Tame realisable classes over Hopf orders
type: Item
instance of: scholarly article
title: Tame realisable classes over Hopf orders
zbMATH Open document ID: 0911.16010
published in: Journal of Algebra
publication date: 1999-02-17T00:00:00Z
review text: 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 \(LA=LG\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1211664
zbMATH Keywords: tame extensions
zbMATH Keywords: class groups
zbMATH Keywords: Hopf orders
zbMATH Keywords: realizable classes
zbMATH Keywords: Galois module structure
zbMATH Keywords: cocommutative separable Hopf algebras
author: Nigel P. Byott
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
full work available at URL: jabr.1997.7247
OpenAlex ID: W2094099506
Wikidata QID: Q57918067
cites work: Toroidal Block Decompositions for Hopf Orders in Group Algebras
cites work: Cleft Extensions of Hopf Algebras, II
cites work: Hopf orders and a generalization of a theorem of L. R. McCulloh
cites work: Tame and Galois extensions with respect to Hopf orders
cites work: Galois structure and elliptic curves
cites work: Taming Wild Extensions with Hopf Algebras
cites work: Tameness and Local Normal Bases for Objects of Finite Hopf Algebras
cites work: Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring
cites work: Hopf Galois theory for separable field extensions
cites work: Galois theories and normal bases
cites work: Galois module structure of elementary abelian extensions
cites work: Galois module structure of abelian extensions.
cites work: When Hopf algebras are Frobenius algebras
cites work: Q5758342
cites work: Group schemes of prime order
cites work: Mordell-Weil groups and the Galois module structure of rings of integers
cites work: Résolvandes et espaces homogènes principaux de schémas en groupe
cites work: \(R\)-Hopf algebra orders in \(KC_{p^ 2}\)
cites work: The valuative condition and R-Hopf algebra orders in KC p3
cites work: Q3884236
DOI: 10.1006/JABR.1997.7247
description: scientific article; zbMATH DE number 1211664