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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.

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    $\begingroup$ SGA 6, Exp. XIII $\endgroup$ Jul 31, 2012 at 21:57
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    $\begingroup$ In the article by B. Kahn, "Sur le groupe des classes d’un schéma arithmétique (avec un appendice de Marc Hindry), Bull. Soc. Math. France 134 (2006), 395–415 in par. 1, p. 400 a survey of the available proofs is given. It seems that Lang and Néron proof was never translated. $\endgroup$ Aug 1, 2012 at 9:21
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    $\begingroup$ Thanks Damian--that's a nice article. The appendix actually contains a (brief) exposition of the essential arithmetic content of the Lang/Neron proof, which is also contained here (math.stanford.edu/~conrad/papers/Kktrace.pdf) in rather more detail (BCnrd pointed me to this article via email). I think I can extract the geometric content from Lang, so this probably suffices. $\endgroup$ Aug 1, 2012 at 17:46

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SGA 6, Exp. XIII

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  • $\begingroup$ Ah, thanks--I also just found a proof in Milne's "Etale Cohomology." I'm still interested in knowing if there's a writeup of Neron's proof relating the theorem to Mordell-Weil. $\endgroup$ Jul 31, 2012 at 21:59

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