Proposition 3

Tag 0D5O, permanent. Cite it as [Bourbaki, Tag 0D5O].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

Let $A$ be a regular noetherian local ring and $M$ a nonzero finitely generated $A$ -module. Its projective dimension is finite, and one has

\mathrm{dp}_A(M) + \mathrm{prof}_A(M) = \dim(A).

In fact, $M$ has finite projective dimension (prop. 1), and one has $\mathrm{prof}(A) = \dim(A)$ since $A$ is a Macaulay ring (§ 2, No. 5, example 7). One then applies th. 1 of § 3, No. 5.

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

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