Let $k$ be a finite field and $G$ a finite type smooth $k$-group scheme. Let $G^0$ and $\Gamma$ be the connected component of identity and the component group of $G$, so there is an exact sequence $1 \rightarrow G^0 \rightarrow G \rightarrow \Gamma \rightarrow 1$. Is the induced $H^1(k, G) \rightarrow H^1(k, \Gamma)$ surjective?


By Lang's theorem, $H^1(k, G^0) = \*$, so the map in question is injective. It is surjective if $G^0$ is central (in particular, if $G$ is commutative), since for commutative $G^0$ one has $H^2(k, G^0) = 0$ for cohomological dimension reasons. It is also surjective if the short exact sequence splits as a semi-direct product. Thus, my question is: does the surjectivity hold in general for noncommutative smooth $G$?



Over a finite field, does a torsor under the component group of G lift to a torsor under G?
This is proved when $G$ is a linear algebraic group in Corollary III.2.4.3 of Serre's book on Galois cohomology (p. 135 in my version). I believe it is only stated here for linear algebraic groups as Serre is at that point looking at general fields of cohomological dimension at most $1$. In the special case of finite fields where one has Lang's theorem, it seems that the proof generalises to your setting.


