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