Proposition 5

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

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

Let X be a topological space of finite dimension. Suppose that all maximal chains of irreducible closed subsets of X have the same length. Then X is catenary; for every irreducible closed subset Z of X, one has

\operatorname{codim}(Z, X) = \dim(X) - \dim(Z);

for every pair (Y, Z) of irreducible closed subsets of X such that Y ⊂ Z, one has

\operatorname{codim}(Y, Z) = \dim(Z) - \dim(Y).

Let Y and Z be two irreducible closed subsets of X such that Y ⊂ Z. Let Y_0 ⊂ ... ⊂ Y_p be a chain such that Y_p = Y and p = \dim(Y), Z_0 ⊂ ... ⊂ Z_q a chain such that Z_0 = Z and q = \operatorname{codim}(Z, X). For every saturated chain T_0 ⊂ ... ⊂ T_r such that T_0 = Y and T_r = Z, the chain

Y_0 \subset ... \subset Y_{p-1} \subset T_0 \subset ... \subset T_r \subset Z_1 ... \subset Z_q

is maximal, and of length p + q + r; by the assumption made on X, one has therefore p + q + r = \dim(X), that is r = \dim(X) - \dim(Y) - \operatorname{codim}(Z, X). It follows that X is catenary and that, for Y and Z as above, one has

\dim(Y) + \operatorname{codim}(Y, Z) = \dim(X) - \operatorname{codim}(Z, X).

Taking Y = Z, one sees that the second member is equal to \dim(Z), whence the proposition.

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

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