Noetherian local domain $A$ with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$

Question

Is there a noetherian local domain $A$ with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$?

This is a follow up question to: Does codimension behave weirdly even in local rings?

Answer 1

As it is pointed out in Matsumura, CRT, page 31, for a noetherian local domain $A$ the equality $\operatorname{ht}\mathfrak p+\dim A/\mathfrak p=\dim A$ holds for any prime ideal $\mathfrak p$ iff $A$ is catenary. So, we are looking for a noetherian local domain which is not catenary. There are no trivial examples, but you can find one at http://stacks.math.columbia.edu/tag/02JE.