Cellular covers of cotorsion-free modules

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$\pi_(\phi)= \pi \phi$foreach$\phi \in \Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free $R$-module of rank $\kappa <\text{Cont}$ 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}.