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Let $G$ be a connected unipotent algebraic (affine) group defined over a perfect field $k$. Then $G$ is $k$-isomorphic as an algebraic $k$-variety to the affine space $\mathbb A^n_k$.

Is there any good reference for this result?

It is probably in Demazure, Michel; Gabriel, Pierre Groupes algébriques, but I do not have an access to this book.

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  • $\begingroup$ its unipotent so you can just use log and exp right? $\endgroup$
    – Daniel Barter
    Commented Dec 4, 2014 at 13:44
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    $\begingroup$ @DanielBarter Only in char equal to 0. I am interested in a positive characteristic. $\endgroup$
    – Andrei Jaikin
    Commented Dec 4, 2014 at 13:46
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    $\begingroup$ Even if you had access to DG, you wouldn't necessarily want to look in there; it is a difficult book to read (though has some gems). $\endgroup$
    – user74230
    Commented Dec 4, 2014 at 15:33

2 Answers 2

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It is, for example, explained in paragraph $8$ (Appendix) of the book Unipotent Algebraic Groups, Lecture Notes in Mathematics Volume 414, 1974, by Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi, which is accessible online. It is also discussed in the first chapter On the theory of unipotent algebraic groups over an arbitrary ground field, with the reference to § $8$, where it is referred to Lazard's original proof.

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  • $\begingroup$ Thank you for your answer. I have a copy of the book by Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi. They consider a more general situation when $k$ is not perfect. Also from mathscinet I can deduce that in Lazard's paper it is proved only the converse implication. $\endgroup$
    – Andrei Jaikin
    Commented Dec 4, 2014 at 13:24
  • $\begingroup$ Yes, you are right I have just found it in the book (it is Remark A.3). Thanks again. $\endgroup$
    – Andrei Jaikin
    Commented Dec 4, 2014 at 13:36
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To add a little perspective to the question and Dietrich's answer, there has been a natural tendency in the literature to go beyond perfect ground fields and traditional algebraic geometry toward formulations for arbitrary fields in scheme language (SGA3, Demazure-Gabriel, Conrad-Gabber-Prasad). Another tendency is to discuss connected solvable groups rather than just unipotent ones. But there are some reasonable textbook treatments besides the Lecture Notes 414 volume.

Much of the work on generalizations to solvable algebraic groups over arbitrary fields was done by M. Rosenlicht in the mid-1950s. This is mostly presented with more "modern" proofs in the expanded second edition of Borel's Linear Algebraic Groups (GTM 126, Springer, 1991), $\S15$. For the structure of a connected unipotent group $G$ over a perfect field $k$ of any characteristic, the key point here is that $G$ "splits" over $k$ (15.5). But then Borel just quotes Rosenlicht without proof in 15.13(a) on the explicit structure of any $k$-split connected solvable group $G$ with $k$ arbitrary: $G$ is $k$-isomorphic as a variety to a product of copies of the additive and multiplicative groups (all being additive groups in the unipotent case).

Springer's Linear Algebraic Groups (2nd ed., Birkhauser, 1998) has a somewhat more explicit treatment along these lines (Chapter 14): see in particular 14.2.6-14.2.7 for unipotent groups over an arbitrary field $F$. Again the group is required to be $F$-split, which is satisfied when $F$ is perfect. (While the ideas in the unipotent case go back to Lazard, the tendency in both of these books is to refer only to Rosenlicht's work on solvable groups.)

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    $\begingroup$ By nilpotence, $U\ne 1$ has a nontrivial smooth connected central $k$-subgroup $C$. Pick maximal $n\ge 0$ so $Z:= p^n(C) \ne 1$, so $U\rightarrow U/Z$ is a torsor (for the etale topology) for the $p$-torsion smooth connected commutative $Z$. Dimension induction gives $U/Z$ is an affine space, so if $Z$ is a vector group then $U=Z\times(U/Z)$ as schemes since torsors for vector groups over affine schemes are trivial. Hence, it suffices to show $Z$ is a vector group. It is only at this final step that perfectness enters; it follows from Tits' work on unipotent groups (see B.2.7 in C-G-P). $\endgroup$
    – user74230
    Commented Dec 4, 2014 at 15:46
  • $\begingroup$ @user74230: Yes, there are definite advantages to an approach like this, though my impression was that the question aimed at something narrower. At one time people tended to work only over characteristic 0 fields, or perhaps all perfect fields, but that's artificial. Tits tried to improve the foundations, for example in his old Yale lectures (though he wasn't satisfied with the mimeographed notes written up by others at the time). $\endgroup$
    – Jim Humphreys
    Commented Dec 4, 2014 at 16:09
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    $\begingroup$ Jim Humphreys, what did Tits tell you about his dissatisfaction with the Yale notes? I wasn't aware of any opinion he had about them. $\endgroup$
    – user74230
    Commented Dec 5, 2014 at 3:55
  • $\begingroup$ @user74230: This goes back many decades, so my recollections may not be 100% clear. But when I got a copy of the mimeographed notes it was made very clear to me (perhaps by Tits himself or by others close to the situation) that he had been unhappy about the way the notes were written up. In any case it's wise to approach these notes with some caution (if copies are still in circulation). I didn't hear his lectures, so I can't judge accurately. $\endgroup$
    – Jim Humphreys
    Commented Dec 5, 2014 at 13:35
  • $\begingroup$ P.S. It may be relevant to observe that the 1966-67 Yale lectures by Tits do not appear in his MathSciNet publication list, whereas the Steinberg lectures from the following year do appear in such a list. In each case the notes were written up by two advanced graduate students (during the two years after I finished at Yale), but Steinberg seems to have had a much more hands-on role. (His lectures have been posted on his UCLA homepage, but after his death earlier this year there seems to be a renewed effort to have them published formally.) $\endgroup$
    – Jim Humphreys
    Commented Dec 5, 2014 at 22:36

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