Illinois Journal of Mathematics

On problems by Baer and Kulikov using $V=L$

Lutz Strüngmann

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Let $T$ be a torsion abelian group and $\lambda$ a cardinal. Among all torsion-free abelian groups $H$ of rank less than or equal to $\lambda$ satisfying $\operatorname{Ext}(H,T)=0$ a group $G$ is called $\lambda$-universal for $T$ if it is universal with respect to group-embedding. We show that in Gödel's constructible universe ($V=L$) there always exists a $\lambda$-universal group for $T$ if $T$ has only finitely many non-trivial bounded $p$-components. This answers a question by Kulikov in the affirmative. Moreover, we prove that in $V=L$ for a large class of torsion-free abelian groups $G$ there exists a completely decomposable group $C$ such that $\operatorname{Ext}(G,T^{\prime})=0$ if and only if $\operatorname{Ext}(C,T^{\prime})=0$ for any torsion abelian group $T^{\prime}$. This is related to a question of Baer.

Article information

Illinois J. Math., Volume 46, Number 2 (2002), 477-490.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20K20: Torsion-free groups, infinite rank
Secondary: 20K15: Torsion-free groups, finite rank 20K40: Homological and categorical methods


Strüngmann, Lutz. On problems by Baer and Kulikov using $V=L$. Illinois J. Math. 46 (2002), no. 2, 477--490. doi:10.1215/ijm/1258136204.

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