# Ampleness under pull-back

Let X be an open subscheme of a scheme Y and denote by j the inclusion j:X->Y. Let L be an ample invertible sheaf on Y, and let L' be the pull-back j*L of L to X. Is is true that L' is ample on X?

I am not 100% sure, it seems very reasonable to me, since X is open.. but I think I know how to prove it if X and Y are both of finite type over some noetherian ring R. Is is true in general?

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If you need to ask then you have to review the definition of ampleness in such generality, such as EGA II, 4.5.3. (Should at least assume $X$, $Y$ are quasi-compact and separated, and EGA II 4.4.2 and 4.5.10 may clarify the relation with other definitions.) Please try harder to figure it out for yourself. –  BCnrd Apr 20 '10 at 4:42
Thanks for the reference. The answer is yes. –  david Apr 20 '10 at 5:08