Corollary

Let A be a noetherian ring and M a finitely generated A-module. Every M-regular sequence is strongly secant for M.

Let x be an M-regular sequence, and J the ideal of A generated by it. For every prime ideal $p \in \operatorname{Supp}(M/JM)$, the image of x in $A_p$ is an $M_p$-regular sequence of elements of $pA_p$, hence a secant sequence for $M_p$ (VIII, § 3, No. 2, cor. to Prop. 3).