Two definitions of a projective morphism

Question

In Hartshorne's Algebraic Geometry, page 103, a morphism $f: X \rightarrow Y$ is said to be projective if it factors as a closed immersion $X \rightarrow {\bf P}^n_Y$ followed by the projection ${\bf P}^n_Y \rightarrow Y$. As noted there, EGA II, 5.5 has another definition, namely $f$ is projective if it factors as a closed immersion $X \rightarrow {\bf P}(\cal E)$ followed by the projection map, where $\cal E$ is finite-type quasi-coherent ${\cal O}_Y$-module.

Hartshorne states without proof nor reference that the two definitions "are equivalent in case $Y$ itself is quasi-projective over an affine scheme".

My question is: does anyone know a proof or a reference for this statement? And if not: is it correct?

It is a good idea to say what is the second definition. – Sasha Nov 20 '18 at 19:07 — Sasha, Nov 20 '18 at 19:07
@sasha: You're right. That's done now. – Visitor Nov 20 '18 at 20:10 — Visitor, Nov 20 '18 at 20:10

score 17 · Accepted Answer · answered Nov 20 '18 at 20:37

Definitely, if $X$ is "Hartshorne-projective" it is also "EGA-projective" (take $\mathcal{E}$ to be a free bundle of rank $n+1$). The opposite direction is true if any coherent sheaf is globally generated by a finite-dimensional vector space of sections after some line bundle twist; indeed, if $V$ generates $\mathcal{E} \otimes L$ then the surjection $V \otimes \mathcal{O}_Y \to \mathcal{E} \otimes L$ induces a closed embedding $$ \mathbb{P}(\mathcal{E}) = \mathbb{P}(\mathcal{E} \otimes L) \to \mathbb{P}(V \otimes \mathcal{O}_Y) = \mathbb{P}^n_Y. $$ So, for instance if $Y$ satisfies reasonable finiteness conditions and admits an ample line bundle, the definitions are equivalent.

Two definitions of a projective morphism

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