Dualities over compact rings
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.
category of right locally compact R-modules
scientific article; zbMATH DE number 4103306