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 $z,z^{\prime }$ of $a$, there exist factorizations $z=z_0,...,z_k=z^{\prime }$ 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.

To appear in Journal of the Australian Mathematical Society

References in corpus (2)

• Factorization theory: From commutative to noncommutative settings
• Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic

Cited by in corpus (6)

• Sets of Arithmetical Invariants in Transfer Krull Monoids
• A Characterization of class groups via sets of lengths {II}
• On the set of catenary degrees of finitely generated cancellative commutative monoids
• Sets of minimal distances and characterizations of class groups of Krull monoids
• On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II
• Minimal relations and catenary degrees in Krull monoids