Torsion-free modules over a Discrete Valuation Ring

[Hurkyl]

Let R be a discrete valuation ring with fraction field F.

I believe it's straightforward to show that any torsion-free module M with the property that M \otimes_R F is a finite dimensional F-vector space is of the form R^m \oplus F^n.

What if M \otimes_R F is infinite dimensional?

 

[Sportin' Life]

My guess is that not much would be known, since the basic criterion of the fundamental theorem of finitely generated R-modules over a PID would not be met.

My way of saying I dunno. It sounds like an interesting question for which I am probably not equipped to help. Good luck.