The catenary degree of Krull monoids II

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G| \ge 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf c (H) \in [3, \mathsf D (G)]$, where $\mathsf D (G)$ denotes the Davenport constant of $G$. It is well-known when $\mathsf c (H) \in \{3,4, \mathsf D (G)\}$ holds true. Based on a characterization of the catenary degree determined in the first paper (The catenary degree of Krull monoids I), we determine the class groups satisfying $\mathsf c (H)= \mathsf D (G)-1$. Apart from the mentioned extremal cases the precise value of $\mathsf c (H)$ is known for no further class groups.