[p] OVER A DISCRETE VALUATION RING

Let R be a discrete valuation ring of characteristic zero with field of fractions K and perfect residue field k of characteristic p> 2. We describe, in terms of Breuil modules, the finite flat abelian local-local R-Hopf algebras H killed by [p] : H → H such that S/R is a Hopf-Galois extension for some discrete valuation ring S whose field of fractions is totally ramified over K. Each such “realizable ” Hopf algebra is necessarily dual to a monogenic Hopf algebra. The classification is obtained by lifting certain k-Hopf algebras to R. We give a criterion for two such Breuil modules to be isomorphic