Between Whitehead groups and uniformization

Márk Poór; Saharon Shelah

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arXiv · 2203.12585

Between Whitehead groups and uniformization

Márk Poór·Saharon Shelah

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Abstract

For a given stationary set $S$ of countable ordinals we prove (in $\mathbf{ZFC}$) that the assertion "every $S$-ladder system has $\aleph_0$-uniformization" is equivalent to "every strongly $\aleph_1$-free abelian group of cardinality $\aleph_1$ with non-freeness invariant $\subseteq S$ is $\aleph_1$-coseparable, i.e. Ext$(G, \oplus_{i=0}^{\infty} \mathbb Z)=0$ (in particular Whitehead, i.e.\ Ext$(G, \mathbb Z)=0$)". This solves problems B3 and B4 from Eklof and Mekler's monograph.

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Márk Poór, Saharon Shelah. 2022-03-23. Between Whitehead groups and uniformization. https://arxiv.org/abs/2203.12585

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