Let $G$ be a group scheme over a scheme $X$ with centre $Z(G)$, automorphism group $\mathrm{Aut}(G)$ and outer automorphism group $\mathrm{Out}(G)$ (viewed as group schemes on $X$).

- If $G$ is
finite flatover $X$, then are $Z(G), \mathrm{Aut}(G)$ and $\mathrm{Out}(G)$ also finite flat over $X$?- If $G$ is
finite étaleover $X$, then are $Z(G), \mathrm{Aut}(G)$ and $\mathrm{Out}(G)$ also finite étale over $X$?

If$Z(G)$ is flat (can fail!) then $G/Z(G)$ exists and $G/Z(G)\to{\rm{Aut}}_{G/X}$ is finite and monic, so a closed immersion, so ${\rm{Out}}_{G/X}$ exists. That being said, #2 is "yes" by reducing to constant $G$, but ${\rm{Aut}}_{\alpha^n_p/\mathbf{F}_p}={\rm{GL}}_n$. $\endgroup$ – nfdc23 Aug 30 '17 at 12:075more comments