Corollary

Tag 0CM6, permanent. Cite it as [Bourbaki, Tag 0CM6].

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

Let A be a finitely generated k-algebra, k' an extension of k, and A' the k'-algebra A \otimesk k'.*

a) We have $\dim(A') = \dim(A)$ .

b) Let p' be a prime ideal of A' and p its inverse image in A; we have $\dim_{p'}(A') = \dim_p(A)$ .

c) Let p' be a minimal prime ideal of A' and p its inverse image in A. Then p is minimal and we have $\dim(A'/p') = \dim(A/p)$ .

The assertions a) and b) follow from prop. 5 by taking $B = k'$ . Let us prove c). The ideal p is minimal (No. 1, prop. 1) and we have

\dim(A'/p') = \dim_{p'}(A') = \dim_p(A) = \dim(A/p).

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

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