Remark 3

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

Commutative Algebra, chapter VIII, § 1. Dimension de Krull d’un anneau

Let $Y$ be a closed subset of the topological space $X$ and $(X_\alpha)_{\alpha \in A}$ (resp. $(Y_\beta)_{\beta \in B}$ ) the family of irreducible components of $X$ (resp. of $Y$ ). For each $\beta \in B$ , let $A(\beta)$ denote the set of $\alpha \in A$ such that $Y_\beta \subset X_\alpha$ . Since every irreducible subset of $X$ is contained in one of the $X_\alpha$ (II, § 4, No. 1, Prop. 5), it follows from Def. 3 that one has:

\operatorname{codim}(Y, X) = \inf_{\beta \in B} \sup_{\alpha \in A(\beta)} \operatorname{codim}(Y_\beta, X_\alpha).

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

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