Background: I am trying to work out some Ext calculations for finite flat group schemes over a ring where p is nilpotent. I know how to do these calculations for finite group schemes over a finite field $k$ using the anti-equivalence with Dieudonne modules. I also know how to do this for finite flat group schemes over the ring of Witt vectors $W(k)$ by using the anti-equivalence with finite Honda systems.

Question: Is there a similar anti-equivalence that classifies finite flat group schemes over a ring where $p$ is nilpotent?

(Although this is probably no simpler, I would be satisfied with knowing the answer when $p^2 = 0$.)

  • $\begingroup$ This is an excellent question! Let's wait and see what Brian Conrad says. I have a feeling he's the go-to-guy with this question. $\endgroup$ Jul 30, 2010 at 18:39
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    $\begingroup$ Crystalline Dieudonne theory (Berthelot-Breen-Messing-De Jong) is possibly of use here. See De Jong's Berlin ICM talk for an overview. It provides a functor (finite flat group schemes) -> (Dieudonne crystals) which is faithful for reasonable base schemes. Fully faithfulness, and identifying the essential image, can delicate (in your case $p=2$ is likely to be very troublesome). Brian Conrad is likely to be able to give a much more informative answer. $\endgroup$ Jul 30, 2010 at 18:42
  • $\begingroup$ The base rings of interest are $W_2(k)$. By the way, I'm perfectly happy assuming $p>2$. $\endgroup$
    – agamzon
    Jul 30, 2010 at 19:54
  • $\begingroup$ OK, so then I will move my comment to an answer. $\endgroup$
    – BCnrd
    Jul 30, 2010 at 20:16

2 Answers 2


Since the case of interest is $W_2(k)$ with perfect $k$ of characteristic $p > 2$, the answer is given by Ioan Berbec's 2009 paper "Group schemes over artinian rings and applications. In that paper (esp. section 3) he defines an essentially surjective additive functor to a certain semi-linear algebra category and proves that it is full (i.e., surjective on Hom's) and that the isomorphism property can be read off on either side. The same method of proof for the isomorphism property (which amounts to passing to a computation on the special fiber, where classical Dieudonne theory applies) works just as well for closed immersions and quotient maps. Thus, it is a simple exercise to deduce that the induced homomorphism between Ext-groups is an isomorphism. So that answers the question in the cases of interest. For more general bases things will be tougher; already for $W_n(k)$ with $n > 2$ I don't know a proved result which is suitable for doing hands-on Ext computations (but maybe Berbec's result can be generalized a bit, possibly imposing a "truncated Barsotti-Tate" condition).


There is an old paper by Oort on Cartier modules of finite flat commutative local group schemes: Oort F., “Dieudonné modules of finite local group schemes”, Indag. Math., 37 (1975), 103–123. There he proves that the category of these group schemes can be fully embedded into the category of Cartier modules. This does not imply that the extensions are the same; yet this is quite possibly true; in my paper I proved this fact for group schemes over p-adic complete discrete valuation rings.


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