= Relative Proj construction
{c}
{title2=$\operatorname{Proj}_X\mathcal A$}
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 $\operatorname{Proj}\mathcal A(U)$ 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>:
$$
\operatorname{Proj}_{X'}(g^*\mathcal A)\cong X'\times_X\operatorname{Proj}_X\mathcal A.
$$
<The Stacks project>[Section 27.16, in particular Lemma 27.16.10, tag 01O3]{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> $\operatorname{Sym}\mathcal E\to\mathcal A$ gives a <closed immersion> into $\operatorname{Proj}_X(\operatorname{Sym}\mathcal E)$, 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>.
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