Extensions in the class of countable torsion-free Abelian groups

Abstract

It is a classical result that for a torsion-free Abelian group A the group $\operatorname {Ext}_{\mathbb {Z}}(A,B)$ is divisible for any Abelian group B. Hence it is of the form for some uniquely determined cardinals r ₀ and r _p . In this paper we clarify when $\operatorname {Ext}_{\mathbb {Z}}(A,B)=0$ and examine the possible values for r ₀ and r _p in case the groups A and B are countable (torsion-free). We also give some methods for constructing torsion-free groups A and B with prescribed cardinals r ₀ and r _p . This is to say that for suitable sequences (r ₀,r _p ∣p∈ℙ) of cardinals we construct torsion-free countable Abelian groups A and B realizing r ₀ and r _p as their invariants of $\operatorname {Ext}_{\mathbb {Z}}(A,B)$.