MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Say $G$ is a reductive group over a field $k$. I usually take $k = \mathbb{C}$ so assume what you want about the field except maybe that its finite. If $X$ is a scheme over $k$ then a principal $G$ bundles over $X$ is a scheme $P$ together with a right action of $G$ and an equivariant projection to $X$ (with trivial action on $X$) such that $P$ is locally trivial in the etale topology. For some groups like $GL_n,SL_n$ and solvable groups principal bundles are locally trivial even in the Zariski topology. These are called special groups and Grothendieck classified them.

I'm curious if $G',G''$ are special groups and $G$ fits into an exact sequence $1 \to G' \to G \to G'' \to 1$, then is it the case that $G$ is special?

There is a paper by Serre that claims this at least for $G',G''$ commutative and its supposed to be a consequence of the exact sequence $\check H^1(X,G') \to \check H^1(X,G) \to \check H^1(X,G'')$. This is \check Cech cohomology in the etale topology. You have $\check H^1(X,G') \cong \check H^1(X_{zar}, G')$ and $\check H^1(X,G'') \cong \check H^1(X_{zar}, G'')$ and a map $\check H^1(X_{zar},G') \to \check H^1(X, G)$ but it seems you are still short of being able to use e.g. the 5-lemma. This can probably be deduced from Grothendieck's thm but I'm wondering if there is a direct argument.

share|cite|improve this question
Let $f\colon X_{et}\to X_{zar}$ be the identity map. For a special group $G$, $R^1f_*G=0$ and so the Leray spectral sequence shows that the map from $H^2_{zar}$ to $H^2_{et}$ is injective in the commutative case, which is what you want in order to apply the 5-lemma. It is surely also true in the noncommutative case, but writing down a proof will be more complicated. – anon Aug 2 '11 at 23:51
Dear anon -- What? – Jason Starr Aug 3 '11 at 1:44
Dear solbap -- There is a long exact sequence, of sorts, for a central extension of a group $G''$ by a group $G'$, cf. Section I.5.7 of Serre's "Galois Cohmology". This is stated only in the case that $X$ is the spectrum of a field (so that \'etale cohomology is Galois cohomology). The generalization to schemes may be contained in Giraud's thesis. – Jason Starr Aug 3 '11 at 1:48
@Jason, sorry, I don't understand your question. The argument Angelo gives is essentially that in my comment, but with the cohomology removed. – anon Aug 3 '11 at 12:58
If you say so. I do know something about non-Abelian cohomology. I certainly understand Angelo's answer. I do not see how what you wrote is "essentially" what he wrote. – Jason Starr Aug 3 '11 at 20:59
up vote 3 down vote accepted

The answer is more or less as Jason says, but the proof is very easy, and does not require any cohomological machinery. If $P \to X$ is a $G$-torsor, then $P/G' \to X$ is a $G''$-torsor, hence it is Zariski-locally trivial. By passing to a cover, we may assume that it is trivial; hence $P$ has a reduction of structure group to $G'$, that is, it comes from a $G'$-torsor $P' \to X$. Such a torsor is Zariski-locally trivial, and this completes the proof.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.