Dualities of locally compact modules over the rationals.
type: Item
instance of: scholarly article
title: Dualities of locally compact modules over the rationals.
zbMATH Open document ID: 1050.13013
DOI: 10.1016/S0021-8693(02)00115-1
author: Dikran Dikranjan
author: Chiara Milan
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: For a (discrete) locally compact commutative ring \(R\) (with unity) denote by \(\mathcal L_{R}\) the category of locally compact topological unitary \(R\)-modules. A contravariant functor \(\sharp:\mathcal L_{R}\to\mathcal L_{R}\) with the property that \(\sharp\circ\sharp\) is naturally equivalent to the identity of \(\mathcal L_{R}\), and, for every morphism \(f\in\mathcal L_{R}\) and every \(r\in R\), \(\sharp(rf)=r\sharp(f)\) has been called a duality by Prodanov. A classical example is the Pontryagin duality. For any such duality and \(X\in\mathcal L_{R}\), the module \(\sharp(X)\) is algebraically isomorphic to CHom\(_R(X,\sharp(R))\) -- the module of continuous \(R\)-module homomorphisms with the compact-open topology. A duality \(\sharp\) is called continuous, if this is also a topological isomorphism, for every \(X\). The authors prove that, when \(R=\mathbb Q\) is the (discrete) field of the rationals, then all the dualities of \(\mathcal L_{R}\) are continuous, however this property does not hold any more when \(R\) represents the discrete fields of the reals or the complex numbers. More generally described are finitely closed subcategories \(\mathcal L\) of \(\mathcal L_{\mathbb Q}\) such that all dualities of \(\mathcal L\) are continuous. In fact, all the dualities of such a category \(\mathcal L\) are naturally equivalent to the Pontryagin duality; this also holds for the cases \(R=\mathbb R\) or \(R=\mathbb C\). Here is a sample result from the paper (\(K\) denotes the (compact) Pontryagin dual of \(\mathbb Q\)): The structure of modules \(X\in\mathcal L_{\mathbb Q}\) is as follows: There are integers \(n\), \(n_p\in \mathbb N\) (\(p\in P\) is a prime) and cardinal numbers \(\alpha\) and \(\beta\) such that \(X\cong\mathbb R^n\times K^\alpha\times\mathbb Q^{(\beta)}\times X_0\), where \(X_0\) is a closed submodule of \(X\) with a compact open essential subgroup isomorphic to \(\prod_{p\in P} \mathbb{Z}_p^{n_p}\). With this identification, \(K^\alpha…
Mathematics Subject Classification ID: 13J20
Mathematics Subject Classification ID: 18A99
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 1868039
zbMATH Keywords: topological module
zbMATH Keywords: Pontryagin duality
zbMATH Keywords: continuous duality
zbMATH Keywords: discontinuous duality
zbMATH Keywords: continuous morphism
zbMATH Keywords: functorial involution
zbMATH Keywords: locally compact commutative ring
zbMATH Keywords: category of locally compact topological unitary modules
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1868039