Self-linearly compact rings and dualities

The study of dualities between categories of modules generalizing Morita duality cannot dispense with the use of topologies (see several references). In this paper we try to give a new representation of dualities between a category of abstract modules and one consisting of topological modules. This method works by endowing the representing bimodule with a ``cotopology'' and by considering ``bounded'' morphisms: let $\mathcal G_S$ be a subcategory of $\rmod S$ and $\RB$ be a category of linearly topologized left modules over $R$ and assume a duality between $\mathcal G_S$ and $\RB$ \[ D_1\mathbin{:}\mathcal G_S\doublearrow\RB\mathbin{:}D_2 \] is given; under suitable conditions on the two categories, there exists a discrete bimodule $_SK_R$ such that $D_1$ is isomorphic to the functor $\Hom_S({-},K)$ (with the topology of pointwise convergence) and $D_2$ is isomorphic to a subfunctor of the functor $\Chom_R({-},K)$. The conditions we impose on $\mathcal G_S$ define on $S$ a right linear topology, which in turn induces on $_RK$ a further structure: a linear topology on $S$ is the same thing as a filter of right ideals and the duality induces a cofilter of submodules of $_RK$; we can look at these submodules as ``bounded'' objects in $_RK$ and the subfunctor we just mentioned is the set of ``bounded'' continuous morphisms, where a morphism to $_RK$ is bounded if its image belongs to the cofilter. We call the pair $_RK$ and cofilter a \emph{cotopological module}. The duality imposes a further condition on the topology on $S$; we call it \emph{self linear compactness}: it is a more general property than linear compactness as introduced by Leptin~\cite{leptin} (see~\ref{exa:selflc-not-lc}); we chose this name to recall that it depends on properties of the finitely generated discrete modules over $S$. We show conversely that every self linearly compact topology induces such a duality with a category of modules over a suitable ring.