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

Question

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 ?

Answer 1

If $M$ is a torsion-free $A$-module, then $M_p$ is a torsion-free $A_p$-module.