Corollary 2

Let A be a noetherian local ring, M a nonzero finitely generated A-module and $\alpha$ a subset of $m_A$, stable under addition and multiplication and such that $\operatorname{long}(M/\alpha M) < +\infty$. There exists a subset of $\alpha$ which is a maximal secant subset for M.

In fact, one has $V(\alpha) \cap \operatorname{Supp}(M) = \operatorname{Supp}(M/\alpha M) = \{m_A\}$ (§ 1, No. 4, Remark 1), hence $\operatorname{codim}(V(\alpha) \cap \operatorname{Supp}(M), \operatorname{Supp}(M)) = \dim(\operatorname{Supp}(M)) = \dim_A(M)$, and Proposition 9 is applied.