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for some n so that pullbacks to ( An immersion is an open immersion followed by a closed immersion.
) Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed.
Conversely, if X is projective, then the pullback of under the closed immersion of X into a projective space is very ample.
That " projective " implies " proper " is deeper: it can be considered as a generalization of the fact that a projective variety is complete ( see below.

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