Submission date: 11 April 2022

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 R_{Ext(C,A)} 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.

Mathematics Subject Classification: 20K35, 54H05 (Primary) 20K40, 20K45 (Secondary)

Keywords and phrases:

Full text arXiv 2204.05431: pdf, ps.