Corollary 2

Let $A$ be a Noetherian ring, $M$ a finitely generated $A$-module and $a$ an ideal of $A$. For $\operatorname{Supp}(M) \subset V(a)$, it is necessary and sufficient that there exist an integer $k$ such that $a^k M = 0$.

If $b$ is the annihilator of $M$, the relation $\operatorname{Supp}(M) \subset V(a)$ is equivalent to $V(b) \subset V(a)$ by Proposition 17 and hence to $a \subset r(b)$, where $r(b)$ is the radical of $b$ (no. 3, Corollary 2 to Proposition 11). Since $A$ is Noetherian, this condition is also equivalent to the existence of an integer $k > 0$ such that $a^k \subset b$ (§ 2, no. 6, Proposition 15).