A Free Module

[Tsakanikas Nickos]

Let $A$ be a noetherian local domain with residue field $k$ and quotient field $K$. If $M$ is a finitely generated $A$-module and if \[ \dim_{k} M \otimes_{A} k = \dim_{K} M \otimes_{A} K = r \]then $M$ is a free $A$-module of rank $r$.