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

Saharon Shelah ¹   ; Lutz Strüngmann ²

¹ Department of Mathematics The Hebrew University of Jerusalem Jerusalem 91904, Israel and} Rutgers University New Brunswick, NJ 08903, U.S.A.
² Department of Mathematics University of Duisburg-Essen 45117 Essen, Germany and Department of Mathematics University of Hawaii 2565 McCarthy Mall Honolulu, HI 96822-2273, U.S.A.

Fundamenta Mathematicae, Tome 193 (2007) no. 2, pp. 141-151

Résumé

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 $\mathop{\rm Ext}(G,\mathbb Z)$.

Détail

DOI : 10.4064/fm193-2-3

Keywords: complete characterization mathop ext mathbb torsion free abelian group assuming dels axiom constructibility plus there weakly compact cardinal particular prove singular cardinal uncountable cofinality which first weakly compact cardinal every sequence varpi cardinals satisfying leq where varpi set primes there torsion free abelian group size equals p rank mathop ext mathbb every prime torsion free rank mathop ext mathbb