TITLE: Modern Proof of the Theorem of the Base
QUESTION [9 upvotes]: I am looking for a modern proof of the so-called "Theorem of the Base"--that the Neron-Severi rank of a smooth projective variety is finite.  One can prove this for varieties over $\mathbb{C}$ easily via transcendental methods---however, the most recent proof I can find which works e.g. in positive characteristic is due to Lang and Neron and is written in the language of Weil's foundations.  (Lang includes a similar proof in his book "Diophantine Geometry.")  

Does anyone know of a proof written in the language of schemes?

My motivation is that I'd like to have a brief reading seminar on this theorem with some graduate students at my institution---while it looks like the Lang/Neron paper is translatable, I think that this sort of translation is a serious burden for seminar participants.  So it would be nice to have a more modern reference.

REPLY [8 votes]: SGA 6, Exp. XIII