# Any local algebraic group is birationally equivalent to an algebraic group

In this paper, page $6$ the authors state the following:

By Weil’s theorem $[17]$, any local algebraic group is birationally equivalent to an algebraic group.

Where

$[17]$ A.Weil. On algebraic groups of transformations. Amer. J. Math. 77, (1955), 355-391.

I would like to know if I can find that Theorem in a textbook about Algebraic Groups.

I would appreciate your help.

PS. I did ask this question in StackExchange, but I got no answer.

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It's not a textbook, but the theorem is proved in the setting of schemes in an expose of Artin in SGA 3. Artin also gives a very sketchy proof of the theorem in his first article in Arithmetic Geometry 1986 (Cornell/Silverman) --- I think I would look there. –  anon Mar 25 '12 at 23:42

## 1 Answer

The answer to your question is basically no. Given the vintage of Weil's paper, you can't expect his statement to occur in this form in later books or even lecture notes on algebraic groups. Weil was then working with a language for algebraic geometry which attempted to add precision to what the Italian geometers had done earlier in the century. But his Foundations book did not last long with the coming of other languages, including the hybrid language developed by Chevalley in the 1950s and the more definitive Grothendieck language. (In his lectures and textbook, Borel took some expository shortcuts to avoid scheme language for the most part; but he didn't claim this was an all-purpose way to deal with algebraic groups.)

Unfortunately it takes quite a bit of work nowadays to read the older papers by Weil, Rosenlicht, and others within a modern context, though the effort sometimes pays off.

That said, it would take me a long time to get far enough inside the context of the papers you cite to say anything sensible about what it all means in current language.

UPDATE: An anonymous but usually reliable source (code name X) refers me to Chapter 5 of the 1990 Springer Ergebnisse volume Neron Models by Bosch, Lutkebohmert, and Raynaud. While this book concerns mainly abelian varieties rather than affine algebraic groups, it seems to provide a complete version of Weil's theorem in more modern language. This can be done most simply over a base field but has wider relevance over a Dedekind domain. In any case, X emphasizes that it's unrealistic to work throughout with affine algebraic groups even if that is one area of potential application. (Those with access to MathSciNet may find the review there by Jim Milne helpful, though he is not X.)

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@Jim What's wrong with the reference in SGA3 given in the comment by anon above? Doesn't it fulfill the goal of explaining Weil's theorem in modern language? Surely, if someone can read Neron Models, he or she can read SGA3 too, right? May I also point out that there is a set of notes by Bas Edixhoven, Ariane Mézard and Matthieu Romagny from a recent summer school at Luminy about SGA3, exactly on the topic of Weil's theorem. These notes can be found here: math.ens.fr/~gille/sem/notes434.html –  Baptiste Calmès Apr 8 '12 at 10:03
@Baptiste: I'm undecided about the relative value of the "modern" accounts mentioned, but should have mentioned an opinion from the source X that the SGA3 treatment is too abstract and general to be a first choice here. At any rate, there do exist some accounts not reliant on Weil's older language. –  Jim Humphreys Apr 8 '12 at 19:43