Explicit classification of Hopf orders in group algebras over local fields of characteristic $p$ beyond the primitively generated case

Let $K$ be a local field of characteristic $p$ with ring of integers $R$. Koch (2017) gave a complete explicit characterization of all $R$-Hopf orders in any finite commutative cocommutative $K$-Hopf algebra of $p$-power rank that is generated by primitive elements, reducing the classification to the single matrix equation $\Theta^{-1}B\Theta^{(p)}\in M_n(R)$. The method stops at the boundary of primitively generated Hopf algebras, and the group algebras themselves lie outside it. The open problem is to extend such an explicit classification - parameterized families with explicit generators, arithmetic conditions on parameters, sameness and isomorphism criteria, dual orders, and parameter counts - to Hopf algebras not generated by primitives, concretely to the elementary abelian group algebras $K[C_{p}^{n}]$ for $n\ge 3$ (classified only for $n\le 2$, and for $n=3$ under restrictions) and the cyclic group algebras $K[C_{p^{n}}]$ for $n\ge 3$ (classified for $n\le 2$; no complete classification is known for any $n\ge 3$), and to settle whether $n(n+1)/2$ parameters suffice for $K[C_{p}^{n}]$, as they do exactly on the dual side $(K[C_{p}^{n}])^{*}$.

Background

Throughout, $p$ is a prime, $K$ is a local field of characteristic $p$ - that is, a field complete with respect to a discrete valuation $\nu:K\to\mathbb{Z}\cup\{\infty\}$ - with ring of integers $R=\{x\in K:\nu(x)\ge 0\}$, a complete discrete valuation ring (DVR) with uniformizer $\pi$, $\nu(\pi)=1$.

A Hopf algebra over a field is simultaneously a commutative algebra and a coalgebra (equipped with comultiplication $\Delta$, counit, and antipode) whose structures are compatible; here all Hopf algebras are finite-dimensional, commutative, cocommutative, and of rank $p^{n}$. An element $t$ is primitive if $\Delta(t)=t\otimes 1+1\otimes t$, and a Hopf algebra is primitively generated if it is generated as an algebra by its primitive elements. The basic examples on the primitively generated side are the linear duals $(K[\Gamma])^{*}$ of group algebras of elementary abelian $p$-groups $\Gamma=C_{p}^{n}$ (where $C_{p}^{n}$ denotes the elementary abelian group of order $p^{n}$), and the truncated polynomial algebras $K[t]/(t^{p^{n}})$ representing Frobenius kernels of the additive group scheme (the Frobenius map being $x\mapsto x^{p}$). The group algebras $K[C_{p}^{n}]$ and $K[C_{p^{n}}]$ with $C_{p^{n}}$ the cyclic group of order $p^{n}$ are, for $n\ge 2$, the fundamental examples of Hopf algebras that are not primitively generated; geometrically, $\mathrm{Spec}\,K[C_{p^{n}}]$ is the group scheme $\mu_{p^{n}}$ of $p^{n}$-th roots of unity.

An $R$-Hopf order in a $K$-Hopf algebra $H$ is an $R$-subalgebra $H_{0}\subset H$, free of finite rank over $R$ with $K\otimes_{R}H_{0}\cong H$, on which the Hopf structure maps of $H$ restrict. Equivalently, $\mathrm{Spec}\,H_{0}$ is a finite flat group scheme over $R$ modelling the group scheme $\mathrm{Spec}\,H$ over $K$, so classifying Hopf orders in $K[C_{p^{n}}]$ is the same as classifying models of $\mu_{p^{n}}$ over $R$. The classification problem has a classical motivation in integral Galois module theory: for a finite Galois (more generally Hopf–Galois) extension $L/K$, a theorem of Childs states that if the associated order $\mathfrak{A}_{H}=\{a\in H:a\,\mathcal{O}_{L}\subseteq\mathcal{O}_{L}\}$ is a Hopf order, then the ring of integers $\mathcal{O}_{L}$ is free over $\mathfrak{A}_{H}$; in characteristic $p$ this bears directly on the Galois module structure of Artin–Schreier–Witt extensions, the cyclic $p^{n}$-extensions built from Witt vectors (a nonstandard ring structure on length-$n$ vectors over $K$ generalizing Artin–Schreier theory).

In characteristic zero, Hopf orders in group rings were classified in low ranks by Larson, Underwood, and Childs–Underwood, who conjectured that a classification of Hopf orders in cyclic group rings of rank $p^{n}$ should involve $n(n+1)/2$ parameters: $n$ valuation parameters and $n(n-1)/2$ unit parameters. In characteristic $p$, the rank-$p$ case is the classical Tate–Oort classification. Koch (2017) settled the primitively generated side in all ranks: choosing primitive generators $t_{1},\dots,t_{n}$ of $H$ with $t_{i}^{p}=\sum_{j}b_{j,i}t_{j}$ gives an $n\times n$ matrix $B=(b_{j,i})$ over $K$, and the $R$-Hopf orders in $H$ are exactly the images $H_{\Theta}=R[\{\sum_{j}\theta_{j,i}t_{j}\}]$ of embeddings parametrized by matrices $\Theta\in\mathrm{GL}_{n}(K)$ satisfying

$\Theta^{-1}B\Theta^{(p)}\in M_{n}(R),$

where $\Theta^{(p)}$ raises each entry of $\Theta$ to the $p$-th power; two choices $\Theta,\Theta'$ give the same order exactly when $\Theta^{-1}\Theta'\in M_{n}(R)^{\times}$. For complete $R$, $\Theta$ may be taken in a lower-triangular diagonal-dominant normal form, yielding $n$ valuation and $n(n-1)/2$ further parameters - for instance in rank $p^{2}$ every Hopf order in such an $H$ is $H_{i,j,\theta}=R[\pi^{i}t_{1}+\theta t_{2},\pi^{j}t_{2}]$ for suitable $i,j,\theta$. The engine of the construction is a Dieudonné theory (a covariant equivalence between primitively generated Hopf algebras and finite free modules over the twisted polynomial ring $R[F]$, $Fa=a^{p}F$) for which the module-to-Hopf-algebra correspondence is completely explicit. Koch emphasizes that this transparency fails beyond the primitively generated world: the available generalized Dieudonné theories for arbitrary finite commutative group schemes in characteristic $p$ (de Jong's Dieudonné modules, Breuil–Kisin modules, frames, Faltings-type strict modules) are more complicated and construct Hopf algebras only up to isomorphism, not as explicit subalgebras, so they do not directly yield Hopf-order embeddings.

On the non-primitively generated side the state of the art is sparse. For elementary abelian group algebras, the classification of $R$-Hopf orders in $K[C_{p}^{2}]$ over a characteristic $p$ DVR is due to Elder–Underwood (2017). Underwood (2022) then computed, via a cohomological analysis of finite group scheme extensions, all Hopf orders in the dual $(K[C_{p}^{3}])^{*}$ (six parameters $i_{1},i_{2},i_{3},\mu,\alpha,\beta$, consistent with the $n(n+1)/2$ count), but could compute the dual orders in $K[C_{p}^{3}]$ - truncated exponential orders $E(i_{1},i_{2},i_{3},\mu,\alpha,\beta)$ - only under restrictions on the parameters ($\nu(\mu)\ge i_{3}-i_{1}$ and $i_{2}\ge i_{3}$, plus a convention of Elder–Underwood), stating explicitly that for $K[C_{p}^{n}]$ the case $n\ge 3$ remains open and that even the sufficiency of $n(n+1)/2$ parameters on the group-ring side is not obvious. For cyclic group algebras, i.e. models of $\mu_{p^{n}}$, the classification is known for $n\le 2$ (Tate–Oort; Tossici); Underwood (J. Algebra 669 (2025), 370–400) treats $K[C_{p^{3}}]$ over a local field, but no complete classification is known for any $n\ge 3$. Adjacent results of Elder–Keating give sufficient conditions, via Galois scaffolds (a computational device for integral Galois module structure), for the associated order of a $p$-extension to be a Hopf order in its group ring, but no classification. The open target is thus the extension of the explicit, embedding-level classification beyond primitively generated Hopf algebras, with the group algebras $K[C_{p}^{n}]$ and $K[C_{p^{n}}]$, $n\ge 3$, as its benchmark instances.

Problem Statement

Let $p$ be prime and let $K$ be a local field of characteristic $p$, complete with respect to the discrete valuation $\nu$, with uniformizer $\pi$, and ring of integers $R$. Let $C_{p}^{n}$ denote the elementary abelian group of order $p^{n}$ and $C_{p^{n}}$ the cyclic group of order $p^{n}$. The problem is to extend the explicit classification of $R$-Hopf orders beyond primitively generated Hopf algebras, with the group algebras as benchmark cases; the common objective is a classification at the same level of explicitness as the Tate–Oort rank-$p$ classification and Koch's matrix classification on the dual side, with all of the following jointly constituting one classification result per family:

1. For the elementary abelian group algebras $K[C_{p}^{n}]$, all $n\ge 3$: produce parameterized families, with explicit $R$-algebra generators and necessary and sufficient arithmetic conditions on the parameters, that exhaust all $R$-Hopf orders; give a criterion for when two parameter choices yield the same order (and describe the isomorphism classes); and compute the dual Hopf orders in $(K[C_{p}^{n}])^{*}$. This must remove the restrictions under which the $n=3$ case is currently known ($\nu(\mu)\ge i_{3}-i_{1}$, $i_{2}\ge i_{3}$, and the Elder–Underwood Convention 1), since those exclusions are precisely the unclassified parameter ranges.

2. For the cyclic group algebras $K[C_{p^{n}}]$, all $n\ge 3$ - equivalently, models of the group scheme $\mu_{p^{n}}$ over $R$: the same exhaustive explicit classification with generators, parameter conditions, sameness/isomorphism criteria, and duals.

3. Settle the parameter-count question: for the elementary abelian family $K[C_{p}^{n}]$, this is the explicit open subquestion of Underwood (2022) - determine whether $n(n+1)/2$ parameters suffice, as they do exactly on the primitively generated dual side $(K[C_{p}^{n}])^{*}$; for the cyclic family $K[C_{p^{n}}]$, a complete classification determines the parameter count, which should then be related or contrasted with the Childs–Underwood $n(n+1)/2$-parameter conjecture for cyclic group rings in characteristic zero, as quoted and discussed in Koch (2017).

Items 1-3 together constitute the classification target, and the two group-algebra families $K[C_{p}^{n}]$ and $K[C_{p^{n}}]$, $n\ge 3$, are jointly posed.

References

1. Alan Koch, "Primitively generated Hopf orders in characteristic $p$", Communications in Algebra 45(6) (2017), 2673–2689. DOI: 10.1080/00927872.2016.1233235; arXiv:1509.07393; https://arxiv.org/abs/1509.07393
2. Robert Underwood, "Hopf orders in $K[C_{p}^{3}]$ in characteristic $p$", Journal of Algebra 595 (2022), 523–550. DOI: 10.1016/j.jalgebra.2021.12.031; arXiv:2012.06540; https://arxiv.org/abs/2012.06540
3. G. Griffith Elder and Robert G. Underwood, "Finite group scheme extensions, and Hopf orders in $KC_{p}^{2}$ over a characteristic $p$ discrete valuation ring", New York Journal of Mathematics 23 (2017), 11–39. zbMATH Open identifier Zbl 1384.16020; https://zbmath.org/6684170; journal page https://nyjm.math.buffalo.edu/j/2017/23_2.html
4. Robert Underwood, "Hopf orders in $K[C_{p^{3}}]$ over a local field", Journal of Algebra 669 (2025), 370–400. DOI: 10.1016/j.jalgebra.2025.02.010; https://doi.org/10.1016/j.jalgebra.2025.02.010
5. G. Griffith Elder and Kevin Keating, "Galois scaffolds for cyclic $p^{n}$-extensions in characteristic $p$", Research in Number Theory 8(4), article 75 (2022). DOI: 10.1007/s40993-022-00381-2; arXiv:2201.08885; https://arxiv.org/abs/2201.08885
6. G. Griffith Elder and Kevin Keating, "Galois scaffolds for $p$-extensions in characteristic $p$", Annales de l'Institut Fourier 76(1) (2026), 397–423. DOI: 10.5802/aif.3712; https://arxiv.org/abs/2308.02775
7. Lindsay Childs, Cornelius Greither, Kevin Keating, Alan Koch, Timothy Kohl, Paul Truman, Robert Underwood, "Hopf Algebras and Galois Module Theory", Mathematical Surveys and Monographs 260, American Mathematical Society (2021). DOI: 10.1090/surv/260; https://doi.org/10.1090/surv/260
8. John Tate and Frans Oort, "Group schemes of prime order", Annales Scientifiques de l'École Normale Supérieure (4) 3 (1970), 1–21. DOI: 10.24033/asens.1186; https://doi.org/10.24033/asens.1186
9. Dajano Tossici, "Models of $\mu_{p^{2},K}$ over a discrete valuation ring", Journal of Algebra 323(7) (2010), 1908–1957. DOI: 10.1016/j.jalgebra.2010.01.012; https://doi.org/10.1016/j.jalgebra.2010.01.012
10. Lindsay N. Childs and Robert G. Underwood, "Cyclic Hopf orders defined by isogenies of formal groups", American Journal of Mathematics 125(6) (2003), 1295–1334. DOI: 10.1353/ajm.2003.0039; https://doi.org/10.1353/ajm.2003.0039
11. A. J. de Jong, "Finite locally free group schemes in characteristic $p$ and Dieudonné modules", Inventiones Mathematicae 114(1) (1993), 89–137. DOI: 10.1007/BF01232664; https://doi.org/10.1007/BF01232664