Every coseparable group may be free

We show that if {Mathematical expression} Cohen reals are added to the universe, then for every reduced non-free torsion-free abelian group A of cardinality less than the continuum, there is a prime p so that Ext _p (A,ℤ)≠0. In particular if it is consistent that there is a supercompact cardinal, then it is consistent (even with weak CH) that every coseparable group is free. The use of some large cardinal hypothesis is needed.