A ring that is catenary but not universally catenary
It is delicate to construct examples of Noetherian rings that are not universally catenary. The first example was found by Masayoshi Nagata (1956, 1962, page 203 example 2), who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary.
Nagata's example is as follows. Choose a field k and a formal power series z=Σi>0aixi in the ring S of formal power series in x over k such that z and x are algebraically independent.
Define z1 = z and zi+1=zi/x–ai.
Let R be the (non-Noetherian) ring generated by x and all the elements zi.
Let m be the ideal (x), and let n be the ideal generated by x–1 and all the elements zi. These are both maximal ideals of R, with residue fields isomorphic to k. The local ring Rm is a regular local ring of dimension 1 (the proof of this uses the fact that z and x are algebraically independent) and the local ring Rn is a regular Noetherian local ring of dimension 2.
Let B be the localization of R with respect to all elements not in either m or n. Then B is a 2-dimensional Noetherian semi-local ring with 2 maximal ideals, mB (of height 1) and nB (of height 2).
Let I be the Jacobson radical of B, and let A = k+I. The ring A is a local domain of dimension 2 with maximal ideal I, so is catenary because all 2-dimensional local domains are catenary. The ring A is Noetherian because B is Noetherian and is a finite A-module. However A is not universally catenary, because if it were then the ideal mB of B would have the same height as mB∩A by the dimension formula for universally catenary rings, but the latter ideal has height equal to dim(A)=2.
Nagata's example is also a quasi-excellent ring, so gives an example of a quasi-excellent ring that is not an excellent ring.