Corollary 1

*Let A be a noetherian ring, M a finitely generated A-module, $p_1, ..., p_n, q_1, ..., q_m$ prime ideals of A and r an integer $\geqslant 1$. Suppose that one has $p_i \not\subset q_j$ for $1 \leqslant i \leqslant n$ and $1 \leqslant j \leqslant m$, and $\dim_{A_{p_i}}(M_{p_i}) \geqslant r$ for $1 \leqslant i \leqslant n$. Then there exists a sequence $(x_1, ..., x_r)$ of elements of A belonging to all the $p_i$ and belonging to none of the $q_j$, such that, for $1 \leqslant i \leqslant n$, the images of $x_1, ..., x_r$ in $A_{p_i}$ form a secant sequence for the module $M_{p_i}$.

Put $\alpha = \bigcap_i p_i$. One has $M_{p_i} \neq \{0\}$, hence $p_i \in \operatorname{Supp}(M)$ for $1 \leqslant i \leqslant n$, whence $V(\alpha) \subset \operatorname{Supp}(M)$. One has

(§ 1, No. 4, Prop. 9), and Proposition 9 is applied.