Pure extensions of locally compact abelian groups

TL;DR

This paper investigates pure extensions and pure injective properties in the category of locally compact abelian groups, providing conditions for when certain extension groups coincide and characterizing pure injectives within a specific class of these groups.

Contribution

It establishes conditions under which Pext coincides with the first Ulm subgroup of Ext and characterizes pure injective groups in a class of LCA groups formed by sums of compactly generated and discrete groups.

Findings

Pext(C,A) coincides with the first Ulm subgroup of Ext(C,A) under certain conditions

Characterization of pure injective groups in the class K of LCA groups

Description of groups G in K where all pure extensions split

Abstract

In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every pure extension of G by a group in K splits and obtain a corresponding dual result.