Bourbaki

Remark 5

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

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

According to prop. 2 and Lemma 1, we therefore have, under the hypotheses of prop. 2, the inequality

dim⁡(Spec(B))≥dim⁡(Spec(A))+inf⁡p∈Spec(A)dim⁡(αρ−1(p)). \dim(\mathrm{Spec}(B)) \geq \dim(\mathrm{Spec}(A)) + \inf_{p \in \mathrm{Spec}(A)} \dim(\alpha\rho^{-1}(p)) .

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

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