= Relative Proj construction
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{title2=$\operatorname{Proj}_X\mathcal A$}
{wiki=Proj_construction#Relative_Proj}
For a quasi-coherent graded $\mathcal O_X$-algebra $\mathcal A$, the relative Proj is obtained by gluing the schemes $\operatorname{Proj}\mathcal A(U)$ over affine open subsets $U\subseteq X$. If $\mathcal A$ is of finite type and generated in degree one, a surjection $\operatorname{Sym}(\mathcal E)\twoheadrightarrow\mathcal A$ realizes $\operatorname{Proj}_X\mathcal A$ as a closed subscheme of the projective bundle $\mathbb P_X(\mathcal E)$.
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