# k-points of an exact sequence of algebraic varieties

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: $$e\rightarrow N(K)\rightarrow G(K)\rightarrow G^\prime(K)\rightarrow e?$$

• 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. – Victor Petrov Jul 22 '19 at 17:56
• 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. – Victor Petrov Jul 22 '19 at 18:02
• Thanks for your comments, but I think that $N=\mu_2$ is not smooth, because it is not geometrically reduced. – tanjia Jul 23 '19 at 1:44
• Oh yes, I missed "geometrically reduced" condition. – 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).