Tame realisable classes over Hopf orders
Let \(K\) be an algebraic number field with algebraic closure \(K^c\) and ring of integers \({\mathcal O}_K\), and let \(G\) be a finite abelian group. A class \(a\) in the class group \(Cl({\mathcal O}_K)\) is realizable if \(a\) is represented by the full ring of integers \({\mathcal O}_L\) of some Galois extension \(L/K\) with Galois group \(G\): then \(L/K\) is necessarily tamely ramified. \textit{L. R. McCulloh} [J. Reine Angew. Math. 375/376, 259-306 (1987; Zbl 0619.12008)] obtained an idelic description of the set \(R(G)\) of realizable classes, and thereby showed that \(R(G)\) is a group. The purpose of this paper is to generalize McCulloh's results to the case where \(A\) is a commutative, cocommutative, separable and finite \(K\)-Hopf algebra and \(\mathcal A\) is a Hopf order over \({\mathcal O}_K\) in \(\mathcal A\). A class in \(Cl(\mathcal A)\) is realizable if it is the class of a strongly \(\mathcal A\)-tame order in a principal homogeneous space \(L\) over \(A^*\). Here \(\mathcal C\) is a strongly \(\mathcal A\)-tame order if for each finite place \(v\) of \(K\), (i) \({\mathcal C}_v\) is the maximal \({\mathcal A}_v\)-stable order in \(L\); (ii) \({\mathcal C}_v\) is \({\mathcal A}_v\)-tame in the sense of the reviewer and \textit{S. Hurley} [Trans. Am. Math. Soc. 298, 763-778 (1986; Zbl 0609.16005)]; and (iii) the \(p\)-part of \({\mathcal C}_v\) is a principal homogeneous space over \({\mathcal A}_v^*\), where \(v\) lies over \(p\). The main result of the paper is an idelic description of the realizable classes in \(Cl(\mathcal A)\) which generalizes McCulloh's for \({\mathcal A}={\mathcal O}_KG\). The author also shows that if \(K\) is a local field containing \(\mathbb{Q}_p\) and \(A\) is a \(K\)-Hopf algebra with \(K^cA=K^cG\) with \(G\) a group of order prime to \(p\), then \(A\) has an \({\mathcal O}_K\)-Hopf order iff there is an unramified extension \(L/K\) so that \(LA=LG\).
scientific article; zbMATH DE number 1211664
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