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.

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  • $\begingroup$ 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 $\endgroup$ – Itachi Oct 22 '15 at 6:43
  • $\begingroup$ @Itachi Almost. $\endgroup$ – user26857 Oct 22 '15 at 6:46
  • $\begingroup$ 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 ? $\endgroup$ – Itachi Oct 22 '15 at 6:48
  • $\begingroup$ @Itachi Exactly. $\endgroup$ – user26857 Oct 22 '15 at 6:53

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