Proposition 7

Tag 0D0N, permanent. Cite it as [Bourbaki, Tag 0D0N].

Commutative Algebra, chapter X, § 1. Profondeur

Let $A$ be a ring, $J$ an ideal of $A$ , $M$ an $A$ -module, and $(x_1, \ldots, x_r)$ an $M$ -regular sequence of elements of $J$ . Then

\operatorname{prof}_A(J; M) = r + \operatorname{prof}_A(J; M/(x_1M + \ldots + x_rM))

and in particular

\operatorname{prof}_A(J; M) \geqslant r

The case $r = 1$ follows from Remark 5 of No. 1, applied to the exact sequence

0 \to M \xrightarrow{(x_1)_M} M \longrightarrow M/x_1M \to 0 .

The general case follows by induction on

r

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

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