Remark 1

Tag 0D9D, permanent. Cite it as [Bourbaki, Tag 0D9D].

Commutative Algebra, chapter X, § 7. Algèbres lisses

When $A$ is noetherian, condition (i) means that the local ring $A_{\mathfrak{m}}$ is regular (VIII, § 5, No. 2, Theorem 1).

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

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