Closed exact categories of modules over generalized adic rings. Part 1: The bounded case

We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining quasi-abelian categories of complete linearly topologized k-modules, which are also closed symmetric monoidal for a suitable choice of tensor product and internal Hom, and have enough projectives or injectives. For k a suitably generalized adic ring, we describe here a few examples of such categories consisting of bounded modules.