here is my question:
I work over the field of complex numbers, and my schemes are separated and of finite type. Let $X$ be a quasi projective scheme (but not projective), let $Y$ be a regular projective scheme, and let $f: X \to Y$ be a monomorphism of schemes (but not necessary an immersion).
Is the pull-back by $f$ of an ample line bundle on $Y$ an ample line bundle on $X$?
And if not, what kind of (not too strong) assumptions I should add on $f$ for this result to be true?
Best Answer
A monomorphism is injective (EGA IV 17.2.6), hence quasi-finite, hence quasi-affine (Zariski main theorem). Now the pull-back of an ample sheaf by a quasi-affine morphism is ample (EGA II 5.1.12).