What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field. Is there a full classification in case of algebraically closed field?
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$\begingroup$ Could you please clarify the following sentence: "By a finite group scheme, I mean $\text{Spec}\ A$ where $A$ is a finite-dimensional algebra over a field"? $\endgroup$– Jason StarrCommented Aug 14, 2015 at 17:59
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$\begingroup$ @Jason Starr I mean that my question is about affine group schemes $Spec A$, where $A$ is a finite dimensional algebra over a field. I would like to know, if there is a classification of such group schemes? $\endgroup$– AZ.Commented Aug 14, 2015 at 18:07
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9$\begingroup$ Over an algebraically closed field $k$ of characteristic $0$, the functor that sends a finite $k$-group scheme to its group of $k$-points is an equivalence of categories from the category of finite $k$-group schemes to the category of finite groups. In characteristic $p$, the story is more involved because there are non-smooth $k$-group schemes such as $\mu_p$ and $\alpha_p$. Over an imperfect field ... well, it is best to wait for grghxy. $\endgroup$– Jason StarrCommented Aug 14, 2015 at 18:19
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2$\begingroup$ As Jason points out, there is a big difference depending on the characteristic of the field. In prime characteristic there are also Frobenius kernels, etc. $\endgroup$– Jim HumphreysCommented Aug 14, 2015 at 20:07
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8$\begingroup$ Over a perfect field $k$ of char. $>0$ the group is a semidirect product of its infinitesimal identity component acted upon by a finite etale group, so any "classification" involves conjugacy classes of embeddings of finite etale groups into Aut-schemes of infinitesimal groups, a hopeless mess. The infinitesimal groups in the commutative case are understood via Dieudonne theory, but that is in no sense a classification (even if $k=\overline{k}$). I could indeed go on and on about the case of imperfect $k$, but the OP only asked about algebraically closed fields; as posed this is hopeless. $\endgroup$– grghxyCommented Aug 14, 2015 at 22:04
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This problem is wide open. Just how wide will depend on your definition of "classification". For example, over algebraically closed fields of characteristic zero, this is equivalent to the problem of classifying finite groups up to isomorphism. We normally say that the problem of classifying finite groups up to isomorphism is a wide open problem that will never be solved, but you might be satisfied to stop once the equivalence is given.
Finite group schemes satisfy the same Jordan-Hölder property as finite groups, so you can reduce the problem to the 2-step Hölder program that we see in the finite group world:
- classifying finite simple group schemes
- the extension problem: If you have $G_1$ arbitrary and $G_3$ simple, classify the groups $G_2$ with $G_1 \triangleleft G_2$ and $G_2/G_1 \cong G_3$.
The first part is done in a weak sense, by a combination of Oort-Tate for order $p$ and the fact that non-abelian simple group schemes are étale. Rather, it is definitely done over algebraically closed fields, where it is basically the classification of finite simple groups, and then one has to classify homomorphisms from the absolute Galois group of $k$ to the automorphism group of a finite simple group (which will never be done for general $k$).
The extension problem is even more hopeless than in the finite group world. In characteristic $p$, you have to deal with the substantial additional complication of infinitesimal structure, and even in the étale case, the descent data seem to mix in some complicated way.
answered Aug 15, 2015 at 19:51