Failure of singular compactness for Hom

TL;DR

Under the assumption of G"odel's constructibility axiom, the paper constructs a singular cardinality abelian group that defies the expected singular compactness property in duality theory, providing a counterexample.

Contribution

It presents the first known counterexample to the singular compactness of the Hom functor in abelian groups under $V=L$.

Findings

Constructed a $oldsymbol{ ext{chi}}$-free abelian group of singular cardinality.

Showed that all smaller subgroups have nontrivial Hom to $oldsymbol{ ext{Z}}$, but the whole group does not.

Provided a consistent counterexample to a long-standing conjecture in duality theory.

Abstract

Assuming G\"odel's axiom of constructibility $V=L$, we construct a $\chi$-free abelian group $G$ of singular cardinality for some suitable cardinal $\chi$ 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 $Hom(-,\mathbb{Z})$.