Breuil-Kisin modules and Hopf orders in cyclic group rings

TL;DR

This paper uses Breuil-Kisin modules to classify Hopf orders in cyclic group rings over p-adic fields, providing explicit descriptions and a Laurent series perspective for these algebraic structures.

Contribution

It introduces a method to determine all Hopf orders in cyclic group rings using Breuil-Kisin modules, including explicit classifications for groups of order p and p^2.

Findings

Classified all Hopf orders in $K ext{}\Gamma$ for cyclic groups of order p and p^2

Identified all cyclic Breuil-Kisin modules relevant to these Hopf orders

Provided a Laurent series interpretation of the modules and orders

Abstract

For $K$ an extension of $\mathbb{Q}_{p}$ with ring of integers $R$ we show how Breuil-Kisin modules can be used to determine Hopf orders in $K$-Hopf algebras of $p$-power dimension. We find all cyclic Breuil-Kisin modules, and use them to compute all of the Hopf orders in the group ring $K\Gamma$ where $\Gamma$ is cyclic of order $p$ or $p^{2}.$ We also give a Laurent series interpretation of the Breuil-Kisin modules that give these Hopf orders.