Theorem 6

Let M be a torsion-free finitely generated A-module. There exists a free submodule L of M such that M/L is isomorphic to an ideal of A.

We shall denote by $n$ the rank of $M$ (rank of $V = M \otimes_A K$ over $K$) and consider $M$ as a lattice of $V$ with respect to $A$. Then, for all $p \in P(A)$, $M_p$ is a lattice of $V$ with respect to $A_p$ (no. 1, Example 6) and, as $A$, is a principal ideal domain, $M$, is a free $A$-module of rank $n$. We shall write:

We shall denote by $k(p)$ the field of fractions of $A/p$ (isomorphic to the residue field of $A_p$); $M(p) = M \otimes_A k(p)$ is therefore a vector space of rank $n$ over $k(p)$. For all $x \in M$, we shall denote by $x(p)$ the canonical image of $x$ in $M(p)$.