Corollary

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

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

Let X be an irreducible topological space of finite dimension. In order that X be catenary, it is necessary and sufficient that, for every pair (Y, Z) of irreducible closed subsets of X such that Y ⊂ Z, one has codim(Y, X) = codim(Y, Z) + codim(Z, X).

The condition is necessary by Prop. 4. Conversely, suppose it satisfied, and denote by c(Z) the integer codim(Z, X) for every irreducible closed subset Z of X. If Y, Z, T are three irreducible closed subsets of X such that Y ⊂ Z ⊂ T, one has

\begin{align*} \operatorname{codim}(Y, Z) + \operatorname{codim}(Z, T) &= (c(Y) - c(Z)) + (c(Z) - c(T)) \\ &= c(Y) - c(T) \\ &= \operatorname{codim}(Y, T), \end{align*}

and X is catenary by Prop. 4.

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

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