Closure properties of $\underset{⟶}{\lim }\mathcal{C}$

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. Our first goal here is to study the closure properties of $\mathcal{L}$ in the general case when $\mathcal{C}\subseteq \mathrm{\text{Mod–}}R$ is arbitrary. Then we concentrate on two important particular cases, when $\mathcal{C}=\mathrm{\text{add}}M$ and $\mathcal{C}=\mathrm{\text{Add}}M$, for an arbitrary module M.

In the first case, we prove that $\underrightarrow{\lim }\mathrm{\text{add}}M=\{N\in \mathrm{\text{Mod–}}R|\exists F\in {\mathcal{F}}_S:N\cong F{\otimes }_{S}M\}$ where $S=\mathrm{\text{End}}M$, and ${\mathcal{F}}_S$ is the class of all flat right S-modules. In the second case, $\underrightarrow{\lim }\mathrm{\text{Add}}M=\{\mathfrak{F}{\odot }_{\mathfrak{S}}M|\mathfrak{F}\in {\mathcal{F}}_{\mathfrak{S}}\}$ 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 $\mathfrak{F}{\odot }_{\mathfrak{S}}M$ is the contratensor product of the right $\mathfrak{S}$-contramodule $\mathfrak{F}$ with the discrete left $\mathfrak{S}$-module M.

For various classes of modules $\mathcal{D}$, we show that if $M\in \mathcal{D}$ then $\underrightarrow{\lim }\mathrm{\text{add}}M=\underrightarrow{\lim }\mathrm{\text{Add}}M$ (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 $\underrightarrow{\lim }\mathrm{\text{Add}}M=\widetilde{\text{Add}M}$ where $\widetilde{\mathrm{\text{Add}}M}$ 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 (e.g., when M is a tilting module over a Dedekind domain), but it is negative in general.

Introduction

Direct limits provide one of the key constructions for forming large modules from families of small ones. In the case when the small modules are taken from a class of finitely presented modules, classic theorems of Lenzing et al. make it possible to describe completely the resulting class of large modules. However, if we start with a class, $\mathcal{C}$, consisting of arbitrary modules, then the structure of the class $\mathcal{L}=\underrightarrow{\lim }\mathcal{C}$ is much less clear: for example, $\mathcal{L}$ need not be closed under direct limits.

Our first goal here is to investigate which closure properties of the class $\mathcal{C}$ carry over to $\mathcal{L}$. Then we will characterize the class $\mathcal{L}$ for two particular instances: when $\mathcal{C}$ is the class of all, and all finite, direct sums of copies of a single (infinitely generated) module M. The first characterization relies on the well-known equivalence between the category $\mathrm{\text{add}}M$ of all direct summands of finite direct sums of copies of M and ${(\mathrm{\text{mod–}}S)}_{\mathrm{proj}}$, the category of all finitely generated projective right S-modules, where S is the endomorphism ring of M. The second characterization is based on a recently discovered equivalence [39] between the category $\mathrm{\text{Add}}M$ of all direct summands of arbitrary direct sums of copies of M and ${(\mathrm{Contra}\mathrm{\text{–}}\mathfrak{S})}_{\mathrm{proj}}$, the category of projective right contramodules over $\mathfrak{S}$, the endomorphism ring of M endowed with the finite topology.

We will prove that in many cases, e.g., when $\mathcal{C}$ consists of small or pure projective modules, particular injective or Prüfer modules, the classes $\underrightarrow{\lim }\mathrm{\text{add}}\mathcal{C}$ and $\underrightarrow{\lim }\mathrm{\text{Add}}\mathcal{C}$ coincide. However, whether this is true in general, remains an open problem.

We will also characterize the class $\underrightarrow{\lim }\mathrm{\text{Add}}P$ when P is a projective module in terms of its trace ideal. Another problem addressed here is the question of whether $\underrightarrow{\lim }\mathrm{\text{Add}}M=\widetilde{\text{Add}M}$ where $\widetilde{\text{Add}M}$ denotes the class of all pure epimorphic images of direct sums of copies of a module M. We will give a positive answer in several particular cases, e.g., when M is an (infinitely generated) tilting module over a Dedekind domain. However, we will show that the answer is negative in general, even if the class $\mathrm{\text{Add}}M$ is closed under direct limits: we will construct an example of a countably generated flat module M such that $\mathrm{\text{Add}}M=\underrightarrow{\lim }\mathrm{\text{Add}}M⊊\widetilde{\text{Add}M}$.

Let us say a few more words about the applications of contramodules to the study of the class $\underrightarrow{\lim }\mathrm{\text{Add}}M$ and to the $\underrightarrow{\lim }\mathrm{\text{add}}M$ versus $\underrightarrow{\lim }\mathrm{\text{Add}}M$ question. The notions of a flat module and a flat contramodule [38], [37], [7] play a key role in the descriptions of the classes $\underrightarrow{\lim }\mathrm{\text{add}}M$ and $\underrightarrow{\lim }\mathrm{\text{Add}}M$, respectively. The classical Govorov–Lazard theorem [22], [27] describes the flat modules as the direct limits of projective modules, or even more precisely, as the direct limits of finitely generated free modules. The analogous assertion is not true for contramodules, generally speaking, and we present a counterexample.

Still it is not known whether every direct limit of projective contramodules is a direct limit of finitely generated projective (or finitely generated free) contramodules. When this holds for the topological endomorphism ring $\mathfrak{S}$ of a module M, it follows that $\underrightarrow{\lim }\mathrm{\text{add}}M=\underrightarrow{\lim }\mathrm{\text{Add}}M$. In particular, this observation is applicable to some Prüfer-type modules M, or more generally, to modules M whose topological endomorphism ring $\mathfrak{S}$ admits a dense left noetherian subring S such that the induced topology on S is a left Gabriel topology with a countable base of ideals generated by central elements. It is important here that ideals generated by central elements in noetherian rings have the Artin–Rees property, which allows to prove that the underlying S-modules of flat $\mathfrak{S}$-contramodules are flat.

We also prove that, for any module M, both the classes $\underrightarrow{\lim }\mathrm{\text{add}}M$ and $\underrightarrow{\lim }\mathrm{\text{Add}}M$ are deconstructible (i.e. every module from the respective class is filtered by modules of bounded size from the same class). The assumption that all flat $\mathfrak{S}$-contramodules are direct limits of projective ones, for $\mathfrak{S}=\mathrm{\text{End}}M$, allows to improve the cardinality estimate for deconstructibility of $\underrightarrow{\lim }\mathrm{\text{Add}}M$. In order to obtain the improved cardinality estimate, we study homological properties of the class of all flat contramodules and its natural subclass of so-called 1-strictly flat contramodules. Under a mild assumption (that all flat contramodules are 1-strictly flat), we show that the class of all flat $\mathfrak{S}$-contramodules is closed under (transfinite) extensions and kernels of epimorphisms, and that it is quasi-deconstructible modulo the class of all so-called contratensor-negligible contramodules.