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

For a quasi-coherent graded ${\mathcal{O}}_X$-algebra $\mathcal{A}$, the relative Proj is obtained by gluing the schemes $\mathrm{Proj}\mathcal{A}(U)$ over affine open subsets $U\subseteq X$. If $\mathcal{A}$ is of finite type and generated in degree one, a surjection $\mathrm{Sym}(\mathcal{E})\twoheadrightarrow \mathcal{A}$ realizes ${\mathrm{Proj}}_X\mathcal{A}$ as a closed subscheme of the projective bundle ${\mathbb{P}}_X(\mathcal{E})$.