Corollary 1

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

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

Suppose that one has $\dim(A) < +\infty$ and that there exists an integer $n$ such that $\dim(B \otimes_A \kappa(p)) \leq n$ for every $p \in \mathrm{Spec}(A)$ . Then one has $\dim(B) < +\infty$ .

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

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