Some families of finite flat group schemes

When people say finite flat group scheme, what exactly do they mean? Sometimes, they just mean finite flat group scheme, presumably over some prefixed base. But often people meant finite flat *commutative* group scheme. It’s confusing. In this blog, I shall always mean the former, and add the adjective “commutative” when I mean it. Anyway, a while back, I was wondering about: can a finite flat commutative group scheme be deformed to a finite flat non-commutative group scheme? Weird random question, I know. But let’s still discuss it, because why not.

Since the moduli of finite flat group scheme of a fixed order is of finite type over \(\mathbb{Z}\), we are basically asking: can there be a finite flat group scheme \(G\) over a DVR, such that the special fibre is commutative whilst the generic fibre is not? There are 3 types of DVR: equal characteristic 0 or p, and mixed characteristic. In the equal characteristic 0 case, we can’t have such an example, as the finite flat group scheme has to be finite étale, and hence the specialization map induces an isomorphism, so the family of groups is essentially constant (appropriately interpreted).

Now let’s discuss the equal characteristic p situation. By the way, p is always a fixed prime for me. Already there are some weird examples in characteristic p setting: one can have a semi-direct product of \(\mu_p\) by \(\alpha_p\) where the action of former on the latter is what you think it is. Said differently, take the Borel in \(PGL_2\) and look at the Frobenius kernel, which exactly gives the said extension. In characteristic 0, there cannot exist such a noncommutative finite group scheme of order \(p^2\). Finally, I shall leave it to the reader to check that the above group scheme can be specialized to the product of two copies of \(\alpha_p\). The idea is that one can deform \(\mu_p\) to \(\alpha_p\) (comultiplication reads: \(x \mapsto t \cdot x \otimes x + x \otimes 1 + 1 \otimes x\)), and we can extend the action of \(\mu_p\) on \(\alpha_p\) to this family (action reads: \(s \mapsto (tx+1) \otimes s)\)). This gives an order \(p^2\) example in equal characteristic p.

Lastly, what if the base has mixed characteristic p? The generic point has characteristic 0. By what I said above, order \(p^2\) group scheme over the generic point is necessarily commutative. The best thing to hope for is to have an example of order \(p^3\) finite flat group scheme over some mixed characteristic \(\mathcal{O}_K\), such that the special fibre is commutative and the generic fibre is not. To that end, let’s start with a central extension of \((\mathbb{Z}/p)^2\) by \(\mathbb{Z}/p\). When \(p \not= 2\), there is exactly one such non-commutative group: commutator map gives rise to an alternating pairing on \((\mathbb{Z}/p)^2\) valued in \(\mathbb{Z}/p\), and we want it to be non-degenerate. When \(p=2\), there are two such groups, one is \(D_4\), the other one is \(\mathcal{O}_H^{\times}\) where \(H\) is the standard Hamiltonian over \(\mathbb{Z}\). In any case, such central non-commutative extension exists. Let \(\mathcal{O}_K = \mathbb{Z}_p[\zeta_p]\), over which we have a tautological character \(\mathbb{Z}/p \to \mu_p\), pushing out the above central extension along this character, gives us a central extension of \((\mathbb{Z}/p)^2\) by \(\mu_p\). The point is that now the special fibre of this extension splits, because the map \(\mathbb{Z}/p \to \mu_p\) is trivial in the special fibre.

I learned the above example from reading Raynaud’s “p-torsion du Schema de Picard”. BTW, in the same paper, he explains why the last example has to be over a base with ramification index at least \((p-1)\). For such a \(G\), if you look at its classifying stack, then its \(Pic^{\tau}\) will have different size on the special and generic fibre. But Raynaud showed that the \(Pic^{\tau}\) of a smooth proper scheme (e.g. a “good enough” approximation of the aforesaid classifying stack) is flat if the ramification index is less than \((p-1)\). Isn’t he amazing??!!

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