2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116622In this paper we complete the characterization of Ext(G,Z) under Godel's axiom of constructibility for any torsion-free abelian group G . In particular, we prove in (V=L) that, for a singular cardinal nu of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence of cardinals (nu_p : p in P) satisfying nu_p <= 2^{nu}, there is a torsion-free abelian group G of size nu such that nu_p equals the p-rank of Ext(G,Z) for every prime p and 2^{nu} is the torsion-free rank of Ext(G,Z).LogicGroup Theorytext