Uniqueness and existence of dualities over compact rings (Q1340391)

Let \((A,\sigma)\) and \((R,\tau)\) be compact topological rings; \({\mathcal L}A_ \sigma\) and \({\mathcal L}R_ \tau\) be the corresponding categories of all right and of all left locally compact topological \((A,\sigma)\)- modules and \((R,\tau)\)-modules with continuous homomorphisms; \(K_ A \in \text{Ob }{\mathcal L}A_ \sigma\) a certain injective cogenerator of finite grade (i.e. it's socle contains only a finite number of summands which are isomorphic to a given simple \(A\)-module); \(M_ A\) be an object of \({\mathcal L}A_ \sigma\). We denote the set of all homomorphisms, equipped with the topology of the uniform convergence on compact submodules of \(M_ A\) by \(\text{Chom}_ A(M_ A, K_ A)\). A new proof of a result obtained in 1983 by Menini and Orsatti is given in \S\S1-4. The theorem asserts that for every duality \(D : {\mathcal L}A_ \sigma \to {\mathcal L}R_ \tau\) there exists an injective cogenerator of finite grade \(K_ A\) such as \((R,\tau) = \text{Chom}_ A(K_ A, K_ A)\) and \(D = \text{Chom}_ A(-,K_ A)\). The converse assertion is proved in \S5, i.e. if \(K_ A\) is an injective cogenerator of finite grade then \((R,\tau) = \text{Chom}_ A(K_ A,K_ A)\) is a topological compact ring and the functor \(D = \text{Chom}_ A(-,K_ A)\), \(D : {\mathcal L}A_ \sigma \to {\mathcal L}R_ \tau\) is a duality.