Cellular covers of cotorsion-free modules

Volume 217 / 2012

Rüdiger Göbel, José L. Rodríguez, Lutz Strüngmann Fundamenta Mathematicae 217 (2012), 211-231 MSC: Primary 20K20, 20K30, 55P60; Secondary 16S60, 16W20. DOI: 10.4064/fm217-3-2

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.

Authors

  • Rüdiger GöbelDepartment of Mathematics
    University of Duisburg-Essen
    Campus Essen, 45117 Essen, Germany
    e-mail
  • José L. RodríguezÁrea de Geometría y Topología
    Facultad de Ciencias Experimentales
    University of Almería
    La cañada de San Urbano
    04120 Almería, Spain
    e-mail
  • Lutz StrüngmannDepartment of Mathematics
    University of Duisburg-Essen
    Campus Essen, 45117 Essen, Germany
    e-mail

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