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 \(\mathsf c(H)\leq 4\).