$x_0$. Suppose that $e\in H^0({{\mathscr{O}}}_X)$ is idempotent and that $x_0^\#(e)=0$. Then, ${\varphi}^\#(e)\in H^0({{\mathscr{O}}}_Y)$ is such that $y_0^\#({\varphi}^\#(e))=0$ and this implies that ${\varphi}^\#(e)=0$. Since ${\varphi}$ is surjective, we conclude that $e=0$ (since $e$ has to vanish on each local ring of $X$). Hence, the only connected component of $X$ must be the one containing the image of $x_0$.

We should now gather material on affine group schemes.

\[22.03.2019–1\]Let $G\in({\mathbf{FGSch}}/A)$. We say that $G$ is pseudo-finite if both its fibres are finite group schemes over the respective residue fields.

We have no examples of pseudo-finite group schemes which are not quasi-finite to offer.

As Lemma \[29.11.2018–1\] below recalls, one advantage of finite group schemes over a field is that the standard criterion for verifying when a morphism is faithfully flat in terms of representation categories [@deligne-milne82 Proposition 2.21, p.139] admits a considerable simplification. This is then transmitted to pseudo-finite group schemes as argued by Lemma \[21.03.2019–1\].

\[29.11.2018–1\] Let $u:H\to G$ be a morphism of group schemes over a field and let $$H\stackrel q{\longrightarrow}I\stackrel i{\longrightarrow}G$$ be its factorisation into a faithfully flat morphism $q$ and a closed immersion $i$. Assuming that $I$ is finite (which is the case if either $G$ or $H$ 