Suppose $0\to F \to G \to H \to 0$ is a short exact sequence of group schemes over a field $k$. Then $0 \to F(k) \to G(k) \to H(k) \to 0$ is not exact ($G(k) \to H(k)$ is not surjective). However, is it true if we look at the $k^{sep}$-points or the $k^{alg}$-points instead of the $k$-points ($k^{sep}$ resp. $k^{alg}$ a separable resp. algebraic closure of $k$)? Is there a reference for this fact?
Group Schemes and rational points
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algebraic-geometry
group-schemes
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0Well, you might not need this information anymore, but I might as well ask: does one not need $H$ to be smooth here? And are your group schemes affine? I have a reference for this in the affine case, provided $H$ is smooth. – 2018-05-28