Failure of singular compactness for Hom

TL;DR

Assuming $V=L$, the paper constructs a $\chi$-free abelian group $G$ of singular size $\lambda$ such that $\mathrm{Hom}(G',\mathbb{Z})\neq 0$ for all nontrivial subgroups $G'\subseteq G$ with $|G'|<\lambda$, while $\mathrm{Hom}(G,\mathbb{Z})=0$, providing a consistent counterexample to singular compactness for the duality $\mathrm{Hom}(-,\mathbb{Z})$. The method uses a four-stage tree-based, diamond-guided construction under $V=L$ and specific chain hypotheses on $\langle \lambda_i:i<\kappa\rangle$, to tightly control duals at singular cardinals. It connects to the Whitehead problem via Ext and discusses how singular-compactness phenomena differ for Ext in this framework. The results illustrate nonreflecting stationary sets and diamond-guided extensions as mechanisms to force nontrivial duals on small subgroups while forcing a vanishing dual at the limit, contributing to the landscape of compactness/incompactness in set-theoretic algebra.

Abstract

Assuming Gödel's axiom of constructibility $V=L$, we construct a $χ$-free abelian group $G$ of singular cardinality for some suitable cardinal $χ$ which is regular and uncountable, equipped with the property that for every nontrivial subgroup $G' \subseteq G$ of smaller cardinality, $Hom(G',\mathbb{Z}) \neq 0$, while $Hom(G,\mathbb{Z}) = 0$. This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor $Hom(-,\mathbb{Z})$.

Theorems & Definitions (20)

• Theorem 1
• Definition 1.2
• Remark 1.3
• proof
• Definition 1.6
• Definition 1.7
• Lemma 1.9
• proof
• Definition 2.2
• Definition 2.5