Quasi-projective morphisms before Proj

On this page, a morphism $f:X\to S$ is quasi-projective if, for some integer $n\ge 0$, it factors over $S$ through a quasi-compact immersion $X\hookrightarrow {\mathbb{P}}_S^n$ followed by the projection ${\mathbb{P}}_S^n\to S$. Projective space and projective morphisms use the finite-dimensional convention of Projective morphisms before Proj, an immersion has the meaning of Immersion of schemes, and quasi-compactness has the meaning of Quasi-compact and quasi-separated morphisms. In Stacks Definition 29.41.1(2), this is called H-quasi-projective; clause (1) separately calls finite type together with a relatively ample invertible sheaf quasi-projective. This page explicitly adopts clause (2)'s convention.

In particular, if $Y\to S$ is projective in the sense of Projective morphisms before Proj and $U\subseteq Y$ is open, then $U\to S$ is quasi-projective on this page whenever the composite immersion $U\hookrightarrow Y\hookrightarrow {\mathbb{P}}_S^n$ is quasi-compact for a projective-space embedding of $Y$.