$\aleph_k$-free cogenerators

Abstract

We prove in ZFC that an abelian group C is cotorsion if and only if Ext(F, C) = 0 for every ℵ k -free group F , and discuss some consequences and related results. This short note includes a condensed overview of the λ-Black Box for ℵ k -free constructions in ZFC. Contents 1. Introduction 1 Acknowledgement 3 2. A characterization of cotorsion groups 3 3. The λ-Black Box 4 3.1. Λ and Λ * 4 3.2. The modules 5 3.3. The prediction 9 4. The proof of Theorem 2 9 5. Final Remark 12 References 13