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 pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$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$|G|\geq 3$. 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$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.