### On the $p$-rank of ${\rm Ext}_{\mathbb Z}(G,{\mathbb Z})$ in certain models of ZFC

by Shelah and Struengmann. [ShSm:874]
Algebra and Logic, 2007
We show that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext_Z (G, Z) is as large as possible for every prime p and any torsion-free abelian group G . Moreover, given an uncountable strong limit cardinal mu of countable cofinality and a partition of Pi (the set of primes) into two disjoint subsets Pi_0 and Pi_1, we show that in some model which is very close to ZFC there is an almost-free abelian group G of size 2^{mu}= mu^+ such that the p-rank of Ext_Z (G, Z) equals 2^{mu}= mu^+ for every p in Pi_0 and 0 otherwise, i.e. for p in Pi_1 .

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