Corollary 1

Tag 0D5P, permanent. Cite it as [Bourbaki, Tag 0D5P].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

One has $\mathrm{dp}_A(M) \geqslant \dim(A) - \dim(M)$ ; for equality to hold, it is necessary and sufficient that $M$ be a Macaulay module.

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

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