Proposition 12

*If a is an ideal $\neq 0$ of A,

Let $p \in P$. Then $aA_p = p^{n_p}A_p$ where $n_p \geq 0$, since $A_p$ is a discrete valuation ring. As $(A/a)_p = A_p/aA_p,\ l_p(A/a) = n_p$, whence $\chi(A/a) = \sum_{p \in P} n_p p = \mathrm{div}\, a$ (§ 1, no. 4, Proposition 7).

On the other hand, $(A : a) = A ; aA_p = p^{-n_p}A_p$, hence $l_p((A : a)/A) = n_p$ and we conclude in the same way.