# Finitely generated torsion free module over $A$ is locally free

Assume that $$A$$ is an integral domain whose local rings $$A_p$$ are principal ideal domains. Show that any finitely generated torsion-free module over $$A$$ is locally free.

I know that finitely generated torsion-free modules over a PID are free. How to apply this here ?

If $$M$$ is a torsion-free $$A$$-module, then $$M_p$$ is a torsion-free $$A_p$$-module.
• is it something like this ? Suppose $m/s \in M_p$. If its torsion then there is $a/s$ such that $a/s \cdot m/s = 0$ so there is $x \in A$ such that $x \cdot am = 0$ so M is not torsion free – Itachi Oct 22 '15 at 6:43
• So because of this, I can localize and $M_p$ would still be finitely generated and torsion free. So $M_p$ is torsion free, fin.gen over PID $A_p$ thus $M_p$ free ? – Itachi Oct 22 '15 at 6:48