Breuil-Kisin modules and Hopf orders in cyclic group rings

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.