Homological algebra with locally compact abelian groups

In this article we study locally compact abelian groups using the language of derived categories. We define a derived Hom-functor on the bounded derived category of LCA groups with values in the derived category of Hausdorff topological abelian groups. We introduce a smallness condition for LCA groups and show that the category of such groups has a natural tensor product and internal Hom. Derived versions of these yield closed tensor triangulated categories which may be of arithmetical interest.