Remark 1

Tag 0D7C, permanent. Cite it as [Bourbaki, Tag 0D7C].

Commutative Algebra, chapter X, § 5. Intersections complètes

Suppose that the equivalent conditions of Prop. 6 are satisfied. Since $B/J$ is flat over $A$ , for every $A$ -algebra $A'$ , the canonical sequence of $A'$ -modules

0 \to A' \otimes_A J \to A' \otimes_A B \to A' \otimes_A (B/J) \to 0

is therefore exact, and the canonical homomorphism

A' \otimes_A J \to J(A' \otimes_A B)

is bijective.

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

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