Homological algebra and set theory

Abstract

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext Λ 1 {\operatorname {Ext}}_\Lambda ^1 for rings Λ \Lambda of global dimension 1. Using Martin’s Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming V = L {\text {V}} = {\text {L}} ) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.