local bases at $q=0$

Question

If $k$ is a field, I'm having troubles in understanding part of the definition of a local base for a $k(q)$-vector space. More precisely, suppose that $V$ is a $k(q)$-vector space, and let $A$ be the localization of $k[q]$ at the ideal $(q)$. We say that an $A$-submodule $L$ of $V$ is an $A$-lattice if $L \otimes_A k(q) $ is isomorphic to $V$ via the obvious map. The reference I'm reading says that an $A$-lattice has to be a free $A$-module. I don't understand this fact, intuitively it seems also false to me because I would't say that if a module is generically free than it's free, but I understand that this is a particular situation since $L$ is naturally embedded in $V$. Can anyone help me please?