Is a Noetherian normal local domain (universally) catenary? Let $R$ be a ring. Then $R$ is $\textit{catenary}$ if for a pair of prime ideal $p \subseteq q$, all maximal chains of prime ideals $p = p_0 \subseteq p_1 \subseteq \dots \subseteq p_n = q$ have the same length. For details and properties, please refer to < Since a Cohen-Macaulay ring is universally catenary, algebras which are essentially of finite type over such ring is again catenary. For me, it is hard to find a non catenary ring, but at the same time not every ring is catenary. I would like to know some good rings are catenary or not. > Is a Noetherian normal local domain $R$ catenary? Of course, the answer is positive if $\dim R \le 2$ (then it is Cohen-Macaulay) or $R$ is complete (by the Cohen structure theorem).

There is a counter-example due to Ogoma in "Non-catenary pseudo-geometric normal rings", which seems to be a modification of one of Nagata's examples of a non-catenary ring. Heitmann has a simpler description of the construction in "A non-catenary, normal, local domain", as does Lech in "Yet another proof of a result of Ogoma".