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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.
16T05 Hopf algebras and their applications
14L15 Group schemes
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers