Modules with Cosupport and Injective Functors

Several authors have studied the filtered colimit closure varinjlimBvarinjlimmathcal{B} of a class Bmathcal{B} of finitely presented modules. Lenzing called varinjlimBvarinjlimmathcal{B} the category of modules with support in Bmathcal{B}, and proved that it is equivalent to the category of flat objects in the functor category (B^op,Ab)(mathcal{B}^mathrm{op},mathsf{Ab}). In this paper, we study the category (Mod-R)^B({mathsf{Mod}textnormal{-}R})^{mathcal{B}} of modules with cosupport in Bmathcal{B}. We show that (Mod-R)^B({mathsf{Mod}textnormal{-}R})^{mathcal{B}} is equivalent to the category of injective objects in (B,Ab)(mathcal{B},mathsf{Ab}), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hügel, Enochs, Krause, Rada, and Saorín make it easy to discuss covering and enveloping properties of (Mod-R)^B({mathsf{Mod}textnormal{-}R})^{mathcal{B}}, and furthermore we compare the naturally associated notions of Bmathcal{B}-coherence and Bmathcal{B}-noetherianness. Finally, we prove a number of stability results for varinjlimBvarinjlimmathcal{B} and (Mod-R)^B({mathsf{Mod}textnormal{-}R})^{mathcal{B}}. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.