I think I came up with the following result. But I'm not 100% sure. Is this correct? If yes, how does one prove this?
Theorem? Let $A$ be a discrete valuation ring, $K$ its field of fractions. Let $L$ be a finite separable extension of $K$. Let $B$ be the integral closure of $A$ in $L$. Let P be the maximal ideal of A. Let $Q_i$, $i$ = $1, ..., r$ be the maximal ideals of $B$ lying over $P$. Let $M$ be a finitely generated torsion-free module over $B$. Let $\hat{M_P}$ be the completion of $M$ with respect to $P$-adic topology. Let $\hat{M_{Q_i}}$ be the completion of $M$ with respect to $(Q_i)$-adic topology. Then $\hat{M_P}$ $\cong$ $\prod_{i}\hat{M_{Q_i}}$
EDIT I need this to prove this theorem.