Proposition 14

If M, M' are two lattices of V such that M' \subset M, M/M' is a finitely generated torsion A-module and:

Clearly M/M' \subset V/M' is a finitely generated torsion module; on the other hand, for all $p \in P$, we know (Algebra, Chapter VII, § 4, no. 2, Theorem 1) that there exist bases $(e_i)_{1 \leq i \leq n}$ of M, and $(e'_i)_{1 \leq i \leq n}$ of $M'_p$ such that $e'_i = \pi^{v_i} e_i$ for $1 \leq i \leq n$ and integers $v_i \geq 0$, $\pi$ being a uniformizer of $A_p$. Therefore (in the notation introduced above) $n'_p - n_p = \sum_{i=1}^n v_i$; and on the other hand,

is isomorphic to the torsion A,-module $\bigoplus_{i=1}^n A_p/p^{v_i} A_p$ and hence its length is $\sum_{i=1}^n v_i$, which proves the proposition, in view of (5) and Definition 4 of no. 5.