Canonicity and completion of Abrashkin's explicit classification of finite flat group schemes of period p: dependence on the p-power lift, functorial reconstruction ideals, and the p = 2 case

Finite flat commutative group schemes killed by p (period p) over the valuation ring of a p-adic field with perfect residue field are classified, for p > 2, by an antiequivalence with Breuil's category of filtered Frobenius modules; Abrashkin re-proved this directly within the theory of finite flat group schemes and supplemented it with a Galois-module criterion, a field-of-norms identification with Faltings's strict modules, and an explicit duality. The classification, however, is non-canonical: it depends on the choice of a uniformizer and a p-power lift K of the base field; its reconstruction of modules from Hopf algebras uses congruence conditions tied to special generators rather than the natural divided-power condition; and at p = 2 the antiequivalence was established by Abrashkin in 2013 only for a modified morphism category over an algebraically closed residue field, with the applications left unwritten. The open questions the source papers state as unresolved are: describing how the antiequivalence varies with the lift K and finding a canonical choice-free formulation; proving or refuting the functorial divided-power characterization of the reconstruction ideals; and completing the p = 2 case by writing out its applications, which the period-2 paper states can be done along the same lines but does not contain.

Background

A finite flat commutative group scheme $G$ over a ring $R$ is a group object among finite flat commutative $R$-schemes; equivalently $G=\operatorname{Spec}A(G)$, where $A(G)$ is a finite flat $R$-algebra carrying a Hopf-algebra structure given by a counit $e_G:A(G)\to R$ and a coaddition $\Delta_G:A(G)\to A(G)\otimes_R A(G)$. A group scheme has period $p$ if it is killed by $p$, i.e. $p\,\mathrm{id}_G=0$. Classifying period-$p$ group schemes over the valuation ring $O_0$ of a $p$-adic field is a foundational problem of integral $p$-adic Hodge theory: the Galois modules of their geometric points are precisely the torsion representations coming from finite flat group schemes, which supply the representations parametrized by flat deformation rings (rings describing families of Galois representations arising from such group schemes) and studied in modularity lifting theorems (results transferring modularity between congruent modular forms), and which encode subtle ramification data. Tate and Oort classified group schemes of prime order $p$ over bases such as $\mathbb{Z}_p$; Breuil later classified all period-$p$ finite flat group schemes over the valuation ring of a $p$-adic field with perfect residue field (a residue field of characteristic $p$ on which the Frobenius $x\mapsto x^p$ is an automorphism) for $p>2$, by an antiequivalence with a category of filtered Frobenius modules obtained from crystalline Dieudonné theory (the classical dictionary between group schemes over perfect fields and linear-algebraic data equipped with Frobenius operators), and Kisin extended the classification to all $p$-power order, still for $p$ odd, the prime $2$ presenting serious obstructions.

The source is Abrashkin's direct proof of Breuil's classification for $p>2$. Let $k$ be a perfect field of characteristic $p>2$, let $K''=\operatorname{Frac}W(k)$ where $W(k)$ is the ring of Witt vectors (the canonical complete discrete valuation ring of characteristic $0$ with residue field $k$), let $K_0/K''$ be totally ramified of degree $e$ with uniformizer $\pi_0$ and valuation ring $O_0$, choose $\pi$ with $\pi^p=\pi_0$, and set $K=K_0(\pi)$, $O=O_0[\pi]$; write $S=k[[t]]$ with Frobenius $\sigma(s)=s^p$. Breuil's category $MF^e_S$ consists of triples $(M^0,M^1,\varphi_1)$ where $M^0$ is a finite free $S$-module, $M^1\subset M^0$ a submodule containing $t^eM^0$, and $\varphi_1:M^1\to M^0$ a $\sigma$-linear map with $\varphi_1(M^1)S=M^0$. Abrashkin constructs an antiequivalence of categories

$G^O_{O_0}:MF^e_S\ \longrightarrow\ \mathcal{G}r_{O_0},$

where $\mathcal{G}r_{O_0}$ is the category of finite flat commutative group schemes over $O_0$ killed by $p$. The proof works entirely inside the theory of finite flat group schemes: essential surjectivity is obtained from Kummer theory (the classification of cyclic field extensions by adjoining roots of elements) and the Lubin–Tate formal group (the formal group with logarithm $l_{LT}(X)=X+X^p/p+X^{p^2}/p^2+\dots$) together with divided-power computations — an ideal carries nilpotent divided powers when for each element $a$ the recursively defined $a_{i+1}=-a_i^p/p$ tends to $0$, and $I^{DP}_A$ denotes the maximal such subideal of the augmentation ideal $I_A=\ker e_G$. As applications, Abrashkin proves a criterion for a finite $\mathbb{F}_p[\Gamma_{K_0}]$-module ($\Gamma_{K_0}$ the absolute Galois group of $K_0$) to arise as the module of geometric points $G(\bar K)$; an identification of these Galois modules with those of characteristic-$p$ analogues coming from Faltings's strict modules — group schemes over $k[[t]]$ with a strict action of the coefficient ring, introduced by Faltings as equicharacteristic substitutes for $p$-divisible groups (Barsotti–Tate groups, i.e. inverse limits of group schemes killed by $p^n$) — via the Fontaine–Wintenberger field-of-norms functor, which identifies the Galois theory of the arithmetically profinite extension $K_\infty=\bigcup_n K_0(\pi_n)$, $\pi_{n+1}^p=\pi_n$ (arithmetically profinite denotes the class of infinite extensions to which this construction applies), with that of a local field of characteristic $p$; and an explicit description of the duality in $\mathcal{G}r_{O_0}$.

The classification, however, is not canonical, and the paper says so. First, the antiequivalence depends essentially on the choice of the $p$-power lift $K=K_0(\pi)$ (equivalently, on the uniformizer $\pi_0$ and the root $\pi$, together with a ring identification $\kappa_{SO}:S/t^{ep}S\to O/pO$); the author states that it would be very interesting to understand how $G^O_{O_0}$ depends on the choice of $K$, and no comparison between different choices is given. Second, in the reconstruction of the filtered module from the Hopf algebra $B$ of a group scheme in the image, one uses the submodule $N^0_1=\{a\in I_B:\delta^+(a)\in I_{B\otimes B}(p)^p\}$, where $\delta^+(a)=\Delta(a)-a\otimes1-1\otimes a$ and $I_{B\otimes B}(p)=\{x:x^p\in p(B\otimes B)\}$; the natural, functorial condition with the divided-power ideal $I^{DP}_{B\otimes B}$ in place of $I_{B\otimes B}(p)^p$ could not be proved, and the usable alternatives all depend on special generators of $B$ and hence are not functorial. Third, everything is carried out only for $p>2$; the paper predicts that at $p=2$ the classification should hold for the modified category $\mathcal{G}r^*_{O_0}$ — same objects as $\mathcal{G}r_{O_0}$, morphisms taken modulo those factoring through the maximal étale quotient of the source (the largest subgroup scheme that becomes a constant group scheme after a finite separable extension) and the maximal multiplicative subgroup of the target (the largest subgroup scheme of multiplicative type, the analogue of a group of roots of unity) — under the assumption that $k$ is algebraically closed. This prediction was later confirmed by Abrashkin's “Group schemes of period 2”, which constructs the antiequivalence $\mathcal{G}r^*_{O_0}\simeq MF^{e,*}_S$ for 2-adic fields with algebraically closed residue field, but leaves the applications unwritten.

Problem Statement

Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $K''=\operatorname{Frac}W(k)$, and $K_0/K''$ a totally ramified extension of degree $e$ with uniformizer $\pi_0$ and valuation ring $O_0$. The problem is to remove the auxiliary choices from Abrashkin's explicit classification of period-$p$ finite flat group schemes over $O_0$ and to complete it at $p=2$; the common objective is a single classification that is explicit, uniform in $p$, and independent of the choices of the uniformizer $\pi_0$, the $p$-power lift $K=K_0(\pi)$ with $\pi^p=\pi_0$, and special generators of Hopf algebras. Concretely, the components stated as unresolved in the sources are:

(a) Dependence on the lift $K$. For $p>2$ (and, in its modified form, at $p=2$), describe how the antiequivalence $G^O_{O_0}:MF^e_S\to\mathcal{G}r_{O_0}$ varies with the choice of $K$, and, separately, determine whether or not the classification admits a canonical, choice-free formulation.

(b) Functorial reconstruction ideals. In the notation of the source, for a group scheme $H=\operatorname{Spec}B$ in the image of $G^O$ with augmentation ideal $I_B$, prove or disprove that the natural map

$\{a\in I_B:\delta^+(a)\in I^{DP}_{B\otimes B}\}\ \longrightarrow\ N^0_1/I^{DP}_B$

is surjective, i.e. that the reconstruction submodules can be characterized by the natural divided-power condition, thereby replacing the non-functorial congruence conditions modulo $I_{B\otimes B}(p)^p$ (or the $J_B$-type conditions depending on special generators of $B$) by a functorial, choice-independent one. In the revised $p=2$ framework of the period-2 paper the reconstruction ideals $J_B$ are already functorial, so this non-functoriality is specific to the $p>2$ framework, where the source states the open question.

(c) Completion of the case $p=2$. In the setting of the period-2 paper (2-adic field with algebraically closed residue field, antiequivalence $\mathcal{G}r^*_{O_0}\simeq MF^{e,*}_S$ of Theorem 0.1), establish the $p=2$ analogues of the applications of the $p>2$ theory, which the period-2 paper asserts can be done along the same lines but does not contain: the criterion for a finite $\mathbb{F}_2[\Gamma_{K_0}]$-module to be of the form $G(\bar K)$ with $G\in\mathcal{G}r^*_{O_0}$; the identification of the Galois modules of geometric points with those of the characteristic-2 analogues coming from Faltings's strict modules via the Fontaine–Wintenberger field-of-norms functor; and the explicit description of the duality in $\mathcal{G}r^*_{O_0}$.

Components (a)–(c) are scientifically linked: (a) and (b) canonicalize the two principal non-canonical ingredients of the same construction (the lift and the reconstruction ideals), and (c) completes the $p=2$ programme by establishing the applications the period-2 paper leaves unwritten. An answer may resolve any component separately; the problem as a whole is resolved when each of (a)–(c) is settled, affirmatively or negatively.

References

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