Corollary 3

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

Commutative Algebra, chapter VIII, § 3. Dimension des anneaux noethériens

Let $A$ be a noetherian ring and $n$ an integer $\geqslant 0$ . One has

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

The ring $A[[X_1, ..., X_n]]$ is the separated completion of the polynomial ring $A[X_1, ..., X_n]$ for the $a$ -adic topology, where $a$ is the ideal generated by $X_1, ..., X_n$ ; we therefore have

\dim(A[[X_1, ..., X_n]]) \leqslant \dim(A[X_1, ..., X_n])

by Corollary 2. On the other hand, we have

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

by Corollary 3 of Proposition 7. Finally, we have

\dim(A) + n \leqslant \dim(A[[X_1, ..., X_n]])

by Example 4 of $§ 1$ , 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.