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Realizable Hopf algebras killed by $$[p]$$ over a discrete valuation ring. (English) Zbl 1186.16024
Summary: 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 $$[p]\colon H\to H$$ 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.
##### MSC:
 16T05 Hopf algebras and their applications 14L15 Group schemes 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers