The p-rank of Extℤ(G, ℤ) in certain models of ZFC

We prove 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 for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π₀ and Π₁, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2^μ = μ⁺ such that the p-rank of Ext _ℤ(G, ℤ) equals 2^μ = μ⁺ for every p ∈ Π₀ and 0 otherwise, that is, for p ∈ Π₁. Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development project No. I-706-54.6/2001. Supported by a grant from the German Research Foundation DFG. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007.