Remark 2

Tag 0D2V, permanent. Cite it as [Bourbaki, Tag 0D2V].

Commutative Algebra, chapter X, § 2. Modules et anneaux macaulayens

If the ring A is local and the nonzero module M, every subset S of $m_A$ strongly secant for M is secant for M. In fact, since the $A$ -module M/SM is nonzero, one has

\text{Card}(S) \leq \operatorname{codim}(\operatorname{Supp}(M/SM), \operatorname{Supp}(M)) \leq \dim(M) - \dim(M/SM)

(VIII, § 1, No. 2, Prop. 3 a)), whence our assertion.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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