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Cellular covers of cotorsion-free modules | Rüdiger Göbel
; José L. Rodríguez
; Lutz Strüngmann
; | Date: |
23 Jun 2009 | Abstract: | 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.
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 | Services: | Forum | Review | PDF | Favorites |
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