Breuil–Kisin Modules and Hopf Orders in Cyclic Group Rings

Abstract

For K, a finite extension of 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 KΓ where Γ 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.