The catenary degree of Krull monoids II

TL;DR

This paper investigates the catenary degree of Krull monoids with finite class groups, specifically characterizing when this degree equals the Davenport constant minus one, extending previous work on extremal cases.

Contribution

It provides a complete characterization of class groups where the catenary degree is exactly one less than the Davenport constant, building on earlier foundational results.

Findings

Characterization of class groups with catenary degree = D(G) - 1

Determination of catenary degree for specific class groups

Extension of previous results on extremal catenary degrees

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.