Remark 1

Tag 0CM1, permanent. Cite it as [Bourbaki, Tag 0CM1].

Commutative Algebra, chapter VIII, § 2. Dimension des algèbres

By cor. 5, the conditions a), b), c) of cor. 6 imply that one has $\dim_p(A) = \dim(A)$ for every prime ideal $p$ of A.

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

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