Remark 2

Tag 0D6W, permanent. Cite it as [Bourbaki, Tag 0D6W].

Commutative Algebra, chapter X, § 5. Intersections complètes

Let A be a ring, J an ideal of finite type of A and p a prime ideal of A containing J. By cor. 2 of th. 1 of § 1, No. 3, one has

\operatorname{prof}_{A_p}(J_p; A_p) \leq [\kappa(p) \otimes_A J : \kappa(p)]

and there is equality if and only if J is completely secant at p.

Suppose that J is distinct from A and generated by a completely secant sequence $(x_1, \ldots, x_r)$ . One then has (prop. 4 of § 1, No. 3, and prop. 1 above)

\operatorname{prof}_A(J; A) = \inf_{p \in V(J)} \operatorname{prof}_{A_p}(J_p; A_p) = r.

If moreover A is noetherian, one has $\operatorname{codim}(V(J), \operatorname{Spec}(A)) \leq r$ (VIII, § 3, No. 3, Prop. 4) and $\operatorname{prof}_A(J; A) \leq \operatorname{codim}(V(J), \operatorname{Spec}(A))$ ( $§ 1$ , No. 7, Prop. 12), whence finally

\operatorname{prof}_A(J; A) = r = \operatorname{codim}(V(J), \operatorname{Spec}(A)) = \operatorname{ht}(J).

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

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