A characterization of $\mathop{\rm Ext}(G,\mathbb Z)$ assuming $(V=L)$

Volume 193 / 2007

Saharon Shelah, Lutz Strüngmann Fundamenta Mathematicae 193 (2007), 141-151 MSC: Primary 20K15, 20K20, 20K35, 20K40; Secondary 18E99, 20J05. DOI: 10.4064/fm193-2-3


We complete the characterization of $\mathop{\rm Ext}(G,\mathbb Z)$ for any torsion-free abelian group $G$ assuming Gödel's axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in $(V=L)$ that, for a singular cardinal $\nu$ of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence $( \nu_p : p \in \varPi )$ of cardinals satisfying $\nu_p \leq 2^{\nu}$ (where $\varPi$ is the set of all primes), there is a torsion-free abelian group $G$ of size $\nu$ such that $\nu_p$ equals the $p$-rank of $\mathop{\rm Ext}(G,\mathbb Z)$ for every prime $p$ and $2^{\nu}$ is the torsion-free rank of $\mathop{\rm Ext}(G,\mathbb Z)$.


  • Saharon ShelahDepartment of Mathematics
    The Hebrew University of Jerusalem
    Jerusalem 91904, Israel
    Rutgers University
    New Brunswick, NJ 08903, U.S.A.
  • Lutz StrüngmannDepartment of Mathematics
    University of Duisburg-Essen
    45117 Essen, Germany
    Department of Mathematics
    University of Hawaii
    2565 McCarthy Mall
    Honolulu, HI 96822-2273, U.S.A.

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