Cellular covers of cotorsion-free modules

Rüdiger Göbel ¹   ; José L. Rodríguez ²   ; Lutz Strüngmann ¹

¹ Department of Mathematics University of Duisburg-Essen Campus Essen, 45117 Essen, Germany
² Área de Geometría y Topología Facultad de Ciencias Experimentales University of Almería La cañada de San Urbano 04120 Almería, Spain

Fundamenta Mathematicae, Tome 217 (2012) no. 3, pp. 211-231

Résumé

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.

Détail

DOI : 10.4064/fm217-3-2

Keywords: paper improve recent results dealing cellular covers r modules cellular covers sometimes called colocalizations come context homotopical localization topological spaces related idempotent cotriples idempotent comonads coreflectors category theory recall homomorphism r modules called cellular cover nbsp induces isomorphism def hom mathop hom nolimits * hom cong hom where * varphi varphi each def hom mathop hom nolimits varphi hom where maps acting every cotorsion free r module rank def aln aleph def cont aln kappa cont realizable kernel cellular cover where rank kappa kappa proof based corners classical idea construct torsion free abelian groups prescribed countable endomorphism rings complements results buckner dugas other prove every cotorsion free r module satisfies rigid conditions admits arbitrarily large cellular covers improves results fuchs bel farjoun bel segev shelah