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Cellular covers of cotorsion-free modules

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Göbel R., Rodríguez J., Strüngmann L.

Fundamenta Mathematicae

2012

tom 217

nr 3

211-231

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.

On localizations of torsion abelian groups

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Rodríguez J., Scherer J., Strüngmann L.

Fundamenta Mathematicae

2004

tom 183

nr 2

123-138

As is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by $|T|^{ℵ₀}$ whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, we completely characterize the relationship between localizations of abelian p-groups and their basic subgroups.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Rodríguez J.

2 Strüngmann L.

1 Göbel R.

1 Scherer J.

1 2012

1 2004

Wyniki wyszukiwania

Cellular covers of cotorsion-free modules

On localizations of torsion abelian groups