The Catenary Degree of Krull Monoids I

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. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf c (H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf c(H)$ and characterize when $\mathsf c(H)\leq 4$.