Corollary 2

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

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

Let A be an integral k-algebra of finite type. For every prime ideal p of A, we have

\operatorname{ht}(p) = \dim(A_p) = \dim(A) - \dim(A/p)

= \dim(A) - \deg.\operatorname{tr}_k\kappa(p).

In particular, we have $\operatorname{ht}(m) = \dim(A_m) = \dim(A)$ for every maximal ideal m of A.

This follows from th. 3 and Remark 4 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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