Tame and Galois extensions with respect to Hopf orders
Let \(K\) be the field of fractions of a Dedekind domain \(R\), let \(G\) be a finite group, let \(\mathfrak G\) be a Hopf order in \(KG\), and let \(S\) be an order admitting \(\mathfrak G\) in a Galois algebra \(L\) over \(K\) with group \(G\). We consider when \(S\) is a Galois or tame \(\mathfrak G\)-extension of \(R\) in the sense of L. N. Childs. We give a criterion for \(S\) to be a Galois \(\mathfrak G\)-extension, involving the inverse differents of various subextensions. We then reduce the study of tame and Galois extensions to a special case, showing that if \(S\) is an integrally closed extension of \(R\), and the corresponding extension of residue class fields is separable, then we may take \(S\) to be a complete discrete valuation ring and \(G\) to be a \(p\)-group. In this case, we give a criterion on the ramification numbers of \(L/K\) for \(S\) to be a Galois \(\mathfrak G\)-extension, and show that if \(S\) is a tame \(\mathfrak G\)-extension then \(S\) is a Galois \(\mathfrak G\)-extension, unless \(G\) is cyclic and \(\mathfrak G\) is the maximal order in \(KG\).
scientific article; zbMATH DE number 856578
scientific article