Corollary 2

Let $M$ be a finitely generated module over a Noetherian ring $A$ and $p$ a prime ideal of $A$. For $M_p$ to be a non-zero $A_p$-module of finite length, it is necessary and sufficient that $p$ be a minimal element of $\operatorname{Ass}(M)$.

By $§ 1$, no. 2, Corollary to Proposition 5, $\operatorname{Ass}_{A_p}(M_p)$ is the set of ideals $q_p$, where $q$ runs through the set of ideals of $\operatorname{Ass}(M)$ which are contained in $p$. On the other hand, $p_p$ is the unique maximal ideal of $A_p$; by Proposition 7, for $M_p$ to be an $A_p$-module of finite length, it is necessary and sufficient that no element of $\operatorname{Ass}(M)$ be strictly contained in $p$. On the other hand, for $M_p \neq 0$, it is necessary and sufficient by definition that $p \in \operatorname{Supp}(M)$ (Chapter 11, § 4, no. 4), that is that $p$ contain an element of $\operatorname{Ass}(M)$ ($§ 1$, no. 3, Proposition 7). This proves the corollary.