Torsion-free abelian groups of finite rank and fields of finite transcendence degree

Abstract

Let $\operatorname{TFAb}_r$ be the class of torsion-free abelian groups of rank $r$, and let $\operatorname{FD}_r$ be the class of fields of characteristic $0$ and transcendence degree~$r$. We compare these classes using various notions. Considering Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the $\operatorname{TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname{FD}_r$. Thomas and Velickovic showed that for sufficiently large $r$, the classes $\operatorname{FD}_r$ are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of $\operatorname{TFAb}_r$ in $\operatorname{FD}_r$, and of $\operatorname{FD}_r$ in $\operatorname{FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname{TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname{TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma_3$, along with the Vitali equivalence relation on $2^\omega$.