Mathematics – Group Theory
Scientific paper
2009-06-23
Mathematics
Group Theory
18 pages. Revised version with some updates and corrections. Introduction improved
Scientific paper
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. 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 {\it cellular cover} over $H$ if $\pi$ induces an isomorphism $\pi_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $\pi_*(\phi)= \pi \phi$ for each $\phi \in \Hom_R(G,G)$ (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 $G\to H$ 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 \cite{BD}. 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-G\"obel \cite{FG} and Farjoun-G\"obel-Segev-Shelah \cite{FGSS07}.
Göbel Rüdiger
Rodriguez Jose L.
Strüngmann Lutz
No associations
LandOfFree
Cellular covers of cotorsion-free modules does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Cellular covers of cotorsion-free modules, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cellular covers of cotorsion-free modules will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-708139