Cellular covers of cotorsion-free modules

DOI10.4064/FM217-3-2zbMATH Open1281.20066arXiv0906.4183OpenAlexW2963387451MaRDI QIDQ2891274

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Publication date: 14 June 2012

Published in: Fundamenta Mathematicae (Search for Journal in Brave)

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. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of $R$-modules $pi:GoH$ is called a {it cellular cover} over $H$ if $pi$ induces an isomorphism $p{i}_{*}:Ho{m}_R(G,G)congHo{m}_R(G,H),$ where $p{i}_{*}(phi)=piphi$ for each $phiinHo{m}_R(G,G)$ (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 $GoH$ 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 $H$ that satisfies some rigid conditions admits arbitrarily large cellular covers $GoH$. This improves results by Fuchs-G"obel cite{FG} and Farjoun-G"obel-Segev-Shelah cite{FGSS07}.

Full work available at URL: https://arxiv.org/abs/0906.4183

zbMATH Keywords

endomorphismsAbelian groupshomomorphismscovers of modulesco-localizationscotorsion-free modulescellular coverscellular covering mapscellular covering modulescellularizations of Eilenberg-Mac Lane spacesShelah black box

Mathematics Subject Classification ID

Associative rings of functions, subdirect products, sheaves of rings (16S60) Automorphisms and endomorphisms (16W20) Localization and completion in homotopy theory (55P60) Torsion-free groups, infinite rank (20K20) Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups (20K30) Other special types of modules and ideals in commutative rings (13C13)

Related Items (4)

Cellular covers of ℵ₁-free abelian groups ⋮ Rigid ℵ1-Free Abelian Groups with Prescribed Factors and Their Role in the Theory of Cellular Covers ⋮ Comparing localizations across adjunctions ⋮ On localizations of quasi-simple groups with given countable center

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