Cellular covers of cotorsion-free modules

Rüdiger Göbel, José L. Rodríguez, and Lutz Strüngmann. "Cellular covers of cotorsion-free modules." Fundamenta Mathematicae 217.3 (2012): 211-231. <http://eudml.org/doc/286084>.

@article{RüdigerGöbel2012,
abstract = {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 π: G → H is called a cellular cover over H if π induces an isomorphism $π⁎: Hom_\{R\}(G,G) ≅ Hom_\{R\}(G,H)$, where π⁎(φ) = πφ for each $φ ∈ Hom_\{R\}(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free R-module of rank $κ < 2^\{ℵ₀\}$ is realizable as the kernel of some cellular cover G → H where the rank of G is 3κ + 1 (or 3, if κ = 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 → H. This improves results by Fuchs-Göbel and Farjoun-Göbel-Segev-Shelah.},
author = {Rüdiger Göbel, José L. Rodríguez, Lutz Strüngmann},
journal = {Fundamenta Mathematicae},
keywords = {cellular covers; co-localizations; cotorsion-free modules; Abelian groups; Shelah black box; cellularizations of Eilenberg-Mac Lane spaces; homomorphisms; endomorphisms; covers of modules; cellular covering modules; cellular covering maps},
language = {eng},
number = {3},
pages = {211-231},
title = {Cellular covers of cotorsion-free modules},
url = {http://eudml.org/doc/286084},
volume = {217},
year = {2012},
}

TY - JOUR
AU - Rüdiger Göbel
AU - José L. Rodríguez
AU - Lutz Strüngmann
TI - Cellular covers of cotorsion-free modules
JO - Fundamenta Mathematicae
PY - 2012
VL - 217
IS - 3
SP - 211
EP - 231
AB - 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 π: G → H is called a cellular cover over H if π induces an isomorphism $π⁎: Hom_{R}(G,G) ≅ Hom_{R}(G,H)$, where π⁎(φ) = πφ for each $φ ∈ Hom_{R}(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free R-module of rank $κ < 2^{ℵ₀}$ is realizable as the kernel of some cellular cover G → H where the rank of G is 3κ + 1 (or 3, if κ = 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 → H. This improves results by Fuchs-Göbel and Farjoun-Göbel-Segev-Shelah.
LA - eng
KW - cellular covers; co-localizations; cotorsion-free modules; Abelian groups; Shelah black box; cellularizations of Eilenberg-Mac Lane spaces; homomorphisms; endomorphisms; covers of modules; cellular covering modules; cellular covering maps
UR - http://eudml.org/doc/286084
ER -