Proposition 9

Let M be a finitely generated A-module. The following conditions are equivalent:

(a) $M_p = 0$ for every prime ideal $p$ of height $\leq 1$. (b) The annihilator $a \triangleleft M$ is an ideal $\neq (0)$ and $A : a = A$ (A: $a$ denoting, as in $§ 1$, no. 1, the set of $x \in K$ such that $xa \in A$).

We know (Chapter II, $§ 2$, no. 2, Corollary 2 to Proposition 4) that the condition $M_p = 0$ is equivalent to $a + p$ and hence to $aA_p = A$, (Chapter II, $§ 2$, no. 5, Remark); on the other hand, for every integral ideal $b \neq 0$ of A, the relation "*$bA_p = A$, for all $p \in P$*" is equivalent to $\operatorname{div} b = \operatorname{div} A = 0$ in $D(A)$ ($§ 1$, no. 4, Proposition 7), or also to $\operatorname{div}(A : b) = 0$ and, as $A : b$ is divisorial ($§ 1$, no. 1, Proposition 1), this relation is also equivalent to $A : b = A$. The proposition then follows by noting that to say that $a \notin p$ for $p = (0)$ means that $a \neq (0)$.