### 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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