# Are integral extensions of a catenary ring still catenary?

A (commutative unitary) Noetherian ring $$R$$ of finite dimension is said to be catenary if for every prime ideal $$\mathfrak{p}$$ of $$R$$ one has $$\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\mathrm{dim}R$$.

Let $$A$$ be a catenary ring and $$B$$ a finite extension of $$A$$ (id est $$A\subseteq B$$ and $$B$$ is a finite $$A$$-module). Is $$B$$ a catenary ring?

• Is your definition of catenary equivalent to the usual one (that requires that for all primes $p \subsetneq q$ all maximal prime chains $p \subsetneq \cdots \subsetneq q$ have the same length)? – tj_ Mar 24 '19 at 10:25

No. Nagata's famous family of examples of non-catenary rings yields a non-catenary finite extension of a catenary noetherian local domain.

Reference: M. Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 51-64.