Closure properties of $\varinjlim\mathcal C$

TL;DR

This paper investigates the closure properties of classes of modules formed by direct limits, especially focusing on classes generated by a single module, and explores their algebraic and topological characterizations.

Contribution

It provides new descriptions of direct limit classes for modules generated by a single module, including flat and contramodule cases, and examines their closure properties and open problems.

Findings

Characterization of direct limit classes via tensor and contratensor products.

Equality of certain classes holds for modules in specific classes like pure projectives.

Counterexamples show the general equality does not always hold.

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.