Corollary 1

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

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

Let $n$ be a positive integer. We have

\dim(k[X_1, ..., X_n]) = n .

For a finite type k-algebra A to have dimension n, it is necessary and sufficient that there exist an injective k-homomorphism $\varphi : k[X_1, ..., X_n] \to A$ making A a finite algebra over $k[X_1, ..., X_n]$ .

This follows from th. 3, from the normalization lemma (V, § 3, No. 1, th. 1) and from th. 1, a) of No. 3.

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

Cited by

Nothing.

Cites

Nothing.