Corollary 2

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

Commutative Algebra, chapter VIII, § 3. Dimension des anneaux noethériens

Let $A$ be a noetherian ring, $a$ an ideal of $A$ , and $\hat{A}$ the separated completion of $A$ for the $a$ -adic topology. One has $\dim(\hat{A}) \leqslant \dim(A)$ , with equality when $A$ is local and $a$ distinct from $A$ .

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

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