Splitting over the base and classification of actions for finite flat group schemes of square free order

Murty and Zong proved that every finite flat finitely presented group scheme $G$ of square free order over a scheme $S$ is a Hochschild extension $1\to G'\to G\to G''\to 1$ of a finite étale $S$-group scheme $G''$ by a commutative direct sum $G'$ of group schemes of prime order, and that the extension splits after a finite étale surjective base change; the classification of square-free-order group schemes is thereby reduced to an action problem, but only up to finite étale descent. The open problem is to remove that caveat: give effective criteria for the splitting to hold over the original base $S$ (the case of a constant finite étale quotient being already settled in the source), and classify explicitly the actions of finite étale $S$-group schemes on the Oort–Tate prime-order summands together with the relevant descent data, so as to obtain a genuine classification of finite flat group schemes of square free order over arbitrary bases, extending the classical Tate–Oort classification of prime-order group schemes.

Background

Let $S$ be a scheme. A finite flat finitely presented group scheme over $S$ is a group object $G$ in the category of $S$-schemes whose algebra of functions $f_*\mathcal{O}_G$ is a locally free $\mathcal{O}_S$-module of finite rank; this rank, a locally constant function on $S$, is called the order of $G$. The order is square free when no prime square divides it at any point of $S$. Over a field of characteristic zero every finite group scheme is étale (Cartier's theorem), but over bases of mixed or residual positive characteristic finite group schemes can have infinitesimal parts, such as $\mu_p$ (the kernel of the $p$-th power map on the multiplicative group, i.e. the group scheme of $p$-th roots of unity) or $\alpha_p$ (the kernel of the Frobenius endomorphism $x\mapsto x^p$), so their structure is a genuinely arithmetic question. A finite étale group scheme over $S$ is one that becomes the constant group scheme attached to a fixed abstract finite group after a finite étale surjective base change; over a connected base it is equivalently given by a finite abstract group together with a monodromy action of the étale fundamental group $\pi_1(S)$ on its group of geometric points.

The prime-order case is classical: Tate and Oort proved that a group scheme of order $p$ over any base is commutative and killed by $p$, and over bases lying over the ring $\Lambda_p$ of $p$-adic integers together with a primitive $(p-1)$-th root of unity they classified all such group schemes by triples $(L,a,b)$ — an invertible sheaf $L$, a section $a$ of $L^{\otimes(1-p)}$ and a section $b$ of $L^{\otimes(p-1)}$ with $ab=w_p$ a fixed unit — a theory that also computes the automorphism functor of each group scheme.

Murty and Zong extended this to all square free orders, but only up to finite étale descent. Their theorem states that every finite flat finitely presented $S$-group scheme $G$ of square free order sits in a Hochschild extension (meaning that $1\to G'(T)\to G(T)\to G''(T)\to 1$ is exact for every $S$-scheme $T$) with $G''$ finite étale over $S$ and $G'$ commutative, a direct sum of group schemes of prime order indexed by the prime factors of the order, and that this extension splits after a finite étale surjective base change $T\to S$: over $T$ the group scheme becomes a semidirect product $G'_T\rtimes G''_T$. The engine is coprimality: since $|G'|$ and $|G''|$ are coprime, multiplication by $|G''|$ is an automorphism of $G'$, which kills the group-cohomological obstruction to splitting whenever $G''$ is constant; indeed the note's own proof of its Lemma 2.9 shows that if $G''$ is already constant over $S$, the splitting holds over $S$ itself. The note concludes that, at the expense of a finite étale descent datum (which encodes how the split product over the cover is glued along the fiber product of the cover with itself), every group scheme of square free order is a semidirect product determined by how $G''$ acts on the prime-order summands of $G'$.

What the note does not provide is the analysis on the original base. For a general $S$ the finite étale quotient $G''$ need not be constant — it carries monodromy — and the note neither determines criteria or obstructions for the splitting to hold already over $S$, nor classifies the possible actions of finite étale $S$-group schemes on the Oort–Tate prime-order summands in explicit cohomological or moduli-theoretic terms. Without these two ingredients there is no classification of square-free-order group schemes over an arbitrary base, only over bases for which a trivializing finite étale cover can be substituted for the base itself.

Problem Statement

Complete the classification of finite flat finitely presented group schemes of square free order over an arbitrary base scheme $S$, removing the 'up to finite étale descent' caveat in the Murty–Zong structure theorem. The objective comprises two linked questions:

(1) Splitting over the original base. For $G$ and a Hochschild extension $1\to G'\to G\to G''\to 1$ as furnished by the Murty–Zong theorem — the extension being part of the data, since such extensions are not canonical: given any two of them for the same $G$, the theorem provides a third with $G'_3=G'_1\cap G'_2$ — determine necessary and sufficient conditions for the extension to split over $S$ itself — that is, for $G\cong G'\rtimes G''$ as $S$-group schemes without any base change — in effective form, computable from the Oort–Tate data of the summands of $G'$ and the monodromy of $G''$ and satisfied if and only if the extension splits.

(2) Classification of the actions. Classify, in explicit cohomological or moduli-theoretic terms over $S$, the possible actions of finite étale $S$-group schemes $G''$ on direct sums of commutative finite flat finitely presented group schemes of prime order, together with the finite étale descent data that arise relative to a finite étale surjective trivializing cover $T\to S$ as provided by the Murty–Zong theorem, so that the isomorphism classes of finite flat finitely presented group schemes of square free order over $S$ are parametrized by computable data, generalizing the Tate–Oort classification from prime order to all square free orders.

The splitting criterion and the action classification are inseparable parts of one classification problem, since the action data determine $G$ only jointly with the descent datum.

References

1. V. Kumar Murty and Ying Zong, Group schemes of square free order, arXiv:1609.02988 [math.AG] (v1 2016, v2 2021), DOI: 10.48550/arXiv.1609.02988, https://arxiv.org/abs/1609.02988
2. John Tate and Frans Oort, Group schemes of prime order, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 1-21, DOI: 10.24033/asens.1186, https://doi.org/10.24033/asens.1186
3. Yu Koide, On torsors for general twisted finite group schemes of prime order, JP J. Algebra Number Theory Appl. 28 (2013), no. 2, 107-127, Zbl 1266.14035, https://zbmath.org/1266.14035