Projectivity via a coherent projective bundle

On a locally Noetherian base $S$, use coherent-projective-bundle projectivity for a global closed immersion $X\hookrightarrow {\mathbb{P}}_S(E)={\mathrm{Proj}}_S\mathrm{Sym}E$ with $E$ a coherent sheaf on $S$. The sheaf $E$ need not be locally free or have a globally bounded number of generators. This is the projectivity convention in this Hilbert packet. H-projectivity from Projective morphisms before Proj is its special case with $E={\mathcal{O}}_S^{n+1}$; no implication from the former to the latter is asserted. Mere local projective embeddings without a global projective-bundle embedding do not define the hypothesis here. A chosen relatively ample polarization $L$ (Relative ampleness over an arbitrary base) need not equal the pullback of $\mathcal{O}(1)$ for the embedding; that pullback gives an auxiliary global relatively very ample bundle $M$. No quasi-compactness assumption on $S$ is part of this definition.