We say that a morphism $f : X \to Y$ is projective if it factors as a closed embedding $i : X \to \mathbb{P}^N_Y$, followed by the projection from $\mathbb{P}^N_Y \to Y$.
Question: Is this property local on the target?
(I know that the more general EGA definition - namely a morphism which is relative proj of the some generated in degree 1 sheaf of graded algebras - does not have this property. I don't understand the reason why this is so.)
At first I thought so, but then I realized that the closed embeddings into relative $\mathbb{P}^n$ don't necessarily glue to a closed embedding into relative projective space, since there maybe nontrivial twisting.
I didn't see any mention of this in Hartshorne, so I guess it is not true. What is the real deal? (Is there some good circumstance when it is target local?)
