Dualities of locally compact modules over the rationals.
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, \(K^\alpha\) is the largest compact \(\mathbb Q\)-submodule of \(X\) and the connected component of \(X\) is \(\mathbb R^n\times K^\alpha\).
category of locally compact topological unitary modules
scientific article; zbMATH DE number 1868039
scientific article