Definition 1

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

Commutative Algebra, chapter VIII, § 3. Dimension des anneaux noethériens

Let $A$ be a noetherian local ring, $M$ a nonzero finitely generated $A$ -module and $S$ a subset of the maximal ideal $m_A$ of $A$ . One says that $S$ is secant for $M$ if one has

(9)

\dim_A(M) = \mathrm{Card}(S) + \dim_A(M/SM) .

If $S$ is secant for $M$ , one has $\mathrm{Card}(S) \leq \dim_A(M)$ , hence $S$ is finite. One says that a family $(x_i)_{i \in I}$ of elements of $m_A$ is secant for $M$ if one has

(10)

\dim_A(M) = \mathrm{Card}(I) + \dim_A(M/\sum_{i \in I} x_i M) ,

that is to say if

i \mapsto x_i

is a bijection of

I

onto a secant subset of

m_A

for

M

An element x of $m_A$ is said to be secant for M if $\{ x \}$ is a secant subset for M, that is to say if one has

\dim_A(M/xM) = \dim_A(M) - 1 .

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

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