Closure properties of $\underrightarrow{\lim }\mathcal{C}$

Abstract

Let $\mathcal{C}$ be a class of modules and $\mathcal{L}=\underrightarrow{\lim }\mathcal{C}$ 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 $\mathcal{C}=\mathrm{add}M$ and $\mathcal{C}=\mathrm{Add}M$, for an arbitrary module $M$. In the first case, we prove that $\underrightarrow{\lim }\mathrm{add}M=\{N\in \mathrm{Mod}-R\mid \exists F\in {\mathcal{F}}_S:N\cong F{\otimes }_{S}M\}$ where $S=\mathrm{End}M$, and ${\mathcal{F}}_S$ is the class of all flat right $S$-modules. In the second case, $\underrightarrow{\lim }\mathrm{Add}M=\{\mathfrak{F}{\odot }_{\mathfrak{S}}M\mid \mathfrak{F}\in {\mathcal{F}}_{\mathfrak{S}}\}$ 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 $\underrightarrow{\lim }\mathrm{add}M=\underrightarrow{\lim }\mathrm{Add}M$ (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 $\underrightarrow{\lim }\mathrm{Add}M=\widetilde{\mathrm{Add}M}$ where $\widetilde{\mathrm{Add}M}$ 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.

59 pages; v.2: a new author joined, major improvements and additions in Sections 5 and 6, new Section 8 inserted, related changes in the final section; v.3: small improvements, misprints corrected

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