Torsion pairs in categories of modules over a preadditive category

Abstract: It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.