Relative Proj construction (${\mathrm{Proj}}_X\mathcal{A}$)

Let $X$ be a scheme and $\mathcal{A}$ a quasi-coherent sheaf of nonnegatively graded algebras with ${\mathcal{A}}_0={\mathcal{O}}_X$. Its relative Proj is obtained by gluing $\mathrm{Proj}\mathcal{A}(U)$ over affine open subschemes $U\subseteq X$. The standard affine opens of Proj and localization identify these constructions on overlaps. It is compatible with arbitrary base change:^[ref]

If $\mathcal{A}$ is generated in degree one and a finite locally free sheaf $\mathcal{E}$ surjects onto ${\mathcal{A}}_1$, the resulting surjective morphism of sheaves $\mathrm{Sym}\mathcal{E}\to \mathcal{A}$ gives a closed immersion into ${\mathrm{Proj}}_X(\mathrm{Sym}\mathcal{E})$, the projective bundle in the quotient-line convention. Local degree-one generators always give such embeddings over suitable affine open subschemes. A global generating vector bundle requires an additional existence hypothesis on ${\mathcal{A}}_1$; it does not follow merely from finite type over an arbitrary scheme.