Let $R$ be a Dedekind domain, $K$ its field of fractions, $P$ a non-zero prime ideal of $R$. Let $\hat R_P$ be the completion of $R$ w.r.t the valuation $v_P$ induced by $P$ and let $L$ be a torsion-free $R$-module.
Is it true that the completion $\hat R_P\otimes_R L$ of $L$ is a torsion-free $\hat R_P$-module?
I'm not sure how to even approach this question. Would the fact that $R$ is dense in $\hat R_P$ w.r.t $v_P$ be helpful?
Many thanks!
We can assume that any torsion element is in the valuation ideal, $PR_P$ which we will just call $P$ by abuse of notation. If not, then we can find an inverse for the torsion number, $n$, in $R_P - P=R_P^\times$, and this would imply $1\cdot v=0$ for some $v\in R_P\otimes_R L$, which implies $v$ was $0$ to begin with.
But then it must be that we need only check if there is $\pi^k$ torsion for some $k$, and $\pi$ a generator for $P$.
But then $\pi^k\in R$ already, so we would have detected it from the start.