Self-duality and torsion Galois modules in number fields
type: Item
instance of: scholarly article
title: Self-duality and torsion Galois modules in number fields
zbMATH Open document ID: 0542.12005
DOI: 10.1016/0021-8693(84)90210-2
author: Maryse Desrochers
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let K/k be a Galois extension of number fields with Galois group \(\Gamma\). If \({\mathcal O}\) is the ring of integers of K, and \(o\) the one of k, then the structure of \({\mathcal O}\) as \(o\Gamma\)-module is investigated by comparing \({\mathcal O}\) with its dual \(Hom_{o}({\mathcal O},o)={\mathcal O}^+\). The trace map \(tr_{K/k}: K\to k\) induces an \(o\Gamma\)-module isomorphism between \({\mathcal O}^+\) and the codifferent C(K/k) of K/k. The \(o\Gamma\)-module \(C(K/k)/{\mathcal O}=T(K,k),\) which is a torsion module, is described explicitly in \(K^{tor,S}(o\Gamma)\), the Grothendieck group of finitely generated torsion modules with finite homological dimension outside (a suitable) S. In the paper the problem is considered in the wildly ramified case. The tame case has been worked out by \textit{S. Chase} in [J. Algebra 91, 207-257 (1984)].
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3861278
zbMATH Keywords: Galois modules
zbMATH Keywords: ring of integers of number fields
zbMATH Keywords: dual module
zbMATH Keywords: codifferent
zbMATH Keywords: torsion module
zbMATH Keywords: Grothendieck group
zbMATH Keywords: finite homological dimension
zbMATH Keywords: wildly ramified case
reviewed by: Jan Van Geel
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90210-2
OpenAlex ID: W2077507601
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description: scientific article; zbMATH DE number 3861278