Closure properties of $\varinjlim\mathcal C$

Abstract

Let $\mathcal C$ be a class of modules and $\mathcal L = \varinjlim \mathcal C$ the class of all direct limits of modules from $\mathcal C$. The class $\mathcal L$ is well understood when $\mathcal C$ consists of finitely presented modules: $\mathcal L$ then enjoys various closure properties. We study the closure properties of $\mathcal L$ in the general case when $\mathcal C \subseteq \mathrm{Mod-}R$ is arbitrary. Then we concentrate on two important particular cases, when $\mathcal C = \operatorname{add} M$ and $\mathcal C = \operatorname{Add} M$, for an arbitrary module $M$. In the first case, we prove that $\varinjlim \operatorname{add} M = \{ N \in \mathrm{Mod-} R \mid \exists F \in \mathcal F_S: N \cong F \otimes_S M \}$ where $S = \operatorname{End} M$, and $\mathcal F_S$ is the class of all flat right $S$-modules. In the second case, $\varinjlim \operatorname{Add} M = \{ \mathfrak F \odot _{\mathfrak S} M \mid \mathfrak F \in \mathcal F_{\mathfrak S} \}$ where $\mathfrak S$ is the endomorphism ring of $M$ endowed with the finite topology, $\mathcal F_{\mathfrak S}$ is the class of all right $\mathfrak S$-contramodules that are direct limits of direct systems of projective right $\mathfrak S$-contramodules, and $\odot_{\mathfrak S}$ denotes the contratensor product. For various classes of modules $\mathcal D$, we show that if $M \in \mathcal D$ then $\varinjlim \operatorname{add} M = \varinjlim \operatorname{Add} M$ (e.g., when $\mathcal D$ consists of pure projective modules), but the equality for an arbitrary module $M$ remains open. Finally, we deal with the question of whether $\varinjlim \operatorname{Add} M = \widetilde{\operatorname{Add} M}$ where $\widetilde{\operatorname{Add} M}$ 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, but it is negative in general.