Universal co-Extensions of torsion abelian groups

TL;DR

This paper explores the properties of Ext-universal objects in abelian categories, providing characterizations for torsion abelian groups and conditions under which categories satisfy certain extension properties.

Contribution

It establishes a characterization of co-Ext-universal objects in torsion abelian groups and links Ext-smallness with the Ab4 condition in Ext-small Ab3 categories.

Findings

Torsion abelian groups with a decomposition into injective and bounded reduced parts are co-Ext-universal.

An Ab3 abelian category is Ab4 if and only if all objects are Ext-universal under Ext-smallness.

Characterization of Ext-universal objects in the category of torsion abelian groups.

Abstract

In [16], a theory of universal extensions in abelian categories is developed; in particular, the notion of Ext-universal object is presented. In the present paper, we show that an Ab3 abelian category which is Ext-small satisfies the Ab4 condition if, and only if, each one of its objects is Ext-universal. We also give a characterization of the co-Ext-universal objects of the category of torsion abelian groups. In particular, we show that such groups are the ones admitting a decomposition $Q\oplus R$, in which $Q$ is injective and $R$ is a reduced group on which each $p$-component is bounded.