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$H$
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 Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux
23 (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.
