Let $N,G,G^\prime$ be three affine algebraic group varieties (i.e geometrically reduced in the sense of J. Milnes) defined over a separably closed field $K$. Suppose that we have the following exact sequence: $$e\rightarrow N\rightarrow G\rightarrow G^\prime\rightarrow e$$

which means $N$ is a subgroup variety of $G$, and the $G\rightarrow G^\prime$ is faithfully flat. My question is considering the $K$-points, do we still have the following exact sequence: $$e\rightarrow N(K)\rightarrow G(K)\rightarrow G^\prime(K)\rightarrow e?$$

  • $\begingroup$ This amounts to the question whether $H^1_{fppf}(K,N)$ is trivial for flat $N$ and separably closed $K$. I am pretty sure that the answer is yes, but can't find a reference right now. $\endgroup$ – Victor Petrov Jul 22 '19 at 17:56
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    $\begingroup$ Hmm, not so sure actually. Try $N=\mu_2$ (and $G=G'={\mathbb G}_m$, say) and $K$ is the separable closure of ${\mathbb F}_2(t)$. However, this certainly works if $N$ is smooth. $\endgroup$ – Victor Petrov Jul 22 '19 at 18:02
  • $\begingroup$ Thanks for your comments, but I think that $N=\mu_2$ is not smooth, because it is not geometrically reduced. $\endgroup$ – tanjia Jul 23 '19 at 1:44
  • $\begingroup$ Oh yes, I missed "geometrically reduced" condition. $\endgroup$ – Victor Petrov Jul 23 '19 at 7:05

Yes, this is true. A group scheme over a field is smooth if and only if it is geometrically reduced, so the hypotheses ensure that $N$ is smooth. You can even allow $G$ and $G'$ to be arbitrary group schemes.

The map $G \rightarrow G'$ is an $N$-torsor, so it is a smooth morphism. This, for any $K$-point $g' \in G'(K)$, the fiber over $g'$ is a smooth $K$-scheme. Since $K$ is separably closed, every smooth $K$-scheme has a $K$-point. (This is an instance of the general fact that smooth morphisms of schemes acquire sections after passing to an étale cover of the base).

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