Corollary 2

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

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

Let $\rho : A \to B$ be a homomorphism of noetherian rings. Then

\dim(B) \leq \dim(A) + \sup_{p \in \mathrm{Spec}(A)} \dim(B \otimes_A \kappa(p)).

Let $q$ be a prime ideal of $B$ and $p = \rho^{-1}(q)$ . By Cor. 1, we have $\dim(B_q) \leq \dim(A_p) + \dim(B_q \otimes_A \kappa(p))$ . It follows (Prop. 6, b) of § 1, No. 3) that we have the inequality $\dim(B_q) \leq \dim(A) + \dim(B \otimes_A \kappa(p))$ , and one concludes by Prop. 8 of § 1, No. 3.

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

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