coordinates $B_U$, the sheaves $\widetilde{B_U(n)}$ on $P_U=\mathrm{Proj}{B}_U$ of Twisting sheaf on Proj; the gluing is compatible for the following reason. Write $V=\mathrm{Spec}C\subseteq U$. The degreewise restriction calculation in Affine-local graded algebras glue their Proj charts gives $B_V=B_U{\otimes }_{R}C$, not an identification with $B_U$ itself. For a homogeneous $f\in (B_U{)}_{+}$, localisation and scalar extension give $B_U(n{)}_{(f)}{\otimes }_{R}C\cong B_V(n{)}_{(f_C)},(b\mathrm{/}{f}^k)\otimes c⟼(b\otimes c)\mathrm{/}{f}_C^k\mathrm{.}$ To check this formula, first localise the graded ring: the universal properties identify $(B_U{)}_f{\otimes }_{R}C$ with $(B_V{)}_{f_C}$ by the displayed fractions. Taking degree $n$ gives the stated shifted degree-zero module isomorphism, because tensor product distributes over homogeneous direct sums. The restriction of the associated module sheaf to the affine open $D_{+}(f){\times }_{U}V$ is associated to this scalar-extended module by An associated sheaf restricts to an associated sheaf on an affine open: tensoring over $(B_U{)}_{(f)}$ with $(B_U{)}_{(f)}{\otimes }_{R}C$ is tensoring over $R$ with $C$. These identifications commute with further localisation and successive restriction, since each sends a fraction to the same fraction with its coefficients restricted. They therefore agree on overlaps and satisfy the triple cocycle, so the twists glue. No invertibility of the twists is needed.

Structural morphism. On $P_U=\mathrm{Proj}{B}_U$ the ring $R=\mathrm{\Gamma }(U,{\mathcal{O}}_S)$ acts on each $B_U(n)$ through the structure map $R\to (B_U{)}_0$, so $P_U$ carries a canonical morphism $P_U\to \mathrm{Spec}R=U$ induced on each chart by the structure map $R\to (B_U{)}_0\to (B_U{)}_{(f)}$. These morphisms are compatible with the identifications $P_U{\mathrm{\mid }}_V\cong P_V$ and glue to a morphism $\pi :{\mathrm{Proj}}_S\mathcal{A}⟶S,$ called the structural morphism, through which ${\mathrm{Proj}}_S\mathcal{A}$ is a scheme over $S$ (Schemes and morphisms over a base).

Absolute case. If $S=\mathrm{Spec}R$ is affine and $\mathcal{A}=\tilde{B}$ for a graded $R$-algebra $B$, then ${\mathrm{Proj}}_S\mathcal{A}=\mathrm{Proj}B$ and $\pi$ is the structural morphism of the absolute Proj; in particular ${\mathrm{Proj}}_{\mathrm{Spec}R}\widetilde{R[x_0,\dots ,x_n]}\cong {\mathbb{P}}_R^n$ by Projective space is Proj of a polynomial ring.

• No finiteness. Existence of ${\mathrm{Proj}}_S\mathcal{A}$ uses only quasi-coherence of the graded pieces and the affine-local Proj construction; no finite generation is required. If, on each affine open $U$ of the base, $B_U$ is generated by $(B_U{)}_1$ over $(B_U{)}_0$, all relative twists are invertible by Invertible twists for degree-one generated rings applied on those affine-base restrictions. This degree-one generation hypothesis does not require finitely many generators and is not part of this definition.
• Choice. The Axiom of Choice is inherited from the affine Proj construction and the affine-local gluing; the definition itself selects no data beyond that interface.