The Stacks project

Lemma 5.20.2. Let $X$ be a topological space. If $X$ is sober and has a dimension function, then $X$ is catenary. Moreover, for any $x \leadsto y$ we have

\[ \delta (x) - \delta (y) = \text{codim}\left(\overline{\{ y\} }, \ \overline{\{ x\} }\right). \]

Proof. Suppose $Y \subset Y' \subset X$ are irreducible closed subsets. Let $\xi \in Y$, $\xi ' \in Y'$ be their generic points. Then we see immediately from the definitions that $\text{codim}(Y, Y') \leq \delta (\xi ) - \delta (\xi ') < \infty $. In fact the first inequality is an equality. Namely, suppose

\[ Y = Y_0 \subset Y_1 \subset \ldots \subset Y_ e = Y' \]

is any maximal chain of irreducible closed subsets. Let $\xi _ i \in Y_ i$ denote the generic point. Then we see that $\xi _ i \leadsto \xi _{i + 1}$ is an immediate specialization. Hence we see that $e = \delta (\xi ) - \delta (\xi ')$ as desired. This also proves the last statement of the lemma. $\square$

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