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29 March 2024
 
  » arxiv » 0906.4183

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Cellular covers of cotorsion-free modules
Rüdiger Göbel ; José L. Rodríguez ; Lutz Strüngmann ;
Date 23 Jun 2009
AbstractIn 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.
Recall that a homomorphism of $R$-modules $pi: G o H$ is called a {it cellular cover} over $H$ if $pi$ induces an isomorphism $pi_*: Hom_R(G,G)cong Hom_R(G,H),$ where $pi_*(varphi)= pi varphi$ for each $varphi in Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free $R$-module of rank $kappa<Cont$ is realizable as the kernel of some cellular cover $G o H$ where the rank of $G$ is $3kappa +1$ (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 $H$ satisfying some rigid conditions admits arbitrarily large cellular covers $G o H$. This improves results by Fuchs--G"obel cite{FG} and Farjoun--G"obel--Segev--Shelah cite{FGSS07}.
Source arXiv, 0906.4183
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