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](http://books.google.com/books?id=vRhtM0UMUh4C&pg=RA1-PA221&lpg=RA1-PA221&dq=%2522theorem+of+the+base%2522+lang&source=bl&ots=YmQSU9ZBZp&sig=Jwm8WbbILU26tVN6bLvIgzdSA9M&hl=en&sa=X&ei=-EcYUIPmCsbL0QH3goD4DQ&ved=0CFUQ6AEwAg#v=onepage&q=%2522theorem%2520of%2520the%2520base%2522%2520lang&f=false) 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.") 



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



Modern Proof of the Theorem of the Base
SGA 6, Exp. XIII


