Let $S$ be a good enough base scheme (say of finite type over an algebraic closed field) and $G\to S$ be a flat group scheme. I want to ask: can we always find a closed embedding $G\to H$ into another flat group scheme $H$, such that $H$ is constant over S locally in the Zariski topology, and that $H/G$ exists as a scheme?
3
-
2$\begingroup$ New contributor, welcome! I am not sure I understand what you mean by "constant". $\endgroup$– Laurent Moret-BaillyCommented Sep 12, 2019 at 20:09
-
1$\begingroup$ I mean that there exists a Zariski cover $S'\to S$ such that the base change $H\times_S S'$ is isomorphic to $H_0 \times S'$ (for an algebraic group $H_0$ over the base field) as group schemes over $S'$. $\endgroup$– Lin ChenCommented Sep 12, 2019 at 21:00
-
$\begingroup$ This is certainly false for families of abelian varieties: subvarieties of abelian varieties cannot be deformed. I am not sure about the affine case. $\endgroup$– AngeloCommented Sep 14, 2019 at 14:10
Add a comment
|