In this talk, I will present an overview of applications of tools from descriptive set theory to commutative and homological algebra, including joint work with Bergfalk-Panagiotopoulos and with Bergfalk-Moraschini-Sarti. In particular, I will explain how the homological invariant Ext for countable abelian groups, parametrizing extensions, can be refined with additional topological information that turns it into a group with a Polish cover and endows its subgroups with a notion of Borel complexity and Borel rank. Furthermore, this invariant admits, for every countable ordinal s, a smallest subgroup of Borel rank 1+s+1. For s=0, this was characterized by Eilenberg and Mac Lane in 1942 as the subgroup parametrizing extensions that split on all the finite subgroups. For s=1, this is shown in the torsion-free case to be the subgroup parametrizing extensions that split on all the finite-rank subgroups.