Corollary 2

Tag 0D4L, permanent. Cite it as [Bourbaki, Tag 0D4L].

Commutative Algebra, chapter X, § 3. Profondeur et dimension homologique

Retain the hypotheses of theorem 1 and suppose moreover that $A$ is a Macaulay ring. Then $dp_A(M)$ is the sum of the two positive integers $\dim(A) - \dim_A(M)$ and $\dim_A(M) - \operatorname{prof}(M)$ .

In particular, $dp_A(M)$ is then greater than $\dim(A) - \dim_A(M)$ , and there is equality if and only if $M$ is Cohen-Macaulay.

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

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