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In the theory of algebraic groups one of the fundamental results is the structure theorem for connected solvable groups. I never understood the proof in any of the standard textbooks, so I just moved on and treated the result like a black box.

Some time later I realized that over $ \mathbb{C} $ you can prove the theorem easily using the exponential map. (side question: Is it true that once you have proved the structure theorem over $ \mathbb{C}$ then you have it over any algebraically closed field of characteristic $ 0 $?)

I still have absolutely no idea why this structure theorem is true over an algebraically closed field of positive characteristic.

Question: What is the main idea in the proof of the structure theorem over an algebraically closed field of positive characteristic?

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  • $\begingroup$ Which structure theorem are you refering to? The fact that any solvable group is a semi-direct product of a unipotent group and an algebraic torus? $\endgroup$
    – Daniel Loughran
    Commented Sep 26, 2014 at 19:21
  • $\begingroup$ yes that is the one. Related statements are things like "you can conjugate any maximal torus to any other one using unipotent group elements" $\endgroup$
    – Daniel Barter
    Commented Sep 26, 2014 at 19:22
  • $\begingroup$ What is it that you want to know, which is not already covered in standard texts, such as Borel/Springer/Humpheys? The "main idea" for me is the Borel fixed point theorem. $\endgroup$
    – Daniel Loughran
    Commented Sep 26, 2014 at 19:24
  • $\begingroup$ Right. The use of the borel fixed point theorem is the same in both the characteristic 0 case and the characteristic p case. What I don't understand is how you split the exact sequence you get when you mod out by the unipotent subgroup $\endgroup$
    – Daniel Barter
    Commented Sep 26, 2014 at 19:28
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    $\begingroup$ @DanielLoughran: This splitting lies way deeper than Jordan decomposition. The proof (see 10.6(3),(4) in Borel's textbook) is very complicated. Building up through a composition series of the unipotent radical, the crux of the matter is to understand how tori act on vector groups in characteristic $p > 0$. The key, which is hidden in the usual treatments, is that if the action is sufficiently nontrivial (only nontrivial weights on Lie algebra) then there must be a linear structure equivariant for the action. Tits gave a direct proof of this fact, yielding a more intuitive approach. $\endgroup$
    – user27920
    Commented Sep 27, 2014 at 3:31

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