From the fact that \(\operatorname{Tor}(A, B) = 0\) if \(A\) is free, deduce that \(\operatorname{Tor}(A, B) = 0\) if \(A\) is torsionfree by applying the previous problem to the directed system of finitely generated subgroups \(A_\alpha\) of \(A\).
Solution
Write a torsionfree abelian group \(A\) as the directed union of its finitely generated subgroups: \[ A=\varinjlim_\alpha A_\alpha. \] Each \(A_\alpha\) is finitely generated and torsionfree, hence is a free abelian group. Therefore \[ \operatorname{Tor}(A_\alpha,B)=0 \] for every \(\alpha\).
By Exercise 4, \(\operatorname{Tor}\) commutes with direct limits in the first variable, so \[ \operatorname{Tor}(A,B) \cong \varinjlim_\alpha\operatorname{Tor}(A_\alpha,B) =0. \]