Hopf orders and a generalization of a theorem of L. R. McCulloh
type: Item
instance of: scholarly article
title: Hopf orders and a generalization of a theorem of L. R. McCulloh
zbMATH Open document ID: 0838.16009
author: Nigel P. Byott
published in: Journal of Algebra
publication date: 1996-04-17T00:00:00Z
review text: Let \(K\) be an algebraic number field with ring of integers \(R\) and \(N\) a tame finite normal extension of \(K\) with Galois group \(G\). Assume that \(G\) is the additive group of a finite field \(F\). The multiplicative group \(C=F\setminus\{0\}\) of \(F\) operates on \(G\), and let \(I\) be the Stickelberger ideal in \(\mathbb{Z} C\). The integral closure \(S\) of \(R\) in \(N\) represents an element \([S]\) of the locally free class group \(\text{Cl}(RG)\), and a theorem of McCulloh says that these elements form a subgroup of \(\text{Cl}(RG)\) which is explicitly given in terms of \(I\). The author generalizes this result to orders \(X\) which he calls ``semilocal principal homogeneous spaces''. He observes that \(N\) is tame over \(K\) if and only if at all primes \(P\ni\text{char }F\) of \(K\), the ring \(S_P\) is a principal homogeneous space over the dual Hopf order \(\text{Map}(G,R)\). Therefore, he starts with any Hopf order \(A\) in \(KG\) and considers \(R\)-orders \(X\) such that the \(X_P\) are principal homogeneous spaces over the dual Hopf order \(B\) of \(A\). The principal homogeneous spaces over \(B\) form an abelian group which coincides with the flat cohomology group \(H^1(\text{Spec }R,\text{Spec }B)\). In analogy with \textit{M. J. Taylor}'s class invariant homomorphism [Ill. J. Math. 32, No. 3, 428-452 (1988; Zbl 0631.14033)], the author defines a class \(\psi(X)\in\text{Cl}(A)\). Assuming that \(A\subset KG\) is invariant under \(C\subset\text{Aut }KG\), he proves that \(\text{Im }\psi=I\cdot\text{Cl}_0 (A)\), where \(\text{Cl}_0(A)\) denotes the kernel of the natural map \(\text{Cl }A \to \text{Cl }R\) induced by the augmentation of \(A\).
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 823511
zbMATH Keywords: semilocal principal homogeneous spaces
zbMATH Keywords: algebraic number fields
zbMATH Keywords: tame finite normal extensions
zbMATH Keywords: Galois groups
zbMATH Keywords: Stickelberger ideals
zbMATH Keywords: locally free class groups
zbMATH Keywords: orders
zbMATH Keywords: dual Hopf orders
zbMATH Keywords: flat cohomology groups
Wikidata QID: Q57918066
MaRDI profile type: Publication
full work available at URL: jabr.1995.1305
OpenAlex ID: W2032705245
DOI: 10.1006/JABR.1995.1305
Recommended article: Taming Wild Extensions with Hopf Algebras
Recommended article: Q4012330
Recommended article: Realisable classes, Stickelberger subgroup and its behaviour under change of the base field
Recommended article: Résolvandes et espaces homogènes principaux de schémas en groupe
Recommended article: Tame Kummer extensions and Stickelberger conditions
Recommended article: Relative Galois module structure of octahedral extensions
Recommended article: Tame realisable classes over Hopf orders
Recommended article: Q4495793
Recommended article: Realizable classes by non-abelian metacyclic extensions and Stickelberger elements
description: scientific article; zbMATH DE number 823511