# Does every finite flat group scheme become constant after finite base change?

Does every finite flat group scheme $G/X$ become constant after finite base change? Which additional properties of the base change morphism can we impose?

Edit: Which conditions do we have to impose on $G/X$ so that the answer becomes "yes"?

-
Perhaps I am mistaken, but is not any finite non-reduced group scheme a counter-example? – Daniel Loughran May 12 '11 at 9:27
The question should be stated more precisely. I assume that "finite" stands for "finite surjective" (at least). Otherwise the answer is yes: take the empty base change. And if "surjective" is intended, Daniel's comment applies. – Laurent Moret-Bailly May 12 '11 at 9:32
I interpreted the question as $\mu_p$ times the base would be an example of a constant group scheme in which case a non-reduced group scheme is not necessarily a counter example. – Torsten Ekedahl May 12 '11 at 9:41
It seems that \'etale group schemes (and only them) become constant after a base change. – Mikhail Bondarko May 12 '11 at 14:32
Does the "only if" follow from fppf descent? – user12832 May 15 '11 at 13:50

Start with a base ring $R$, say $R=k[t]$, where $k$ is a field of positive characteristic and an $R$-algebra $A$ which is finitely generated projective as $R$-module. As example we take $A=R[s]/(s^2-ts)$. Then the invertible elements of $A$ is a smooth algebraic group scheme $A^\ast$ over $R$. This construction commutes with base change so in our example it is $\mathbb G_m^2$ over any point of $\mathrm{Spec}R$ different from the origon and ias $\mathbb G_m\times\mathbb G_a$ over the origin. To get a finite group scheme we simply take the kernel of the relative Frobenius map to get $\mu_p^2$ outside of the origin and $\mu_p\times\alpha_p$ over the origin.
Another class of examples is to look at an abelian scheme $B$ over $R$ and look at the kernel of multiplication on $B$ by some prime (say) $p$. Over points of $R$ over which $p$ is invertible this kernel will have a non-trivial infinitesimal part and over points where $p$ is invertible it will be étale. Hence we can let $R$ be a mixed characteristic DVR with residue field characteristic $p$. We can also let $R$ be characteristic $p$ DVR with $B$ being a supersingular elliptic curve over the special point and ordinary over the generic.
Here is a very similar example written out explicitly: Let $k$ be characteristic $p$; let $R=k[t]$; let $A=R[u]/u^p$. Put a group structure on $\mathrm{Spec} \ A$ by $u \ast u' = u+u'+tu u'$. For $t \neq 0$, this is isomorphic to $\mu_p$; to see this, write $v=1+tu$, then $v \ast v' = v v'$ and note that $v^p=1$. For $t=0$, this is obviously $\alpha_p$. – David Speyer May 12 '11 at 14:14