Let $V/K$ be a vector space where $K = Frac(A)$ for some Dedekind domain $A$, $X$ a lattice (and hence an $A$- submodule) of $V$, and $u$ a $K$-automorphism of $V$. The claim is that $\chi_A (X, uX) = (det(u))$.
Here for lattices $X_1$ and $X_2$ of $V$, we define $\chi_A(X_1, X_2) = \chi_A(X_1/X_3) \chi_A(X_2/X_3)^{-1}$ for some lattice $X_3$ inside of $X_1$ and $X_2$.
For a finite length $A$- module $M$ we define $\chi_A(M) = \prod \mathfrak{p}_i$ where each $\mathfrak{p}_i$ corresponds to a maximal ideal of $A$ such that $A/(\mathfrak{p}_i) \cong M_i/M_{i+1}$ for $M_i$ in the composition series of $M$.
For the proof Serre claims that "we are reduced the case where $X$ is free by localizing and multiplying $u$ by a constant." I can see that essentially what he is saying is that we can reduce to the case where $A$ is a DVR via localization and then get that $X$ is free since we have a torsion free module over a PID, but I don't see why we can localize here to reduce to this case and any help in understanding this would be greatly appreciated.