Cellular covers of cotorsion-free modules

TL;DR

This paper advances the understanding of cellular covers in module theory, showing how cotorsion-free modules can be realized as kernels of such covers and establishing conditions for their existence, with implications for homotopical localization.

Contribution

It provides new constructions of cellular covers for cotorsion-free modules and extends the classes of modules known to admit large cellular covers, improving prior results.

Findings

Cotorsion-free modules of rank less than continuum are kernels of cellular covers with controlled rank.

Every cotorsion-free module satisfying certain conditions admits arbitrarily large cellular covers.

The results build on classical torsion-free group constructions and improve existing theorems.

Abstract

In this paper we improve recent results dealing with cellular covers of $R$-modules. Cellular covers (sometimes called co-localizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of $R$-modules $\pi: G\to H$ is called a {\it cellular cover} over $H$ if $\pi$ induces an isomorphism $\pi_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $\pi_*(\phi)= \pi \phi$ for each $\phi \in \Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free $R$-module of rank $\kappa<\Cont$ is realizable as the kernel of some cellular cover $G\to H$ where the rank of $G$ is $3\kappa +1$ (or 3, if $\kappa=1$). The proof is based on Corner's classical idea of how to construct torsion-free abelian groups with prescribed countable endomorphism rings. This complements results by Buckner--Dugas \cite{BD}. On the other hand, we prove that every cotorsion-free $R$-module $H$ that satisfies some rigid conditions admits arbitrarily large cellular covers $G\to H$. This improves results by Fuchs-G\"obel \cite{FG} and Farjoun-G\"obel-Segev-Shelah \cite{FGSS07}.