Dualities over compact rings
type: Item
instance of: scholarly article
title: Dualities over compact rings
zbMATH Open document ID: 0674.16027
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108115
review text: Let R be a compact ring with unity, M a right local compact R-module and \(\Gamma\) (M) the Pontrjagin dual of M. For every \(r\in R\) and every \(\xi\in \Gamma (M)\) the character \(r\xi\) is defined by the relation \((r\xi)(m)=\xi (mr)\) for all \(m\in M\). It is proved that \(\Gamma\) (M) is a left topological R-module. So one can define a functor \(\Gamma_ R: {\mathcal L}\)-R\(\to R\)-\({\mathcal L}\) between the category of right locally compact R-modules and the category of left locally compact R-modules, which induces a duality. The functor \({}_ R\Gamma\) is defined in the analogous way. Theorem. If R is a compact ring with unity and \(\Delta_ 1: {\mathcal L}\)-R\(\to R\)-\({\mathcal L}\), \(\Delta_ 2: R\)-\({\mathcal L}\to {\mathcal L}\)-R is a duality then there exists two natural isomorphisms \(\Delta_ 1\cong \Gamma_ R\), \(\Delta_ 2\cong_ R\Gamma\). This result was proved earlier by L. Stoyanov in the case when R is compact and commutative.
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 4103306
zbMATH Keywords: Pontrjagin dual
zbMATH Keywords: category of right locally compact R-modules
zbMATH Keywords: compact ring
zbMATH Keywords: duality
zbMATH Keywords: natural isomorphisms
author: Enrico Gregorio
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4103306