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"?
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1$\begingroup$ Perhaps I am mistaken, but is not any finite non-reduced group scheme a counter-example? $\endgroup$– Daniel LoughranCommented May 12, 2011 at 9:27
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4$\begingroup$ 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. $\endgroup$– Laurent Moret-BaillyCommented May 12, 2011 at 9:32
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3$\begingroup$ 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. $\endgroup$– Torsten EkedahlCommented May 12, 2011 at 9:41
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$\begingroup$ It seems that \'etale group schemes (and only them) become constant after a base change. $\endgroup$– Mikhail BondarkoCommented May 12, 2011 at 14:32
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$\begingroup$ Does the "only if" follow from fppf descent? $\endgroup$– user12832Commented May 15, 2011 at 13:50
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1 Answer
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The answer to the first question is no. It is definitely possible to write down an example directly but here is one systematic way of giving examples (in positive characteristic).
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.
answered May 12, 2011 at 9:37
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$\begingroup$ 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 $\alpha_p$. $\endgroup$ Commented May 12, 2011 at 14:14