My question comes from the reading of Tate's paper $p$-divisible groups. In the last few pages there is an argument which gives as trivial the following fact. If we take a $p$-divisible group over a CDVR with algebraically closed residue field then, if the characteristic of the residue field is different from $p$, the group has to be étale. This means that the representing Hopf algebra at each spot of the $p$-divisible group becomes separable when tensored with $k$. This essentially can be rephrased saying that an Artinian $k$-algebra with rank different from the characteristic of $k$ is reduced. This can be proved reducing to local Artinian algebras, just using the structure theorem, but now I really don't know how can I prove this fact. Do you have any hint?
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$\begingroup$ I assume that by reduced you mean no nonzero nilpotents. In a local artin ring, the maximal ideal is the nilradical. So the only time a local artin ring is reduced is when it is in fact a field. $\endgroup$– rghthndsdApr 28, 2015 at 13:01
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$\begingroup$ Yes! I agree that we have now to prove that a finite local artinian Hopf algebra of rank a power of a prime $p$ over an algebraically closed field of characteristic $q\neq p$ is a field. In particular it's the base field itself, by algebraically closure. How can we prove this now? $\endgroup$– rimeApr 28, 2015 at 18:06
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