Corollary 2

Tag 0D5Q, permanent. Cite it as [Bourbaki, Tag 0D5Q].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

For the $A$ -module $M$ to be free, it is necessary and sufficient that it be a Macaulay module and of dimension $\dim(A)$ , or again that it be of depth $\geqslant \dim(A)$ .

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

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