The classification problem for extensions of torsion-free abelian groups, I

TL;DR

This paper analyzes the complexity of classifying extensions of certain countable abelian groups, using Borel complexity theory and homological algebra, to determine when such extensions are classifiable by countable structures.

Contribution

It provides a detailed complexity classification of extensions of torsion-free abelian groups by various types of groups, expanding previous results and introducing Borel-definable homological algebra methods.

Findings

Classifying extensions C by A has a specific Borel complexity depending on C and A.

Extensions split if and only if they split on all finite-rank subgroups, for certain groups.

The paper introduces Borel-definable homological algebra as a framework for these classifications.

Abstract

Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free $R$-module for some subring $R\subseteq \mathbb{Q}$. Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation $\mathcal{R}_{\mathbf{Ext}\left( C,A\right) }$ of isomorphism of extensions of $C$ by $A$. This complements a previous result by the same author, settling the case when $C$ is torsion and $A$ is arbitrary. We establish the main result within the framework of Borel-definable homological algebra, recently introduced in collaboration with Bergfalk and Panagiotopoulos. As a consequence of our main results, we will obtain that if $C$ is torsion-free and $A$ is either a free $R$-module or a torsion group with bounded components, then an extension of $C$ by $A$ splits if and only if it splits on all finite-rank subgroups of $C$. This is a purely algebraic statements obtained with methods from Borel-definable homological algebra.