Remark 3

Tag 0D1R, permanent. Cite it as [Bourbaki, Tag 0D1R].

Commutative Algebra, chapter X, § 1. Profondeur

$\mathrm{ht}(p) = 1$ and $\mathrm{ht}(q) \geq 2$ . Then, by the relation

\dim(B_q) = \dim(A_p) + \dim(\kappa(p) \otimes_A B_q)

(VIII, § 3, No. 4, Cor. 1 of Prop. 7), the ring

\kappa(p) \otimes_A B_q

is of dimension

\geq 1

. It is reduced by assumption, hence of depth

\geq 1

(No. 1, Remark 3). We then have (No. 6, Cor. of Prop. 11)

\mathrm{prof}(B_q) = \mathrm{prof}(A_p) + \mathrm{prof}(\kappa(p) \otimes_A B_q) \geq 1 + 1 = 2 .

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

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