Proposition 5

Let A be a noetherian ring, M a finitely generated A-module, and S a finite subset of A. The following conditions are equivalent:

(i) the subset S of A is strongly secant for M; (ii) for every element $p$ of Supp(M/SM), the canonical mapping $A \to A_p$ induces a bijection of S onto a subset of $pA_p$ secant for $M_p$.

(i) $\Rightarrow$ (ii) : Let $p \in \operatorname{Supp}(M/SM)$ and let $S'$ be the image of S in $A_p$. The set $S'$ is contained in the maximal ideal $pA_p$, and one has

(VIII, § 1, No. 4, Prop. 9). The inequality Card(S) $\leq \operatorname{codim}(\operatorname{Supp}(M/SM), \operatorname{Supp}(M))$ and Prop. 3 b) of VIII, § 1, No. 2 imply the relations

Since $M_p$ is nonzero, one has on the other hand $\dim(M_p) \leq \text{Card}(S') + \dim(M_p/S'M_p)$ (VIII, § 3, No. 2, formula (8)). Condition (ii) then follows from the inequality $\text{Card}(S') \leq \text{Card}(S)$.

(ii) $\Rightarrow$ (i) : One may suppose SM $\neq$ M. If condition (ii) is satisfied, one has for every prime ideal $p$ of Supp(M/SM)

which implies (i) by passing to the greatest lower bound.