Closure properties of [formula omitted

Abstract

Let C be a class of modules and L=lim→C the class of all direct limits of modules from C. The class L is well understood when C consists of finitely presented modules: L then enjoys various closure properties. Our first goal here is to study the closure properties of L in the general case when C⊆Mod–R is arbitrary. Then we concentrate on two important particular cases, when C=addM and C=AddM, for an arbitrary module M.In the first case, we prove that lim→addM={N∈Mod–R|∃F∈FS:N≅F⊗SM} where S=EndM, and FS is the class of all flat right S-modules. In the second case, lim→AddM={F⊙SM|F∈FS} where S is the endomorphism ring of M endowed with the finite topology, FS is the class of all right S-contramodules that are direct limits of direct systems of projective right S-contramodules, and F⊙SM is the contratensor product of the right S-contramodule F with the discrete left S-module M.For various classes of modules D, we show that if M∈D then lim→addM=lim→AddM (e.g., when D consists of pure projective modules), but the equality for an arbitrary module M remains open. Finally, we deal with the question of whether lim→AddM=AddM˜ where AddM˜ is the class of all pure epimorphic images of direct sums of copies of a module M. We show that the answer is positive in several particular cases (e.g., when M is a tilting module over a Dedekind domain), but it is negative in general.