Cellular covers of cotorsion-free modules
Volume 217 / 2012
Abstract
In this paper we improve recent results dealing with cellular covers of $R$-modules. Cellular covers (sometimes called colocalizations) 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 cellular cover over $H$ if $\pi$ induces an isomorphism $\def\Hom{\mathop{\rm Hom}\nolimits}\pi_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $\pi_*(\varphi)= \pi \varphi$ for each $\def\Hom{\mathop{\rm Hom}\nolimits}\varphi \in \Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free $R$-module of rank $\def\aln{{\aleph_0}}\def\Cont{2^{\aln}}\kappa<\Cont$ 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. 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–Göbel and Farjoun–Göbel–Segev–Shelah.
