This is a question related to local fields, particularly on page 48 of Local fields by Serre. Let $A$ be a Dedekind domain with field of fractions $K$. Let $V$ be a finite dimensional vector space over $K$. A lattice of $V$ (with respect to $A$) is a sub-$A$-module $X$ of $V$ that is finitely generated and spans $V$. In the first section of chapter 3, Serre proceeds to define what $\chi(X_1, X_2)$, $X_1,X_2$ lattices in $V$.
The question is regarding a statement at the very end of Section 1 (on page 48) after proposition 2. Let $n= [V :K]$ and let $W = \wedge^nV$ ; it is a one dimensional vector space over $K$. To each lattice $X$ of $V$, let us associate $X_W = \wedge^n X$, which may be identified with a lattice of $W$.
I am unable to understand the next statement which says `` as [W:K] =1, if $D$ and $D'$ are two lattices of $W$, there is a unique nonzero fractional ideal $\mathfrak{a}$ of $K$ such that $D' = \mathfrak{a}D$.
Please, can anyone clarify this?
Thank you
I don't have Serre's book in hand but if $D$ is a lattice in $W$, who has dimension $1$ over $K$, then $D$ can be thought as a fractional ideal of $A$, since it's a finitely generated $A$-submodule of $K$, so $ \frak a = \frak d'\frak d^{-1}$, where $\frak d$ and $\frak d'$ denote the fractional ideals of $D$ and $D'$, respectively.