Definition 111.18.1. A Noetherian ring $A$ is said to be catenary if for any triple of prime ideals ${\mathfrak p}_1 \subset {\mathfrak p}_2 \subset {\mathfrak p}_3$ we have
\[ ht({\mathfrak p}_3 / {\mathfrak p}_1) = ht({\mathfrak p}_3/{\mathfrak p}_2) + ht({\mathfrak p}_2/{\mathfrak p}_1). \]
Here $ht(\mathfrak p/\mathfrak q)$ means the height of $\mathfrak p/\mathfrak q$ in the ring $A/\mathfrak q$. In a formula
\[ ht(\mathfrak p/\mathfrak q) = \dim (A_\mathfrak p/\mathfrak qA_\mathfrak p) = \dim ((A/\mathfrak q)_\mathfrak p) = \dim ((A/\mathfrak q)_{\mathfrak p/\mathfrak q}) \]
A topological space $X$ is catenary, if given $T \subset T' \subset X$ with $T$ and $T'$ closed and irreducible, then there exists a maximal chain of irreducible closed subsets
\[ T = T_0 \subset T_1 \subset \ldots \subset T_ n = T' \]
and every such chain has the same (finite) length.
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