On localizations of torsion abelian groups

Abstract

Abstract. 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 | ℵ 0 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.1. Introduction. Localization functors have a long history and were extensively studied in many fields of mathematics. However, during the last decade, new applications of these functors in homotopy theory due to Bousfield, Casacuberta, Dror Farjoun and others (see e.g. [2], [5], [11]) have pushed several authors to investigate the effect of homotopical localization functors on homotopy or homology groups. For instance, the effect on the fundamental group can often be described by means of group-theoretical localization functors. Motivated by this relationship, important advances have recently been achieved in the study of group localization functors, especially related to their behavior on certain classes of groups, like for example finite or nilpotent groups ([6] Roughly speaking, a localization functor in the category Grp of all groups is an idempotent functor L : Grp → Grp together with a natural transformation from the identity functor Id into L. We will recall in Section 2 the basic definitions and properties of such objects.

Cellular Covers of Local Groups

“…We believe that the answer to Farjoun's question is negative as well in the category of abelian groups. It is tempting to use the inclusion n Z/p n ֒→ n Z/p n , which is a localization by [21,Example 6.5], but we have not found an adequate cellularization functor to continue.…”

Idempotent functors and localizations in categories of modules and Abelian groups

“…Orthogonality, f -local objects, and f -equivalences have appeared in category theory; then they began to be considered for groups and modules (see [17,18,41,47]). Localization functors have a long history, and the functors were intensively studied in many fields of mathematics.…”

“…Following [47], we partition all idempotent functors to the following four classes. The class I consists of all functors L such that LZ(p ∞ ) = Z(p ∞ ).…”

Localizations of torsion-free abelian groups II

“…[2,4], or [8]. See [10] for localizations of torsion abelian groups. In [4] it was shown that localizations of torsion-free abelian groups exist in abundance.…”